diff --git a/CHANGELOG.md b/CHANGELOG.md
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1,5 +1,46 @@
 # Changelog
 
+## 0.3.0.0
+
+- Updated the `Transition` class definition with a defaulted internal dense
+  backend method and relaxed finite-state constraints on cached queries.
+  Existing custom instances can continue to implement only `transitionLaw`.
+  The major version bump accounts for these class and signature changes.
+
+- Updated the package maintainer contact email.
+- Made `evolveVectorN` choose between repeated matrix-vector multiplication
+  and matrix powering, avoiding unnecessary full matrix powers for moderate
+  step counts.
+- Added `simulateMatrix`, which lazily prepares and shares cumulative sampling
+  rows during a trajectory instead of rebuilding map-backed distributions on
+  every transition.
+- Reworked the GTH stationary solver around strict indexed mutable-vector
+  loops, substantially reducing allocation in large irreducible classes.
+- Cached complete communicating-class classifications lazily in transition
+  matrices, allowing repeated whole-chain queries to share public results.
+- Split warm-class benchmarks into cached access and fully consumed cases,
+  with equivalent state traversal in Haskell and Python.
+- Reduced support-graph construction allocation by scanning the matrix's
+  storable row-major vector directly instead of materialising boxed row and
+  value lists first.
+- Centralised adaptive dense finite-horizon paths shared by finite-time,
+  hitting-time, return-time, and visit-count analyses. Branching
+  transition-matrix rows use BLAS-backed vector propagation, while
+  deterministic matrices and general locally finite kernels retain the
+  existing sparse recurrences and public API.
+- Added a batched absorption-probability matrix operation that solves every
+  recurrent destination with one shared factorisation, while scalar
+  absorption queries now solve their single right-hand side directly instead
+  of constructing the complete fundamental inverse.
+- Reduced shared @(I - Q)@ solve overhead by proving well-conditioned DTMC
+  systems from an extra right-hand side in the existing LU factorisation,
+  avoiding a separate singular-value decomposition except near the numerical
+  rejection threshold. Matrix validation and residual norms now scan
+  unboxed storage rather than materialising boxed element lists.
+- Represented dense bounded-hitting targets by their canonical indices rather
+  than two full floating-point masks, reducing target-mass collection to the
+  target coordinates and clearing only those coordinates after each step.
+
 ## 0.2.0.0
 
 First Hackage release.
diff --git a/README.md b/README.md
--- a/README.md
+++ b/README.md
@@ -26,7 +26,7 @@
 Add the package to your Cabal file:
 
 ```cabal
-build-depends: dtmc ^>=0.2.0.0
+build-depends: dtmc ^>=0.3.0.0
 ```
 
 The package requires GHC 9.6 or newer and a BLAS/LAPACK implementation for
diff --git a/benchmark/README.md b/benchmark/README.md
new file mode 100644
--- /dev/null
+++ b/benchmark/README.md
@@ -0,0 +1,174 @@
+# `dtmc` versus PyDTMC benchmark
+
+This directory contains a public-API benchmark of `dtmc` against PyDTMC
+9.0.0. It deliberately measures equivalent mathematical results rather than
+similarly named functions.
+
+## Quick start
+
+Python 3.12, GHC 9.10.3, Cabal, and a BLAS/LAPACK installation are required.
+Using another compiler supported by `dtmc` requires regenerating the Cabal
+freeze file and records a different environment.
+
+Haskell benchmark dependencies are pinned by `cabal.project.freeze`; direct
+and transitive Python package versions are pinned by
+`python/requirements.lock` and captured with the results.
+
+```console
+python3 benchmark/run.py all --mode smoke
+```
+
+The smoke workflow creates an isolated Python environment, builds the Haskell
+benchmark, generates five 10-state fixtures, verifies deterministic outputs,
+runs short Criterion and pyperf measurements, aggregates them, and creates
+plots. The full reproducible experiment is:
+
+```console
+python3 benchmark/run.py all --mode full
+```
+
+The main overview is `ratio-summary.csv`; ratios above one favor `dtmc`.
+
+The full suite is intentionally expensive. Individual phases are available as
+`bootstrap`, `generate`, `verify`, `benchmark`, `profile`, and `analyse`
+commands. `profile` records Haskell RTS allocation/GC statistics and Python
+`cProfile` data for stationary, hitting, committor, mean-recurrence, bounded
+visit, and long-simulation representatives.
+
+## Fairness contract
+
+- Both implementations read the same little-endian IEEE-754 payload. SHA-256
+  hashes live in the generated `data/generated/manifest.json`; input decoding
+  is not timed. The committed experiment definition is `data/spec.json`.
+- Inputs are valid stochastic matrices away from each library's validation
+  tolerance. Invalid-input behavior is not compared.
+- Finite matrices are dense in both libraries. The `low-outdegree` family has
+  exact zeros but is not described as sparse storage.
+- Construction measures each public constructor as defined. PyDTMC eagerly
+  builds its NetworkX graph, while `dtmc` leaves its support graph lazy.
+- `structure/classes-lifecycle` measures construction through the first SCC
+  answer. `classes-cold` excludes transition-matrix construction and graph
+  materialization. `classes-warm/access` measures cached property access;
+  `classes-warm/consumed` traverses every returned state in both libraries.
+- Cached PyDTMC properties and lazy `dtmc` graph/classification fields are
+  recreated for each cold sample. Fixture setup is excluded from those
+  samples.
+- `hitting-*/cold-all-states` computes and consumes the complete all-state
+  result. The `dtmc`-only `hitting-*/warm-lookup` rows measure one scalar
+  lookup through the same partially applied function after its shared solve
+  has been forced; PyDTMC has no equivalent reusable lookup object.
+- Bounded return cases ask for the probability of returning to state zero by
+  step 10 or 100. PyDTMC exposes the individual first-passage masses, so its
+  timed adapter sums them. Bounded visit cases ask for the expected visits to
+  the first manifest target, starting in state zero. PyDTMC's reward horizon
+  is one less because it includes both time zero and the named final step.
+- Scalar finite-time cases use state zero as the source and the first manifest
+  target as the destination. Timed-observation cases start from the complete
+  manifest initial distribution. PyDTMC's redistribution result is indexed at
+  the destination so both sides return and consume one scalar.
+- Bounded hitting cases start from state zero and use the complete manifest
+  target set. They compare exact, at-most, and greater-than events at steps 10
+  and 100. PyDTMC computes first passage to each destination separately, so
+  its fixture makes every target absorbing before calling the public
+  multi-target operation. This leaves first-hit probabilities unchanged while
+  making the target events disjoint. The matrix transformation and chain
+  construction are outside the timed operation. The adapters then select the
+  last mass, sum the masses, or subtract their sum from one respectively.
+- Pyperf records 30 fresh full-operation values per normal cell (three worker
+  processes, ten values each). Warm-cache lookups are internally batched;
+  Criterion uses its calibrated sample schedule. Smoke mode intentionally uses
+  fewer values and is not performance evidence.
+- Numeric results are consumed with deterministic checksums. This prevents
+  Haskell laziness from turning a benchmark into wrapper construction alone.
+- BLAS thread environment variables are fixed to one. The actual Haskell and
+  NumPy linkage is captured in `results/environment.json`.
+- Simulation compares throughput and validates behavior statistically; equal
+  seeds do not imply equal paths because the RNG algorithms differ. PyDTMC's
+  public simulation call creates its generator during the timed operation;
+  `dtmc` receives the generator required by its public API before timing. The
+  `dtmc` case uses its public matrix-specialized API; its lazy row preparation
+  and cache are created inside the timed operation.
+
+## Dataset families
+
+- `dense`: every entry is positive; irreducible and aperiodic.
+- `low-outdegree`: four or fewer exact-positive entries per row, including a
+  self-loop and the next ring state; irreducible and aperiodic.
+- `absorbing`: two singleton absorbing states and a transient dense block from
+  which absorption is certain.
+- `reducible`: a transient communicating block feeding two closed irreducible
+  classes, with probabilities kept away from either library's near-zero
+  threshold.
+- `periodic`: a deterministic directed cycle, included so period and cyclic
+  decomposition do nontrivial work.
+
+Full runs use sizes 10, 25, 50, 100, 250, 500, and 1000 with seeds 1729, 2718,
+and 31415. Cubic operations stop at 500, occupation matrices at 250, bounded
+finite-horizon cases at 100, and dense NetworkX structural cases at 500. These
+caps are symmetric.
+
+## Measured mappings
+
+| Benchmark | `dtmc` | PyDTMC |
+|---|---|---|
+| `construction/public-consumed` | `fromRows` | `MarkovChain(P)` |
+| `evolution/one-step` | `evolveVector` | `redistribute(1)` |
+| `evolution/k-*` | `evolveVectorN` | `redistribute(k)` |
+| `finite-time/step` | `stepProbability` | matrix entry lookup |
+| `finite-time/n-step/k-*` | `nStepProbability` | point-mass `redistribute(k)` lookup |
+| `finite-time/observation/k-*` | `probability [At k target]` | initial-law `redistribute(k)` lookup |
+| `power/*` | `power` | `to_nth_order` |
+| `structure/classes-*` | `communicatingClasses` | `communicating_classes` |
+| `structure/irreducible-*` | `irreducible` | `is_irreducible` |
+| `structure/period-*` | `chainPeriod` | `period` |
+| `structure/cyclic-classes-*` | `cyclicClasses` | `cyclic_classes` |
+| `stationary` | `stationaryDistributions` | `pi` |
+| `hitting-probability/cold-all-states` | all-state eventual hitting | `hitting_probabilities` |
+| `hitting-time/cold-all-states` | all-state expected hitting | `hitting_times` |
+| `hitting/bounded/exact/k-*` | `probabilityGivenInitialState (EqualTo k)` | final `first_passage_probabilities` mass |
+| `hitting/bounded/at-most/k-*` | `probabilityGivenInitialState (AtMost k)` | summed `first_passage_probabilities` |
+| `hitting/bounded/greater-than/k-*` | `probabilityGivenInitialState (GreaterThan k)` | one minus summed `first_passage_probabilities` |
+| `race/forward-committor` | `raceProbabilityGivenInitialState` | `committor_probabilities("forward", ...)` |
+| `return/mean-recurrence` | `expectationGivenInitialState` | `mean_recurrence_times` |
+| `return/bounded/k-*` | `probabilityGivenInitialState (AtMost k)` | summed `first_passage_probabilities` |
+| `visits/bounded-expectation/k-*` | `boundedExpectationGivenInitialState` | `expected_rewards(k - 1, ...)` |
+| `fundamental-matrix` | `fundamentalMatrix` | `fundamental_matrix` |
+| `absorption-time` | all-state expectation, transient part | `mean_absorption_times` |
+| `absorption/probabilities` | `probabilityMatrix` | `absorption_probabilities` |
+| `occupation-matrix` | `occupationMatrix` | `mean_number_visits` |
+| `simulation/*` | `simulateMatrix` | `simulate` |
+
+`evolveVectorN` chooses between `k` matrix-vector steps and matrix powering
+using the matrix dimension and requested step count; PyDTMC performs `k`
+matrix-vector steps. The comparison is a public end-to-end workload, not a
+claim that their internal primitives are identical.
+
+Race/committor and mean-recurrence comparisons use only the dense and
+low-outdegree aperiodic irreducible families because PyDTMC returns `None` for
+non-ergodic chains. Absorption-probability rows use the manifest absorber order
+and canonical transient-state order.
+
+PyDTMC's `mean_number_visits` excludes the visit at time zero, while
+`occupationMatrix` includes it. The Python case therefore adds the identity
+matrix to the PyDTMC result. That semantic adapter is included in the public
+workload timing and in correctness verification.
+
+## Correctness
+
+`benchmark/python/verify.py` compares complete outputs through size 100 by
+default. It checks structural equality, exact infinity masks, and numeric
+values using `atol=1e-10` and `rtol=1e-8`. It records maximum absolute and
+relative discrepancies in `results/verification.json` and exits unsuccessfully
+on a mismatch.
+
+## Interpreting output
+
+Raw engine output is retained under `results/raw/`. `summary.csv` contains
+median, MAD, quartiles, and sample counts. `ratios.csv` reports
+`PyDTMC time / dtmc time`, so values above one favor `dtmc`. Do not compare
+cached and cold rows or infer sparse-storage performance from low-outdegree
+fixtures. The generated SVGs include both runtime scaling and speed-ratio
+plots; ratio confidence bands summarize the three deterministic seeds.
+Analysis reads only raw JSON files corresponding to datasets in the current
+generated manifest. Raw files from another mode are retained but ignored, so a
+smoke run can safely follow a full run without deleting the full results.
diff --git a/benchmark/REPORT.md b/benchmark/REPORT.md
new file mode 100644
--- /dev/null
+++ b/benchmark/REPORT.md
@@ -0,0 +1,301 @@
+# Benchmark report: dtmc versus PyDTMC 9.0.0
+
+## Result
+
+The full benchmark completed successfully on 14 September 2026. It compares
+equivalent public-API workloads in dtmc and PyDTMC 9.0.0 on the same generated
+transition matrices.
+
+Across 932 family-size-operation aggregates:
+
+- 917 aggregates (98.4%) have a median ratio above one and therefore favour
+  dtmc.
+- 913 aggregates (98.0%) have a complete 95% seed-bootstrap interval above
+  one.
+- 13 aggregates (1.4%) have a complete interval below one and favour PyDTMC.
+- 6 aggregates (0.6%) have an interval that crosses one and are inconclusive.
+- The unweighted median ratio across all aggregates is 14.52.
+
+The ratio is:
+
+~~~text
+PyDTMC elapsed time / dtmc elapsed time
+~~~
+
+A ratio of 14.52 means that PyDTMC took 14.52 times as long as dtmc for that
+benchmark case. Ratios below one favour PyDTMC. The overall median gives every
+family-size-operation aggregate equal weight. It is a summary of this
+benchmark suite, not a prediction that every dtmc operation is 14.52 times
+faster.
+
+The source of truth for these results is
+[results/ratio-summary.csv](results/ratio-summary.csv). It contains one
+aggregate for each family, size, and operation, calculated from three fixture
+seeds. The underlying 2,796 seed-level ratios are in
+[results/ratios.csv](results/ratios.csv).
+
+## Results by operation group
+
+“Clear” means that the complete bootstrap interval across the three fixture
+seeds lies on one side of one. “Inconclusive” means that it crosses one.
+
+| Operation group | Aggregates | Median ratio | Clear dtmc wins | Clear PyDTMC wins | Inconclusive |
+|---|---:|---:|---:|---:|---:|
+| Absorption probabilities | 6 | 9.47 | 6 | 0 | 0 |
+| Absorption time | 6 | 7.92 | 6 | 0 | 0 |
+| Bounded hitting | 120 | 16.34 | 120 | 0 | 0 |
+| Bounded return | 40 | 16.76 | 40 | 0 | 0 |
+| Bounded visits | 40 | 3.85 | 40 | 0 | 0 |
+| Construction | 35 | 16.94 | 35 | 0 | 0 |
+| Evolution | 105 | 20.03 | 103 | 0 | 2 |
+| Eventual hitting probability | 30 | 2.62 | 29 | 0 | 1 |
+| Expected hitting time | 30 | 5.23 | 29 | 0 | 1 |
+| Finite-time analysis | 100 | 9.51 | 92 | 6 | 2 |
+| Fundamental matrix | 6 | 5.56 | 6 | 0 | 0 |
+| Matrix power | 90 | 25.80 | 90 | 0 | 0 |
+| Mean recurrence time | 12 | 20.83 | 12 | 0 | 0 |
+| Occupation matrix | 10 | 26.49 | 10 | 0 | 0 |
+| Race probability | 12 | 15.50 | 12 | 0 | 0 |
+| Simulation | 28 | 12.93 | 28 | 0 | 0 |
+| Stationary distributions | 30 | 11.94 | 28 | 2 | 0 |
+| Structure | 232 | 18.60 | 227 | 5 | 0 |
+
+The recently optimised bounded-hitting, bounded-return, bounded-visit,
+absorption-probability, and mean-recurrence paths all have clear wins in every
+aggregate included in the full run.
+
+## Representative plots
+
+The plots report the median PyDTMC/dtmc runtime ratio across three fixture
+seeds. Values above one favour dtmc, values below one favour PyDTMC, and the
+shaded region is the 95% seed-bootstrap interval.
+
+### Dense one-step evolution
+
+![Dense one-step evolution speed ratio](results/plots/dense--evolution-one-step--speed-ratio.svg)
+
+### Exact bounded hitting at 100 steps
+
+![Absorbing-chain bounded hitting speed ratio](results/plots/absorbing--hitting-bounded-exact-k-100--speed-ratio.svg)
+
+### Dense expected hitting time
+
+![Dense expected hitting-time speed ratio](results/plots/dense--hitting-time-cold-all-states--speed-ratio.svg)
+
+### Known exception: periodic observations at 100 steps
+
+![Periodic finite-time observation speed ratio](results/plots/periodic--finite-time-observation-k-100--speed-ratio.svg)
+
+## Cases that favour PyDTMC
+
+The 13 clear PyDTMC wins are concentrated in three behaviours.
+
+### Periodic finite-time observations
+
+The full-initial-distribution observation path is slower on deterministic
+periodic chains at sizes 25, 50, and 100.
+
+| Operation | Size | Ratio | Equivalent PyDTMC advantage |
+|---|---:|---:|---:|
+| finite-time/observation/k-10 | 25 | 0.601 | 1.66 times faster |
+| finite-time/observation/k-10 | 50 | 0.286 | 3.50 times faster |
+| finite-time/observation/k-10 | 100 | 0.138 | 7.26 times faster |
+| finite-time/observation/k-100 | 25 | 0.275 | 3.64 times faster |
+| finite-time/observation/k-100 | 50 | 0.110 | 9.12 times faster |
+| finite-time/observation/k-100 | 100 | 0.049 | 20.44 times faster |
+
+This weakness is specific to timed observations from the complete initial
+distribution. The scalar finite-time/n-step comparisons are clear dtmc wins
+in all 40 aggregates.
+
+### Warm cyclic-class access
+
+Five periodic warm-access aggregates favour PyDTMC:
+
+| Sizes | Ratio range | Equivalent PyDTMC advantage |
+|---|---:|---:|
+| 50, 100, 250, 500, and 1000 | 0.890–0.949 | 1.05–1.12 times faster |
+
+The difference is small but consistent across the three seeds. Cold cyclic
+decomposition strongly favours dtmc, so this result concerns repeated access
+to an already-computed value rather than the decomposition algorithm.
+
+### Stationary distributions
+
+Two 500-state stationary cases favour PyDTMC:
+
+| Family | Ratio | Equivalent PyDTMC advantage |
+|---|---:|---:|
+| Periodic | 0.920 | 1.09 times faster |
+| Low-outdegree | 0.934 | 1.07 times faster |
+
+These are narrow losses. The other 28 stationary aggregates clearly favour
+dtmc.
+
+## Inconclusive cases
+
+Six intervals cross one:
+
+| Family | Size | Operation | Median ratio | 95% interval |
+|---|---:|---|---:|---:|
+| Absorbing | 1000 | evolution/k-100 | 0.888 | 0.886–1.081 |
+| Low-outdegree | 1000 | evolution/k-100 | 0.968 | 0.924–1.084 |
+| Periodic | 10 | finite-time/observation/k-100 | 1.018 | 0.984–1.033 |
+| Dense | 25 | finite-time/step | 2.998 | 0.465–3.861 |
+| Dense | 25 | hitting-probability/cold-all-states | 1.753 | 0.768–2.081 |
+| Dense | 25 | hitting-time/cold-all-states | 3.920 | 0.626–4.101 |
+
+The median direction alone should not be used to declare a winner in these
+cases.
+
+## Scaling
+
+Fifty-two family-operation combinations occur at every tested size. Holding
+this panel constant avoids comparing a different mixture of operations at
+each size.
+
+| States | Median ratio in common panel | Aggregates favouring dtmc |
+|---:|---:|---:|
+| 10 | 27.34 | 52 of 52 |
+| 25 | 23.72 | 52 of 52 |
+| 50 | 18.20 | 51 of 52 |
+| 100 | 15.22 | 51 of 52 |
+| 250 | 12.81 | 51 of 52 |
+| 500 | 11.28 | 51 of 52 |
+| 1000 | 9.70 | 49 of 52 |
+
+dtmc remains faster in nearly every common-panel aggregate at 1000 states,
+but its relative advantage generally narrows as the matrices grow.
+
+The log-log fits in [results/scaling.csv](results/scaling.csv) contain 172
+operation-family pairs with both engines present. The fitted dtmc exponent is
+higher in 135 pairs and lower in 37. These finite-range empirical fits are not
+proofs of asymptotic complexity. They do indicate that several dtmc paths grow
+faster over the tested size range even though they retain a lower absolute
+runtime.
+
+Large-input scaling is therefore a more important optimisation target than
+small-operation overhead. The clearest candidates are long evolution,
+stationary distributions, and the periodic finite-time observation path.
+
+## Correctness
+
+[results/verification.json](results/verification.json) reports:
+
+- 1,392 successful output comparisons;
+- zero failed comparisons;
+- a maximum absolute discrepancy of 7.276e-11; and
+- configured tolerances of atol=1e-10 and rtol=1e-8.
+
+One near-zero bounded-hitting result has a maximum relative discrepancy of
+one, but its absolute discrepancy is only 1.542e-20; it passes the stated
+absolute tolerance.
+
+Verification checks complete deterministic outputs through 100 states. The 45
+datasets at sizes 250, 500, and 1000 are intentionally skipped by the
+verification-size cap. Their timing measurements are present, but this run
+does not provide the same element-by-element correctness check at those
+sizes.
+
+## Benchmark scope and method
+
+The full run uses:
+
+- five chain families: dense, low-outdegree, absorbing, reducible, and
+  periodic;
+- sizes 10, 25, 50, 100, 250, 500, and 1000;
+- deterministic seeds 1729, 2718, and 31415;
+- symmetric size caps for cubic, occupation-matrix, and bounded-horizon
+  operations; and
+- public end-to-end operations that return equivalent mathematical results.
+
+Both implementations read identical binary fixtures, and input decoding is
+outside the timed operation. Finite matrices use dense storage in both
+libraries. The low-outdegree family tests matrices containing many exact
+zeros, not a sparse-storage representation.
+
+Pyperf records 30 fresh values per normal PyDTMC cell. Criterion uses its
+calibrated sampling schedule for dtmc. Each engine's median runtime is first
+calculated for each concrete dataset. Seed-level ratios are then grouped by
+family, size, and operation. The reported 95% intervals bootstrap the three
+seed-level log ratios. They measure variation across those three fixtures;
+they do not measure variation across machines or separate overnight runs.
+
+Some equivalent workloads require documented adapters. Examples include
+summing PyDTMC first-passage masses for bounded return, aligning its reward
+horizon for bounded visits, and making bounded-hitting targets absorbing
+outside the timed region. These mappings are part of the benchmark's fairness
+contract and are documented in [README.md](README.md).
+
+## Environment
+
+[results/environment.json](results/environment.json) records:
+
+- commit daf5d56d8b7967ced2274dfee0c73f2bacb997ff;
+- Apple M1, arm64;
+- macOS 15.6.1;
+- GHC 9.10.3;
+- optimisation flag -O2, with -threaded and -rtsopts;
+- Python 3.13.9;
+- PyDTMC 9.0.0;
+- NumPy 2.3.1 and SciPy 1.16.1;
+- Apple Accelerate BLAS and LAPACK for both engines; and
+- single-thread limits for BLAS-related runtimes.
+
+The working tree contained only untracked benchmark logs and backups when the
+environment was captured.
+
+The benchmark documentation specifies Python 3.12, but this run used Python
+3.13.9. This does not invalidate a same-machine comparison in which both
+engines use the recorded environment. A publication-oriented reproducibility
+run should use the documented Python version or update and justify the
+requirement.
+
+## Limits on interpretation
+
+The results support a claim that dtmc is faster for almost all tested
+public-API workloads on this Apple M1 configuration. They do not establish:
+
+- identical speedups on Linux, x86-64, or different BLAS implementations;
+- a general Haskell-versus-Python language result;
+- sparse-matrix performance;
+- correctness above 100 states with the same verification coverage;
+- performance for operations without equivalent PyDTMC APIs; or
+- one universal speedup applicable to every user workload.
+
+The largest ratios occur in cold structural operations and should not be
+presented as typical speedups for the entire library. The 14.52 median is less
+sensitive to those large outliers, but it still reflects the operation mix in
+this suite.
+
+## Recommended next work
+
+1. Profile finite-time/observation on deterministic periodic chains. Its
+   disadvantage increases sharply with both size and horizon.
+2. Profile the 100-step evolution policy at 1000 states. Two families are
+   currently inconclusive and the empirical scaling trend favours PyDTMC.
+3. Inspect warm cyclic-class access only after the larger finite-time issue.
+   Its loss is stable but small.
+4. Repeat a focused subset after each optimisation before committing to
+   another full run.
+5. Run a second full experiment on a quiet Linux x86-64 machine with the
+   documented Python version before publishing hardware-independent claims.
+
+## Result files
+
+- [results/ratio-summary.csv](results/ratio-summary.csv): primary three-seed
+  comparison.
+- [results/ratios.csv](results/ratios.csv): seed-level paired ratios.
+- [results/summary.csv](results/summary.csv): per-engine medians, dispersion,
+  confidence intervals, and sample counts.
+- [results/scaling.csv](results/scaling.csv): empirical log-log scaling fits.
+- [results/verification.json](results/verification.json): correctness
+  comparisons.
+- [results/environment.json](results/environment.json): compiler, packages,
+  linkage, flags, commit, and thread controls.
+- [results/plots/](results/plots/): selected publication plots.
+
+The remaining 350 generated plots and the raw Criterion and pyperf output
+remain local build artefacts and are intentionally ignored to avoid adding
+approximately 15 MB of generated charts and much larger raw timing files to
+the repository.
diff --git a/benchmark/analyse.py b/benchmark/analyse.py
new file mode 100644
--- /dev/null
+++ b/benchmark/analyse.py
@@ -0,0 +1,315 @@
+#!/usr/bin/env python3
+"""Aggregate Criterion and pyperf JSON into comparable robust statistics."""
+
+from __future__ import annotations
+
+import argparse
+import csv
+import hashlib
+import json
+import math
+import random
+import statistics
+from collections import defaultdict
+from pathlib import Path
+from typing import Any, Iterable
+
+
+ROOT = Path(__file__).resolve().parent
+RESULTS = ROOT / "results"
+
+
+def criterion_samples(path: Path) -> Iterable[tuple[str, float]]:
+    payload = json.loads(path.read_text(encoding="utf-8"))
+    if (
+        isinstance(payload, list)
+        and len(payload) == 3
+        and payload[0] == "criterion"
+        and isinstance(payload[2], list)
+    ):
+        reports = payload[2]
+    else:
+        reports = payload if isinstance(payload, list) else payload.get("reports", [])
+    for report in reports:
+        name = report.get("reportName") or report.get("name")
+        measured = report.get("reportMeasured", [])
+        samples: list[float] = []
+        keys = report.get("reportKeys", [])
+        for measurement in measured:
+            if isinstance(measurement, dict):
+                elapsed = measurement.get("measTime", measurement.get("time"))
+                iterations = measurement.get("measIters", measurement.get("iters", 1))
+                if elapsed is not None:
+                    samples.append(float(elapsed) / max(1.0, float(iterations)))
+            elif isinstance(measurement, list) and keys:
+                values = dict(zip(keys, measurement))
+                elapsed = values.get("time")
+                iterations = values.get("iters", 1)
+                if elapsed is not None:
+                    samples.append(float(elapsed) / max(1.0, float(iterations)))
+        if not samples:
+            analysis = report.get("reportAnalysis", report.get("analysis", {}))
+            estimate = analysis.get("anMean", analysis.get("mean", {}))
+            point = estimate.get("estPoint", estimate.get("point")) if isinstance(estimate, dict) else None
+            if point is not None:
+                samples = [float(point)]
+        for sample in samples:
+            yield str(name), sample
+
+
+def pyperf_samples(path: Path) -> Iterable[tuple[str, float]]:
+    payload = json.loads(path.read_text(encoding="utf-8"))
+    for benchmark in payload.get("benchmarks", []):
+        metadata = benchmark.get("metadata", {})
+        name = metadata.get("name") or benchmark.get("name")
+        for run in benchmark.get("runs", []):
+            for value in run.get("values", []):
+                yield str(name), float(value)
+
+
+def quartiles(values: list[float]) -> tuple[float, float]:
+    if len(values) < 2:
+        return values[0], values[0]
+    cuts = statistics.quantiles(values, n=4, method="inclusive")
+    return cuts[0], cuts[2]
+
+
+def percentile(values: list[float], probability: float) -> float:
+    ordered = sorted(values)
+    position = probability * (len(ordered) - 1)
+    lower = int(position)
+    upper = min(lower + 1, len(ordered) - 1)
+    fraction = position - lower
+    return ordered[lower] * (1.0 - fraction) + ordered[upper] * fraction
+
+
+def bootstrap_median_interval(values: list[float], seed: int) -> tuple[float, float]:
+    if len(values) == 1:
+        return values[0], values[0]
+    rng = random.Random(seed)
+    estimates = [
+        statistics.median(rng.choices(values, k=len(values)))
+        for _ in range(2000)
+    ]
+    return percentile(estimates, 0.025), percentile(estimates, 0.975)
+
+
+def stable_seed(value: str) -> int:
+    return int.from_bytes(hashlib.sha256(value.encode("utf-8")).digest()[:8], "big")
+
+
+def summarize(values: list[float], seed_key: str) -> dict[str, float | int]:
+    median = statistics.median(values)
+    deviations = [abs(value - median) for value in values]
+    q1, q3 = quartiles(values)
+    ci_low, ci_high = bootstrap_median_interval(values, stable_seed(seed_key))
+    return {
+        "median_seconds": median,
+        "mad_seconds": statistics.median(deviations),
+        "q1_seconds": q1,
+        "q3_seconds": q3,
+        "median_ci95_low_seconds": ci_low,
+        "median_ci95_high_seconds": ci_high,
+        "samples": len(values),
+    }
+
+
+def split_name(name: str, metadata: dict[str, dict[str, Any]]) -> tuple[str, str]:
+    for dataset in sorted(metadata, key=len, reverse=True):
+        prefix = dataset + "/"
+        if name.startswith(prefix):
+            return dataset, name[len(prefix) :]
+    raise ValueError(f"benchmark name does not begin with a manifest dataset id: {name}")
+
+
+def active_result_paths(
+    results: Path, engine: str, datasets: Iterable[str]
+) -> Iterable[Path]:
+    directory = results / "raw" / engine
+    for dataset in sorted(datasets):
+        path = directory / f"{dataset}.json"
+        if path.is_file():
+            yield path
+
+
+def main() -> None:
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument(
+        "--data-root", type=Path, default=ROOT / "data" / "generated"
+    )
+    parser.add_argument("--results-root", type=Path, default=RESULTS)
+    args = parser.parse_args()
+    data_root = args.data_root.resolve()
+    results = args.results_root.resolve()
+
+    manifest = json.loads(
+        (data_root / "manifest.json").read_text(encoding="utf-8")
+    )
+    metadata = {entry["id"]: entry for entry in manifest["datasets"]}
+    grouped: dict[tuple[str, str, str], list[float]] = defaultdict(list)
+
+    for path in active_result_paths(results, "haskell", metadata):
+        for name, value in criterion_samples(path):
+            dataset, operation = split_name(name, metadata)
+            grouped[("dtmc", dataset, operation)].append(value)
+    for path in active_result_paths(results, "python", metadata):
+        for name, value in pyperf_samples(path):
+            dataset, operation = split_name(name, metadata)
+            grouped[("pydtmc", dataset, operation)].append(value)
+
+    rows: list[dict[str, Any]] = []
+    for (engine, dataset, operation), values in sorted(grouped.items()):
+        entry = metadata[dataset]
+        rows.append(
+            {
+                "engine": engine,
+                "dataset": dataset,
+                "family": entry["family"],
+                "size": entry["size"],
+                "seed": entry["seed"],
+                "operation": operation,
+                **summarize(values, f"{engine}/{dataset}/{operation}"),
+            }
+        )
+
+    results.mkdir(parents=True, exist_ok=True)
+    fields = [
+        "engine",
+        "dataset",
+        "family",
+        "size",
+        "seed",
+        "operation",
+        "median_seconds",
+        "mad_seconds",
+        "q1_seconds",
+        "q3_seconds",
+        "median_ci95_low_seconds",
+        "median_ci95_high_seconds",
+        "samples",
+    ]
+    with (results / "summary.csv").open("w", newline="", encoding="utf-8") as stream:
+        writer = csv.DictWriter(stream, fieldnames=fields)
+        writer.writeheader()
+        writer.writerows(rows)
+
+    by_key = {(row["engine"], row["dataset"], row["operation"]): row for row in rows}
+    ratios: list[dict[str, Any]] = []
+    for row in rows:
+        if row["engine"] != "dtmc":
+            continue
+        peer = by_key.get(("pydtmc", row["dataset"], row["operation"]))
+        if peer is None:
+            continue
+        ratios.append(
+            {
+                "dataset": row["dataset"],
+                "family": row["family"],
+                "size": row["size"],
+                "seed": row["seed"],
+                "operation": row["operation"],
+                "pydtmc_over_dtmc": peer["median_seconds"] / row["median_seconds"],
+            }
+        )
+    ratio_fields = ["dataset", "family", "size", "seed", "operation", "pydtmc_over_dtmc"]
+    with (results / "ratios.csv").open("w", newline="", encoding="utf-8") as stream:
+        writer = csv.DictWriter(stream, fieldnames=ratio_fields)
+        writer.writeheader()
+        writer.writerows(ratios)
+
+    ratio_groups: dict[tuple[str, int, str], list[float]] = defaultdict(list)
+    for row in ratios:
+        ratio_groups[(row["family"], int(row["size"]), row["operation"])].append(
+            float(row["pydtmc_over_dtmc"])
+        )
+    ratio_summary: list[dict[str, Any]] = []
+    for (family, size, operation), values in sorted(ratio_groups.items()):
+        logs = [math.log(value) for value in values]
+        log_low, log_high = bootstrap_median_interval(
+            logs, stable_seed(f"ratio/{family}/{size}/{operation}")
+        )
+        ratio_summary.append(
+            {
+                "family": family,
+                "size": size,
+                "operation": operation,
+                "geometric_median_pydtmc_over_dtmc": math.exp(statistics.median(logs)),
+                "ci95_low": math.exp(log_low),
+                "ci95_high": math.exp(log_high),
+                "datasets": len(values),
+            }
+        )
+    ratio_summary_fields = [
+        "family",
+        "size",
+        "operation",
+        "geometric_median_pydtmc_over_dtmc",
+        "ci95_low",
+        "ci95_high",
+        "datasets",
+    ]
+    with (results / "ratio-summary.csv").open("w", newline="", encoding="utf-8") as stream:
+        writer = csv.DictWriter(stream, fieldnames=ratio_summary_fields)
+        writer.writeheader()
+        writer.writerows(ratio_summary)
+
+    scaling_points: dict[tuple[str, str, str, int], list[float]] = defaultdict(list)
+    for row in rows:
+        scaling_points[
+            (row["engine"], row["family"], row["operation"], int(row["size"]))
+        ].append(float(row["median_seconds"]))
+    scaling_groups: dict[tuple[str, str, str], list[tuple[int, float]]] = defaultdict(list)
+    for (engine, family, operation, size), values in scaling_points.items():
+        scaling_groups[(engine, family, operation)].append(
+            (size, statistics.median(values))
+        )
+    scaling_rows: list[dict[str, Any]] = []
+    for (engine, family, operation), points in sorted(scaling_groups.items()):
+        if len(points) < 3:
+            continue
+        points.sort()
+        xs = [math.log(float(size)) for size, _ in points]
+        ys = [math.log(seconds) for _, seconds in points]
+        x_mean = statistics.mean(xs)
+        y_mean = statistics.mean(ys)
+        denominator = sum((value - x_mean) ** 2 for value in xs)
+        slope = sum((x - x_mean) * (y - y_mean) for x, y in zip(xs, ys)) / denominator
+        intercept = y_mean - slope * x_mean
+        fitted = [intercept + slope * value for value in xs]
+        total = sum((value - y_mean) ** 2 for value in ys)
+        residual = sum((actual - predicted) ** 2 for actual, predicted in zip(ys, fitted))
+        scaling_rows.append(
+            {
+                "engine": engine,
+                "family": family,
+                "operation": operation,
+                "slope": slope,
+                "r_squared": 1.0 - residual / total if total > 0 else 1.0,
+                "minimum_size": points[0][0],
+                "maximum_size": points[-1][0],
+                "points": len(points),
+            }
+        )
+    scaling_fields = [
+        "engine",
+        "family",
+        "operation",
+        "slope",
+        "r_squared",
+        "minimum_size",
+        "maximum_size",
+        "points",
+    ]
+    with (results / "scaling.csv").open("w", newline="", encoding="utf-8") as stream:
+        writer = csv.DictWriter(stream, fieldnames=scaling_fields)
+        writer.writeheader()
+        writer.writerows(scaling_rows)
+
+    print(
+        f"wrote {len(rows)} summaries, {len(ratios)} paired ratios, "
+        f"and {len(scaling_rows)} scaling fits"
+    )
+
+
+if __name__ == "__main__":
+    main()
diff --git a/benchmark/cabal.project b/benchmark/cabal.project
new file mode 100644
--- /dev/null
+++ b/benchmark/cabal.project
@@ -0,0 +1,7 @@
+packages: .
+
+benchmarks: True
+tests: False
+
+package dtmc
+  optimization: 2
diff --git a/benchmark/cabal.project.freeze b/benchmark/cabal.project.freeze
new file mode 100644
--- /dev/null
+++ b/benchmark/cabal.project.freeze
@@ -0,0 +1,141 @@
+active-repositories: hackage.haskell.org:merge
+constraints: any.Glob ==0.10.2,
+             any.OneTuple ==0.4.3,
+             OneTuple +base-ge-4-15 +base-ge-4-16,
+             any.Only ==0.1,
+             any.QuickCheck ==2.18.0.0,
+             QuickCheck -old-random +templatehaskell,
+             any.StateVar ==1.2.2,
+             any.aeson ==2.2.5.0,
+             aeson +ordered-keymap,
+             any.ansi-terminal ==1.1.5,
+             ansi-terminal -example,
+             any.ansi-terminal-types ==1.1.3,
+             any.array ==0.5.8.0,
+             any.assoc ==1.1.1,
+             assoc -tagged,
+             any.async ==2.2.6,
+             async -bench -debug-auto-label,
+             any.attoparsec ==0.14.4,
+             attoparsec -developer,
+             any.base ==4.20.2.0,
+             any.base-compat ==0.15.0,
+             any.base-compat-batteries ==0.15.0,
+             any.base-orphans ==0.9.4,
+             any.bifunctors ==5.6.3,
+             bifunctors +tagged,
+             any.binary ==0.8.9.3,
+             any.binary-orphans ==1.0.6,
+             binary-orphans +base-ge-4-16 +base-ge-4-17,
+             any.bitvec ==1.1.6.0,
+             bitvec +simd,
+             any.bytestring ==0.12.2.0,
+             any.cassava ==0.5.4.1,
+             any.character-ps ==0.1,
+             any.code-page ==0.2.1,
+             any.colour ==2.3.7,
+             any.comonad ==5.0.10,
+             comonad +containers +distributive +indexed-traversable,
+             any.containers ==0.7,
+             any.contravariant ==1.5.6,
+             contravariant +statevar,
+             any.criterion ==1.6.5.0,
+             criterion -embed-data-files -fast,
+             any.criterion-measurement ==0.2.5.0,
+             criterion-measurement -fast,
+             any.data-default ==0.8.0.2,
+             any.data-default-class ==0.2.0.0,
+             any.data-fix ==0.3.4,
+             any.deepseq ==1.5.0.0,
+             any.dense-linear-algebra ==0.1.0.0,
+             any.directory ==1.3.8.5,
+             any.distributive ==0.6.3,
+             distributive +tagged,
+             any.dlist ==1.0,
+             dlist -werror,
+             any.exceptions ==0.10.9,
+             any.filepath ==1.5.4.0,
+             any.finite-typelits ==0.2.1.0,
+             any.ghc-bignum ==1.3,
+             any.ghc-boot-th ==9.10.3,
+             any.ghc-internal ==9.1003.0,
+             any.ghc-prim ==0.12.0,
+             any.hashable ==1.5.1.0,
+             hashable -arch-native -random-initial-seed,
+             any.hmatrix ==0.20.2,
+             hmatrix -disable-default-paths -no-random_r -openblas,
+             any.hsc2hs ==0.68.10,
+             hsc2hs -in-ghc-tree,
+             any.indexed-traversable ==0.1.5,
+             indexed-traversable +base-ge-4-18,
+             any.indexed-traversable-instances ==0.1.2.1,
+             any.integer-conversion ==0.1.1,
+             any.integer-logarithms ==1.0.5,
+             integer-logarithms -check-bounds +integer-gmp,
+             any.js-chart ==2.9.4.1,
+             any.math-functions ==0.3.4.4,
+             math-functions +system-erf +system-expm1,
+             any.microstache ==1.0.3.1,
+             any.mtl ==2.3.1,
+             any.mwc-random ==0.15.3.0,
+             mwc-random -benchpapi,
+             any.network-uri ==2.6.4.2,
+             any.optparse-applicative ==0.19.0.0,
+             optparse-applicative +process,
+             any.os-string ==2.0.7,
+             any.parallel ==3.3.0.0,
+             any.parsec ==3.1.18.0,
+             any.pretty ==1.1.3.6,
+             any.prettyprinter ==1.7.2,
+             prettyprinter -buildreadme +text,
+             any.prettyprinter-ansi-terminal ==1.1.4,
+             prettyprinter-ansi-terminal +text,
+             any.primitive ==0.9.1.0,
+             any.process ==1.6.26.1,
+             any.random ==1.3.1,
+             any.rts ==1.0.2,
+             any.scientific ==0.3.8.1,
+             scientific -integer-simple,
+             any.semialign ==1.4,
+             semialign +semigroupoids,
+             any.semigroupoids ==6.0.2,
+             semigroupoids +comonad +containers +contravariant +tagged +unordered-containers,
+             any.semigroups ==0.20.1,
+             any.split ==0.2.5,
+             any.splitmix ==0.1.3.2,
+             splitmix -optimised-mixer,
+             any.statistics ==0.16.5.0,
+             statistics -benchpapi,
+             any.stm ==2.5.3.1,
+             any.storable-complex ==0.2.3.0,
+             any.strict ==0.5.1,
+             any.tagged ==0.8.10,
+             tagged +deepseq +template-haskell,
+             any.tasty ==1.5.4,
+             tasty +unix,
+             any.template-haskell ==2.22.0.0,
+             any.text ==2.1.3,
+             any.text-iso8601 ==0.1.1.1,
+             any.text-short ==0.1.6.1,
+             text-short -asserts,
+             any.th-abstraction ==0.7.2.0,
+             any.th-compat ==0.1.7,
+             any.these ==1.2.1,
+             any.time ==1.12.2,
+             any.time-compat ==1.9.9,
+             any.transformers ==0.6.1.1,
+             any.transformers-compat ==0.8,
+             transformers-compat -five +five-three +generic-deriving +mtl,
+             any.unix ==2.8.7.0,
+             any.unordered-containers ==0.2.21,
+             unordered-containers -debug,
+             any.uuid-types ==1.0.6.1,
+             any.vector ==0.13.2.0,
+             vector +boundschecks -internalchecks -unsafechecks -wall,
+             any.vector-algorithms ==0.9.1.0,
+             vector-algorithms +bench +boundschecks -internalchecks -llvm -unsafechecks,
+             any.vector-binary-instances ==0.2.5.2,
+             any.vector-stream ==0.1.0.1,
+             any.vector-th-unbox ==0.2.2,
+             any.witherable ==0.5
+index-state: hackage.haskell.org 2026-08-27T21:15:46Z
diff --git a/benchmark/data/generate.py b/benchmark/data/generate.py
new file mode 100644
--- /dev/null
+++ b/benchmark/data/generate.py
@@ -0,0 +1,227 @@
+#!/usr/bin/env python3
+"""Generate byte-identical benchmark inputs for dtmc and PyDTMC.
+
+Each payload is little-endian float64 data in row-major order.  The n*n
+transition probabilities are followed by the n coordinates of the initial
+distribution.  Metadata and SHA-256 digests are stored in manifest.json.
+"""
+
+from __future__ import annotations
+
+import argparse
+import hashlib
+import json
+from pathlib import Path
+from typing import Callable
+
+import numpy as np
+
+
+DEFAULT_SIZES = (10, 25, 50, 100, 250, 500, 1000)
+DEFAULT_SEEDS = (1729, 2718, 31415)
+FAMILY_OFFSETS = {
+    "dense": 10_000_000,
+    "low-outdegree": 20_000_000,
+    "absorbing": 30_000_000,
+    "reducible": 40_000_000,
+    "periodic": 50_000_000,
+}
+
+
+def normalized_integer_weights(rng: np.random.Generator, size: int) -> np.ndarray:
+    values = rng.integers(1, 1001, size=size, dtype=np.int64).astype(np.float64)
+    return values / np.sum(values, dtype=np.float64)
+
+
+def make_dense(n: int, rng: np.random.Generator) -> tuple[np.ndarray, list[int], list[int]]:
+    matrix = np.vstack([normalized_integer_weights(rng, n) for _ in range(n)])
+    width = max(1, n // 20)
+    targets = list(range(n // 2, min(n, n // 2 + width)))
+    return matrix, targets, []
+
+
+def make_low_outdegree(
+    n: int, rng: np.random.Generator
+) -> tuple[np.ndarray, list[int], list[int]]:
+    matrix = np.zeros((n, n), dtype=np.float64)
+    degree = min(4, n)
+    for source in range(n):
+        destinations = {source, (source + 1) % n}
+        while len(destinations) < degree:
+            destinations.add(int(rng.integers(0, n)))
+        ordered = sorted(destinations)
+        matrix[source, ordered] = normalized_integer_weights(rng, len(ordered))
+    width = max(1, n // 20)
+    targets = list(range(n // 2, min(n, n // 2 + width)))
+    return matrix, targets, []
+
+
+def make_absorbing(
+    n: int, rng: np.random.Generator
+) -> tuple[np.ndarray, list[int], list[int]]:
+    if n < 4:
+        raise ValueError("absorbing fixtures require at least four states")
+    absorbing = [n - 2, n - 1]
+    matrix = np.zeros((n, n), dtype=np.float64)
+    for source in range(n - 2):
+        # Every transient row has a positive edge to each absorber, so the
+        # transient block has spectral radius strictly below one.
+        matrix[source, :] = normalized_integer_weights(rng, n)
+    matrix[n - 2, n - 2] = 1.0
+    matrix[n - 1, n - 1] = 1.0
+    return matrix, absorbing.copy(), absorbing
+
+
+def _closed_ring(
+    matrix: np.ndarray,
+    members: list[int],
+    rng: np.random.Generator,
+) -> None:
+    for offset, source in enumerate(members):
+        destinations = sorted({source, members[(offset + 1) % len(members)]})
+        matrix[source, destinations] = normalized_integer_weights(rng, len(destinations))
+
+
+def make_reducible(
+    n: int, rng: np.random.Generator
+) -> tuple[np.ndarray, list[int], list[int]]:
+    if n < 6:
+        raise ValueError("reducible fixtures require at least six states")
+    class_size = max(2, n // 10)
+    transient_count = n - 2 * class_size
+    class_a = list(range(transient_count, transient_count + class_size))
+    class_b = list(range(transient_count + class_size, n))
+    matrix = np.zeros((n, n), dtype=np.float64)
+
+    for source in range(transient_count):
+        # A dense transient SCC avoids exponentially tiny reachability values
+        # that fall on PyDTMC's internal np.isclose threshold. Threshold-stress
+        # behavior is intentionally outside this ordinary performance suite.
+        ordered = list(range(transient_count)) + [class_a[0], class_b[0]]
+        matrix[source, ordered] = normalized_integer_weights(rng, len(ordered))
+
+    _closed_ring(matrix, class_a, rng)
+    _closed_ring(matrix, class_b, rng)
+    return matrix, class_a, []
+
+
+def make_periodic(
+    n: int, rng: np.random.Generator
+) -> tuple[np.ndarray, list[int], list[int]]:
+    del rng
+    matrix = np.zeros((n, n), dtype=np.float64)
+    for source in range(n):
+        matrix[source, (source + 1) % n] = 1.0
+    width = max(1, n // 20)
+    targets = list(range(n // 2, min(n, n // 2 + width)))
+    return matrix, targets, []
+
+
+GENERATORS: dict[
+    str, Callable[[int, np.random.Generator], tuple[np.ndarray, list[int], list[int]]]
+] = {
+    "dense": make_dense,
+    "low-outdegree": make_low_outdegree,
+    "absorbing": make_absorbing,
+    "reducible": make_reducible,
+    "periodic": make_periodic,
+}
+
+
+def make_initial(n: int, rng: np.random.Generator) -> np.ndarray:
+    return normalized_integer_weights(rng, n)
+
+
+def dataset_id(family: str, size: int, seed: int) -> str:
+    return f"{family}-n{size:04d}-s{seed:05d}"
+
+
+def generate(output_dir: Path, sizes: list[int], seeds: list[int]) -> dict[str, object]:
+    output_dir.mkdir(parents=True, exist_ok=True)
+    records: list[dict[str, object]] = []
+
+    for family, family_generator in GENERATORS.items():
+        for size in sizes:
+            for seed in seeds:
+                stream_seed = FAMILY_OFFSETS[family] + size * 1009 + seed
+                rng = np.random.Generator(np.random.PCG64(stream_seed))
+                matrix, targets, absorbing = family_generator(size, rng)
+                initial = make_initial(size, rng)
+
+                if matrix.shape != (size, size) or initial.shape != (size,):
+                    raise AssertionError("generator returned an invalid shape")
+                if not np.all(np.isfinite(matrix)) or not np.all(matrix >= 0.0):
+                    raise AssertionError("generator returned invalid probabilities")
+                if not np.allclose(matrix.sum(axis=1), 1.0, atol=1e-14, rtol=0.0):
+                    raise AssertionError("matrix rows are not stochastic")
+                if not np.isclose(initial.sum(), 1.0, atol=1e-14, rtol=0.0):
+                    raise AssertionError("initial distribution is not stochastic")
+
+                identifier = dataset_id(family, size, seed)
+                file_name = f"{identifier}.f64"
+                path = output_dir / file_name
+                payload = np.concatenate((matrix.ravel(order="C"), initial)).astype("<f8")
+                payload.tofile(path)
+                digest = hashlib.sha256(path.read_bytes()).hexdigest()
+                records.append(
+                    {
+                        "id": identifier,
+                        "family": family,
+                        "size": size,
+                        "seed": seed,
+                        "stream_seed": stream_seed,
+                        "file": file_name,
+                        "sha256": digest,
+                        "targets": targets,
+                        "absorbing": absorbing,
+                    }
+                )
+
+    return {
+        "schema_version": 1,
+        "format": {
+            "dtype": "float64",
+            "byte_order": "little",
+            "layout": "row-major",
+            "contents": ["transition_matrix[size*size]", "initial_distribution[size]"],
+        },
+        "generator": {
+            "name": "benchmark/data/generate.py",
+            "numpy_version": np.__version__,
+            "bit_generator": "PCG64",
+        },
+        "datasets": records,
+    }
+
+
+def parse_csv_ints(value: str) -> list[int]:
+    return [int(item) for item in value.split(",") if item]
+
+
+def main() -> None:
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument(
+        "--output-dir",
+        type=Path,
+        default=Path(__file__).resolve().parent / "generated",
+    )
+    parser.add_argument("--sizes", default=",".join(map(str, DEFAULT_SIZES)))
+    parser.add_argument("--seeds", default=",".join(map(str, DEFAULT_SEEDS)))
+    args = parser.parse_args()
+
+    sizes = parse_csv_ints(args.sizes)
+    seeds = parse_csv_ints(args.seeds)
+    if not sizes or any(size < 6 for size in sizes):
+        parser.error("all sizes must be integers greater than or equal to six")
+    if not seeds:
+        parser.error("at least one seed is required")
+
+    manifest = generate(args.output_dir, sizes, seeds)
+    manifest_path = args.output_dir / "manifest.json"
+    manifest_path.write_text(json.dumps(manifest, indent=2) + "\n", encoding="utf-8")
+    print(f"generated {len(manifest['datasets'])} datasets in {args.output_dir}")
+    print(f"manifest: {manifest_path}")
+
+
+if __name__ == "__main__":
+    main()
diff --git a/benchmark/data/spec.json b/benchmark/data/spec.json
new file mode 100644
--- /dev/null
+++ b/benchmark/data/spec.json
@@ -0,0 +1,21 @@
+{
+  "schema_version": 1,
+  "families": [
+    "dense",
+    "low-outdegree",
+    "absorbing",
+    "reducible",
+    "periodic"
+  ],
+  "sizes": [10, 25, 50, 100, 250, 500, 1000],
+  "seeds": [1729, 2718, 31415],
+  "power_exponents": [2, 10, 100],
+  "evolution_steps": [1, 10, 100],
+  "bounded_steps": [10, 100],
+  "simulation_steps": [10000, 100000],
+  "verification_maximum_size": 100,
+  "numeric_tolerances": {
+    "absolute": 1e-10,
+    "relative": 1e-8
+  }
+}
diff --git a/benchmark/haskell/Cases.hs b/benchmark/haskell/Cases.hs
new file mode 100644
--- /dev/null
+++ b/benchmark/haskell/Cases.hs
@@ -0,0 +1,573 @@
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE TypeApplications #-}
+
+module Cases (benchmarksFor) where
+
+import Control.DeepSeq (NFData (rnf))
+import Control.Exception (evaluate)
+import Criterion.Main
+import Data.Finite (Finite)
+import Data.Proxy (Proxy (Proxy))
+import Data.Vector.Unboxed qualified as U
+import Dataset
+import Dtmc.Analysis.Absorption qualified as Absorption
+import Dtmc.Analysis.Classification qualified as Classification
+import Dtmc.Analysis.Event (DiscreteEvent (AtMost, EqualTo, GreaterThan))
+import Dtmc.Analysis.FiniteTime qualified as FiniteTime
+import Dtmc.Analysis.HittingTime qualified as Hitting
+import Dtmc.Analysis.ReturnTime qualified as Return
+import Dtmc.Analysis.Stationary qualified as Stationary
+import Dtmc.Analysis.VisitCount qualified as VisitCount
+import Dtmc.Distribution.Vector qualified as Vector
+import Dtmc.Dynamics qualified as Dynamics
+import Dtmc.Simulation qualified as Simulation
+import Dtmc.State (finiteStates)
+import Dtmc.Transition.Matrix qualified as Matrix
+import Force
+import GHC.Exts (RealWorld)
+import GHC.TypeNats (KnownNat)
+import Numeric.Natural (Natural)
+import System.Random.MWC qualified as MWC
+
+newtype Prepared n = Prepared {unPrepared :: Matrix.TransitionMatrix (Finite n)}
+
+instance NFData (Prepared n) where
+    rnf (Prepared matrix) = checksumMatrix matrix `seq` ()
+
+data PreparedDynamics n = PreparedDynamics
+    { dynamicsMatrix :: Matrix.TransitionMatrix (Finite n)
+    , dynamicsInitial :: Vector.DistributionVector (Finite n)
+    }
+
+instance NFData (PreparedDynamics n) where
+    rnf prepared =
+        checksumMatrix (dynamicsMatrix prepared)
+            `seq` checksumVector (dynamicsInitial prepared)
+            `seq` ()
+
+newtype PreparedGen = PreparedGen {unPreparedGen :: MWC.Gen RealWorld}
+
+instance NFData PreparedGen where
+    rnf (PreparedGen generator) = generator `seq` ()
+
+data PreparedScalarLookup n = PreparedScalarLookup
+    { scalarLookup :: Finite n -> Double
+    , scalarLookupState :: Finite n
+    }
+
+instance NFData (PreparedScalarLookup n) where
+    rnf prepared =
+        scalarLookup prepared
+            `seq` scalarLookupState prepared
+            `seq` ()
+
+benchmarksFor ::
+    forall n.
+    (KnownNat n) =>
+    Proxy n ->
+    FilePath ->
+    Entry ->
+    Benchmark
+benchmarksFor _ dataRoot entry =
+    env (loadDataset @n dataRoot entry) $ \dataset ->
+        bgroup
+            (entryId entry)
+            ( commonBenchmarks entry dataset
+                ++ finiteTimeBenchmarks entry dataset
+                ++ boundedHittingBenchmarks entry dataset
+                ++ absorptionBenchmarks entry dataset
+                ++ occupationBenchmarks entry dataset
+                ++ returnBenchmarks entry dataset
+                ++ visitBenchmarks entry dataset
+                ++ simulationBenchmarks entry dataset
+            )
+
+commonBenchmarks :: forall n. (KnownNat n) => Entry -> Dataset n -> [Benchmark]
+commonBenchmarks entry dataset =
+    [ bgroup
+        "construction"
+        [ bench "public-consumed" $
+            nf (checksumMatrix . matrixFromRows @n) (datasetRows dataset)
+        ]
+    , env (preparedDynamics dataset) $ \ ~(PreparedDynamics matrix initial) ->
+        bgroup
+            "evolution"
+            [ bench "one-step" $
+                nf (checksumVector . Dynamics.evolveVector initial) matrix
+            , bench "k-10" $
+                nf (checksumVector . Dynamics.evolveVectorN 10 initial) matrix
+            , bench "k-100" $
+                nf (checksumVector . Dynamics.evolveVectorN 100 initial) matrix
+            ]
+    ]
+        ++ powerCases
+        ++ structureCases
+        ++ numericalCases
+  where
+    targets = datasetTargets dataset
+    competing = datasetCompeting dataset
+    powerCases
+        | entrySize entry > 500 = []
+        | otherwise =
+            [ env (preparedMatrix dataset) $ \ ~(Prepared matrix) ->
+                bgroup
+                    "power"
+                    [ bench (show powerValue) $
+                        nf (checksumMatrix . Matrix.power powerValue) matrix
+                    | powerValue <- [2, 10, 100]
+                    ]
+            ]
+    structureCases
+        | entryFamily entry == "dense" && entrySize entry > 500 = []
+        | otherwise =
+            [ bgroup
+                "structure"
+                ( [ bench "classes-lifecycle" $
+                    nf (classesLifecycle (Proxy @n)) (datasetRows dataset)
+                , bench "classes-cold" $
+                    perRunEnv (preparedGraph dataset) $ \prepared ->
+                        evaluate $!
+                            checksumClasses
+                                (Classification.communicatingClasses (unPrepared prepared))
+                , env (preparedWarmClasses dataset) $ \ ~(Prepared matrix) ->
+                    bgroup
+                        "classes-warm"
+                        [ bench "access" $
+                            whnf Classification.communicatingClasses matrix
+                        , bench "consumed" $
+                            nf
+                                (checksumClasses . Classification.communicatingClasses)
+                                matrix
+                        ]
+                , bench "irreducible-cold" $
+                    perRunEnv (preparedGraph dataset) $ \prepared ->
+                        evaluate $! Classification.irreducible (unPrepared prepared)
+                , env (preparedWarmIrreducible dataset) $ \ ~(Prepared matrix) ->
+                    bench "irreducible-warm" $
+                        nf Classification.irreducible matrix
+                  ]
+                    ++ periodicStructureCases
+                )
+            ]
+      where
+        periodicStructureCases
+            | entryFamily entry /= "periodic" = []
+            | otherwise =
+                [ bench "period-cold" $
+                    perRunEnv (preparedGraph dataset) $ \prepared ->
+                        evaluate $!
+                            checksumPeriod
+                                (Classification.chainPeriod (unPrepared prepared))
+                , env (preparedWarmPeriod dataset) $ \ ~(Prepared matrix) ->
+                    bench "period-warm" $
+                        nf (checksumPeriod . Classification.chainPeriod) matrix
+                , bench "cyclic-classes-cold" $
+                    perRunEnv (preparedGraph dataset) $ \prepared ->
+                        evaluate $!
+                            checksumCyclicClasses
+                                (Classification.cyclicClasses (unPrepared prepared))
+                , env (preparedWarmCyclicClasses dataset) $ \ ~(Prepared matrix) ->
+                    bench "cyclic-classes-warm" $
+                        nf
+                            (checksumCyclicClasses . Classification.cyclicClasses)
+                            matrix
+                ]
+    numericalCases
+        | entrySize entry > 500 = []
+        | otherwise =
+            [ bench "stationary" $
+                perRunEnv (preparedGraph dataset) $ \prepared ->
+                    evaluate $!
+                        checksumStationary
+                            ( eitherOrFail
+                                (Stationary.stationaryDistributions (unPrepared prepared))
+                            )
+            , bgroup
+                "hitting-probability"
+                [ bench "cold-all-states" $
+                    perRunEnv (preparedGraph dataset) $ \prepared ->
+                        evaluate $!
+                            hittingProbabilityChecksum targets (unPrepared prepared)
+                , env (preparedWarmHittingProbability dataset) $ \prepared ->
+                    bench "warm-lookup" $
+                        whnf
+                            (\fixture -> scalarLookup fixture (scalarLookupState fixture))
+                            prepared
+                ]
+            , bgroup
+                "hitting-time"
+                [ bench "cold-all-states" $
+                    perRunEnv (preparedGraph dataset) $ \prepared ->
+                        evaluate $!
+                            hittingExpectationChecksum targets (unPrepared prepared)
+                , env (preparedWarmHittingExpectation dataset) $ \prepared ->
+                    bench "warm-lookup" $
+                        whnf
+                            (\fixture -> scalarLookup fixture (scalarLookupState fixture))
+                            prepared
+                ]
+            ]
+                ++ raceCases
+      where
+        raceCases
+            | entryFamily entry `notElem` ["dense", "low-outdegree"] = []
+            | otherwise =
+                [ bench "race/forward-committor" $
+                    perRunEnv (preparedGraph dataset) $ \prepared ->
+                        evaluate $!
+                            raceProbabilityChecksum
+                                targets
+                                competing
+                                (unPrepared prepared)
+                ]
+
+finiteTimeBenchmarks :: forall n. (KnownNat n) => Entry -> Dataset n -> [Benchmark]
+finiteTimeBenchmarks entry dataset
+    | entrySize entry > 100 = []
+    | otherwise =
+        [ bench "finite-time/step" $
+            perRunEnv (preparedMatrix dataset) $ \prepared ->
+                evaluate $!
+                    FiniteTime.stepProbability
+                        (unPrepared prepared)
+                        source
+                        target
+        ]
+            ++ [ bench ("finite-time/n-step/k-" ++ show steps) $
+                    perRunEnv (preparedMatrix dataset) $ \prepared ->
+                        evaluate $!
+                            FiniteTime.nStepProbability
+                                steps
+                                (unPrepared prepared)
+                                source
+                                target
+               | steps <- [10, 100]
+               ]
+            ++ [ bench ("finite-time/observation/k-" ++ show time) $
+                    perRunEnv (preparedDynamics dataset) $ \prepared ->
+                        evaluate $!
+                            FiniteTime.probability
+                                (dynamicsInitial prepared)
+                                (dynamicsMatrix prepared)
+                                [FiniteTime.At time target]
+               | time <- [10, 100]
+               ]
+  where
+    source = benchmarkState @n
+    target = benchmarkTarget dataset
+
+boundedHittingBenchmarks :: forall n. (KnownNat n) => Entry -> Dataset n -> [Benchmark]
+boundedHittingBenchmarks entry dataset
+    | entrySize entry > 100 = []
+    | otherwise =
+        [ bench ("hitting/bounded/" ++ label ++ "/k-" ++ show horizon) $
+            perRunEnv (preparedMatrix dataset) $ \prepared ->
+                evaluate $!
+                    Hitting.probabilityGivenInitialState
+                        (event horizon)
+                        (unPrepared prepared)
+                        isTarget
+                        initial
+        | (label, event) <-
+            [ ("exact", EqualTo)
+            , ("at-most", AtMost)
+            , ("greater-than", GreaterThan)
+            ]
+        , horizon <- [10, 100]
+        ]
+  where
+    initial = benchmarkState @n
+    targets = datasetTargets dataset
+    isTarget state = state `elem` targets
+
+absorptionBenchmarks :: forall n. (KnownNat n) => Entry -> Dataset n -> [Benchmark]
+absorptionBenchmarks entry dataset
+    | entryFamily entry /= "absorbing" = []
+    | entrySize entry > 500 = []
+    | otherwise =
+        [ bench "fundamental-matrix" $
+            perRunEnv (preparedGraph dataset) $ \prepared ->
+                evaluate $!
+                    checksumFundamental
+                        ( eitherOrFail
+                            (Absorption.fundamentalMatrix (unPrepared prepared))
+                        )
+        , bench "absorption-time" $
+            perRunEnv (preparedGraph dataset) $ \prepared ->
+                evaluate $! absorptionExpectationChecksum (unPrepared prepared)
+        , bench "absorption/probabilities" $
+            perRunEnv (preparedGraph dataset) $ \prepared ->
+                evaluate $!
+                    absorptionProbabilityChecksum (unPrepared prepared)
+        ]
+
+occupationBenchmarks :: forall n. (KnownNat n) => Entry -> Dataset n -> [Benchmark]
+occupationBenchmarks entry dataset
+    | entryFamily entry `notElem` ["absorbing", "reducible"] = []
+    | entrySize entry > 250 = []
+    | otherwise =
+        [ bench "occupation-matrix" $
+            perRunEnv (preparedGraph dataset) $ \prepared ->
+                evaluate $!
+                    checksumOccupation
+                        ( eitherOrFail
+                            (VisitCount.occupationMatrix (unPrepared prepared))
+                        )
+        ]
+
+returnBenchmarks :: forall n. (KnownNat n) => Entry -> Dataset n -> [Benchmark]
+returnBenchmarks entry dataset = meanRecurrenceCase ++ boundedCases
+  where
+    meanRecurrenceCase
+        | entryFamily entry `notElem` ["dense", "low-outdegree"] = []
+        | entrySize entry > 500 = []
+        | otherwise =
+            [ bench "return/mean-recurrence" $
+                perRunEnv (preparedGraph dataset) $ \prepared ->
+                    evaluate $! returnExpectationChecksum (unPrepared prepared)
+            ]
+    boundedCases
+        | entrySize entry > 100 = []
+        | otherwise =
+            [ bench ("return/bounded/k-" ++ show bound) $
+                perRunEnv (preparedMatrix dataset) $ \prepared ->
+                    evaluate $!
+                        Return.probabilityGivenInitialState
+                            (AtMost bound)
+                            (unPrepared prepared)
+                            (benchmarkState @n)
+            | bound <- [10, 100]
+            ]
+
+visitBenchmarks :: forall n. (KnownNat n) => Entry -> Dataset n -> [Benchmark]
+visitBenchmarks entry dataset
+    | entrySize entry > 100 = []
+    | otherwise =
+        [ bench ("visits/bounded-expectation/k-" ++ show bound) $
+            perRunEnv (preparedMatrix dataset) $ \prepared ->
+                evaluate $!
+                    VisitCount.boundedExpectationGivenInitialState
+                        bound
+                        (benchmarkState @n)
+                        (unPrepared prepared)
+                        (== target)
+        | bound <- [10, 100]
+        ]
+  where
+    target = benchmarkTarget dataset
+
+simulationBenchmarks :: forall n. (KnownNat n) => Entry -> Dataset n -> [Benchmark]
+simulationBenchmarks entry dataset
+    | entryFamily entry `notElem` ["dense", "low-outdegree"] = []
+    | otherwise =
+        [ env (preparedMatrix dataset) $ \ ~(Prepared matrix) ->
+            bgroup
+                "simulation"
+                [ bench (show transitions) $
+                    perRunEnv (preparedGenerator (entrySeed entry)) $ \generator ->
+                        simulationChecksum
+                            transitions
+                            matrix
+                            (unPreparedGen generator)
+                | transitions <- [10000, 100000]
+                ]
+        ]
+
+matrixFromRows :: forall n. (KnownNat n) => [[Double]] -> Matrix.TransitionMatrix (Finite n)
+matrixFromRows = eitherOrFail . Matrix.fromRows
+
+vectorFromWeights :: forall n. (KnownNat n) => [Double] -> Vector.DistributionVector (Finite n)
+vectorFromWeights = eitherOrFail . Vector.fromList
+
+preparedMatrix :: forall n. (KnownNat n) => Dataset n -> IO (Prepared n)
+preparedMatrix dataset = do
+    let matrix = matrixFromRows @n (datasetRows dataset)
+    _ <- evaluate (checksumMatrix matrix)
+    pure (Prepared matrix)
+
+preparedDynamics :: forall n. (KnownNat n) => Dataset n -> IO (PreparedDynamics n)
+preparedDynamics dataset = do
+    Prepared matrix <- preparedMatrix dataset
+    let initial = vectorFromWeights @n (datasetInitialWeights dataset)
+    _ <- evaluate (checksumVector initial)
+    pure (PreparedDynamics matrix initial)
+
+preparedGraph :: forall n. (KnownNat n) => Dataset n -> IO (Prepared n)
+preparedGraph dataset = do
+    prepared <- preparedMatrix dataset
+    case finiteStates :: [Finite n] of
+        [] -> pure prepared
+        first : _ -> do
+            _ <- evaluate (Classification.supportEdge (unPrepared prepared) first first)
+            pure prepared
+
+preparedWarmClasses :: forall n. (KnownNat n) => Dataset n -> IO (Prepared n)
+preparedWarmClasses dataset = do
+    prepared <- preparedGraph dataset
+    _ <- evaluate (checksumClasses (Classification.communicatingClasses (unPrepared prepared)))
+    pure prepared
+
+preparedWarmIrreducible :: forall n. (KnownNat n) => Dataset n -> IO (Prepared n)
+preparedWarmIrreducible dataset = do
+    prepared <- preparedGraph dataset
+    _ <- evaluate (Classification.irreducible (unPrepared prepared))
+    pure prepared
+
+preparedWarmPeriod :: forall n. (KnownNat n) => Dataset n -> IO (Prepared n)
+preparedWarmPeriod dataset = do
+    prepared <- preparedGraph dataset
+    _ <- evaluate (checksumPeriod (Classification.chainPeriod (unPrepared prepared)))
+    pure prepared
+
+preparedWarmCyclicClasses :: forall n. (KnownNat n) => Dataset n -> IO (Prepared n)
+preparedWarmCyclicClasses dataset = do
+    prepared <- preparedGraph dataset
+    _ <- evaluate (checksumCyclicClasses (Classification.cyclicClasses (unPrepared prepared)))
+    pure prepared
+
+preparedWarmHittingProbability ::
+    forall n.
+    (KnownNat n) =>
+    Dataset n ->
+    IO (PreparedScalarLookup n)
+preparedWarmHittingProbability dataset = do
+    Prepared matrix <- preparedGraph dataset
+    let rawLookup = Hitting.eventualProbabilityGivenInitialState matrix (datasetTargets dataset)
+        lookupValue = eitherOrFail . rawLookup
+        state = benchmarkState @n
+    _ <- evaluate (lookupValue state)
+    pure (PreparedScalarLookup lookupValue state)
+
+preparedWarmHittingExpectation ::
+    forall n.
+    (KnownNat n) =>
+    Dataset n ->
+    IO (PreparedScalarLookup n)
+preparedWarmHittingExpectation dataset = do
+    Prepared matrix <- preparedGraph dataset
+    let rawLookup = Hitting.expectationGivenInitialState matrix (datasetTargets dataset)
+        lookupValue queryState = checksumExpectations [eitherOrFail (rawLookup queryState)]
+        state = benchmarkState @n
+    _ <- evaluate (lookupValue state)
+    pure (PreparedScalarLookup lookupValue state)
+
+preparedGenerator :: Int -> IO PreparedGen
+preparedGenerator seed =
+    PreparedGen
+        <$> MWC.initialize
+            ( U.fromList
+                [ fromIntegral seed
+                , fromIntegral (seed * 1664525 + 1013904223)
+                , 0x9e3779b9
+                , 0x243f6a88
+                ]
+            )
+
+classesLifecycle :: forall n. (KnownNat n) => Proxy n -> [[Double]] -> Double
+classesLifecycle _ = checksumClasses . Classification.communicatingClasses . matrixFromRows @n
+
+hittingProbabilityChecksum ::
+    forall n.
+    (KnownNat n) =>
+    [Finite n] ->
+    Matrix.TransitionMatrix (Finite n) ->
+    Double
+hittingProbabilityChecksum targets matrix =
+    weightedEither
+        [lookupProbability state | state <- finiteStates]
+  where
+    lookupProbability = Hitting.eventualProbabilityGivenInitialState matrix targets
+
+hittingExpectationChecksum ::
+    forall n.
+    (KnownNat n) =>
+    [Finite n] ->
+    Matrix.TransitionMatrix (Finite n) ->
+    Double
+hittingExpectationChecksum targets matrix =
+    checksumExpectations
+        (eitherOrFail (sequence [lookupExpectation state | state <- finiteStates]))
+  where
+    lookupExpectation = Hitting.expectationGivenInitialState matrix targets
+
+raceProbabilityChecksum ::
+    forall n.
+    (KnownNat n) =>
+    [Finite n] ->
+    [Finite n] ->
+    Matrix.TransitionMatrix (Finite n) ->
+    Double
+raceProbabilityChecksum successful competing matrix =
+    weightedEither
+        [lookupProbability state | state <- finiteStates]
+  where
+    lookupProbability =
+        Hitting.raceProbabilityGivenInitialState matrix successful competing
+
+returnExpectationChecksum ::
+    forall n.
+    (KnownNat n) =>
+    Matrix.TransitionMatrix (Finite n) ->
+    Double
+returnExpectationChecksum matrix =
+    checksumExpectations
+        (eitherOrFail (sequence [lookupExpectation state | state <- finiteStates]))
+  where
+    lookupExpectation = Return.expectationGivenInitialState matrix
+
+absorptionExpectationChecksum ::
+    forall n.
+    (KnownNat n) =>
+    Matrix.TransitionMatrix (Finite n) ->
+    Double
+absorptionExpectationChecksum matrix =
+    checksumExpectations
+        (eitherOrFail (sequence [lookupExpectation state | state <- finiteStates]))
+  where
+    lookupExpectation = Absorption.expectationGivenInitialState matrix
+
+absorptionProbabilityChecksum ::
+    forall n.
+    (KnownNat n) =>
+    Matrix.TransitionMatrix (Finite n) ->
+    Double
+absorptionProbabilityChecksum =
+    checksumAbsorption . eitherOrFail . Absorption.probabilityMatrix
+
+benchmarkState :: forall n. (KnownNat n) => Finite n
+benchmarkState =
+    case finiteStates of
+        [] -> error "benchmark requires a non-empty state space"
+        first : _ -> first
+
+benchmarkTarget :: forall n. (KnownNat n) => Dataset n -> Finite n
+benchmarkTarget dataset =
+    case datasetTargets dataset of
+        [] -> error "benchmark dataset requires at least one target"
+        target : _ -> target
+
+checksumPeriod :: Maybe Natural -> Double
+checksumPeriod = maybe 0 fromIntegral
+
+checksumCyclicClasses :: forall n. Maybe [[Finite n]] -> Double
+checksumCyclicClasses = maybe 0 (checksumStates . concat)
+
+weightedEither :: (Show problem) => [Either problem Double] -> Double
+weightedEither =
+    foldl'
+        (\total value -> total + eitherOrFail value)
+        0
+
+simulationChecksum ::
+    forall n.
+    (KnownNat n) =>
+    Natural ->
+    Matrix.TransitionMatrix (Finite n) ->
+    MWC.Gen RealWorld ->
+    IO Double
+simulationChecksum transitions matrix generator = do
+    case finiteStates :: [Finite n] of
+        [] -> error "benchmark simulation requires a non-empty state space"
+        initial : _ -> do
+            result <- Simulation.simulateMatrix transitions matrix initial generator
+            pure (checksumStates (eitherOrFail result))
diff --git a/benchmark/haskell/Dataset.hs b/benchmark/haskell/Dataset.hs
new file mode 100644
--- /dev/null
+++ b/benchmark/haskell/Dataset.hs
@@ -0,0 +1,167 @@
+{-# LANGUAGE DeriveAnyClass #-}
+{-# LANGUAGE DerivingStrategies #-}
+{-# LANGUAGE GADTs #-}
+{-# LANGUAGE OverloadedStrings #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE TypeApplications #-}
+
+module Dataset (
+    Dataset (..),
+    Entry (..),
+    Manifest (..),
+    SomeEntry (..),
+    datasetAbsorbing,
+    datasetCompeting,
+    datasetTargets,
+    loadDataset,
+    loadManifest,
+    selectEntries,
+    toSomeEntry,
+) where
+
+import Control.DeepSeq (NFData)
+import Control.Monad (replicateM, unless)
+import Data.Aeson (FromJSON (parseJSON), eitherDecodeFileStrict', withObject, (.:))
+import Data.Binary.Get (Get, getDoublele, isEmpty, runGetOrFail)
+import Data.ByteString.Lazy qualified as BL
+import Data.Finite (Finite)
+import Data.List (sortOn)
+import Data.Proxy (Proxy (Proxy))
+import Dtmc.State (finiteStates)
+import GHC.Generics (Generic)
+import GHC.TypeNats (KnownNat)
+import System.FilePath ((</>))
+
+data Entry = Entry
+    { entryId :: String
+    , entryFamily :: String
+    , entrySize :: Int
+    , entrySeed :: Int
+    , entryFile :: FilePath
+    , entrySha256 :: String
+    , entryTargetIndices :: [Int]
+    , entryAbsorbingIndices :: [Int]
+    }
+    deriving stock (Eq, Show, Generic)
+    deriving anyclass (NFData)
+
+instance FromJSON Entry where
+    parseJSON = withObject "benchmark dataset" $ \value ->
+        Entry
+            <$> value .: "id"
+            <*> value .: "family"
+            <*> value .: "size"
+            <*> value .: "seed"
+            <*> value .: "file"
+            <*> value .: "sha256"
+            <*> value .: "targets"
+            <*> value .: "absorbing"
+
+newtype Manifest = Manifest {manifestEntries :: [Entry]}
+    deriving stock (Eq, Show)
+
+instance FromJSON Manifest where
+    parseJSON = withObject "benchmark manifest" $ \value ->
+        Manifest <$> value .: "datasets"
+
+data Dataset n = Dataset
+    { datasetEntry :: Entry
+    , datasetRows :: [[Double]]
+    , datasetInitialWeights :: [Double]
+    }
+    deriving stock (Generic)
+    deriving anyclass (NFData)
+
+data SomeEntry where
+    SomeEntry :: (KnownNat n) => Proxy n -> Entry -> SomeEntry
+
+loadManifest :: FilePath -> IO Manifest
+loadManifest path = do
+    decoded <- eitherDecodeFileStrict' path
+    case decoded of
+        Left problem -> fail ("cannot decode benchmark manifest: " ++ problem)
+        Right manifest -> pure manifest
+
+selectEntries ::
+    Maybe String ->
+    Maybe Int ->
+    Maybe Int ->
+    Manifest ->
+    [Entry]
+selectEntries family size seed =
+    sortOn (\entry -> (entryFamily entry, entrySize entry, entrySeed entry))
+        . filter matches
+        . manifestEntries
+  where
+    matches entry =
+        maybe True (== entryFamily entry) family
+            && maybe True (== entrySize entry) size
+            && maybe True (== entrySeed entry) seed
+
+toSomeEntry :: Entry -> Either String SomeEntry
+toSomeEntry entry =
+    case entrySize entry of
+        10 -> Right (SomeEntry (Proxy @10) entry)
+        25 -> Right (SomeEntry (Proxy @25) entry)
+        50 -> Right (SomeEntry (Proxy @50) entry)
+        100 -> Right (SomeEntry (Proxy @100) entry)
+        250 -> Right (SomeEntry (Proxy @250) entry)
+        500 -> Right (SomeEntry (Proxy @500) entry)
+        1000 -> Right (SomeEntry (Proxy @1000) entry)
+        unsupported -> Left ("unsupported type-level benchmark size: " ++ show unsupported)
+
+loadDataset :: forall n. FilePath -> Entry -> IO (Dataset n)
+loadDataset dataRoot entry = do
+    let size = entrySize entry
+        expected = size * size + size
+        path = dataRoot </> entryFile entry
+    bytes <- BL.readFile path
+    values <-
+        case runGetOrFail (payload expected) bytes of
+            Left (_, offset, problem) ->
+                fail ("cannot decode " ++ path ++ " at byte " ++ show offset ++ ": " ++ problem)
+            Right (_, _, decoded) -> pure decoded
+    let (matrixValues, initial) = splitAt (size * size) values
+        rows = chunksOf size matrixValues
+    unless (length rows == size && all ((== size) . length) rows) $
+        fail ("invalid matrix dimensions in " ++ path)
+    pure
+        Dataset
+            { datasetEntry = entry
+            , datasetRows = rows
+            , datasetInitialWeights = initial
+            }
+
+datasetTargets :: forall n. (KnownNat n) => Dataset n -> [Finite n]
+datasetTargets dataset = statesAtIndices (entryTargetIndices (datasetEntry dataset))
+
+datasetAbsorbing :: forall n. (KnownNat n) => Dataset n -> [Finite n]
+datasetAbsorbing dataset = statesAtIndices (entryAbsorbingIndices (datasetEntry dataset))
+
+datasetCompeting :: forall n. (KnownNat n) => Dataset n -> [Finite n]
+datasetCompeting dataset =
+    take (max 1 (length targets)) [state | state <- finiteStates, state `notElem` targets]
+  where
+    targets = datasetTargets dataset
+
+statesAtIndices :: forall n. (KnownNat n) => [Int] -> [Finite n]
+statesAtIndices = map stateAtIndex
+  where
+    states = finiteStates :: [Finite n]
+    stateAtIndex index
+        | index < 0 || index >= length states =
+            error ("target index outside state space: " ++ show index)
+        | otherwise = states !! index
+
+payload :: Int -> Get [Double]
+payload count = do
+    values <- replicateM count getDoublele
+    empty <- isEmpty
+    unless empty (fail "trailing bytes")
+    pure values
+
+chunksOf :: Int -> [value] -> [[value]]
+chunksOf _ [] = []
+chunksOf width values =
+    let (prefix, suffix) = splitAt width values
+     in prefix : chunksOf width suffix
diff --git a/benchmark/haskell/Force.hs b/benchmark/haskell/Force.hs
new file mode 100644
--- /dev/null
+++ b/benchmark/haskell/Force.hs
@@ -0,0 +1,77 @@
+{-# LANGUAGE ScopedTypeVariables #-}
+
+module Force (
+    checksumClasses,
+    checksumAbsorption,
+    checksumExpectations,
+    checksumFundamental,
+    checksumMatrix,
+    checksumOccupation,
+    checksumStationary,
+    checksumStates,
+    checksumVector,
+    eitherOrFail,
+) where
+
+import Data.Finite (Finite, getFinite)
+import Dtmc.Analysis.Classification (CommClass (..))
+import Dtmc.Analysis.Expectation (Expectation (..))
+import Dtmc.Distribution.Vector qualified as Vector
+import Dtmc.State (FiniteState, stateIndex)
+import Dtmc.Transition.Matrix qualified as Matrix
+
+checksumValues :: [Double] -> Double
+checksumValues = foldl' (+) 0
+
+checksumVector :: Vector.DistributionVector state -> Double
+checksumVector = checksumValues . Vector.toList
+
+checksumMatrix :: Matrix.TransitionMatrix state -> Double
+checksumMatrix = checksumValues . concat . Matrix.toRows
+
+checksumClasses :: (FiniteState state) => [CommClass state] -> Double
+checksumClasses classes =
+    checksumValues
+        [ fromIntegral (getFinite (stateIndex member))
+        | communicatingClass <- classes
+        , member <- classMembers communicatingClass
+        ]
+
+checksumStates :: [Finite n] -> Double
+checksumStates =
+    checksumValues . map (fromIntegral . getFinite)
+
+checksumExpectations :: [Expectation] -> Double
+checksumExpectations = foldl' step 0
+  where
+    step total expectation =
+        let value = case expectation of
+                FiniteExpectation finite -> finite
+                InfiniteExpectation -> 1
+            next = total + value
+         in next
+
+checksumFundamental :: ([state], [[Double]]) -> Double
+checksumFundamental (_, rows) = checksumValues (concat rows)
+
+checksumAbsorption :: ([state], [state], [[Double]]) -> Double
+checksumAbsorption (_, _, rows) = checksumValues (concat rows)
+
+checksumOccupation :: [[Expectation]] -> Double
+checksumOccupation = checksumExpectations . concat
+
+checksumStationary ::
+    (FiniteState state) =>
+    [([state], Vector.DistributionVector state)] ->
+    Double
+checksumStationary = foldl' step 0
+  where
+    step total (members, distribution) =
+        let memberTotal =
+                checksumValues
+                    (map (fromIntegral . getFinite . stateIndex) members)
+            next = total + memberTotal + checksumVector distribution
+         in next
+
+eitherOrFail :: (Show problem) => Either problem value -> value
+eitherOrFail = either (error . ("benchmark operation failed: " ++) . show) id
diff --git a/benchmark/haskell/Main.hs b/benchmark/haskell/Main.hs
new file mode 100644
--- /dev/null
+++ b/benchmark/haskell/Main.hs
@@ -0,0 +1,45 @@
+{-# LANGUAGE GADTs #-}
+
+module Main (main) where
+
+import Cases (benchmarksFor)
+import Control.Monad (when)
+import Criterion.Main (defaultMain)
+import Criterion.Types (Benchmark)
+import Data.Maybe (fromMaybe)
+import Dataset
+import System.Environment (getArgs, lookupEnv)
+import Text.Read (readMaybe)
+import Verification (writeVerification)
+
+main :: IO ()
+main = do
+    arguments <- getArgs
+    dataRoot <- fromMaybe "benchmark/data/generated" <$> lookupEnv "DTMC_BENCH_DATA"
+    manifest <- loadManifest (dataRoot ++ "/manifest.json")
+    family <- lookupEnv "DTMC_BENCH_FAMILY"
+    size <- readEnvironment "DTMC_BENCH_SIZE"
+    seed <- readEnvironment "DTMC_BENCH_SEED"
+    let selected = selectEntries family size seed manifest
+    when (null selected) $
+        fail "no datasets matched DTMC_BENCH_FAMILY/DTMC_BENCH_SIZE/DTMC_BENCH_SEED"
+    typed <- traverse (either fail pure . toSomeEntry) selected
+    case arguments of
+        ["--verify-json", outputPath] -> do
+            maximumSize <- fromMaybe 100 <$> readEnvironment "DTMC_VERIFY_MAX_SIZE"
+            writeVerification outputPath dataRoot maximumSize typed
+        _ -> defaultMain (map (benchmarkFor dataRoot) typed)
+
+benchmarkFor :: FilePath -> SomeEntry -> Benchmark
+benchmarkFor dataRoot (SomeEntry proxy entry) =
+    benchmarksFor proxy dataRoot entry
+
+readEnvironment :: (Read value) => String -> IO (Maybe value)
+readEnvironment name = do
+    raw <- lookupEnv name
+    case raw of
+        Nothing -> pure Nothing
+        Just value ->
+            case readMaybe value of
+                Nothing -> fail ("environment variable " ++ name ++ " is invalid: " ++ value)
+                Just parsed -> pure (Just parsed)
diff --git a/benchmark/haskell/Verification.hs b/benchmark/haskell/Verification.hs
new file mode 100644
--- /dev/null
+++ b/benchmark/haskell/Verification.hs
@@ -0,0 +1,213 @@
+{-# LANGUAGE GADTs #-}
+{-# LANGUAGE OverloadedStrings #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE TypeApplications #-}
+
+module Verification (writeVerification) where
+
+import Data.Aeson (Value (Null), encode, object, toJSON, (.=))
+import Data.ByteString.Lazy qualified as BL
+import Data.Finite (Finite, getFinite)
+import Data.List (transpose)
+import Data.Proxy (Proxy)
+import Dataset
+import Dtmc.Analysis.Absorption qualified as Absorption
+import Dtmc.Analysis.Classification qualified as Classification
+import Dtmc.Analysis.Event (DiscreteEvent (AtMost, EqualTo, GreaterThan))
+import Dtmc.Analysis.Expectation (Expectation (..))
+import Dtmc.Analysis.FiniteTime qualified as FiniteTime
+import Dtmc.Analysis.HittingTime qualified as Hitting
+import Dtmc.Analysis.ReturnTime qualified as Return
+import Dtmc.Analysis.Stationary qualified as Stationary
+import Dtmc.Analysis.VisitCount qualified as VisitCount
+import Dtmc.Distribution.Vector qualified as Vector
+import Dtmc.Dynamics qualified as Dynamics
+import Dtmc.State (finiteStates)
+import Dtmc.Transition.Matrix qualified as Matrix
+import Force (eitherOrFail)
+import GHC.TypeNats (KnownNat)
+
+writeVerification :: FilePath -> FilePath -> Int -> [SomeEntry] -> IO ()
+writeVerification outputPath dataRoot maxSize entries = do
+    results <- traverse (verifySome dataRoot maxSize) entries
+    BL.writeFile
+        outputPath
+        ( encode
+            ( object
+                [ "schema_version" .= (1 :: Int)
+                , "maximum_size" .= maxSize
+                , "datasets" .= results
+                ]
+            )
+        )
+
+verifySome :: FilePath -> Int -> SomeEntry -> IO Value
+verifySome dataRoot maxSize (SomeEntry (_ :: Proxy n) entry)
+    | entrySize entry > maxSize =
+        pure
+            ( object
+                [ "id" .= entryId entry
+                , "size" .= entrySize entry
+                , "skipped" .= True
+                ]
+            )
+    | otherwise = verifyDataset <$> loadDataset @n dataRoot entry
+
+verifyDataset :: forall n. (KnownNat n) => Dataset n -> Value
+verifyDataset dataset =
+    object
+        [ "id" .= entryId entry
+        , "family" .= family
+        , "size" .= entrySize entry
+        , "seed" .= entrySeed entry
+        , "sha256" .= entrySha256 entry
+        , "evolve_1" .= Vector.toList (Dynamics.evolveVector initial matrix)
+        , "evolve_10" .= Vector.toList (Dynamics.evolveVectorN 10 initial matrix)
+        , "power_10" .= Matrix.toRows (Matrix.power 10 matrix)
+        , "finite_time_step" .= FiniteTime.stepProbability matrix initialState visitTarget
+        , "finite_time_n_step_10" .= finiteTimeNStep 10
+        , "finite_time_n_step_100" .= finiteTimeNStep 100
+        , "finite_time_observation_10" .= finiteTimeObservation 10
+        , "finite_time_observation_100" .= finiteTimeObservation 100
+        , "classes" .= classes
+        , "irreducible" .= Classification.irreducible matrix
+        , "stationary" .= stationary
+        , "hitting_probability" .= hittingProbabilities
+        , "hitting_time" .= map expectationValue hittingTimes
+        , "hitting_bounded_exact_10" .= hittingBounded EqualTo 10
+        , "hitting_bounded_exact_100" .= hittingBounded EqualTo 100
+        , "hitting_bounded_at_most_10" .= hittingBounded AtMost 10
+        , "hitting_bounded_at_most_100" .= hittingBounded AtMost 100
+        , "hitting_bounded_greater_than_10" .= hittingBounded GreaterThan 10
+        , "hitting_bounded_greater_than_100" .= hittingBounded GreaterThan 100
+        , "race_probability" .= raceProbabilities
+        , "return_mean" .= map expectationValue returnMeans
+        , "return_bounded_10" .= returnBounded 10
+        , "return_bounded_100" .= returnBounded 100
+        , "visits_bounded_expectation_10" .= visitsBounded 10
+        , "visits_bounded_expectation_100" .= visitsBounded 100
+        , "chain_period" .= chainPeriod
+        , "cyclic_classes" .= cyclicClasses
+        , "fundamental" .= fundamental
+        , "absorption_time" .= absorptionTimes
+        , "absorption_probability" .= absorptionProbabilities
+        , "occupation" .= occupation
+        ]
+  where
+    entry = datasetEntry dataset
+    family = entryFamily entry
+    matrix :: Matrix.TransitionMatrix (Finite n)
+    matrix = eitherOrFail (Matrix.fromRows (datasetRows dataset))
+    initial :: Vector.DistributionVector (Finite n)
+    initial = eitherOrFail (Vector.fromList (datasetInitialWeights dataset))
+    targets = datasetTargets dataset
+    competing = datasetCompeting dataset
+    states = finiteStates :: [Finite n]
+    initialState =
+        case states of
+            [] -> error "verification requires a non-empty state space"
+            first : _ -> first
+    visitTarget =
+        case targets of
+            [] -> error "verification requires at least one target"
+            target : _ -> target
+    finiteTimeNStep steps =
+        FiniteTime.nStepProbability steps matrix initialState visitTarget
+    finiteTimeObservation time =
+        FiniteTime.probability
+            initial
+            matrix
+            [FiniteTime.At time visitTarget]
+    hittingBounded event horizon =
+        Hitting.probabilityGivenInitialState
+            (event horizon)
+            matrix
+            isTarget
+            initialState
+    isTarget state = state `elem` targets
+    index :: Finite n -> Int
+    index = fromIntegral . getFinite
+    classes =
+        [ map index (Classification.classMembers communicatingClass)
+        | communicatingClass <- Classification.communicatingClasses matrix
+        ]
+    stationary =
+        [ object
+            [ "members" .= map index members
+            , "weights" .= Vector.toList distribution
+            ]
+        | (members, distribution) <-
+            eitherOrFail (Stationary.stationaryDistributions matrix)
+        ]
+    hittingProbabilityAt = Hitting.eventualProbabilityGivenInitialState matrix targets
+    hittingProbabilities =
+        eitherOrFail (sequence [hittingProbabilityAt state | state <- states])
+    hittingTimeAt = Hitting.expectationGivenInitialState matrix targets
+    hittingTimes = eitherOrFail (sequence [hittingTimeAt state | state <- states])
+    raceProbabilities
+        | family `notElem` ["dense", "low-outdegree"] = Null
+        | otherwise =
+            let atState = Hitting.raceProbabilityGivenInitialState matrix targets competing
+             in toJSON (eitherOrFail (sequence [atState state | state <- states]))
+    returnMeanAt = Return.expectationGivenInitialState matrix
+    returnMeans = eitherOrFail (sequence [returnMeanAt state | state <- states])
+    returnBounded bound =
+        Return.probabilityGivenInitialState (AtMost bound) matrix initialState
+    visitsBounded bound =
+        VisitCount.boundedExpectationGivenInitialState
+            bound
+            initialState
+            matrix
+            (== visitTarget)
+    chainPeriod
+        | family /= "periodic" = Null
+        | otherwise = toJSON (Classification.chainPeriod matrix)
+    cyclicClasses
+        | family /= "periodic" = Null
+        | otherwise =
+            case Classification.cyclicClasses matrix of
+                Nothing -> Null
+                Just groups -> toJSON (map (map index) groups)
+    fundamental
+        | family /= "absorbing" = Null
+        | otherwise =
+            let (transient, values) = eitherOrFail (Absorption.fundamentalMatrix matrix)
+             in object ["states" .= map index transient, "values" .= values]
+    absorptionTimes
+        | family /= "absorbing" = Null
+        | otherwise =
+            let atState = Absorption.expectationGivenInitialState matrix
+                values = eitherOrFail (sequence [atState state | state <- states])
+             in toJSONExpectations values
+    absorptionProbabilities
+        | family /= "absorbing" = Null
+        | otherwise =
+            let (transient, absorbing, rows) =
+                    eitherOrFail (Absorption.probabilityMatrix matrix)
+                columns
+                    | null rows = replicate (length absorbing) []
+                    | otherwise = transpose rows
+             in object
+                    [ "absorbing" .= map index absorbing
+                    , "transient" .= map index transient
+                    , "values" .= columns
+                    ]
+    occupation
+        | family `notElem` ["absorbing", "reducible"] = Null
+        | otherwise =
+            let values = eitherOrFail (VisitCount.occupationMatrix matrix)
+             in toJSONExpectationMatrix values
+
+expectationValue :: Expectation -> Value
+expectationValue expectation =
+    case expectation of
+        FiniteExpectation value -> object ["finite" .= value]
+        InfiniteExpectation -> Null
+
+toJSONExpectations :: [Expectation] -> Value
+toJSONExpectations =
+    toJSON . map expectationValue
+
+toJSONExpectationMatrix :: [[Expectation]] -> Value
+toJSONExpectationMatrix =
+    toJSON . map (map expectationValue)
diff --git a/benchmark/plot.py b/benchmark/plot.py
new file mode 100644
--- /dev/null
+++ b/benchmark/plot.py
@@ -0,0 +1,128 @@
+#!/usr/bin/env python3
+"""Create runtime and speed-ratio plots from aggregated benchmark CSV files."""
+
+from __future__ import annotations
+
+import argparse
+import csv
+import statistics
+from collections import defaultdict
+from pathlib import Path
+
+import matplotlib
+
+matplotlib.use("Agg")
+import matplotlib.pyplot as plt
+
+
+ROOT = Path(__file__).resolve().parent
+RESULTS = ROOT / "results"
+
+
+def safe_name(value: str) -> str:
+    return "".join(character if character.isalnum() else "-" for character in value).strip("-")
+
+
+def plot_runtime(rows: list[dict[str, str]], plots: Path) -> int:
+    grouped: dict[tuple[str, str, str, int], list[float]] = defaultdict(list)
+    for row in rows:
+        grouped[
+            (row["family"], row["operation"], row["engine"], int(row["size"]))
+        ].append(float(row["median_seconds"]))
+
+    combinations = sorted({(family, operation) for family, operation, _, _ in grouped})
+    for family, operation in combinations:
+        fig, axis = plt.subplots(figsize=(7.2, 4.5), constrained_layout=True)
+        drawn = False
+        for engine, color in (("dtmc", "#3366cc"), ("pydtmc", "#dc3912")):
+            points = sorted(
+                (
+                    size,
+                    statistics.median(values),
+                )
+                for (candidate_family, candidate_operation, candidate_engine, size), values in grouped.items()
+                if candidate_family == family
+                and candidate_operation == operation
+                and candidate_engine == engine
+            )
+            if points:
+                axis.plot(
+                    [point[0] for point in points],
+                    [point[1] for point in points],
+                    marker="o",
+                    color=color,
+                    label=engine,
+                )
+                drawn = True
+        if not drawn:
+            plt.close(fig)
+            continue
+        axis.set_xscale("log")
+        axis.set_yscale("log")
+        axis.set_xlabel("states")
+        axis.set_ylabel("median seconds")
+        axis.set_title(f"{family}: {operation}")
+        axis.grid(True, which="both", alpha=0.25)
+        axis.legend()
+        fig.savefig(plots / f"{safe_name(family)}--{safe_name(operation)}.svg")
+        plt.close(fig)
+    return len(combinations)
+
+
+def plot_ratios(rows: list[dict[str, str]], plots: Path) -> int:
+    grouped: dict[tuple[str, str], list[dict[str, str]]] = defaultdict(list)
+    for row in rows:
+        grouped[(row["family"], row["operation"])].append(row)
+
+    for (family, operation), group in sorted(grouped.items()):
+        ordered = sorted(group, key=lambda row: int(row["size"]))
+        sizes = [int(row["size"]) for row in ordered]
+        ratios = [float(row["geometric_median_pydtmc_over_dtmc"]) for row in ordered]
+        lows = [float(row["ci95_low"]) for row in ordered]
+        highs = [float(row["ci95_high"]) for row in ordered]
+
+        fig, axis = plt.subplots(figsize=(7.2, 4.5), constrained_layout=True)
+        axis.plot(sizes, ratios, marker="o", color="#6a3d9a", label="PyDTMC / dtmc")
+        axis.fill_between(sizes, lows, highs, color="#6a3d9a", alpha=0.18, label="95% CI")
+        axis.axhline(1.0, color="#555555", linestyle="--", linewidth=1.0)
+        axis.set_xscale("log")
+        axis.set_yscale("log")
+        axis.set_xlabel("states")
+        axis.set_ylabel("median speed ratio")
+        axis.set_title(f"{family}: {operation} speed ratio")
+        axis.grid(True, which="both", alpha=0.25)
+        axis.legend()
+        fig.savefig(
+            plots / f"{safe_name(family)}--{safe_name(operation)}--speed-ratio.svg"
+        )
+        plt.close(fig)
+    return len(grouped)
+
+
+def read_csv(path: Path) -> list[dict[str, str]]:
+    with path.open(newline="", encoding="utf-8") as stream:
+        return list(csv.DictReader(stream))
+
+
+def main() -> None:
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument("--results-root", type=Path, default=RESULTS)
+    args = parser.parse_args()
+    results = args.results_root.resolve()
+
+    summary_path = results / "summary.csv"
+    ratio_path = results / "ratio-summary.csv"
+    if not summary_path.exists() or not ratio_path.exists():
+        raise SystemExit("run benchmark/analyse.py before plotting")
+
+    plots = results / "plots"
+    plots.mkdir(parents=True, exist_ok=True)
+    for stale in plots.glob("*.svg"):
+        stale.unlink()
+    runtime_count = plot_runtime(read_csv(summary_path), plots)
+    ratio_count = plot_ratios(read_csv(ratio_path), plots)
+    print(f"wrote {runtime_count} runtime and {ratio_count} ratio plots to {plots}")
+
+
+if __name__ == "__main__":
+    main()
diff --git a/benchmark/python/bench.py b/benchmark/python/bench.py
new file mode 100644
--- /dev/null
+++ b/benchmark/python/bench.py
@@ -0,0 +1,585 @@
+#!/usr/bin/env python3
+"""Pyperf benchmarks matching the public dtmc benchmark operations."""
+
+from __future__ import annotations
+
+import hashlib
+import json
+import os
+from dataclasses import dataclass
+from pathlib import Path
+from time import perf_counter
+from typing import Any, Callable
+
+import numpy as np
+import pyperf
+from pydtmc import MarkovChain
+
+
+@dataclass(frozen=True)
+class Dataset:
+    metadata: dict[str, Any]
+    matrix: np.ndarray
+    initial: np.ndarray
+
+    @property
+    def name(self) -> str:
+        return str(self.metadata["id"])
+
+    @property
+    def family(self) -> str:
+        return str(self.metadata["family"])
+
+    @property
+    def size(self) -> int:
+        return int(self.metadata["size"])
+
+    @property
+    def seed(self) -> int:
+        return int(self.metadata["seed"])
+
+    @property
+    def targets(self) -> list[int]:
+        return [int(value) for value in self.metadata["targets"]]
+
+    @property
+    def absorbing(self) -> list[int]:
+        return [int(value) for value in self.metadata["absorbing"]]
+
+    @property
+    def competing(self) -> list[int]:
+        targets = set(self.targets)
+        return [state for state in range(self.size) if state not in targets][
+            : max(1, len(targets))
+        ]
+
+
+def data_root() -> Path:
+    return Path(os.environ.get("DTMC_BENCH_DATA", "benchmark/data/generated")).resolve()
+
+
+def load_selected_dataset() -> Dataset:
+    root = data_root()
+    manifest = json.loads((root / "manifest.json").read_text(encoding="utf-8"))
+    family = os.environ.get("DTMC_BENCH_FAMILY")
+    size = os.environ.get("DTMC_BENCH_SIZE")
+    seed = os.environ.get("DTMC_BENCH_SEED")
+    matches = [
+        entry
+        for entry in manifest["datasets"]
+        if (family is None or entry["family"] == family)
+        and (size is None or entry["size"] == int(size))
+        and (seed is None or entry["seed"] == int(seed))
+    ]
+    if len(matches) != 1:
+        raise RuntimeError(
+            "select exactly one dataset with DTMC_BENCH_FAMILY, "
+            "DTMC_BENCH_SIZE, and DTMC_BENCH_SEED"
+        )
+    entry = matches[0]
+    payload_path = root / entry["file"]
+    digest = hashlib.sha256(payload_path.read_bytes()).hexdigest()
+    if digest != entry["sha256"]:
+        raise RuntimeError(f"SHA-256 mismatch for {payload_path}")
+    n = int(entry["size"])
+    values = np.fromfile(payload_path, dtype="<f8")
+    expected = n * n + n
+    if values.size != expected:
+        raise RuntimeError(f"{payload_path} contains {values.size} values, expected {expected}")
+    matrix = values[: n * n].reshape((n, n)).copy()
+    initial = values[n * n :].copy()
+    matrix.flags.writeable = False
+    initial.flags.writeable = False
+    return Dataset(entry, matrix, initial)
+
+
+def checksum_array(value: Any) -> float:
+    array = np.asarray(value, dtype=np.float64).ravel(order="C")
+    return float(np.sum(array, dtype=np.float64))
+
+
+def consume_chain(chain: MarkovChain) -> float:
+    return checksum_array(chain.p)
+
+
+def consume_classes(classes: list[list[str]]) -> int:
+    return sum(int(state) - 1 for group in classes for state in group)
+
+
+def consume_classes_access(classes: list[list[str]]) -> bool:
+    return classes is not None
+
+
+def consume_bool(value: bool) -> int:
+    return int(value)
+
+
+def consume_float(value: float) -> float:
+    return float(value)
+
+
+def consume_sequence(value: list[int]) -> int:
+    return sum(map(int, value))
+
+
+def consume_optional_array(value: Any) -> float:
+    if value is None:
+        raise RuntimeError("PyDTMC operation unexpectedly returned None")
+    return checksum_array(value)
+
+
+def occupation_including_initial(chain: MarkovChain) -> np.ndarray:
+    """Adapt PyDTMC's positive-time visit counts to dtmc's time-zero convention."""
+    values = chain.mean_number_visits()
+    if values is None:
+        raise RuntimeError("PyDTMC operation unexpectedly returned None")
+    adjusted = np.asarray(values, dtype=np.float64).copy()
+    adjusted.flat[:: adjusted.shape[0] + 1] += 1.0
+    return adjusted
+
+
+def bounded_return_probability(chain: MarkovChain, steps: int) -> float:
+    probabilities = chain.first_passage_probabilities(steps, 0, [0])
+    return float(np.sum(probabilities, dtype=np.float64))
+
+
+def finite_time_probability(
+    chain: MarkovChain,
+    steps: int,
+    initial: np.ndarray,
+    destination: int,
+) -> float:
+    distribution = chain.redistribute(
+        steps,
+        initial_status=initial,
+        output_last=True,
+    )
+    return float(distribution[destination])
+
+
+def bounded_hitting_masses(
+    chain: MarkovChain,
+    steps: int,
+    initial_state: int,
+    targets: list[int],
+) -> np.ndarray:
+    return np.asarray(
+        chain.first_passage_probabilities(steps, initial_state, targets),
+        dtype=np.float64,
+    )
+
+
+def target_absorbing_matrix(
+    matrix: np.ndarray,
+    targets: list[int],
+) -> np.ndarray:
+    """Preserve first hits while making PyDTMC's target events disjoint."""
+    adjusted = matrix.copy()
+    adjusted[targets, :] = 0.0
+    adjusted[targets, targets] = 1.0
+    adjusted.flags.writeable = False
+    return adjusted
+
+
+def exact_hitting_probability(
+    chain: MarkovChain,
+    steps: int,
+    initial_state: int,
+    targets: list[int],
+) -> float:
+    masses = bounded_hitting_masses(chain, steps, initial_state, targets)
+    return float(masses[-1])
+
+
+def bounded_hitting_probability(
+    chain: MarkovChain,
+    steps: int,
+    initial_state: int,
+    targets: list[int],
+) -> float:
+    masses = bounded_hitting_masses(chain, steps, initial_state, targets)
+    return float(np.sum(masses, dtype=np.float64))
+
+
+def upper_hitting_probability(
+    chain: MarkovChain,
+    steps: int,
+    initial_state: int,
+    targets: list[int],
+) -> float:
+    return 1.0 - bounded_hitting_probability(
+        chain,
+        steps,
+        initial_state,
+        targets,
+    )
+
+
+def bounded_visit_expectation(
+    chain: MarkovChain, steps: int, rewards: np.ndarray
+) -> float:
+    return float(chain.expected_rewards(steps - 1, rewards)[0])
+
+
+def fresh_chain(dataset: Dataset) -> MarkovChain:
+    return MarkovChain(dataset.matrix)
+
+
+def operation_selected(name: str) -> bool:
+    selected_many = os.environ.get("DTMC_BENCH_OPERATIONS")
+    if selected_many is not None:
+        operations = [item for item in selected_many.split(",") if item]
+        return any(name.endswith(f"/{operation}") for operation in operations)
+
+    selected = os.environ.get("DTMC_BENCH_OPERATION")
+    return selected is None or selected in name
+
+
+def register_fresh(
+    runner: pyperf.Runner,
+    name: str,
+    setup: Callable[[], Any],
+    operation: Callable[[Any], Any],
+    consume: Callable[[Any], Any],
+) -> None:
+    """Exclude fixture setup while still using a fresh cache for every sample."""
+    if not operation_selected(name):
+        return
+
+    def timed(loops: int) -> float:
+        elapsed = 0.0
+        for _ in range(loops):
+            fixture = setup()
+            started = perf_counter()
+            result = operation(fixture)
+            consume(result)
+            elapsed += perf_counter() - started
+        return elapsed
+
+    runner.bench_time_func(name, timed)
+
+
+def register_warm(
+    runner: pyperf.Runner,
+    name: str,
+    fixture: Any,
+    operation: Callable[[Any], Any],
+    consume: Callable[[Any], Any],
+) -> None:
+    if not operation_selected(name):
+        return
+    inner_loops = 10_000
+
+    def timed(loops: int) -> float:
+        started = perf_counter()
+        for _ in range(loops * inner_loops):
+            consume(operation(fixture))
+        return perf_counter() - started
+
+    runner.bench_time_func(name, timed, inner_loops=inner_loops)
+
+
+def register_benchmarks(runner: pyperf.Runner, dataset: Dataset) -> None:
+    prefix = dataset.name
+    matrix = dataset.matrix
+    initial = dataset.initial
+    targets = dataset.targets
+    competing = dataset.competing
+    target = targets[0]
+    hitting_matrix = target_absorbing_matrix(matrix, targets)
+    point_initial = np.zeros(dataset.size, dtype=np.float64)
+    point_initial[0] = 1.0
+    point_initial.flags.writeable = False
+
+    register_fresh(
+        runner,
+        f"{prefix}/construction/public-consumed",
+        lambda: matrix,
+        MarkovChain,
+        consume_chain,
+    )
+
+    for steps in (1, 10, 100):
+        label = "one-step" if steps == 1 else f"k-{steps}"
+        register_fresh(
+            runner,
+            f"{prefix}/evolution/{label}",
+            lambda: fresh_chain(dataset),
+            lambda chain, count=steps: chain.redistribute(
+                count, initial_status=initial, output_last=True
+            ),
+            checksum_array,
+        )
+
+    if dataset.size <= 100:
+        register_fresh(
+            runner,
+            f"{prefix}/finite-time/step",
+            lambda: fresh_chain(dataset),
+            lambda chain: chain.p[0, target],
+            consume_float,
+        )
+        for steps in (10, 100):
+            register_fresh(
+                runner,
+                f"{prefix}/finite-time/n-step/k-{steps}",
+                lambda: fresh_chain(dataset),
+                lambda chain, count=steps: finite_time_probability(
+                    chain,
+                    count,
+                    point_initial,
+                    target,
+                ),
+                consume_float,
+            )
+            register_fresh(
+                runner,
+                f"{prefix}/finite-time/observation/k-{steps}",
+                lambda: fresh_chain(dataset),
+                lambda chain, count=steps: finite_time_probability(
+                    chain,
+                    count,
+                    initial,
+                    target,
+                ),
+                consume_float,
+            )
+
+    if dataset.size <= 500:
+        for exponent in (2, 10, 100):
+            register_fresh(
+                runner,
+                f"{prefix}/power/{exponent}",
+                lambda: fresh_chain(dataset),
+                lambda chain, value=exponent: chain.to_nth_order(value),
+                consume_chain,
+            )
+
+    if dataset.family != "dense" or dataset.size <= 500:
+        register_fresh(
+            runner,
+            f"{prefix}/structure/classes-lifecycle",
+            lambda: matrix,
+            lambda value: MarkovChain(value).communicating_classes,
+            consume_classes,
+        )
+        register_fresh(
+            runner,
+            f"{prefix}/structure/classes-cold",
+            lambda: fresh_chain(dataset),
+            lambda chain: chain.communicating_classes,
+            consume_classes,
+        )
+
+        warm_classes = fresh_chain(dataset)
+        consume_classes(warm_classes.communicating_classes)
+        register_warm(
+            runner,
+            f"{prefix}/structure/classes-warm/access",
+            warm_classes,
+            lambda chain: chain.communicating_classes,
+            consume_classes_access,
+        )
+        register_warm(
+            runner,
+            f"{prefix}/structure/classes-warm/consumed",
+            warm_classes,
+            lambda chain: chain.communicating_classes,
+            consume_classes,
+        )
+        register_fresh(
+            runner,
+            f"{prefix}/structure/irreducible-cold",
+            lambda: fresh_chain(dataset),
+            lambda chain: chain.is_irreducible,
+            consume_bool,
+        )
+
+        if dataset.family == "periodic":
+            register_fresh(
+                runner,
+                f"{prefix}/structure/period-cold",
+                lambda: fresh_chain(dataset),
+                lambda chain: chain.period,
+                consume_float,
+            )
+            warm_period = fresh_chain(dataset)
+            consume_float(warm_period.period)
+            register_warm(
+                runner,
+                f"{prefix}/structure/period-warm",
+                warm_period,
+                lambda chain: chain.period,
+                consume_float,
+            )
+            register_fresh(
+                runner,
+                f"{prefix}/structure/cyclic-classes-cold",
+                lambda: fresh_chain(dataset),
+                lambda chain: chain.cyclic_classes,
+                consume_classes,
+            )
+            warm_cyclic_classes = fresh_chain(dataset)
+            consume_classes(warm_cyclic_classes.cyclic_classes)
+            register_warm(
+                runner,
+                f"{prefix}/structure/cyclic-classes-warm",
+                warm_cyclic_classes,
+                lambda chain: chain.cyclic_classes,
+                consume_classes,
+            )
+
+        warm_irreducible = fresh_chain(dataset)
+        consume_bool(warm_irreducible.is_irreducible)
+        register_warm(
+            runner,
+            f"{prefix}/structure/irreducible-warm",
+            warm_irreducible,
+            lambda chain: chain.is_irreducible,
+            consume_bool,
+        )
+
+    if dataset.size <= 500:
+        register_fresh(
+            runner,
+            f"{prefix}/stationary",
+            lambda: fresh_chain(dataset),
+            lambda chain: chain.pi,
+            checksum_array,
+        )
+        register_fresh(
+            runner,
+            f"{prefix}/hitting-probability/cold-all-states",
+            lambda: fresh_chain(dataset),
+            lambda chain: chain.hitting_probabilities(targets),
+            checksum_array,
+        )
+        register_fresh(
+            runner,
+            f"{prefix}/hitting-time/cold-all-states",
+            lambda: fresh_chain(dataset),
+            lambda chain: chain.hitting_times(targets),
+            checksum_array,
+        )
+
+        if dataset.family in {"dense", "low-outdegree"}:
+            register_fresh(
+                runner,
+                f"{prefix}/race/forward-committor",
+                lambda: fresh_chain(dataset),
+                lambda chain: chain.committor_probabilities(
+                    "forward", competing, targets
+                ),
+                consume_optional_array,
+            )
+
+    if dataset.family in {"dense", "low-outdegree"} and dataset.size <= 500:
+        register_fresh(
+            runner,
+            f"{prefix}/return/mean-recurrence",
+            lambda: fresh_chain(dataset),
+            lambda chain: chain.mean_recurrence_times(),
+            consume_optional_array,
+        )
+
+    if dataset.size <= 100:
+        rewards = np.zeros(dataset.size, dtype=np.float64)
+        rewards[targets[0]] = 1.0
+        rewards.flags.writeable = False
+        for steps in (10, 100):
+            for label, operation in (
+                ("exact", exact_hitting_probability),
+                ("at-most", bounded_hitting_probability),
+                ("greater-than", upper_hitting_probability),
+            ):
+                register_fresh(
+                    runner,
+                    f"{prefix}/hitting/bounded/{label}/k-{steps}",
+                    lambda: MarkovChain(hitting_matrix),
+                    lambda chain, count=steps, query=operation: query(
+                        chain,
+                        count,
+                        0,
+                        targets,
+                    ),
+                    consume_float,
+                )
+            register_fresh(
+                runner,
+                f"{prefix}/return/bounded/k-{steps}",
+                lambda: fresh_chain(dataset),
+                lambda chain, count=steps: bounded_return_probability(chain, count),
+                consume_float,
+            )
+            register_fresh(
+                runner,
+                f"{prefix}/visits/bounded-expectation/k-{steps}",
+                lambda: fresh_chain(dataset),
+                lambda chain, count=steps: bounded_visit_expectation(
+                    chain, count, rewards
+                ),
+                consume_float,
+            )
+
+    if dataset.family == "absorbing" and dataset.size <= 500:
+        register_fresh(
+            runner,
+            f"{prefix}/fundamental-matrix",
+            lambda: fresh_chain(dataset),
+            lambda chain: chain.fundamental_matrix,
+            consume_optional_array,
+        )
+        register_fresh(
+            runner,
+            f"{prefix}/absorption/probabilities",
+            lambda: fresh_chain(dataset),
+            lambda chain: chain.absorption_probabilities(),
+            consume_optional_array,
+        )
+        register_fresh(
+            runner,
+            f"{prefix}/absorption-time",
+            lambda: fresh_chain(dataset),
+            lambda chain: chain.mean_absorption_times(),
+            consume_optional_array,
+        )
+
+    if dataset.family in {"absorbing", "reducible"} and dataset.size <= 250:
+        register_fresh(
+            runner,
+            f"{prefix}/occupation-matrix",
+            lambda: fresh_chain(dataset),
+            occupation_including_initial,
+            consume_optional_array,
+        )
+
+    if dataset.family in {"dense", "low-outdegree"}:
+        for transitions in (10_000, 100_000):
+            register_fresh(
+                runner,
+                f"{prefix}/simulation/{transitions}",
+                lambda: fresh_chain(dataset),
+                lambda chain, count=transitions: chain.simulate(
+                    count,
+                    initial_state=0,
+                    output_indices=True,
+                    seed=dataset.seed,
+                ),
+                consume_sequence,
+            )
+
+
+def main() -> None:
+    dataset = load_selected_dataset()
+    # One cold operation per value prevents pyperf calibration from recreating
+    # an expensive untimed fixture thousands of times. Three processes with ten
+    # values give 30 independent full-operation samples in the normal suite.
+    runner = pyperf.Runner(processes=3, values=10, warmups=3, loops=1)
+    runner.metadata["dataset"] = dataset.name
+    runner.metadata["dataset_sha256"] = dataset.metadata["sha256"]
+    runner.metadata["pydtmc_version"] = "9.0.0"
+    register_benchmarks(runner, dataset)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/benchmark/python/pyproject.toml b/benchmark/python/pyproject.toml
new file mode 100644
--- /dev/null
+++ b/benchmark/python/pyproject.toml
@@ -0,0 +1,15 @@
+[project]
+name = "dtmc-pydtmc-benchmark"
+version = "0.1.0"
+requires-python = ">=3.12,<3.13"
+dependencies = [
+  "matplotlib==3.10.6",
+  "networkx==3.5",
+  "numpy==2.3.1",
+  "pyperf==2.9.0",
+  "PyDTMC==9.0.0",
+  "scipy==1.16.1",
+]
+
+[tool.dtmc-benchmark]
+pydtmc-sdist-sha256 = "e3cbe0e192bf85e1941b0b8b85314a5ae273df04aaa7c168667d0bb4fee5c8ef"
diff --git a/benchmark/python/requirements.lock b/benchmark/python/requirements.lock
new file mode 100644
--- /dev/null
+++ b/benchmark/python/requirements.lock
@@ -0,0 +1,18 @@
+# Exact Python 3.12 environment resolved on 2026-09-08. PyDTMC's
+# source-distribution hash is recorded in pyproject.toml.
+contourpy==1.3.3
+cycler==0.12.1
+fonttools==4.64.0
+kiwisolver==1.5.1
+matplotlib==3.10.6
+networkx==3.5
+numpy==2.3.1
+packaging==26.3
+pillow==12.3.0
+psutil==7.2.2
+PyDTMC==9.0.0
+pyparsing==3.3.2
+pyperf==2.9.0
+python-dateutil==2.9.0.post0
+scipy==1.16.1
+six==1.17.0
diff --git a/benchmark/python/verify.py b/benchmark/python/verify.py
new file mode 100644
--- /dev/null
+++ b/benchmark/python/verify.py
@@ -0,0 +1,463 @@
+#!/usr/bin/env python3
+"""Compare untimed dtmc outputs with PyDTMC on the shared fixtures."""
+
+from __future__ import annotations
+
+import argparse
+import json
+import math
+from dataclasses import dataclass, asdict
+from pathlib import Path
+from typing import Any
+
+import numpy as np
+from pydtmc import MarkovChain
+
+from bench import (
+    Dataset,
+    bounded_hitting_probability,
+    bounded_return_probability,
+    bounded_visit_expectation,
+    data_root,
+    exact_hitting_probability,
+    finite_time_probability,
+    occupation_including_initial,
+    target_absorbing_matrix,
+    upper_hitting_probability,
+)
+
+
+@dataclass
+class Comparison:
+    dataset: str
+    operation: str
+    max_absolute_error: float
+    max_relative_error: float
+    passed: bool
+
+
+def load_dataset(root: Path, entry: dict[str, Any]) -> Dataset:
+    n = int(entry["size"])
+    values = np.fromfile(root / entry["file"], dtype="<f8")
+    if values.size != n * n + n:
+        raise RuntimeError(f"invalid payload size for {entry['id']}")
+    return Dataset(
+        entry,
+        values[: n * n].reshape((n, n)).copy(),
+        values[n * n :].copy(),
+    )
+
+
+def expectation_array(value: Any) -> np.ndarray:
+    def decode(item: Any) -> float:
+        if item is None:
+            return math.inf
+        if isinstance(item, dict) and "finite" in item:
+            return float(item["finite"])
+        if isinstance(item, list):
+            raise TypeError("nested expectation list reached scalar decoder")
+        return float(item)
+
+    if isinstance(value, list) and value and isinstance(value[0], list):
+        return np.asarray([[decode(item) for item in row] for row in value], dtype=float)
+    return np.asarray([decode(item) for item in value], dtype=float)
+
+
+def compare_numeric(
+    dataset: str,
+    operation: str,
+    haskell: Any,
+    python: Any,
+    atol: float,
+    rtol: float,
+) -> Comparison:
+    left = np.asarray(haskell, dtype=float)
+    right = np.asarray(python, dtype=float)
+    if left.shape != right.shape:
+        raise AssertionError(
+            f"{dataset}/{operation}: shape mismatch {left.shape} != {right.shape}"
+        )
+    if not np.array_equal(np.isinf(left), np.isinf(right)):
+        raise AssertionError(f"{dataset}/{operation}: infinity masks differ")
+    finite = np.isfinite(left) & np.isfinite(right)
+    if not np.array_equal(np.isnan(left), np.isnan(right)):
+        raise AssertionError(f"{dataset}/{operation}: NaN masks differ")
+    if np.any(finite):
+        absolute = np.abs(left[finite] - right[finite])
+        scale = np.maximum(np.maximum(np.abs(left[finite]), np.abs(right[finite])), 1e-300)
+        relative = absolute / scale
+        max_absolute = float(np.max(absolute))
+        max_relative = float(np.max(relative))
+        passed = bool(np.all(absolute <= atol + rtol * np.abs(right[finite])))
+    else:
+        max_absolute = 0.0
+        max_relative = 0.0
+        passed = True
+    return Comparison(dataset, operation, max_absolute, max_relative, passed)
+
+
+def state_index(label: str) -> int:
+    # Default PyDTMC labels are one-based decimal strings.
+    return int(label) - 1
+
+
+def normalized_partition(groups: list[list[int]]) -> list[list[int]]:
+    return sorted(sorted(group) for group in groups)
+
+
+def compare_dataset(
+    record: dict[str, Any],
+    dataset: Dataset,
+    atol: float,
+    rtol: float,
+) -> list[Comparison]:
+    name = dataset.name
+    chain = MarkovChain(dataset.matrix)
+    targets = dataset.targets
+    hitting_chain = MarkovChain(target_absorbing_matrix(dataset.matrix, targets))
+    rewards = np.zeros(dataset.size, dtype=np.float64)
+    rewards[targets[0]] = 1.0
+    point_initial = np.zeros(dataset.size, dtype=np.float64)
+    point_initial[0] = 1.0
+    comparisons = [
+        compare_numeric(
+            name,
+            "evolve_1",
+            record["evolve_1"],
+            chain.redistribute(1, initial_status=dataset.initial, output_last=True),
+            atol,
+            rtol,
+        ),
+        compare_numeric(
+            name,
+            "evolve_10",
+            record["evolve_10"],
+            chain.redistribute(10, initial_status=dataset.initial, output_last=True),
+            atol,
+            rtol,
+        ),
+        compare_numeric(
+            name,
+            "power_10",
+            record["power_10"],
+            chain.to_nth_order(10).p,
+            atol,
+            rtol,
+        ),
+        compare_numeric(
+            name,
+            "finite_time_step",
+            [record["finite_time_step"]],
+            [chain.p[0, targets[0]]],
+            atol,
+            rtol,
+        ),
+        compare_numeric(
+            name,
+            "finite_time_n_step_10",
+            [record["finite_time_n_step_10"]],
+            [finite_time_probability(chain, 10, point_initial, targets[0])],
+            atol,
+            rtol,
+        ),
+        compare_numeric(
+            name,
+            "finite_time_n_step_100",
+            [record["finite_time_n_step_100"]],
+            [finite_time_probability(chain, 100, point_initial, targets[0])],
+            atol,
+            rtol,
+        ),
+        compare_numeric(
+            name,
+            "finite_time_observation_10",
+            [record["finite_time_observation_10"]],
+            [finite_time_probability(chain, 10, dataset.initial, targets[0])],
+            atol,
+            rtol,
+        ),
+        compare_numeric(
+            name,
+            "finite_time_observation_100",
+            [record["finite_time_observation_100"]],
+            [finite_time_probability(chain, 100, dataset.initial, targets[0])],
+            atol,
+            rtol,
+        ),
+        compare_numeric(
+            name,
+            "hitting_probability",
+            record["hitting_probability"],
+            chain.hitting_probabilities(targets),
+            atol,
+            rtol,
+        ),
+        compare_numeric(
+            name,
+            "hitting_time",
+            expectation_array(record["hitting_time"]),
+            chain.hitting_times(targets),
+            atol,
+            rtol,
+        ),
+        compare_numeric(
+            name,
+            "hitting_bounded_exact_10",
+            [record["hitting_bounded_exact_10"]],
+            [exact_hitting_probability(hitting_chain, 10, 0, targets)],
+            atol,
+            rtol,
+        ),
+        compare_numeric(
+            name,
+            "hitting_bounded_exact_100",
+            [record["hitting_bounded_exact_100"]],
+            [exact_hitting_probability(hitting_chain, 100, 0, targets)],
+            atol,
+            rtol,
+        ),
+        compare_numeric(
+            name,
+            "hitting_bounded_at_most_10",
+            [record["hitting_bounded_at_most_10"]],
+            [bounded_hitting_probability(hitting_chain, 10, 0, targets)],
+            atol,
+            rtol,
+        ),
+        compare_numeric(
+            name,
+            "hitting_bounded_at_most_100",
+            [record["hitting_bounded_at_most_100"]],
+            [bounded_hitting_probability(hitting_chain, 100, 0, targets)],
+            atol,
+            rtol,
+        ),
+        compare_numeric(
+            name,
+            "hitting_bounded_greater_than_10",
+            [record["hitting_bounded_greater_than_10"]],
+            [upper_hitting_probability(hitting_chain, 10, 0, targets)],
+            atol,
+            rtol,
+        ),
+        compare_numeric(
+            name,
+            "hitting_bounded_greater_than_100",
+            [record["hitting_bounded_greater_than_100"]],
+            [upper_hitting_probability(hitting_chain, 100, 0, targets)],
+            atol,
+            rtol,
+        ),
+        compare_numeric(
+            name,
+            "return_bounded_10",
+            [record["return_bounded_10"]],
+            [bounded_return_probability(chain, 10)],
+            atol,
+            rtol,
+        ),
+        compare_numeric(
+            name,
+            "return_bounded_100",
+            [record["return_bounded_100"]],
+            [bounded_return_probability(chain, 100)],
+            atol,
+            rtol,
+        ),
+        compare_numeric(
+            name,
+            "visits_bounded_expectation_10",
+            [record["visits_bounded_expectation_10"]],
+            [bounded_visit_expectation(chain, 10, rewards)],
+            atol,
+            rtol,
+        ),
+        compare_numeric(
+            name,
+            "visits_bounded_expectation_100",
+            [record["visits_bounded_expectation_100"]],
+            [bounded_visit_expectation(chain, 100, rewards)],
+            atol,
+            rtol,
+        ),
+    ]
+
+    if dataset.family in {"dense", "low-outdegree"}:
+        py_race = chain.committor_probabilities(
+            "forward", dataset.competing, targets
+        )
+        if py_race is None:
+            raise AssertionError(f"{name}: PyDTMC returned no committor probabilities")
+        comparisons.append(
+            compare_numeric(
+                name,
+                "race_probability",
+                record["race_probability"],
+                py_race,
+                atol,
+                rtol,
+            )
+        )
+        py_return_mean = chain.mean_recurrence_times()
+        if py_return_mean is None:
+            raise AssertionError(f"{name}: PyDTMC returned no mean recurrence times")
+        comparisons.append(
+            compare_numeric(
+                name,
+                "return_mean",
+                expectation_array(record["return_mean"]),
+                py_return_mean,
+                atol,
+                rtol,
+            )
+        )
+
+    haskell_classes = sorted(sorted(group) for group in record["classes"])
+    python_classes = sorted(sorted(state_index(state) for state in group) for group in chain.communicating_classes)
+    if haskell_classes != python_classes:
+        raise AssertionError(f"{name}: communicating classes differ")
+    if bool(record["irreducible"]) != bool(chain.is_irreducible):
+        raise AssertionError(f"{name}: irreducibility differs")
+
+    if dataset.family == "periodic":
+        if int(record["chain_period"]) != int(chain.period):
+            raise AssertionError(f"{name}: chain periods differ")
+        haskell_cyclic = normalized_partition(record["cyclic_classes"])
+        python_cyclic = normalized_partition(
+            [[state_index(state) for state in group] for group in chain.cyclic_classes]
+        )
+        if haskell_cyclic != python_cyclic:
+            raise AssertionError(f"{name}: cyclic classes differ")
+
+    haskell_stationary = {
+        tuple(item["members"]): np.asarray(item["weights"], dtype=float)
+        for item in record["stationary"]
+    }
+    python_stationary: dict[tuple[int, ...], np.ndarray] = {}
+    for vector in chain.pi:
+        array = np.asarray(vector, dtype=float)
+        support = tuple(int(i) for i in np.flatnonzero(array > 0.0))
+        python_stationary[support] = array
+    if set(haskell_stationary) != set(python_stationary):
+        raise AssertionError(f"{name}: stationary supports differ")
+    for support, haskell_vector in haskell_stationary.items():
+        comparisons.append(
+            compare_numeric(
+                name,
+                f"stationary[{','.join(map(str, support))}]",
+                haskell_vector,
+                python_stationary[support],
+                atol,
+                rtol,
+            )
+        )
+
+    if dataset.family == "absorbing":
+        fundamental = record["fundamental"]
+        py_fundamental = chain.fundamental_matrix
+        if py_fundamental is None:
+            raise AssertionError(f"{name}: PyDTMC did not classify fixture as absorbing")
+        comparisons.append(
+            compare_numeric(
+                name,
+                "fundamental",
+                fundamental["values"],
+                py_fundamental,
+                atol,
+                rtol,
+            )
+        )
+        absorption_probability = record["absorption_probability"]
+        py_absorbing = [state_index(state) for state in chain.absorbing_states]
+        py_transient = [state_index(state) for state in chain.transient_states]
+        if absorption_probability["absorbing"] != py_absorbing:
+            raise AssertionError(f"{name}: absorbing-state order differs")
+        if absorption_probability["transient"] != py_transient:
+            raise AssertionError(f"{name}: transient-state order differs")
+        py_absorption_probability = chain.absorption_probabilities()
+        if py_absorption_probability is None:
+            raise AssertionError(f"{name}: PyDTMC returned no absorption probabilities")
+        comparisons.append(
+            compare_numeric(
+                name,
+                "absorption_probability",
+                absorption_probability["values"],
+                py_absorption_probability,
+                atol,
+                rtol,
+            )
+        )
+        transient = [int(index) for index in fundamental["states"]]
+        py_absorption = chain.mean_absorption_times()
+        if py_absorption is None:
+            raise AssertionError(f"{name}: PyDTMC returned no absorption times")
+        hs_absorption = expectation_array(record["absorption_time"])[transient]
+        comparisons.append(
+            compare_numeric(
+                name,
+                "absorption_time",
+                hs_absorption,
+                py_absorption,
+                atol,
+                rtol,
+            )
+        )
+
+    if dataset.family in {"absorbing", "reducible"}:
+        py_occupation = occupation_including_initial(chain)
+        comparisons.append(
+            compare_numeric(
+                name,
+                "occupation",
+                expectation_array(record["occupation"]),
+                py_occupation,
+                atol,
+                rtol,
+            )
+        )
+
+    return comparisons
+
+
+def main() -> None:
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument("haskell_json", type=Path)
+    parser.add_argument("--output", type=Path, required=True)
+    parser.add_argument("--atol", type=float, default=1e-10)
+    parser.add_argument("--rtol", type=float, default=1e-8)
+    args = parser.parse_args()
+
+    root = data_root()
+    manifest = json.loads((root / "manifest.json").read_text(encoding="utf-8"))
+    entries = {entry["id"]: entry for entry in manifest["datasets"]}
+    haskell = json.loads(args.haskell_json.read_text(encoding="utf-8"))
+    comparisons: list[Comparison] = []
+    skipped: list[str] = []
+    for record in haskell["datasets"]:
+        if record.get("skipped"):
+            skipped.append(record["id"])
+            continue
+        entry = entries[record["id"]]
+        comparisons.extend(
+            compare_dataset(record, load_dataset(root, entry), args.atol, args.rtol)
+        )
+
+    failed = [comparison for comparison in comparisons if not comparison.passed]
+    output = {
+        "schema_version": 1,
+        "absolute_tolerance": args.atol,
+        "relative_tolerance": args.rtol,
+        "comparisons": [asdict(comparison) for comparison in comparisons],
+        "skipped": skipped,
+        "passed": not failed,
+    }
+    args.output.parent.mkdir(parents=True, exist_ok=True)
+    args.output.write_text(json.dumps(output, indent=2) + "\n", encoding="utf-8")
+    if failed:
+        names = ", ".join(f"{item.dataset}/{item.operation}" for item in failed)
+        raise SystemExit(f"numerical verification failed: {names}")
+    print(f"verified {len(comparisons)} operation results; maximum size {haskell['maximum_size']}")
+
+
+if __name__ == "__main__":
+    main()
diff --git a/benchmark/results/environment.json b/benchmark/results/environment.json
new file mode 100644
--- /dev/null
+++ b/benchmark/results/environment.json
@@ -0,0 +1,36 @@
+{
+  "captured_at_utc": "2026-09-14T09:16:40Z",
+  "platform": "macOS-15.6.1-arm64-arm-64bit",
+  "machine": "arm64",
+  "processor": "arm",
+  "cpu_brand": "Apple M1",
+  "python": "Python 3.13.9",
+  "python_packages": "contourpy==1.3.3\ncycler==0.12.1\nfonttools==4.64.0\nkiwisolver==1.5.1\nmatplotlib==3.10.6\nnetworkx==3.5\nnumpy==2.3.1\npackaging==26.3\npillow==12.3.0\npsutil==7.2.2\nPyDTMC==9.0.0\npyparsing==3.3.2\npyperf==2.9.0\npython-dateutil==2.9.0.post0\nscipy==1.16.1\nsix==1.17.0",
+  "numpy_configuration": "/Users/a123/CodingProjects/dtmc/benchmark/.venv/lib/python3.13/site-packages/numpy/__config__.py:155: UserWarning: Install `pyyaml` for better output\n  warnings.warn(\"Install `pyyaml` for better output\", stacklevel=1)\n{\n  \"Compilers\": {\n    \"c\": {\n      \"name\": \"clang\",\n      \"linker\": \"ld64\",\n      \"version\": \"15.0.0\",\n      \"commands\": \"cc\"\n    },\n    \"cython\": {\n      \"name\": \"cython\",\n      \"linker\": \"cython\",\n      \"version\": \"3.1.2\",\n      \"commands\": \"cython\"\n    },\n    \"c++\": {\n      \"name\": \"clang\",\n      \"linker\": \"ld64\",\n      \"version\": \"15.0.0\",\n      \"commands\": \"c++\"\n    }\n  },\n  \"Machine Information\": {\n    \"host\": {\n      \"cpu\": \"aarch64\",\n      \"family\": \"aarch64\",\n      \"endian\": \"little\",\n      \"system\": \"darwin\"\n    },\n    \"build\": {\n      \"cpu\": \"aarch64\",\n      \"family\": \"aarch64\",\n      \"endian\": \"little\",\n      \"system\": \"darwin\"\n    }\n  },\n  \"Build Dependencies\": {\n    \"blas\": {\n      \"name\": \"accelerate\",\n      \"found\": true,\n      \"version\": \"unknown\",\n      \"detection method\": \"system\",\n      \"include directory\": \"unknown\",\n      \"lib directory\": \"unknown\",\n      \"openblas configuration\": \"unknown\",\n      \"pc file directory\": \"unknown\"\n    },\n    \"lapack\": {\n      \"name\": \"accelerate\",\n      \"found\": true,\n      \"version\": \"unknown\",\n      \"detection method\": \"system\",\n      \"include directory\": \"unknown\",\n      \"lib directory\": \"unknown\",\n      \"openblas configuration\": \"unknown\",\n      \"pc file directory\": \"unknown\"\n    }\n  },\n  \"Python Information\": {\n    \"path\": \"/private/var/folders/y6/nj790rtn62lfktb1sh__79hc0000gn/T/build-env-oawxqquo/bin/python\",\n    \"version\": \"3.13\"\n  },\n  \"SIMD Extensions\": {\n    \"baseline\": [\n      \"NEON\",\n      \"NEON_FP16\",\n      \"NEON_VFPV4\",\n      \"ASIMD\"\n    ],\n    \"found\": [\n      \"ASIMDHP\"\n    ],\n    \"not found\": [\n      \"ASIMDFHM\"\n    ]\n  }\n}",
+  "ghc": "The Glorious Glasgow Haskell Compilation System, version 9.10.3",
+  "cabal": "cabal-install version 3.16.1.0 \ncompiled using version 3.16.1.0 of the Cabal library (in-tree)",
+  "haskell_benchmark_binary": "/Users/a123/CodingProjects/dtmc/dist-newstyle/build/aarch64-osx/ghc-9.10.3/dtmc-0.2.0.0/b/dtmc-bench/opt/build/dtmc-bench/dtmc-bench",
+  "haskell_benchmark_linkage": "/Users/a123/CodingProjects/dtmc/dist-newstyle/build/aarch64-osx/ghc-9.10.3/dtmc-0.2.0.0/b/dtmc-bench/opt/build/dtmc-bench/dtmc-bench:\n\t/usr/lib/libSystem.B.dylib (compatibility version 1.0.0, current version 1351.0.0)\n\t/System/Library/Frameworks/Accelerate.framework/Versions/A/Frameworks/vecLib.framework/Versions/A/libBLAS.dylib (compatibility version 1.0.0, current version 1.0.0)\n\t/System/Library/Frameworks/Accelerate.framework/Versions/A/Frameworks/vecLib.framework/Versions/A/libLAPACK.dylib (compatibility version 1.0.0, current version 1.0.0)\n\t/usr/lib/libiconv.2.dylib (compatibility version 7.0.0, current version 7.0.0)\n\t/usr/lib/libffi.dylib (compatibility version 1.0.0, current version 40.0.0)\n\t/System/Library/Frameworks/Security.framework/Versions/A/Security (compatibility version 1.0.0, current version 61439.120.27)\n\t/usr/lib/libcharset.1.dylib (compatibility version 1.0.0, current version 1.0.0)",
+  "haskell_options": [
+    "-O2",
+    "-threaded",
+    "-rtsopts"
+  ],
+  "cabal_freeze_sha256": "a5defccf9dd5c343d47bcc3b149f66470c14bdf6e9c6ebc1f01b925b7ac036cd",
+  "python_lock_sha256": "e86309826e1e02112e05208e9cc26ad65c958855fcb0e74461e49c2b922d048e",
+  "dataset_spec_sha256": "e1be2e0ff4d0ac1618aaa434e2676aa5af6c04a76ee556e286eba16214c7eda1",
+  "dataset_manifest_sha256": "8e127b4d9565e66db49843b1e9dc3673ed6fc38026a9f8edc0875b629dd6af1d",
+  "git_commit": "daf5d56d8b7967ced2274dfee0c73f2bacb997ff",
+  "git_status": "?? benchmark-full-20260913-000643.log\n?? benchmark-smoke-20260913-000643.log\n?? benchmark/backups/",
+  "thread_environment": {
+    "OPENBLAS_NUM_THREADS": "1",
+    "OMP_NUM_THREADS": "1",
+    "MKL_NUM_THREADS": "1",
+    "VECLIB_MAXIMUM_THREADS": "1",
+    "NUMEXPR_NUM_THREADS": "1",
+    "MPLCONFIGDIR": "/Users/a123/CodingProjects/dtmc/benchmark/.cache/matplotlib",
+    "XDG_CACHE_HOME": "/Users/a123/CodingProjects/dtmc/benchmark/.cache",
+    "MPLBACKEND": "Agg"
+  },
+  "benchmark_scope": "complete suite"
+}
diff --git a/benchmark/results/plots/absorbing--hitting-bounded-exact-k-100--speed-ratio.svg b/benchmark/results/plots/absorbing--hitting-bounded-exact-k-100--speed-ratio.svg
new file mode 100644
--- /dev/null
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diff --git a/benchmark/results/ratio-summary.csv b/benchmark/results/ratio-summary.csv
new file mode 100644
--- /dev/null
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+family,size,operation,geometric_median_pydtmc_over_dtmc,ci95_low,ci95_high,datasets
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+reducible,10,hitting/bounded/at-most/k-10,13.35051138872153,13.025728837875244,13.419947719672866,3
+reducible,10,hitting/bounded/at-most/k-100,18.025597362113828,17.817467922760724,18.038804208362272,3
+reducible,10,hitting/bounded/exact/k-10,13.002200403127075,12.992941193203054,13.169268515099036,3
+reducible,10,hitting/bounded/exact/k-100,18.186504590415062,18.136931924928746,18.356850933983985,3
+reducible,10,hitting/bounded/greater-than/k-10,13.453891822261916,13.453547786391535,13.463136451488948,3
+reducible,10,hitting/bounded/greater-than/k-100,18.044895538736608,17.910541116769238,18.535185330090307,3
+reducible,10,occupation-matrix,28.563901859542195,28.492146543045283,28.624233519636274,3
+reducible,10,power/10,70.03445317157728,69.42570409253402,70.368517580796,3
+reducible,10,power/100,54.4917328640181,54.46515904622537,56.59566920786062,3
+reducible,10,power/2,90.82994980051836,90.27745148298409,91.45220674339774,3
+reducible,10,return/bounded/k-10,14.610591869489815,14.473849867428614,14.835492485148105,3
+reducible,10,return/bounded/k-100,18.37056942140311,18.296873771936763,18.94219404923155,3
+reducible,10,stationary,15.047305231097665,14.92711672423652,15.535597631922718,3
+reducible,10,structure/classes-cold,8.213893347013029,7.998095397562365,8.247227937874376,3
+reducible,10,structure/classes-lifecycle,49.81682097505722,48.42248051632651,49.86830774177793,3
+reducible,10,structure/classes-warm/access,25.36420575222952,23.375511303695713,25.603563388138078,3
+reducible,10,structure/classes-warm/consumed,22.723389093559582,22.574683847027064,22.855777423477946,3
+reducible,10,structure/irreducible-cold,11.779514660576615,11.4949738891004,11.827688064479418,3
+reducible,10,structure/irreducible-warm,23.83346982694304,23.740406146321607,25.155848712891572,3
+reducible,10,visits/bounded-expectation/k-10,3.871449263199117,3.8397869759431575,3.8772889653960774,3
+reducible,10,visits/bounded-expectation/k-100,2.125773776013471,2.1121657928465924,2.144018109493355,3
+reducible,25,construction/public-consumed,26.357176691098218,26.345184296705202,26.38100785760586,3
+reducible,25,evolution/k-10,27.891657759051594,27.80314109182467,28.03184848663048,3
+reducible,25,evolution/k-100,32.48754358417514,32.348502106939314,32.72838117710219,3
+reducible,25,evolution/one-step,108.3667423501173,107.08104766805118,111.49671024667472,3
+reducible,25,finite-time/n-step/k-10,17.119144010951235,16.563856974713936,17.153045743368526,3
+reducible,25,finite-time/n-step/k-100,11.45627444129713,11.417011115038454,11.482876501844947,3
+reducible,25,finite-time/observation/k-10,14.109618448827007,14.011162880110634,14.28253007220505,3
+reducible,25,finite-time/observation/k-100,11.21382368375022,11.177682941229223,11.268100553221492,3
+reducible,25,finite-time/step,1.5632248586173647,1.5493211961143787,1.5749144512743452,3
+reducible,25,hitting-probability/cold-all-states,1.697280392512117,1.6796685766869714,1.74740576556614,3
+reducible,25,hitting-time/cold-all-states,6.099430975933869,6.097691142837171,6.287736867654803,3
+reducible,25,hitting/bounded/at-most/k-10,14.247125573387708,13.859923300557588,14.620463724835158,3
+reducible,25,hitting/bounded/at-most/k-100,20.97686306678311,20.06282529436679,22.129965629229936,3
+reducible,25,hitting/bounded/exact/k-10,14.04998534636816,13.962389616171214,14.598680458591447,3
+reducible,25,hitting/bounded/exact/k-100,20.825935857223453,20.209469376944284,20.970040233186378,3
+reducible,25,hitting/bounded/greater-than/k-10,14.282647705284447,14.018607872487824,14.356803946394342,3
+reducible,25,hitting/bounded/greater-than/k-100,20.290589543019777,20.109939286646824,21.484576990542372,3
+reducible,25,occupation-matrix,28.47087518610547,28.18553987741487,28.53719397763619,3
+reducible,25,power/10,31.747071411223416,31.498132965112703,36.24348440684096,3
+reducible,25,power/100,26.363601044759626,25.889296754707374,27.118893109421723,3
+reducible,25,power/2,39.43673598192472,38.54197543359135,40.550772886422756,3
+reducible,25,return/bounded/k-10,16.321191006704794,15.751131454150258,16.561798922753432,3
+reducible,25,return/bounded/k-100,20.40242472683203,20.352065367370987,22.183974175787423,3
+reducible,25,stationary,20.913400603880188,20.830408508608496,20.94437144779975,3
+reducible,25,structure/classes-cold,12.059964835325342,11.72242729197371,13.169210164323154,3
+reducible,25,structure/classes-lifecycle,23.417850183842376,23.1155764935474,23.6085304830412,3
+reducible,25,structure/classes-warm/access,25.415136048438576,25.11609032948314,25.808685083366434,3
+reducible,25,structure/classes-warm/consumed,15.852846079492123,15.802828816938852,15.918475507628163,3
+reducible,25,structure/irreducible-cold,15.339271247807385,14.824081812073626,15.915893228351107,3
+reducible,25,structure/irreducible-warm,23.79558509962253,23.738520281726657,23.855589871107163,3
+reducible,25,visits/bounded-expectation/k-10,5.009113843514018,4.886918478901888,5.227023610326794,3
+reducible,25,visits/bounded-expectation/k-100,2.1947018358345094,2.1912654521890502,2.2342039991460156,3
+reducible,50,construction/public-consumed,20.82726275247997,20.29884748305302,20.920208403026663,3
+reducible,50,evolution/k-10,20.030135493000348,19.36954447244667,20.15305455932836,3
+reducible,50,evolution/k-100,15.489005799822555,15.381582773481723,15.50873032273386,3
+reducible,50,evolution/one-step,84.67560921112383,82.4605920663524,92.43873531763603,3
+reducible,50,finite-time/n-step/k-10,14.866914114478748,14.838720263410664,15.262037175529963,3
+reducible,50,finite-time/n-step/k-100,7.314601629495305,7.269914329631151,7.343496126220264,3
+reducible,50,finite-time/observation/k-10,12.002983788561371,11.893808427879998,12.084917703980285,3
+reducible,50,finite-time/observation/k-100,7.149161907646442,7.09839018455373,7.18901514293235,3
+reducible,50,finite-time/step,1.9357053016338006,1.9267490521083674,2.1841724937026252,3
+reducible,50,hitting-probability/cold-all-states,1.4967795034404752,1.4767336395898578,1.7664910600181025,3
+reducible,50,hitting-time/cold-all-states,3.802021711322215,3.7874431709623635,3.857587604492423,3
+reducible,50,hitting/bounded/at-most/k-10,11.887658175375599,11.742427960599372,20.629474806128453,3
+reducible,50,hitting/bounded/at-most/k-100,17.05236057487154,17.020045254504954,17.086826250835195,3
+reducible,50,hitting/bounded/exact/k-10,11.65473013565086,11.629871120916611,11.657449279071031,3
+reducible,50,hitting/bounded/exact/k-100,17.062143144763144,17.03348325796674,17.18113149832192,3
+reducible,50,hitting/bounded/greater-than/k-10,11.747156076318408,11.554702747489738,11.769474466862226,3
+reducible,50,hitting/bounded/greater-than/k-100,17.06999928796173,16.997600183446,17.092856838389114,3
+reducible,50,occupation-matrix,34.23901995286661,33.93535541733749,35.9809842788031,3
+reducible,50,power/10,26.45253352379304,25.865331575004063,26.504157748415974,3
+reducible,50,power/100,22.077410984723546,21.522648525065293,22.280882455898343,3
+reducible,50,power/2,27.875454246679936,27.515053197408818,29.475809750469658,3
+reducible,50,return/bounded/k-10,14.729431201105866,14.080488164937055,14.8704891423317,3
+reducible,50,return/bounded/k-100,17.58569400050675,17.5771103927894,18.799856379755404,3
+reducible,50,stationary,23.16725431971863,23.0337607195079,24.041890711172275,3
+reducible,50,structure/classes-cold,12.826102901561802,12.354136897766391,13.037261520019449,3
+reducible,50,structure/classes-lifecycle,18.311176511516663,18.17080041212987,18.51824173116355,3
+reducible,50,structure/classes-warm/access,24.561991827776612,24.00745254281556,25.22002314631249,3
+reducible,50,structure/classes-warm/consumed,12.977414288220814,12.702699361089053,12.98001507567072,3
+reducible,50,structure/irreducible-cold,15.30089752646157,14.879288130513388,15.520755314731609,3
+reducible,50,structure/irreducible-warm,23.502866085023115,21.832546024426687,24.226964549666256,3
+reducible,50,visits/bounded-expectation/k-10,6.226586655305317,6.217226284780255,6.276987409901662,3
+reducible,50,visits/bounded-expectation/k-100,2.0707712194654224,2.0013305532055257,2.153320936430698,3
+reducible,100,construction/public-consumed,19.381552124886216,19.106166746402817,19.628251198412354,3
+reducible,100,evolution/k-10,13.779391439708153,13.639686061077766,15.708754870832202,3
+reducible,100,evolution/k-100,4.638883403351589,4.594203115141249,4.660119713801407,3
+reducible,100,evolution/one-step,67.98227201706514,66.86911470013567,77.84749882320605,3
+reducible,100,finite-time/n-step/k-10,11.450984303259395,11.278915956700343,12.611247316020437,3
+reducible,100,finite-time/n-step/k-100,4.5481767341222685,4.523724797265196,5.121241528877919,3
+reducible,100,finite-time/observation/k-10,11.262310351070925,9.55106190248719,13.536804047032954,3
+reducible,100,finite-time/observation/k-100,4.373487274589755,4.3487825777891285,4.685250991399412,3
+reducible,100,finite-time/step,3.4287946053254648,2.524223131409796,6.873327619624457,3
+reducible,100,hitting-probability/cold-all-states,1.8941235735796267,1.8304870371052788,2.096340780434679,3
+reducible,100,hitting-time/cold-all-states,3.5601029061707035,3.5390372318605205,3.884197913839694,3
+reducible,100,hitting/bounded/at-most/k-10,9.605715995999567,9.491421789951778,10.062526126176705,3
+reducible,100,hitting/bounded/at-most/k-100,15.934529564014818,15.816495546563203,16.194573397549938,3
+reducible,100,hitting/bounded/exact/k-10,9.699932639253094,9.687631735433445,11.331489858461522,3
+reducible,100,hitting/bounded/exact/k-100,16.074404558324243,16.07381646525996,16.73579630535169,3
+reducible,100,hitting/bounded/greater-than/k-10,9.6422484104744,8.992432438801712,11.026650758621852,3
+reducible,100,hitting/bounded/greater-than/k-100,16.013248860483085,15.777525294144246,16.05844478271325,3
+reducible,100,occupation-matrix,31.48344156831675,30.437584659271486,31.49516306170555,3
+reducible,100,power/10,24.953281681903718,24.511536454129885,25.777010432675258,3
+reducible,100,power/100,19.992281019450914,19.019767190909366,20.739364875762377,3
+reducible,100,power/2,26.41987247808198,26.247435578115585,27.047772028810684,3
+reducible,100,return/bounded/k-10,14.12431813488494,13.983633362865309,17.324068213811334,3
+reducible,100,return/bounded/k-100,16.835671058163303,16.76853477779501,16.83618717413807,3
+reducible,100,stationary,24.25105246076073,23.602295238089926,24.587559050851382,3
+reducible,100,structure/classes-cold,13.172085015848191,13.154220191770126,13.725938419601029,3
+reducible,100,structure/classes-lifecycle,16.133432619832647,15.643214501142161,16.384589289615874,3
+reducible,100,structure/classes-warm/access,25.09263475247289,24.105136808188924,25.245203472175437,3
+reducible,100,structure/classes-warm/consumed,11.46126394900063,11.438240066393,11.556665168631666,3
+reducible,100,structure/irreducible-cold,14.795072809646388,14.392070825102955,15.08038555881801,3
+reducible,100,structure/irreducible-warm,23.90749140101455,23.67563893667323,25.938008496194104,3
+reducible,100,visits/bounded-expectation/k-10,6.3224881076540065,6.018439596175496,6.491637452517361,3
+reducible,100,visits/bounded-expectation/k-100,1.8276155957752203,1.8162050611522556,1.922378952364169,3
+reducible,250,construction/public-consumed,13.440836077125379,13.246987442709997,13.74259298479228,3
+reducible,250,evolution/k-10,5.038410540910749,4.981991676677521,5.150900511740963,3
+reducible,250,evolution/k-100,1.747997641404126,1.7437790003381604,1.7480465209839593,3
+reducible,250,evolution/one-step,31.79513042787878,31.59296300219302,32.86195958051851,3
+reducible,250,hitting-probability/cold-all-states,3.3296818137474737,3.3279488847289462,3.348540567812139,3
+reducible,250,hitting-time/cold-all-states,4.660237937613108,4.6233826236516755,4.672393436375036,3
+reducible,250,occupation-matrix,34.48297481422739,32.48165615405256,34.825077239810796,3
+reducible,250,power/10,19.61542531068868,18.045000002698245,19.66042556970387,3
+reducible,250,power/100,15.870212789796609,15.763678335503593,15.925858386561288,3
+reducible,250,power/2,25.57198156047058,25.537956815127856,25.581311380458306,3
+reducible,250,stationary,22.118030670401815,22.109331453631274,22.24052633187675,3
+reducible,250,structure/classes-cold,12.6312412432041,12.577133517127452,12.695654894160947,3
+reducible,250,structure/classes-lifecycle,10.673484965118693,10.608058139935263,10.775450691742968,3
+reducible,250,structure/classes-warm/access,25.112076018672386,23.273346279726127,25.426521886894587,3
+reducible,250,structure/classes-warm/consumed,10.796519094437194,10.717811020038482,10.835632248918573,3
+reducible,250,structure/irreducible-cold,12.995005457480257,12.916162199830387,13.037715613721025,3
+reducible,250,structure/irreducible-warm,23.43946065059647,23.395415933536086,23.539452090291316,3
+reducible,500,construction/public-consumed,10.000113024132133,9.848383966335492,10.049870138090665,3
+reducible,500,evolution/k-10,3.4418447289764385,3.3820043160861375,3.4548952712150003,3
+reducible,500,evolution/k-100,1.399572365785616,1.3983557384905259,1.4012982391578208,3
+reducible,500,evolution/one-step,21.925190524130972,21.72474801255448,22.01234844222421,3
+reducible,500,hitting-probability/cold-all-states,3.753334919264832,3.687348291629801,3.8822890031130726,3
+reducible,500,hitting-time/cold-all-states,5.361123953532604,5.257589200001232,5.411729091069608,3
+reducible,500,power/10,13.406505494658584,13.160688946702583,13.59162430348565,3
+reducible,500,power/100,9.602696675402854,9.563403034177048,9.613041372549699,3
+reducible,500,power/2,19.96685765116789,19.86014138451214,20.160287364110104,3
+reducible,500,stationary,18.714914769161194,18.6049523927407,18.997318719689495,3
+reducible,500,structure/classes-cold,10.65405615416232,10.466644869967219,10.770244323804949,3
+reducible,500,structure/classes-lifecycle,8.163676610477374,8.139277836537254,8.253883179736729,3
+reducible,500,structure/classes-warm/access,24.873805004594974,24.64314837570111,25.07920225798261,3
+reducible,500,structure/classes-warm/consumed,11.36480378416838,11.351246733369287,11.374177256777505,3
+reducible,500,structure/irreducible-cold,10.906249053038815,10.846067849561598,10.924206318707416,3
+reducible,500,structure/irreducible-warm,23.02981500438876,22.77973475165827,23.365344803336527,3
+reducible,1000,construction/public-consumed,8.925409172603157,8.8162932772307,9.029575207014526,3
+reducible,1000,evolution/k-10,1.8309296745883208,1.8135892520400727,1.8672945712283702,3
+reducible,1000,evolution/k-100,1.0590482114856477,1.0497962530255964,1.0638168348168835,3
+reducible,1000,evolution/one-step,9.521285495368177,9.403272459174374,9.56546152779292,3
+reducible,1000,structure/classes-cold,3.9871219599423755,3.8644079633483566,4.058687884542632,3
+reducible,1000,structure/classes-lifecycle,6.530642877748845,6.506691942526172,6.546840780218748,3
+reducible,1000,structure/classes-warm/access,25.103430488202427,25.082683289671888,25.273717724968364,3
+reducible,1000,structure/classes-warm/consumed,11.627063837438694,11.485707177372738,11.64173677501544,3
+reducible,1000,structure/irreducible-cold,4.180041596964045,4.134915523707161,4.1859207246114805,3
+reducible,1000,structure/irreducible-warm,23.600489900299664,23.28831082368587,23.602142676140183,3
diff --git a/benchmark/results/ratios.csv b/benchmark/results/ratios.csv
new file mode 100644
--- /dev/null
+++ b/benchmark/results/ratios.csv
@@ -0,0 +1,2797 @@
+dataset,family,size,seed,operation,pydtmc_over_dtmc
+absorbing-n0010-s01729,absorbing,10,1729,absorption-time,9.45558475240783
+absorbing-n0010-s01729,absorbing,10,1729,absorption/probabilities,9.42038873375433
+absorbing-n0010-s01729,absorbing,10,1729,construction/public-consumed,58.47297183810736
+absorbing-n0010-s01729,absorbing,10,1729,evolution/k-10,31.392611387358116
+absorbing-n0010-s01729,absorbing,10,1729,evolution/k-100,84.41113959175361
+absorbing-n0010-s01729,absorbing,10,1729,evolution/one-step,121.14601585774416
+absorbing-n0010-s01729,absorbing,10,1729,finite-time/n-step/k-10,16.960427866568512
+absorbing-n0010-s01729,absorbing,10,1729,finite-time/n-step/k-100,14.91672512616429
+absorbing-n0010-s01729,absorbing,10,1729,finite-time/observation/k-10,14.930784011106324
+absorbing-n0010-s01729,absorbing,10,1729,finite-time/observation/k-100,14.562716805870394
+absorbing-n0010-s01729,absorbing,10,1729,finite-time/step,1.1561888459873055
+absorbing-n0010-s01729,absorbing,10,1729,fundamental-matrix,7.755665271720604
+absorbing-n0010-s01729,absorbing,10,1729,hitting-probability/cold-all-states,2.661509405507161
+absorbing-n0010-s01729,absorbing,10,1729,hitting-time/cold-all-states,6.195325126923157
+absorbing-n0010-s01729,absorbing,10,1729,hitting/bounded/at-most/k-10,13.19613056121106
+absorbing-n0010-s01729,absorbing,10,1729,hitting/bounded/at-most/k-100,17.87981840076335
+absorbing-n0010-s01729,absorbing,10,1729,hitting/bounded/exact/k-10,12.868563912709723
+absorbing-n0010-s01729,absorbing,10,1729,hitting/bounded/exact/k-100,18.45238213406581
+absorbing-n0010-s01729,absorbing,10,1729,hitting/bounded/greater-than/k-10,13.676738300121352
+absorbing-n0010-s01729,absorbing,10,1729,hitting/bounded/greater-than/k-100,18.171290015447376
+absorbing-n0010-s01729,absorbing,10,1729,occupation-matrix,20.90160785592371
+absorbing-n0010-s01729,absorbing,10,1729,power/10,71.36010332259025
+absorbing-n0010-s01729,absorbing,10,1729,power/100,55.90959343988362
+absorbing-n0010-s01729,absorbing,10,1729,power/2,93.31775901401524
+absorbing-n0010-s01729,absorbing,10,1729,return/bounded/k-10,14.399008266080672
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diff --git a/benchmark/results/scaling.csv b/benchmark/results/scaling.csv
new file mode 100644
--- /dev/null
+++ b/benchmark/results/scaling.csv
@@ -0,0 +1,355 @@
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diff --git a/benchmark/results/summary.csv b/benchmark/results/summary.csv
new file mode 100644
--- /dev/null
+++ b/benchmark/results/summary.csv
@@ -0,0 +1,5773 @@
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diff --git a/benchmark/results/verification.json b/benchmark/results/verification.json
new file mode 100644
--- /dev/null
+++ b/benchmark/results/verification.json
@@ -0,0 +1,9799 @@
+{
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+      "max_absolute_error": 2.0816681711721685e-17,
+      "max_relative_error": 6.800978306100704e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s02718",
+      "operation": "finite_time_observation_10",
+      "max_absolute_error": 0.0,
+      "max_relative_error": 0.0,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s02718",
+      "operation": "finite_time_observation_100",
+      "max_absolute_error": 1.3877787807814457e-17,
+      "max_relative_error": 4.546288386315261e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s02718",
+      "operation": "hitting_probability",
+      "max_absolute_error": 2.886579864025407e-15,
+      "max_relative_error": 5.938340622596604e-15,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s02718",
+      "operation": "hitting_time",
+      "max_absolute_error": 0.0,
+      "max_relative_error": 0.0,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s02718",
+      "operation": "hitting_bounded_exact_10",
+      "max_absolute_error": 6.938893903907228e-18,
+      "max_relative_error": 7.682633307051576e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s02718",
+      "operation": "hitting_bounded_exact_100",
+      "max_absolute_error": 1.951563910473908e-18,
+      "max_relative_error": 1.7449084574866569e-15,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s02718",
+      "operation": "hitting_bounded_at_most_10",
+      "max_absolute_error": 1.3877787807814457e-17,
+      "max_relative_error": 1.3224204808198229e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s02718",
+      "operation": "hitting_bounded_at_most_100",
+      "max_absolute_error": 5.551115123125783e-16,
+      "max_relative_error": 1.2560136544239755e-15,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s02718",
+      "operation": "hitting_bounded_greater_than_10",
+      "max_absolute_error": 0.0,
+      "max_relative_error": 0.0,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s02718",
+      "operation": "hitting_bounded_greater_than_100",
+      "max_absolute_error": 1.1102230246251565e-15,
+      "max_relative_error": 1.98951493311809e-15,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s02718",
+      "operation": "return_bounded_10",
+      "max_absolute_error": 0.0,
+      "max_relative_error": 0.0,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s02718",
+      "operation": "return_bounded_100",
+      "max_absolute_error": 0.0,
+      "max_relative_error": 0.0,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s02718",
+      "operation": "visits_bounded_expectation_10",
+      "max_absolute_error": 2.7755575615628914e-17,
+      "max_relative_error": 2.144047147570049e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s02718",
+      "operation": "visits_bounded_expectation_100",
+      "max_absolute_error": 1.3322676295501878e-15,
+      "max_relative_error": 5.723608572197463e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s02718",
+      "operation": "stationary[80,81,82,83,84,85,86,87,88,89]",
+      "max_absolute_error": 2.7755575615628914e-17,
+      "max_relative_error": 2.7093684537214094e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s02718",
+      "operation": "stationary[90,91,92,93,94,95,96,97,98,99]",
+      "max_absolute_error": 5.551115123125783e-17,
+      "max_relative_error": 1.481815290563897e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s02718",
+      "operation": "occupation",
+      "max_absolute_error": 1.9984014443252818e-15,
+      "max_relative_error": 3.6255716770403634e-15,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "evolve_1",
+      "max_absolute_error": 1.0408340855860843e-17,
+      "max_relative_error": 6.485929672625364e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "evolve_10",
+      "max_absolute_error": 2.7755575615628914e-17,
+      "max_relative_error": 6.452583728390348e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "power_10",
+      "max_absolute_error": 1.1102230246251565e-16,
+      "max_relative_error": 1.8212588610360833e-15,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "finite_time_step",
+      "max_absolute_error": 2.168404344971009e-19,
+      "max_relative_error": 1.1307325157213405e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "finite_time_n_step_10",
+      "max_absolute_error": 0.0,
+      "max_relative_error": 0.0,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "finite_time_n_step_100",
+      "max_absolute_error": 2.7755575615628914e-17,
+      "max_relative_error": 5.164018886561277e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "finite_time_observation_10",
+      "max_absolute_error": 1.3877787807814457e-17,
+      "max_relative_error": 4.603177159391941e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "finite_time_observation_100",
+      "max_absolute_error": 3.469446951953614e-17,
+      "max_relative_error": 6.185761416407708e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "hitting_probability",
+      "max_absolute_error": 9.992007221626409e-16,
+      "max_relative_error": 2.0031675980760458e-15,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "hitting_time",
+      "max_absolute_error": 0.0,
+      "max_relative_error": 0.0,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "hitting_bounded_exact_10",
+      "max_absolute_error": 3.469446951953614e-18,
+      "max_relative_error": 3.2981324421698183e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "hitting_bounded_exact_100",
+      "max_absolute_error": 1.734723475976807e-18,
+      "max_relative_error": 1.7510904295187668e-15,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "hitting_bounded_at_most_10",
+      "max_absolute_error": 0.0,
+      "max_relative_error": 0.0,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "hitting_bounded_at_most_100",
+      "max_absolute_error": 2.220446049250313e-16,
+      "max_relative_error": 4.769651406674546e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "hitting_bounded_greater_than_10",
+      "max_absolute_error": 0.0,
+      "max_relative_error": 0.0,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "hitting_bounded_greater_than_100",
+      "max_absolute_error": 1.1102230246251565e-16,
+      "max_relative_error": 2.0772656842665662e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "return_bounded_10",
+      "max_absolute_error": 0.0,
+      "max_relative_error": 0.0,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "return_bounded_100",
+      "max_absolute_error": 1.6653345369377348e-16,
+      "max_relative_error": 5.261529905048627e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "visits_bounded_expectation_10",
+      "max_absolute_error": 2.7755575615628914e-17,
+      "max_relative_error": 1.797477902512475e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "visits_bounded_expectation_100",
+      "max_absolute_error": 1.3322676295501878e-15,
+      "max_relative_error": 3.339532272428933e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "stationary[80,81,82,83,84,85,86,87,88,89]",
+      "max_absolute_error": 1.3877787807814457e-17,
+      "max_relative_error": 2.0183346120152875e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "stationary[90,91,92,93,94,95,96,97,98,99]",
+      "max_absolute_error": 1.1102230246251565e-16,
+      "max_relative_error": 2.2174142292479812e-16,
+      "passed": true
+    },
+    {
+      "dataset": "reducible-n0100-s31415",
+      "operation": "occupation",
+      "max_absolute_error": 2.886579864025407e-15,
+      "max_relative_error": 4.042264149537729e-15,
+      "passed": true
+    }
+  ],
+  "skipped": [
+    "absorbing-n0250-s01729",
+    "absorbing-n0250-s02718",
+    "absorbing-n0250-s31415",
+    "absorbing-n0500-s01729",
+    "absorbing-n0500-s02718",
+    "absorbing-n0500-s31415",
+    "absorbing-n1000-s01729",
+    "absorbing-n1000-s02718",
+    "absorbing-n1000-s31415",
+    "dense-n0250-s01729",
+    "dense-n0250-s02718",
+    "dense-n0250-s31415",
+    "dense-n0500-s01729",
+    "dense-n0500-s02718",
+    "dense-n0500-s31415",
+    "dense-n1000-s01729",
+    "dense-n1000-s02718",
+    "dense-n1000-s31415",
+    "low-outdegree-n0250-s01729",
+    "low-outdegree-n0250-s02718",
+    "low-outdegree-n0250-s31415",
+    "low-outdegree-n0500-s01729",
+    "low-outdegree-n0500-s02718",
+    "low-outdegree-n0500-s31415",
+    "low-outdegree-n1000-s01729",
+    "low-outdegree-n1000-s02718",
+    "low-outdegree-n1000-s31415",
+    "periodic-n0250-s01729",
+    "periodic-n0250-s02718",
+    "periodic-n0250-s31415",
+    "periodic-n0500-s01729",
+    "periodic-n0500-s02718",
+    "periodic-n0500-s31415",
+    "periodic-n1000-s01729",
+    "periodic-n1000-s02718",
+    "periodic-n1000-s31415",
+    "reducible-n0250-s01729",
+    "reducible-n0250-s02718",
+    "reducible-n0250-s31415",
+    "reducible-n0500-s01729",
+    "reducible-n0500-s02718",
+    "reducible-n0500-s31415",
+    "reducible-n1000-s01729",
+    "reducible-n1000-s02718",
+    "reducible-n1000-s31415"
+  ],
+  "passed": true
+}
diff --git a/benchmark/run.py b/benchmark/run.py
new file mode 100644
--- /dev/null
+++ b/benchmark/run.py
@@ -0,0 +1,364 @@
+#!/usr/bin/env python3
+"""Reproducible orchestration for the dtmc versus PyDTMC benchmark."""
+
+from __future__ import annotations
+
+import argparse
+import hashlib
+import json
+import os
+import platform
+import subprocess
+import sys
+from pathlib import Path
+from typing import Any
+
+
+BENCHMARK_ROOT = Path(__file__).resolve().parent
+PROJECT_ROOT = BENCHMARK_ROOT.parent
+DATA_DIR = BENCHMARK_ROOT / "data"
+DATA_ROOT = DATA_DIR / "generated"
+RESULTS_ROOT = BENCHMARK_ROOT / "results"
+VENV = BENCHMARK_ROOT / ".venv"
+CABAL_PROJECT = "benchmark/cabal.project"
+SPEC = json.loads((DATA_DIR / "spec.json").read_text(encoding="utf-8"))
+SIZES = tuple(map(int, SPEC["sizes"]))
+SEEDS = tuple(map(int, SPEC["seeds"]))
+FAMILIES = tuple(map(str, SPEC["families"]))
+THREAD_ENV = {
+    "OPENBLAS_NUM_THREADS": "1",
+    "OMP_NUM_THREADS": "1",
+    "MKL_NUM_THREADS": "1",
+    "VECLIB_MAXIMUM_THREADS": "1",
+    "NUMEXPR_NUM_THREADS": "1",
+    "MPLCONFIGDIR": str(BENCHMARK_ROOT / ".cache" / "matplotlib"),
+    "XDG_CACHE_HOME": str(BENCHMARK_ROOT / ".cache"),
+    "MPLBACKEND": "Agg",
+}
+
+
+def run(command: list[str], *, env: dict[str, str] | None = None) -> None:
+    printable = " ".join(command)
+    print(f"+ {printable}", flush=True)
+    subprocess.run(command, cwd=PROJECT_ROOT, env=env, check=True)
+
+
+def capture(command: list[str], *, env: dict[str, str] | None = None) -> str:
+    try:
+        return subprocess.run(
+            command,
+            cwd=PROJECT_ROOT,
+            env=env,
+            check=True,
+            text=True,
+            stdout=subprocess.PIPE,
+            stderr=subprocess.STDOUT,
+        ).stdout.strip()
+    except (OSError, subprocess.CalledProcessError) as problem:
+        return f"unavailable: {problem}"
+
+
+def python_executable() -> Path:
+    name = "python.exe" if os.name == "nt" else "python"
+    return VENV / ("Scripts" if os.name == "nt" else "bin") / name
+
+
+def haskell_executable() -> Path:
+    result = subprocess.run(
+        ["cabal", "list-bin", f"--project-file={CABAL_PROJECT}", "dtmc-bench"],
+        cwd=PROJECT_ROOT,
+        check=True,
+        text=True,
+        stdout=subprocess.PIPE,
+    )
+    path = Path(result.stdout.strip())
+    if not path.exists():
+        raise RuntimeError("dtmc-bench is not built; run the bootstrap command first")
+    return path
+
+
+def bootstrap() -> None:
+    (BENCHMARK_ROOT / ".cache" / "matplotlib").mkdir(parents=True, exist_ok=True)
+    if not python_executable().exists():
+        run([sys.executable, "-m", "venv", str(VENV)])
+    run(
+        [
+            str(python_executable()),
+            "-m",
+            "pip",
+            "install",
+            "--disable-pip-version-check",
+            "-r",
+            str(BENCHMARK_ROOT / "python" / "requirements.lock"),
+        ]
+    )
+    run(["cabal", "build", f"--project-file={CABAL_PROJECT}", "dtmc-bench"])
+
+
+def generate(mode: str) -> None:
+    sizes = SIZES if mode == "full" else (10,)
+    seeds = SEEDS if mode == "full" else (SEEDS[0],)
+    python = str(python_executable()) if python_executable().exists() else sys.executable
+    run(
+        [
+            python,
+            str(DATA_DIR / "generate.py"),
+            "--sizes",
+            ",".join(map(str, sizes)),
+            "--seeds",
+            ",".join(map(str, seeds)),
+        ]
+    )
+
+
+def benchmark_environment(
+    entry: dict[str, Any], data_root: Path = DATA_ROOT
+) -> dict[str, str]:
+    environment = os.environ.copy()
+    environment.update(THREAD_ENV)
+    environment.update(
+        {
+            "DTMC_BENCH_DATA": str(data_root),
+            "DTMC_BENCH_FAMILY": str(entry["family"]),
+            "DTMC_BENCH_SIZE": str(entry["size"]),
+            "DTMC_BENCH_SEED": str(entry["seed"]),
+        }
+    )
+    return environment
+
+
+def manifest_entries(mode: str, data_root: Path = DATA_ROOT) -> list[dict[str, Any]]:
+    manifest = json.loads((data_root / "manifest.json").read_text(encoding="utf-8"))
+    entries = list(manifest["datasets"])
+    if mode == "smoke":
+        entries = [
+            entry
+            for entry in entries
+            if entry["size"] == 10 and entry["seed"] == SEEDS[0]
+        ]
+    return entries
+
+
+def verify(mode: str) -> None:
+    RESULTS_ROOT.mkdir(parents=True, exist_ok=True)
+    (BENCHMARK_ROOT / ".cache" / "matplotlib").mkdir(parents=True, exist_ok=True)
+    haskell_output = RESULTS_ROOT / "haskell-verification.json"
+    verification_output = RESULTS_ROOT / "verification.json"
+    environment = os.environ.copy()
+    environment.update(THREAD_ENV)
+    environment["DTMC_BENCH_DATA"] = str(DATA_ROOT)
+    environment["DTMC_VERIFY_MAX_SIZE"] = "100" if mode == "full" else "10"
+    run(
+        [
+            str(haskell_executable()),
+            "--verify-json",
+            str(haskell_output),
+        ],
+        env=environment,
+    )
+    run(
+        [
+            str(python_executable()),
+            str(BENCHMARK_ROOT / "python" / "verify.py"),
+            str(haskell_output),
+            "--output",
+            str(verification_output),
+        ],
+        env=environment,
+    )
+
+
+def benchmark(mode: str) -> None:
+    raw_haskell = RESULTS_ROOT / "raw" / "haskell"
+    raw_python = RESULTS_ROOT / "raw" / "python"
+    raw_haskell.mkdir(parents=True, exist_ok=True)
+    raw_python.mkdir(parents=True, exist_ok=True)
+
+    for entry in manifest_entries(mode):
+        name = str(entry["id"])
+        environment = benchmark_environment(entry)
+        haskell_json = raw_haskell / f"{name}.json"
+        haskell_csv = raw_haskell / f"{name}.csv"
+        haskell_raw = raw_haskell / f"{name}.raw"
+        python_json = raw_python / f"{name}.json"
+        for output in (haskell_json, haskell_csv, haskell_raw, python_json):
+            output.unlink(missing_ok=True)
+        criterion_command = [
+            str(haskell_executable()),
+            "--json",
+            str(haskell_json),
+            "--csv",
+            str(haskell_csv),
+            "--raw",
+            str(haskell_raw),
+            "--time-limit",
+            "0.1" if mode == "smoke" else "5",
+            "--resamples",
+            "100" if mode == "smoke" else "10000",
+        ]
+        run(criterion_command, env=environment)
+
+        pyperf_command = [
+            str(python_executable()),
+            str(BENCHMARK_ROOT / "python" / "bench.py"),
+            "--copy-env",
+            "-o",
+            str(python_json),
+        ]
+        if mode == "smoke":
+            pyperf_command.append("--fast")
+        run(pyperf_command, env=environment)
+
+
+def record_environment(
+    results_root: Path = RESULTS_ROOT,
+    data_root: Path = DATA_ROOT,
+    benchmark_scope: list[str] | None = None,
+) -> None:
+    results_root.mkdir(parents=True, exist_ok=True)
+    environment = os.environ.copy()
+    environment.update(THREAD_ENV)
+    python = str(python_executable()) if python_executable().exists() else sys.executable
+    haskell_binary = haskell_executable()
+
+    def digest(path: Path) -> str:
+        return hashlib.sha256(path.read_bytes()).hexdigest()
+
+    record = {
+        "captured_at_utc": capture(["date", "-u", "+%Y-%m-%dT%H:%M:%SZ"]),
+        "platform": platform.platform(),
+        "machine": platform.machine(),
+        "processor": platform.processor(),
+        "cpu_brand": capture(["sysctl", "-n", "machdep.cpu.brand_string"]),
+        "python": capture([python, "--version"]),
+        "python_packages": capture([python, "-m", "pip", "freeze"]),
+        "numpy_configuration": capture(
+            [python, "-c", "import numpy as n; n.show_config()"], env=environment
+        ),
+        "ghc": capture(["ghc", "--version"]),
+        "cabal": capture(["cabal", "--version"]),
+        "haskell_benchmark_binary": str(haskell_binary),
+        "haskell_benchmark_linkage": capture(["otool", "-L", str(haskell_binary)]),
+        "haskell_options": ["-O2", "-threaded", "-rtsopts"],
+        "cabal_freeze_sha256": digest(BENCHMARK_ROOT / "cabal.project.freeze"),
+        "python_lock_sha256": digest(BENCHMARK_ROOT / "python" / "requirements.lock"),
+        "dataset_spec_sha256": digest(DATA_DIR / "spec.json"),
+        "dataset_manifest_sha256": digest(data_root / "manifest.json"),
+        "git_commit": capture(["git", "rev-parse", "HEAD"]),
+        "git_status": capture(["git", "status", "--short"]),
+        "thread_environment": THREAD_ENV,
+        "benchmark_scope": benchmark_scope or "complete suite",
+    }
+    (results_root / "environment.json").write_text(
+        json.dumps(record, indent=2) + "\n", encoding="utf-8"
+    )
+
+
+def profile() -> None:
+    entries = manifest_entries("full")
+    candidates = [entry for entry in entries if entry["family"] == "dense"]
+    entry = next((item for item in candidates if item["size"] == 100), candidates[0])
+    environment = benchmark_environment(entry)
+    profile_root = BENCHMARK_ROOT / "profiles"
+    profile_root.mkdir(parents=True, exist_ok=True)
+    for operation in (
+        "stationary",
+        "hitting-probability/cold-all-states",
+        "race/forward-committor",
+        "return/mean-recurrence",
+        "visits/bounded-expectation/k-100",
+        "simulation/100000",
+    ):
+        safe = operation.replace("/", "-")
+        haskell_path = profile_root / f"haskell-{safe}.txt"
+        with haskell_path.open("w", encoding="utf-8") as stream:
+            subprocess.run(
+                [
+                    str(haskell_executable()),
+                    "--time-limit",
+                    "0.2",
+                    "--resamples",
+                    "100",
+                    f"{entry['id']}/{operation}",
+                    "+RTS",
+                    "-s",
+                    "-RTS",
+                ],
+                cwd=PROJECT_ROOT,
+                env=environment,
+                check=True,
+                stdout=stream,
+                stderr=subprocess.STDOUT,
+            )
+        python_environment = environment.copy()
+        python_environment["DTMC_BENCH_OPERATION"] = operation
+        run(
+            [
+                str(python_executable()),
+                str(BENCHMARK_ROOT / "python" / "bench.py"),
+                "--copy-env",
+                "--debug-single-value",
+                "--profile",
+                str(profile_root / f"python-{safe}.prof"),
+            ],
+            env=python_environment,
+        )
+
+
+def analyse(
+    results_root: Path = RESULTS_ROOT, data_root: Path = DATA_ROOT
+) -> None:
+    run(
+        [
+            str(python_executable()),
+            str(BENCHMARK_ROOT / "analyse.py"),
+            "--data-root",
+            str(data_root),
+            "--results-root",
+            str(results_root),
+        ]
+    )
+    run(
+        [
+            str(python_executable()),
+            str(BENCHMARK_ROOT / "plot.py"),
+            "--results-root",
+            str(results_root),
+        ]
+    )
+
+
+def main() -> None:
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument(
+        "command",
+        choices=(
+            "bootstrap",
+            "generate",
+            "verify",
+            "benchmark",
+            "profile",
+            "analyse",
+            "all",
+        ),
+    )
+    parser.add_argument("--mode", choices=("smoke", "full"), default="smoke")
+    args = parser.parse_args()
+
+    if args.command in {"bootstrap", "all"}:
+        bootstrap()
+    if args.command in {"generate", "all"}:
+        generate(args.mode)
+    if args.command in {"verify", "all"}:
+        verify(args.mode)
+    if args.command in {"benchmark", "all"}:
+        benchmark(args.mode)
+    if args.command == "profile":
+        profile()
+    if args.command in {"analyse", "all"}:
+        record_environment()
+        analyse()
+
+
+if __name__ == "__main__":
+    main()
diff --git a/cabal.project b/cabal.project
new file mode 100644
--- /dev/null
+++ b/cabal.project
@@ -0,0 +1,7 @@
+packages: .
+
+-- Haddock includes private backend methods in public instance details.
+-- Generate their internal documentation so those links have destinations;
+-- this does not expose the module to library users.
+program-options
+  haddock-options: --show=Dtmc.Transition.Internal
diff --git a/dtmc.cabal b/dtmc.cabal
--- a/dtmc.cabal
+++ b/dtmc.cabal
@@ -1,6 +1,6 @@
 cabal-version:      3.0
 name:               dtmc
-version:            0.2.0.0
+version:            0.3.0.0
 synopsis:           Type-safe discrete-time Markov chains
 description:
   Type-safe finite discrete-time Markov chains with matrix and kernel
@@ -16,7 +16,7 @@
 license:            BSD-3-Clause
 license-file:       LICENSE
 author:             Arkadii Kholmetskii
-maintainer:         Arkadii Kholmetskii <373321aa@gmail.com>
+maintainer:         Arkadii Kholmetskii <kholmetskiia@gmail.com>
 copyright:          2026 Arkadii Kholmetskii
 build-type:         Simple
 tested-with:
@@ -29,6 +29,32 @@
     README.md
     CHANGELOG.md
 
+extra-source-files:
+    cabal.project
+    benchmark/README.md
+    benchmark/REPORT.md
+    benchmark/results/environment.json
+    benchmark/results/ratio-summary.csv
+    benchmark/results/ratios.csv
+    benchmark/results/scaling.csv
+    benchmark/results/summary.csv
+    benchmark/results/verification.json
+    benchmark/results/plots/absorbing--hitting-bounded-exact-k-100--speed-ratio.svg
+    benchmark/results/plots/dense--evolution-one-step--speed-ratio.svg
+    benchmark/results/plots/dense--hitting-time-cold-all-states--speed-ratio.svg
+    benchmark/results/plots/periodic--finite-time-observation-k-100--speed-ratio.svg
+    benchmark/cabal.project
+    benchmark/cabal.project.freeze
+    benchmark/analyse.py
+    benchmark/plot.py
+    benchmark/run.py
+    benchmark/data/generate.py
+    benchmark/data/spec.json
+    benchmark/python/bench.py
+    benchmark/python/pyproject.toml
+    benchmark/python/requirements.lock
+    benchmark/python/verify.py
+
 common lang
   default-language: GHC2021
 
@@ -74,10 +100,13 @@
   other-modules:
       Dtmc.Distribution.Vector.Internal
       Dtmc.Distribution.Map.Internal
+      Dtmc.Transition.Internal
       Dtmc.Transition.Matrix.Internal
       Dtmc.Simplex.Internal
+      Dtmc.Analysis.FiniteHorizon.Internal
       Dtmc.Analysis.FiniteTime.Internal
       Dtmc.Analysis.Classification.Internal
+      Dtmc.Analysis.Classification.Reachability.Internal
       Dtmc.Analysis.LinearSystem.Internal
       Dtmc.Analysis.Initial.Internal
       Dtmc.Transition.Matrix.Internal.Graph
@@ -92,6 +121,7 @@
     , hmatrix >=0.20.2 && <0.21
     , mwc-random >=0.15.0.1 && <0.16
     , primitive >=0.9 && <0.10
+    , vector >=0.13 && <0.14
 
 test-suite spec
   import:           lang
@@ -141,6 +171,35 @@
   build-tool-depends:
       hspec-discover:hspec-discover >=2.11.17 && <2.12
 
+benchmark dtmc-bench
+  import:           lang
+  type:             exitcode-stdio-1.0
+  hs-source-dirs:   benchmark/haskell
+  main-is:          Main.hs
+
+  other-modules:
+      Cases
+      Dataset
+      Force
+      Verification
+
+  ghc-options:
+      -threaded
+      -rtsopts
+
+  build-depends:
+      aeson >=2.2 && <2.3
+    , base
+    , binary >=0.8.9 && <0.9
+    , bytestring >=0.11 && <0.13
+    , criterion >=1.6.4 && <1.7
+    , deepseq >=1.4.8 && <1.6
+    , dtmc
+    , finite-typelits
+    , filepath >=1.4 && <1.6
+    , mwc-random
+    , vector >=0.13 && <0.14
+
 source-repository head
   type:             git
   location:         https://github.com/kholmetskii/dtmc.git
@@ -148,4 +207,4 @@
 source-repository this
   type:             git
   location:         https://github.com/kholmetskii/dtmc.git
-  tag:              v0.2.0.0
+  tag:              v0.3.0.0
diff --git a/src/Dtmc/Analysis/Absorption.hs b/src/Dtmc/Analysis/Absorption.hs
--- a/src/Dtmc/Analysis/Absorption.hs
+++ b/src/Dtmc/Analysis/Absorption.hs
@@ -42,6 +42,7 @@
     fundamentalMatrix,
 
     -- * Absorption probabilities
+    probabilityMatrix,
     probability,
     probabilityGivenInitialState,
 
@@ -69,6 +70,8 @@
  )
 import Dtmc.Analysis.LinearSystem.Internal (
     fundamental,
+    solveIminusQ,
+    solveIminusQVector,
     subMatrix,
  )
 import Dtmc.Distribution (
@@ -103,7 +106,6 @@
 retains @O(n + E)@ graph-cache space.
 -}
 canonicalOrder ::
-    (FiniteState state) =>
     TransitionMatrix state ->
     ([state], [state])
 canonicalOrder p =
@@ -142,6 +144,42 @@
     transientIdx = map toIndex transient
     matrix = unTransitionMatrix p
 
+{- | Compute the complete transient-to-recurrent absorption-probability matrix
+@B = (I - Q)^-1 R'@.
+
+The returned triple contains the transient row states, the recurrent column
+states, and the matrix rows in those respective canonical orders. Entry
+@(i,k)@ is the probability that recurrent state @k@ is the first recurrent
+state visited when starting from transient state @i@.
+
+All recurrent columns are solved together, sharing one factorisation of
+@I - Q@. A chain with no transient states returns empty matrix rows without a
+solve.
+
+Complexity: including first-time graph classification,
+@O(n^2 + (n + E) log(n + 1) + t^3 + t^2 r)@ time,
+@O(n^2 + n + E + t^2 + t r)@ temporary space, @O(n + E)@ retained
+graph-cache space, and @O(t r)@ result space for @r@ recurrent states.
+-}
+probabilityMatrix ::
+    (FiniteState state) =>
+    TransitionMatrix state ->
+    Either LinearSystemError ([state], [state], [[Double]])
+probabilityMatrix p
+    | null transient = Right ([], recurrent, [])
+    | otherwise = do
+        values <-
+            solveIminusQ
+                (subMatrix transientIdx transientIdx matrix)
+                (subMatrix transientIdx recurrentIdx matrix)
+        pure (transient, recurrent, LA.toLists values)
+  where
+    transient = transientStates p
+    recurrent = recurrentStates p
+    transientIdx = map toIndex transient
+    recurrentIdx = map toIndex recurrent
+    matrix = unTransitionMatrix p
+
 {- | Compute absorption probabilities into one recurrent state in canonical
 state order. Coordinate @i@ is the probability that the supplied target is
 the first recurrent state visited when starting from @i@.
@@ -152,8 +190,8 @@
 * a recurrent starting state has already arrived, so its coordinate is @1@
   when it is the target and @0@ otherwise.
 
-Transient coordinates are the corresponding column of @B = G R'@ and inherit
-the numerical behaviour and errors of 'fundamentalMatrix'.
+Transient coordinates solve @(I - Q) b = R'(:,k)@ directly and inherit the
+numerical behaviour and errors of the shared linear solver.
 
 Complexity: including first-time graph classification,
 @O(n^2 + (n + E) log(n + 1) + t^3)@ worst-case time,
@@ -173,9 +211,12 @@
     | null transientIdx =
         Right (LA.fromList [arrived i | i <- [0 .. dim - 1]])
     | otherwise = do
-        g <- fundamental (subMatrix transientIdx transientIdx matrix)
         let exits = LA.flatten (subMatrix transientIdx [targetIdx] matrix)
-            solved = LA.toList (g LA.#> exits)
+        transientProbabilities <-
+            solveIminusQVector
+                (subMatrix transientIdx transientIdx matrix)
+                exits
+        let solved = LA.toList transientProbabilities
             interior :: Unboxed.UArray Int Double
             interior =
                 Unboxed.accumArray
@@ -226,8 +267,10 @@
 state visited, conditioned on @X_0 = i@. A recurrent initial state is already
 absorbed at time zero.
 
-Partial application shares one lazy all-state table. A non-recurrent target
-produces exact zeros without a numerical solve.
+Partial application shares one lazy all-state table for transient starting
+states. Recurrent starting states return their exact boundary values without
+forcing that table. A non-recurrent target produces exact zeros without a
+numerical solve.
 
 Complexity: the first forced query takes @O(n^3)@ worst-case time and
 @O(n^2)@ temporary space and may retain an @O(n)@ all-state result and
@@ -241,8 +284,14 @@
     state ->
     Either LinearSystemError Double
 probabilityGivenInitialState p target =
-    \i -> (`LA.atIndex` toIndex i) <$> values
+    atInitial
   where
+    atInitial initial
+        | not targetIsRecurrent = Right 0
+        | recurrentState p initial =
+            Right (if initial == target then 1 else 0)
+        | otherwise = (`LA.atIndex` toIndex initial) <$> values
+    targetIsRecurrent = recurrentState p target
     values = probabilityByState p target
 
 {- | Compute the expected number of transitions until the chain first enters
diff --git a/src/Dtmc/Analysis/Classification.hs b/src/Dtmc/Analysis/Classification.hs
--- a/src/Dtmc/Analysis/Classification.hs
+++ b/src/Dtmc/Analysis/Classification.hs
@@ -17,7 +17,8 @@
 'FiniteState' method costs and construction of the shared support graph. Its
 first use adds @O(n^2)@ time and temporary space and retains @O(n + E)@ cache
 space. Strong components, closedness, periods, and phases are also computed
-lazily and shared by later queries on the same matrix.
+lazily. The complete typed classification is retained after first use, so
+whole-chain queries share public classes and state lists as well as the graph.
 
 A @0 x 0@ matrix has no communicating classes and is neither irreducible nor
 aperiodic here.
@@ -66,7 +67,11 @@
     stateFromInt,
     stateIndexInt,
  )
-import Dtmc.Transition.Matrix.Internal (TransitionMatrix, tmSupport)
+import Dtmc.Transition.Matrix.Internal (
+    TransitionMatrix,
+    tmClassification,
+    tmSupport,
+ )
 import Dtmc.Transition.Matrix.Internal.Graph qualified as G
 import Numeric.Natural (Natural)
 
@@ -152,11 +157,11 @@
 
 Complexity: excluding shared support-graph construction, the first full
 evaluation takes @O(n + E + n log(n + 1))@ time and @O(n + E)@ temporary
-space and retains @O(n)@ component cache; subsequent evaluations take
-@O(n)@ time and temporary space. Result space is @O(n)@.
+space and retains @O(n)@ classification space. Subsequent calls take @O(1)@
+time and space before traversal of the shared result; traversing every member
+still takes @O(n)@ time. The @O(n)@ result is retained by the matrix.
 -}
 communicatingClasses ::
-    (FiniteState state) =>
     TransitionMatrix state ->
     [CommClass state]
 communicatingClasses = classesOf . classify
@@ -284,10 +289,10 @@
 
 Complexity: excluding shared support-graph construction, the first full
 evaluation takes @O((n + E) log(n + 1))@ time and @O(n + E)@ temporary space
-and retains @O(n)@ component and closedness cache; later evaluations take
-@O(n)@ time and temporary space. Result space is @O(n)@.
+and retains @O(n)@ graph and classification space. Later calls take @O(1)@
+time and space before traversal of the shared @O(n)@ result.
 -}
-recurrentStates :: (FiniteState state) => TransitionMatrix state -> [state]
+recurrentStates :: TransitionMatrix state -> [state]
 recurrentStates = recurrentStatesOf . classify
 
 {- | Return the members of non-closed communicating classes, ordered by class
@@ -297,10 +302,10 @@
 
 Complexity: excluding shared support-graph construction, the first full
 evaluation takes @O((n + E) log(n + 1))@ time and @O(n + E)@ temporary space
-and retains @O(n)@ component and closedness cache; later evaluations take
-@O(n)@ time and temporary space. Result space is @O(n)@.
+and retains @O(n)@ graph and classification space. Later calls take @O(1)@
+time and space before traversal of the shared @O(n)@ result.
 -}
-transientStates :: (FiniteState state) => TransitionMatrix state -> [state]
+transientStates :: TransitionMatrix state -> [state]
 transientStates = transientStatesOf . classify
 
 {- | Return the states that form a communicating class on their own and cannot
@@ -312,10 +317,10 @@
 
 Complexity: excluding shared support-graph construction, the first full
 evaluation takes @O((n + E) log(n + 1))@ time and @O(n + E)@ temporary space
-and retains @O(n)@ component and closedness cache; later evaluations take
-@O(n)@ time and temporary space. Result space is @O(n)@.
+and retains @O(n)@ graph and classification space. Later calls take @O(1)@
+time and space before traversal of the shared result.
 -}
-absorbingStates :: (FiniteState state) => TransitionMatrix state -> [state]
+absorbingStates :: TransitionMatrix state -> [state]
 absorbingStates = absorbingStatesOf . classify
 
 {- | Return the period shared by every state of an irreducible chain. Returns
@@ -330,7 +335,7 @@
 component and period cache; later queries take @O(1)@ time. Temporary and
 result space per cached query are @O(1)@.
 -}
-chainPeriod :: (FiniteState state) => TransitionMatrix state -> Maybe Natural
+chainPeriod :: TransitionMatrix state -> Maybe Natural
 chainPeriod = chainPeriodOf . classify
 
 {- | Test whether the chain is irreducible and aperiodic. For a finite DTMC
@@ -351,34 +356,8 @@
 
 Complexity: full evaluation on an unforced matrix takes
 @O(n^2 + (n + E) log(n + 1))@ time, @O(n^2 + n + E)@ temporary space,
-@O(n + E)@ retained graph-cache space, and @O(n)@ result space. With all
-graph facts cached, it takes @O(n)@ time and @O(n)@ temporary and result
-space.
+and @O(n + E)@ retained graph and classification space. Later calls take
+@O(1)@ time and space before traversal of the shared result.
 -}
-classify :: (FiniteState state) => TransitionMatrix state -> Classification state
-classify p =
-    Classification
-        { classesOf = cs
-        , isIrreducible = irreducible'
-        , isAperiodic = aperiodic'
-        , isErgodic = irreducible' && aperiodic'
-        , chainPeriodOf = chainPeriodOf'
-        , recurrentStatesOf = concatMap classMembers (filter classClosed cs)
-        , transientStatesOf = concatMap classMembers (filter (not . classClosed) cs)
-        , absorbingStatesOf = [i | cc <- cs, classClosed cc, [i] <- [classMembers cc]]
-        }
-  where
-    g = tmSupport p
-    cs =
-        [ CommClass
-            { classMembers = map toState c
-            , classPeriod = G.periodOf g v
-            , classClosed = G.inClosedComponent g v
-            }
-        | c@(v : _) <- G.components g
-        ]
-    irreducible' = graphIrreducible g
-    aperiodic' = graphAperiodic g
-    chainPeriodOf' = case cs of
-        [c] -> classPeriod c
-        _ -> Nothing
+classify :: TransitionMatrix state -> Classification state
+classify = tmClassification
diff --git a/src/Dtmc/Analysis/Classification/Internal.hs b/src/Dtmc/Analysis/Classification/Internal.hs
--- a/src/Dtmc/Analysis/Classification/Internal.hs
+++ b/src/Dtmc/Analysis/Classification/Internal.hs
@@ -2,34 +2,18 @@
 
 {- |
 Module      : Dtmc.Analysis.Classification.Internal
-Description : Internal carriers and graph operations for chain classification.
+Description : Internal carriers and construction for chain classification.
 
-Raw carrier types and solver-oriented graph operations behind
-"Dtmc.Analysis.Classification": the per-class summary t'CommClass' and the
-whole-chain structural report t'Classification'. This module exposes the
-report constructor for trusted internal use; constructing it here may produce
-summary fields inconsistent with its communicating classes.
+Raw carrier types behind "Dtmc.Analysis.Classification": the per-class summary
+@CommClass@ and whole-chain structural report @Classification@. The report
+builder is shared with the transition-matrix cache.
 -}
 module Dtmc.Analysis.Classification.Internal (
     type CommClass (..),
     type Classification (..),
-    backwardReachable,
+    classificationFromGraph,
 ) where
 
-import Data.Maybe (
-    fromMaybe,
- )
-import Dtmc.State (
-    FiniteState,
- )
-import Dtmc.State.Internal (
-    stateFromInt,
-    stateIndexInt,
- )
-import Dtmc.Transition.Matrix.Internal (
-    TransitionMatrix,
-    tmSupport,
- )
 import Dtmc.Transition.Matrix.Internal.Graph qualified as G
 import Numeric.Natural (
     Natural,
@@ -89,32 +73,52 @@
 
 deriving instance (Show state) => Show (Classification state)
 
-toState :: (FiniteState state) => Int -> state
-toState index =
-    fromMaybe
-        (error "Dtmc.Analysis.Classification.Internal: graph vertex out of bounds")
-        (stateFromInt index)
-
-toIndex :: (FiniteState state) => state -> Int
-toIndex = stateIndexInt
-
-{- | Return states from which an allowed seed is reachable along a support
-path containing only states accepted by @allowed@. Disallowed seeds are
-ignored; the result is duplicate-free and ordered by state index.
+{- | Build a complete typed classification from an index-to-state conversion
+and a shared support graph. Supplying the canonical conversion of a
+'Dtmc.State.FiniteState' instance produces the public classification.
 
-For the complexity bounds, @n@ is the state count, @E@ the support-edge count,
-@s@ the number of supplied seeds, and @r@ the number of returned states.
+The result is deliberately lazy so a transition matrix can carry it without
+forcing graph construction. Once forced, all whole-chain projections share
+the same classes and state lists.
 
-Complexity: excluding @n@ evaluations of @allowed@, 'FiniteState' method
-costs, and shared support-graph construction, @O(n + E + s)@ time,
-@O(n + E + s)@ temporary space, and @O(r)@ result space. The first reverse
-traversal also retains @O(n + E)@ predecessor-cache space.
+Complexity: @O(n + E)@ time after the graph's component and period caches are
+available, @O(n)@ temporary space, and @O(n)@ retained result space.
 -}
-backwardReachable ::
-    (FiniteState state) =>
-    TransitionMatrix state ->
-    (state -> Bool) ->
-    [state] ->
-    [state]
-backwardReachable p allowed seeds =
-    map toState (G.backwardReachable (tmSupport p) (allowed . toState) (map toIndex seeds))
+classificationFromGraph :: (Int -> state) -> G.Graph -> Classification state
+classificationFromGraph toState graph =
+    Classification
+        { classesOf = communicating
+        , isIrreducible = irreducible'
+        , isAperiodic = aperiodic'
+        , isErgodic = irreducible' && aperiodic'
+        , chainPeriodOf = case communicating of
+            [communicatingClass] -> classPeriod communicatingClass
+            _ -> Nothing
+        , recurrentStatesOf =
+            concatMap classMembers (filter classClosed communicating)
+        , transientStatesOf =
+            concatMap classMembers (filter (not . classClosed) communicating)
+        , absorbingStatesOf =
+            [ state
+            | communicatingClass <- communicating
+            , classClosed communicatingClass
+            , [state] <- [classMembers communicatingClass]
+            ]
+        }
+  where
+    components = G.components graph
+    communicating =
+        [ CommClass
+            { classMembers = map toState component
+            , classPeriod = G.periodOf graph first
+            , classClosed = G.inClosedComponent graph first
+            }
+        | component@(first : _) <- components
+        ]
+    irreducible' = case components of
+        [component] -> not (null component)
+        _ -> False
+    aperiodic' =
+        not (null components)
+            && all ((== Just 1) . G.componentPeriod graph) components
+{-# INLINE classificationFromGraph #-}
diff --git a/src/Dtmc/Analysis/Classification/Reachability/Internal.hs b/src/Dtmc/Analysis/Classification/Reachability/Internal.hs
new file mode 100644
--- /dev/null
+++ b/src/Dtmc/Analysis/Classification/Reachability/Internal.hs
@@ -0,0 +1,47 @@
+{- |
+Module      : Dtmc.Analysis.Classification.Reachability.Internal
+Description : Solver-oriented typed support-graph reachability.
+
+Internal adapter between finite-state values and the integer support graph.
+-}
+module Dtmc.Analysis.Classification.Reachability.Internal (
+    backwardReachable,
+) where
+
+import Data.Maybe (fromMaybe)
+import Dtmc.State (FiniteState)
+import Dtmc.State.Internal (stateFromInt, stateIndexInt)
+import Dtmc.Transition.Matrix.Internal (TransitionMatrix, tmSupport)
+import Dtmc.Transition.Matrix.Internal.Graph qualified as G
+
+toState :: (FiniteState state) => Int -> state
+toState index =
+    fromMaybe
+        (error "Dtmc.Analysis.Classification: graph vertex out of bounds")
+        (stateFromInt index)
+
+{- | Return states from which an allowed seed is reachable along a support
+path containing only states accepted by @allowed@. Disallowed seeds are
+ignored; the result is duplicate-free and ordered by state index.
+
+For the complexity bounds, @n@ is the state count, @E@ the support-edge count,
+@s@ the number of supplied seeds, and @r@ the number of returned states.
+
+Complexity: excluding @n@ evaluations of @allowed@, 'FiniteState' method
+costs, and shared support-graph construction, @O(n + E + s)@ time,
+@O(n + E + s)@ temporary space, and @O(r)@ result space. The first reverse
+traversal also retains @O(n + E)@ predecessor-cache space.
+-}
+backwardReachable ::
+    (FiniteState state) =>
+    TransitionMatrix state ->
+    (state -> Bool) ->
+    [state] ->
+    [state]
+backwardReachable matrix allowed seeds =
+    map toState
+        ( G.backwardReachable
+            (tmSupport matrix)
+            (allowed . toState)
+            (map stateIndexInt seeds)
+        )
diff --git a/src/Dtmc/Analysis/FiniteHorizon/Internal.hs b/src/Dtmc/Analysis/FiniteHorizon/Internal.hs
new file mode 100644
--- /dev/null
+++ b/src/Dtmc/Analysis/FiniteHorizon/Internal.hs
@@ -0,0 +1,296 @@
+{-# LANGUAGE TypeFamilies #-}
+
+{- |
+Module      : Dtmc.Analysis.FiniteHorizon.Internal
+Description : Dense finite-state primitives for finite-horizon analyses.
+
+Private BLAS-backed recurrences shared by finite-time, hitting-time,
+return-time, and bounded-visit analyses. Public analysis modules retain their
+locally finite map recurrences and select these implementations only when a
+transition representation supplies a suitable dense backend.
+-}
+module Dtmc.Analysis.FiniteHorizon.Internal (
+    availableDenseBackend,
+    branchingDenseBackend,
+    denseStepProbability,
+    denseProbabilityAfter,
+    denseExactHittingProbability,
+    denseLowerHittingProbability,
+    denseUpperHittingProbability,
+    denseExactReturnProbability,
+    denseLowerReturnProbability,
+    denseUpperReturnProbability,
+    denseBoundedVisitExpectation,
+) where
+
+import Data.Map.Strict qualified as Map
+import Data.Vector.Storable qualified as Storable
+import Data.Vector.Storable.Mutable qualified as Mutable
+import Dtmc.Transition.Internal (
+    DenseTransitionBackend (..),
+    Transition (..),
+    denseRowBranches,
+ )
+import Numeric.LinearAlgebra qualified as LA
+import Numeric.Natural (
+    Natural,
+ )
+
+availableDenseBackend ::
+    (Transition transition) =>
+    transition ->
+    Maybe (DenseTransitionBackend (TransitionState transition))
+availableDenseBackend = transitionDenseBackend
+
+branchingDenseBackend ::
+    (Transition transition) =>
+    transition ->
+    [TransitionState transition] ->
+    Maybe (DenseTransitionBackend (TransitionState transition))
+branchingDenseBackend transition sources = do
+    backend <- availableDenseBackend transition
+    if any (denseRowBranches backend) sources then Just backend else Nothing
+
+denseStepProbability ::
+    DenseTransitionBackend state ->
+    state ->
+    state ->
+    Double
+denseStepProbability backend source destination =
+    denseTransitionMatrix backend
+        `LA.atIndex` ( denseTransitionIndex backend source
+                     , denseTransitionIndex backend destination
+                     )
+
+denseProbabilityAfter ::
+    Natural ->
+    [(state, Double)] ->
+    DenseTransitionBackend state ->
+    state ->
+    Double
+denseProbabilityAfter steps weights backend destination =
+    evolved Storable.! denseTransitionIndex backend destination
+  where
+    evolved = iterateDense steps backend (weightsVector backend weights)
+
+denseExactHittingProbability ::
+    Natural ->
+    DenseTransitionBackend state ->
+    (state -> Bool) ->
+    state ->
+    Double
+denseExactHittingProbability time backend isTarget initial =
+    go time (pointVector backend initial)
+  where
+    targets = denseTargetSet backend initial isTarget
+
+    go 0 _ = 0
+    go remaining survivors =
+        let advanced = advanceDense backend survivors
+            hitMass = targetMass targets advanced
+         in if remaining == 1
+                then hitMass
+                else
+                    let next = clearTargets targets advanced
+                     in next `seq`
+                            if Storable.all (== 0) next
+                                then 0
+                                else go (remaining - 1) next
+
+denseLowerHittingProbability ::
+    Natural ->
+    DenseTransitionBackend state ->
+    (state -> Bool) ->
+    state ->
+    Double
+denseLowerHittingProbability steps backend isTarget initial =
+    go steps (pointVector backend initial) 0
+  where
+    targets = denseTargetSet backend initial isTarget
+
+    go 0 _ total = total
+    go remaining survivors total =
+        let advanced = advanceDense backend survivors
+            hitMass = targetMass targets advanced
+            cumulative = total + hitMass
+            next = clearTargets targets advanced
+         in cumulative `seq`
+                next `seq`
+                    if Storable.all (== 0) next
+                        then cumulative
+                        else go (remaining - 1) next cumulative
+
+denseUpperHittingProbability ::
+    Natural ->
+    DenseTransitionBackend state ->
+    (state -> Bool) ->
+    state ->
+    Double
+denseUpperHittingProbability time backend isTarget initial =
+    go time (pointVector backend initial)
+  where
+    targets = denseTargetSet backend initial isTarget
+
+    go 0 survivors = Storable.sum survivors
+    go remaining survivors =
+        let advanced = advanceDense backend survivors
+            next = clearTargets targets advanced
+         in next `seq`
+                if Storable.all (== 0) next
+                    then 0
+                    else go (remaining - 1) next
+
+denseExactReturnProbability ::
+    Natural ->
+    DenseTransitionBackend state ->
+    state ->
+    Double
+denseExactReturnProbability time backend initial =
+    go time (pointVector backend initial)
+  where
+    initialIndex = denseTransitionIndex backend initial
+
+    go 0 _ = 0
+    go remaining survivors =
+        let advanced = advanceDense backend survivors
+            returnMass = advanced Storable.! initialIndex
+         in if remaining == 1
+                then returnMass
+                else go (remaining - 1) (clearCoordinate initialIndex advanced)
+
+denseLowerReturnProbability ::
+    Natural ->
+    DenseTransitionBackend state ->
+    state ->
+    Double
+denseLowerReturnProbability bound backend initial =
+    go bound (pointVector backend initial) 0
+  where
+    initialIndex = denseTransitionIndex backend initial
+
+    go remaining _ total | remaining <= 1 = total
+    go remaining survivors total =
+        let advanced = advanceDense backend survivors
+            returnMass = advanced Storable.! initialIndex
+            cumulative = total + returnMass
+            next = clearCoordinate initialIndex advanced
+         in cumulative `seq` next `seq` go (remaining - 1) next cumulative
+
+denseUpperReturnProbability ::
+    Natural ->
+    DenseTransitionBackend state ->
+    state ->
+    Double
+denseUpperReturnProbability time backend initial =
+    go time (pointVector backend initial)
+  where
+    initialIndex = denseTransitionIndex backend initial
+
+    go 0 survivors = Storable.sum survivors
+    go remaining survivors =
+        let advanced = advanceDense backend survivors
+            next = clearCoordinate initialIndex advanced
+         in next `seq` go (remaining - 1) next
+
+denseBoundedVisitExpectation ::
+    Natural ->
+    Map.Map state Double ->
+    DenseTransitionBackend state ->
+    (state -> Bool) ->
+    Double
+denseBoundedVisitExpectation 0 _ _ _ = 0
+denseBoundedVisitExpectation bound weights backend isVisited =
+    go bound (weightsVector backend (Map.toList weights)) 0
+  where
+    visitedMask =
+        Storable.fromList
+            [if isVisited state then 1 else 0 | state <- denseTransitionStates backend]
+
+    go remaining current expectation =
+        let visitProbability = visitedMask LA.<.> current
+            cumulative = expectation + visitProbability
+         in if remaining == 1
+                then cumulative
+                else
+                    let next = advanceDense backend current
+                     in cumulative `seq` next `seq` go (remaining - 1) next cumulative
+
+advanceDense ::
+    DenseTransitionBackend state ->
+    LA.Vector Double ->
+    LA.Vector Double
+advanceDense backend = (LA.tr (denseTransitionMatrix backend) LA.#>)
+
+iterateDense ::
+    Natural ->
+    DenseTransitionBackend state ->
+    LA.Vector Double ->
+    LA.Vector Double
+iterateDense 0 _ weights = weights
+iterateDense remaining backend weights =
+    let next = advanceDense backend weights
+     in next `seq` iterateDense (remaining - 1) backend next
+
+pointVector :: DenseTransitionBackend state -> state -> LA.Vector Double
+pointVector backend selected =
+    Storable.generate dimension $ \index ->
+        if index == selectedIndex then 1 else 0
+  where
+    dimension = LA.rows (denseTransitionMatrix backend)
+    selectedIndex = denseTransitionIndex backend selected
+
+weightsVector ::
+    DenseTransitionBackend state ->
+    [(state, Double)] ->
+    LA.Vector Double
+weightsVector backend weights =
+    Storable.accum
+        (+)
+        (Storable.replicate dimension 0)
+        [ (denseTransitionIndex backend state, weight)
+        | (state, weight) <- weights
+        , weight /= 0
+        ]
+  where
+    dimension = LA.rows (denseTransitionMatrix backend)
+
+-- Target coordinates in dense-backend order. Keeping only matching indices
+-- avoids full floating-point target and survivor masks.
+newtype DenseTargetSet = DenseTargetSet
+    { denseTargetIndices :: Storable.Vector Int
+    }
+
+denseTargetSet ::
+    DenseTransitionBackend state ->
+    state ->
+    (state -> Bool) ->
+    DenseTargetSet
+denseTargetSet backend initial isTarget =
+    DenseTargetSet
+        ( Storable.fromList
+            [ index
+            | (index, state) <- zip [0 ..] (denseTransitionStates backend)
+            , index /= initialIndex
+            , isTarget state
+            ]
+        )
+  where
+    initialIndex = denseTransitionIndex backend initial
+
+targetMass :: DenseTargetSet -> LA.Vector Double -> Double
+targetMass targets values =
+    Storable.foldl'
+        (\total index -> total + Storable.unsafeIndex values index)
+        0
+        (denseTargetIndices targets)
+
+clearTargets :: DenseTargetSet -> LA.Vector Double -> LA.Vector Double
+clearTargets targets =
+    Storable.modify $ \mutable ->
+        Storable.forM_
+            (denseTargetIndices targets)
+            (\index -> Mutable.unsafeWrite mutable index 0)
+
+clearCoordinate :: Int -> LA.Vector Double -> LA.Vector Double
+clearCoordinate index =
+    Storable.modify $ \mutable -> Mutable.unsafeWrite mutable index 0
diff --git a/src/Dtmc/Analysis/FiniteTime.hs b/src/Dtmc/Analysis/FiniteTime.hs
--- a/src/Dtmc/Analysis/FiniteTime.hs
+++ b/src/Dtmc/Analysis/FiniteTime.hs
@@ -17,6 +17,15 @@
     ConditionalProbabilityError (..),
 ) where
 
+import Data.Maybe (
+    fromMaybe,
+ )
+import Dtmc.Analysis.FiniteHorizon.Internal (
+    availableDenseBackend,
+    branchingDenseBackend,
+    denseProbabilityAfter,
+    denseStepProbability,
+ )
 import Dtmc.Analysis.FiniteTime.Internal (
     NormalisedObservations (..),
     normalise,
@@ -41,7 +50,8 @@
 observations denotes their conjunction; list order has no meaning.
 -}
 data Observation state
-    = At Natural state -- ^ Require the supplied state at the specified time.
+    = -- | Require the supplied state at the specified time.
+      At Natural state
     deriving (Eq, Show)
 
 -- | Why a conditional probability query has no defined value.
@@ -53,8 +63,10 @@
 {- | Return the one-step transition probability
 @P(X_1 = j | X_0 = i)@ through any locally finite 'Transition'.
 
-Complexity: excluding 'transitionLaw', @O(log(s + 1))@ time and @O(1)@
-temporary and result space for returned law support size @s@.
+For a transition matrix, the dense backend reads the matrix entry directly in
+@O(1)@ time and space. Otherwise, excluding 'transitionLaw', complexity is
+@O(log(s + 1))@ time and @O(1)@ temporary and result space for returned law
+support size @s@.
 -}
 stepProbability ::
     (Transition kernel, Ord (TransitionState kernel)) =>
@@ -62,8 +74,14 @@
     TransitionState kernel ->
     TransitionState kernel ->
     Double
-stepProbability kernel source =
-    probabilityAt (transitionLaw kernel source)
+stepProbability kernel source destination =
+    fromMaybe genericProbability optimizedProbability
+  where
+    optimizedProbability = do
+        backend <- availableDenseBackend kernel
+        pure (denseStepProbability backend source destination)
+    genericProbability =
+        probabilityAt (transitionLaw kernel source) destination
 
 {- | Return the @k@-step transition probability @P(X_k = j | X_0 = i)@. At
 @k = 0@ this is the Kronecker delta.
@@ -75,6 +93,9 @@
 Complexity: excluding 'transitionLaw',
 @O(k (w + e log(u + 1) + u) + log(r + 1) + 1)@ time, @O(w + u)@ temporary
 space, and @O(1)@ result space.
+
+For a transition matrix whose source row branches, the adaptive dense path
+takes @O(k n^2 + n)@ time and @O(n)@ temporary space for @n@ states.
 -}
 nStepProbability ::
     (Transition kernel, Ord (TransitionState kernel)) =>
@@ -83,8 +104,18 @@
     TransitionState kernel ->
     TransitionState kernel ->
     Double
-nStepProbability steps kernel source =
-    probabilityAt (evolveN steps (pointMass source) kernel)
+nStepProbability 0 _ source destination =
+    if source == destination then 1 else 0
+nStepProbability 1 kernel source destination =
+    stepProbability kernel source destination
+nStepProbability steps kernel source destination =
+    fromMaybe genericProbability optimizedProbability
+  where
+    optimizedProbability = do
+        backend <- branchingDenseBackend kernel [source]
+        pure (denseProbabilityAfter steps [(source, 1)] backend destination)
+    genericProbability =
+        probabilityAt (evolveN steps (pointMass source) kernel) destination
 
 {- | Compute the probability of a conjunction of timed observations.
 Observation order has no meaning, duplicates collapse, and an empty
@@ -105,6 +136,10 @@
 'transitionLaw' evaluations,
 @O(m log(m + 1) + s_0 + k C + q log(r + 1) + 1)@ time,
 @O(m + w + u)@ temporary space, and @O(1)@ result space.
+
+When the initial support contains a branching transition-matrix row, the
+first observation uses dense propagation; later gaps use the same adaptive
+matrix path described by 'nStepProbability'.
 -}
 probability ::
     ( Distribution distribution
@@ -121,9 +156,20 @@
         Impossible -> 0
         Consistent [] -> 1
         Consistent ((firstTime, firstState) : rest) ->
-            probabilityAt (evolveN firstTime initial kernel) firstState
+            firstProbability firstTime firstState
                 * gaps (firstTime, firstState) rest
   where
+    firstProbability firstTime firstState
+        | firstTime == 0 = genericProbability
+        | otherwise = fromMaybe genericProbability optimizedProbability
+      where
+        weights = distributionWeights initial
+        optimizedProbability = do
+            backend <- branchingDenseBackend kernel (map fst weights)
+            pure (denseProbabilityAfter firstTime weights backend firstState)
+        genericProbability =
+            probabilityAt (evolveN firstTime initial kernel) firstState
+
     gaps _ [] = 1
     gaps (previousTime, previousState) ((time, state) : more) =
         nStepProbability
diff --git a/src/Dtmc/Analysis/HittingTime.hs b/src/Dtmc/Analysis/HittingTime.hs
--- a/src/Dtmc/Analysis/HittingTime.hs
+++ b/src/Dtmc/Analysis/HittingTime.hs
@@ -50,7 +50,7 @@
 import Data.Maybe (
     fromMaybe,
  )
-import Dtmc.Analysis.Classification.Internal (
+import Dtmc.Analysis.Classification.Reachability.Internal (
     backwardReachable,
  )
 import Dtmc.Analysis.Event (
@@ -59,6 +59,12 @@
 import Dtmc.Analysis.Expectation (
     Expectation (..),
  )
+import Dtmc.Analysis.FiniteHorizon.Internal (
+    branchingDenseBackend,
+    denseExactHittingProbability,
+    denseLowerHittingProbability,
+    denseUpperHittingProbability,
+ )
 import Dtmc.Analysis.Initial.Internal (
     expectationUnderEither,
     probabilityUnder,
@@ -147,8 +153,13 @@
 exactProbabilityAt time kernel isTarget initialState
     | time == 0 = if isTarget initialState then 1 else 0
     | isTarget initialState = 0
-    | otherwise = go time (Map.singleton initialState 1)
+    | otherwise = fromMaybe genericProbability optimizedProbability
   where
+    optimizedProbability = do
+        backend <- branchingDenseBackend kernel [initialState]
+        pure (denseExactHittingProbability time backend isTarget initialState)
+    genericProbability = go time (Map.singleton initialState 1)
+
     go 0 _ = 0
     go _ survivors | Map.null survivors = 0
     go remaining survivors =
@@ -178,8 +189,19 @@
 lowerTailProbability bound kernel isTarget initialState
     | bound == 0 = 0
     | isTarget initialState = 1
-    | otherwise = go (bound - 1) (Map.singleton initialState 1) 0
+    | otherwise = fromMaybe genericProbability optimizedProbability
   where
+    optimizedProbability = do
+        backend <- branchingDenseBackend kernel [initialState]
+        pure
+            ( denseLowerHittingProbability
+                (bound - 1)
+                backend
+                isTarget
+                initialState
+            )
+    genericProbability = go (bound - 1) (Map.singleton initialState 1) 0
+
     go 0 _ total = total
     go _ survivors total | Map.null survivors = total
     go remaining survivors total =
@@ -469,8 +491,14 @@
     Double
 upperTailProbability time kernel isTarget initialState
     | isTarget initialState = 0
-    | otherwise = go time (Map.singleton initialState 1)
+    | time == 0 = 1
+    | otherwise = fromMaybe genericProbability optimizedProbability
   where
+    optimizedProbability = do
+        backend <- branchingDenseBackend kernel [initialState]
+        pure (denseUpperHittingProbability time backend isTarget initialState)
+    genericProbability = go time (Map.singleton initialState 1)
+
     go 0 survivors = sum (Map.elems survivors)
     go _ survivors | Map.null survivors = 0
     go remaining survivors =
@@ -524,6 +552,10 @@
 Complexity: excluding 'transitionLaw' and predicate evaluation,
 @O(k (w + e log(u + 1) + u) + 1)@ time, @O(w + u)@ temporary space, and
 @O(1)@ result space.
+
+For a transition matrix whose initial row branches, the adaptive dense path
+evaluates the target predicate once per finite state and takes
+@O(n + k n^2)@ time and @O(n)@ temporary space for @n@ states.
 -}
 probabilityGivenInitialState ::
     ( Transition kernel
diff --git a/src/Dtmc/Analysis/Limiting.hs b/src/Dtmc/Analysis/Limiting.hs
--- a/src/Dtmc/Analysis/Limiting.hs
+++ b/src/Dtmc/Analysis/Limiting.hs
@@ -91,7 +91,7 @@
 retains @O(n + E)@ graph and classification caches. With those facts cached,
 it takes @O(c)@ time, @O(c)@ temporary space, and @O(1)@ result space.
 -}
-converges :: (FiniteState state) => TransitionMatrix state -> Bool
+converges :: TransitionMatrix state -> Bool
 converges p =
     all
         ((== Just 1) . classPeriod)
diff --git a/src/Dtmc/Analysis/LinearSystem/Internal.hs b/src/Dtmc/Analysis/LinearSystem/Internal.hs
--- a/src/Dtmc/Analysis/LinearSystem/Internal.hs
+++ b/src/Dtmc/Analysis/LinearSystem/Internal.hs
@@ -20,6 +20,8 @@
     fundamental,
 ) where
 
+import Data.List qualified as List
+import Data.Vector.Storable qualified as Storable
 import Dtmc.Analysis.LinearSystem (
     LinearSystemError (..),
  )
@@ -32,14 +34,49 @@
 residualLimitValue = 1e-9
 
 allFinite :: LA.Matrix Double -> Bool
-allFinite = all isFinite . LA.toList . LA.flatten
+allFinite = Storable.all isFinite . LA.flatten
 
 isFinite :: Double -> Bool
 isFinite value = not (isNaN value || isInfinite value)
 
 infinityNorm :: LA.Matrix Double -> Double
-infinityNorm = maximum . (0 :) . map (sum . map abs) . LA.toLists
+infinityNorm = maximumAbsoluteSum . LA.toRows
 
+oneNorm :: LA.Matrix Double -> Double
+oneNorm = maximumAbsoluteSum . LA.toColumns
+
+maximumAbsoluteSum :: [LA.Vector Double] -> Double
+maximumAbsoluteSum =
+    List.foldl' max 0
+        . map (Storable.foldl' (\acc value -> acc + abs value) 0)
+
+maximumAbsolute :: LA.Vector Double -> Double
+maximumAbsolute = Storable.foldl' (\acc value -> max acc (abs value)) 0
+
+reciprocalConditionForIminusQ ::
+    LA.Matrix Double ->
+    LA.Vector Double ->
+    Double
+reciprocalConditionForIminusQ coefficient inverseTimesOne
+    | Storable.all isFinite inverseTimesOne
+        && isFinite provenLowerBound
+        && provenLowerBound >= conditionLimit =
+        provenLowerBound
+    | otherwise = LA.rcond coefficient
+  where
+    inverseInfinityNorm = maximumAbsolute inverseTimesOne
+    -- The inverse one-norm is at most n times its infinity norm, and every
+    -- spectral norm is at most the square root of one-norm times infinity norm.
+    provenLowerBound =
+        1
+            / ( inverseInfinityNorm
+                    * sqrt
+                        ( fromIntegral (LA.rows coefficient)
+                            * oneNorm coefficient
+                            * infinityNorm coefficient
+                        )
+              )
+
 relativeSystemResidual ::
     LA.Matrix Double ->
     LA.Matrix Double ->
@@ -53,6 +90,28 @@
                 + infinityNorm rightHandSide
             )
 
+validateSolution ::
+    LA.Matrix Double ->
+    LA.Matrix Double ->
+    LA.Matrix Double ->
+    Double ->
+    Either LinearSystemError (LA.Matrix Double)
+validateSolution coefficient rightHandSide solution reciprocalCondition
+    | not (allFinite solution) = Left NonFiniteSolution
+    | not (isFinite reciprocalCondition) = Left NonFiniteSystem
+    | reciprocalCondition < conditionLimit =
+        Left (IllConditionedSystem reciprocalCondition)
+    | residual > residualLimitValue =
+        Left
+            ( ResidualTooLarge
+                { relativeResidual = residual
+                , residualLimit = residualLimitValue
+                }
+            )
+    | otherwise = Right solution
+  where
+    residual = relativeSystemResidual coefficient solution rightHandSide
+
 {- | Solve @A X = B@ by LU decomposition. @A@ must be a non-empty square
 matrix, @B@ must have the same number of rows, and all entries must be finite;
 incompatible dimensions raise a backend error.
@@ -75,22 +134,12 @@
     | otherwise =
         case LA.linearSolve coefficient rightHandSide of
             Nothing -> Left SingularSystem
-            Just solution
-                | not (allFinite solution) -> Left NonFiniteSolution
-                | not (isFinite reciprocalCondition) -> Left NonFiniteSystem
-                | reciprocalCondition < conditionLimit ->
-                    Left (IllConditionedSystem reciprocalCondition)
-                | residual > residualLimitValue ->
-                    Left
-                        ( ResidualTooLarge
-                            { relativeResidual = residual
-                            , residualLimit = residualLimitValue
-                            }
-                        )
-                | otherwise -> Right solution
-              where
-                residual =
-                    relativeSystemResidual coefficient solution rightHandSide
+            Just solution ->
+                validateSolution
+                    coefficient
+                    rightHandSide
+                    solution
+                    reciprocalCondition
   where
     reciprocalCondition = LA.rcond coefficient
 
@@ -120,16 +169,23 @@
 rowSums :: LA.Matrix Double -> LA.Vector Double
 rowSums m = m LA.#> LA.konst 1 (LA.cols m)
 
-{- | Solve @(I - Q) X = B@ by LU decomposition. @Q@ must be @n x n@, @B@
-must be @n x r@, and all entries must be finite; incompatible dimensions
-raise a backend error.
+{- | Solve @(I - Q) X = B@ by one LU decomposition. @Q@ must be a
+transient-to-transient, substochastic @n x n@ matrix, @B@ must be @n x r@,
+and all entries must be finite; incompatible dimensions raise a backend
+error.
 
 Returns a 'LinearSystemError' when the input or result is non-finite, the
 backend reports singularity, the reciprocal condition estimate is below
-@1e-12@, or the scaled infinity-norm residual exceeds @1e-9@. Current DTMC
-callers choose @Q@ with spectral radius below one, which guarantees
-invertibility in exact arithmetic but not a reliable 'Double' result.
+@1e-12@, or the scaled infinity-norm residual exceeds @1e-9@.
 
+The solve appends a column of ones to @B@. Since @(I - Q)^-1@ is
+non-negative, the extra result gives its exact infinity norm in exact
+arithmetic. Matrix-norm inequalities then provide a conservative lower bound
+on the reciprocal 2-norm condition number. A bound above the acceptance limit
+proves that the solution is sufficiently conditioned without a separate
+singular-value decomposition; uncertain cases retain the backend condition
+estimate and therefore the same rejection threshold.
+
 Complexity: @O(n^3 + n^2 r)@ time, @O(n^2 + n r)@ temporary space, and
 @O(n r)@ result space for @r@ right-hand sides.
 -}
@@ -138,10 +194,34 @@
     LA.Matrix Double ->
     Either LinearSystemError (LA.Matrix Double)
 solveIminusQ q rightHandSide
-    | not (allFinite q) = Left NonFiniteSystem
-    | otherwise = solveLinearSystem coefficient rightHandSide
+    | not (allFinite q && allFinite rightHandSide) = Left NonFiniteSystem
+    | otherwise =
+        case LA.linearSolve coefficient augmentedRightHandSide of
+            Nothing -> Left SingularSystem
+            Just augmentedSolution ->
+                let solution =
+                        LA.takeColumns
+                            rightHandSideColumns
+                            augmentedSolution
+                    inverseTimesOne =
+                        LA.flatten
+                            ( LA.dropColumns
+                                rightHandSideColumns
+                                augmentedSolution
+                            )
+                 in validateSolution
+                        coefficient
+                        rightHandSide
+                        solution
+                        ( reciprocalConditionForIminusQ
+                            coefficient
+                            inverseTimesOne
+                        )
   where
     coefficient = LA.ident (LA.rows q) - q
+    rightHandSideColumns = LA.cols rightHandSide
+    augmentedRightHandSide =
+        rightHandSide LA.||| LA.asColumn (LA.konst 1 (LA.rows q))
 
 {- | Solve @(I - Q) x = b@ for a vector of length @n@. This has the same
 shape requirements, numerical behaviour, and errors as 'solveIminusQ'.
diff --git a/src/Dtmc/Analysis/ReturnTime.hs b/src/Dtmc/Analysis/ReturnTime.hs
--- a/src/Dtmc/Analysis/ReturnTime.hs
+++ b/src/Dtmc/Analysis/ReturnTime.hs
@@ -43,6 +43,9 @@
     Map,
  )
 import Data.Map.Strict qualified as Map
+import Data.Maybe (
+    fromMaybe,
+ )
 import Dtmc.Analysis.Classification (
     recurrentState,
     transientStates,
@@ -53,6 +56,12 @@
 import Dtmc.Analysis.Expectation (
     Expectation (..),
  )
+import Dtmc.Analysis.FiniteHorizon.Internal (
+    branchingDenseBackend,
+    denseExactReturnProbability,
+    denseLowerReturnProbability,
+    denseUpperReturnProbability,
+ )
 import Dtmc.Analysis.Initial.Internal (
     expectationUnderEither,
     probabilityUnder,
@@ -130,8 +139,12 @@
     Double
 exactProbabilityAt 0 _ _ = 0
 exactProbabilityAt time kernel initialState =
-    go time (Map.singleton initialState 1)
+    fromMaybe genericProbability optimizedProbability
   where
+    optimizedProbability = do
+        backend <- branchingDenseBackend kernel [initialState]
+        pure (denseExactReturnProbability time backend initialState)
+    genericProbability = go time (Map.singleton initialState 1)
     isInitial state = state == initialState
 
     go 0 _ = 0
@@ -157,9 +170,14 @@
     kernel ->
     TransitionState kernel ->
     Double
+lowerTailProbability bound _ _ | bound <= 1 = 0
 lowerTailProbability bound kernel initialState =
-    go bound (Map.singleton initialState 1) 0
+    fromMaybe genericProbability optimizedProbability
   where
+    optimizedProbability = do
+        backend <- branchingDenseBackend kernel [initialState]
+        pure (denseLowerReturnProbability bound backend initialState)
+    genericProbability = go bound (Map.singleton initialState 1) 0
     isInitial state = state == initialState
 
     go remaining _ total | remaining <= 1 = total
@@ -319,9 +337,14 @@
     kernel ->
     TransitionState kernel ->
     Double
+upperTailProbability 0 _ _ = 1
 upperTailProbability time kernel initialState =
-    go time (Map.singleton initialState 1)
+    fromMaybe genericProbability optimizedProbability
   where
+    optimizedProbability = do
+        backend <- branchingDenseBackend kernel [initialState]
+        pure (denseUpperReturnProbability time backend initialState)
+    genericProbability = go time (Map.singleton initialState 1)
     isInitial state = state == initialState
     go 0 survivors = sum (Map.elems survivors)
     go _ survivors | Map.null survivors = 0
@@ -372,7 +395,11 @@
 are per-step upper bounds on survivor states, traversed transition edges, and
 accumulated destinations.
 
-Complexity: excluding 'transitionLaw',
+For a 'TransitionMatrix' whose initial row branches, an internal adaptive path
+uses dense matrix-vector multiplication in @O(k n^2)@ time and @O(n)@
+temporary space. A deterministic initial row retains the sparse recurrence.
+
+Generic complexity: excluding 'transitionLaw',
 @O(k (w + e log(u + 1) + u) + 1)@ time, @O(w + u)@ temporary space, and
 @O(1)@ result space.
 -}
diff --git a/src/Dtmc/Analysis/Stationary.hs b/src/Dtmc/Analysis/Stationary.hs
--- a/src/Dtmc/Analysis/Stationary.hs
+++ b/src/Dtmc/Analysis/Stationary.hs
@@ -29,24 +29,11 @@
     stationaryDistributions,
 ) where
 
-import Control.Monad.ST (
-    ST,
-    runST,
- )
-import Data.Array.MArray (
-    newListArray,
-    readArray,
-    writeArray,
- )
-import Data.Array.ST (
-    STUArray,
- )
+import Control.Monad.ST (runST)
 import Data.Array.Unboxed qualified as Unboxed
-import Dtmc.Analysis.Classification (
-    classClosed,
-    classMembers,
-    communicatingClasses,
- )
+import Data.Maybe (fromMaybe)
+import Data.Vector.Storable qualified as Storable
+import Data.Vector.Storable.Mutable qualified as Mutable
 import Dtmc.Analysis.LinearSystem (
     LinearSystemError (..),
  )
@@ -61,18 +48,23 @@
  )
 import Dtmc.State.Internal (
     stateCardinalityInt,
-    stateIndexInt,
+    stateFromInt,
  )
 import Dtmc.Transition.Matrix (
     TransitionMatrix,
  )
 import Dtmc.Transition.Matrix.Internal (
+    tmSupport,
     unTransitionMatrix,
  )
+import Dtmc.Transition.Matrix.Internal.Graph qualified as G
 import Numeric.LinearAlgebra qualified as LA
 
-toIndex :: (FiniteState state) => state -> Int
-toIndex = stateIndexInt
+toState :: (FiniteState state) => Int -> state
+toState index =
+    fromMaybe
+        (error "Dtmc.Analysis.Stationary: graph vertex out of bounds")
+        (stateFromInt index)
 
 {- | Compute the stationary vector of a non-empty irreducible stochastic
 block, in the block's own ordering, by Grassmann-Taksar-Heyman state
@@ -111,19 +103,33 @@
     Either LinearSystemError (LA.Vector Double)
 stationaryOfBlock block
     | dimension == 0 = Left SingularSystem
-    | not (all isFinite (LA.toList (LA.flatten block))) = Left NonFiniteSystem
+    | not (Storable.all isFinite flattened) = Left NonFiniteSystem
+    | dimension == 1 = singletonSolution
     | otherwise =
-        case reduceGth dimension (LA.toList (LA.flatten block)) of
+        case reduceGth dimension flattened of
             Nothing -> Left SingularSystem
             Just weights -> normalise weights
   where
     dimension = LA.rows block
+    flattened = LA.flatten block
+    singletonSolution
+        | residual > limit =
+            Left
+                ( ResidualTooLarge
+                    { relativeResidual = residual
+                    , residualLimit = limit
+                    }
+                )
+        | otherwise = Right (Storable.singleton 1)
+      where
+        limit = 1e-9
+        residual = abs (flattened Storable.! 0 - 1)
     normalise weights
-        | not (all isFinite weights) = Left NonFiniteSolution
+        | not (Storable.all isFinite weights) = Left NonFiniteSolution
         | scale <= 0 = Left SingularSystem
         | not (isFinite total) = Left NonFiniteSolution
         | total <= 0 = Left SingularSystem
-        | not (all isFinite stationaryWeights) = Left NonFiniteSolution
+        | not (Storable.all isFinite stationary) = Left NonFiniteSolution
         | not (isFinite residual) = Left NonFiniteSolution
         | residual > limit =
             Left
@@ -135,22 +141,19 @@
         | otherwise = Right stationary
       where
         limit = 1e-9
-        scale = maximum weights
-        scaledWeights = map (/ scale) weights
-        total = sum scaledWeights
-        stationaryWeights = map (/ total) scaledWeights
-        stationary = LA.fromList stationaryWeights
+        scale = Storable.maximum weights
+        scaledWeights = Storable.map (/ scale) weights
+        total = Storable.sum scaledWeights
+        stationary = Storable.map (/ total) scaledWeights
         residual =
-            foldr (max . abs) 0 (LA.toList (LA.tr block LA.#> stationary - stationary))
+            Storable.foldl'
+                (\largest value -> max largest (abs value))
+                0
+                (LA.tr block LA.#> stationary - stationary)
 
 isFinite :: Double -> Bool
 isFinite value = not (isNaN value || isInfinite value)
 
--- Allocating through a signature that quantifies the state thread keeps the
--- array type unambiguous without local annotations inside 'runST'.
-newFlatArray :: (Int, Int) -> [Double] -> ST s (STUArray s Int Double)
-newFlatArray = newListArray
-
 {- | Compute the unnormalised GTH weights of an @n x n@ block supplied in
 row-major order. Return 'Nothing' when an elimination step finds no positive
 way out of the state being removed.
@@ -161,26 +164,42 @@
 Complexity: @O(n^3)@ time, @O(n^2)@ temporary space, and @O(n)@ result
 space.
 -}
-reduceGth :: Int -> [Double] -> Maybe [Double]
+reduceGth :: Int -> Storable.Vector Double -> Maybe (Storable.Vector Double)
 reduceGth n entries = runST $ do
-    a <- newFlatArray (0, n * n - 1) entries
+    a <- Storable.thaw entries
     let index i j = i * n + j
 
-        exitMass k =
-            sum <$> mapM (readArray a . index k) [0 .. k - 1]
+        exitMass k = sumRow 0 0
+          where
+            sumRow j total
+                | j >= k = pure total
+                | otherwise = do
+                    value <- Mutable.unsafeRead a (index k j)
+                    let next = total + value
+                    next `seq` sumRow (j + 1) next
 
         absorbRow k s i = do
-            entering <- readArray a (index i k)
+            entering <- Mutable.unsafeRead a (index i k)
             let scaled = entering / s
-            writeArray a (index i k) scaled
-            mapM_
-                ( \j -> do
-                    leaving <- readArray a (index k j)
-                    current <- readArray a (index i j)
-                    writeArray a (index i j) (current + scaled * leaving)
-                )
-                [0 .. k - 1]
+            Mutable.unsafeWrite a (index i k) scaled
+            updateColumns 0 scaled
+          where
+            updateColumns j scaled
+                | j >= k = pure ()
+                | otherwise = do
+                    leaving <- Mutable.unsafeRead a (index k j)
+                    current <- Mutable.unsafeRead a (index i j)
+                    Mutable.unsafeWrite a (index i j) (current + scaled * leaving)
+                    updateColumns (j + 1) scaled
 
+        absorbRows k s = updateRow 0
+          where
+            updateRow i
+                | i >= k = pure ()
+                | otherwise = do
+                    absorbRow k s i
+                    updateRow (i + 1)
+
         eliminate k
             | k < 1 = pure True
             | otherwise = do
@@ -188,22 +207,33 @@
                 if s <= 0
                     then pure False
                     else do
-                        mapM_ (absorbRow k s) [0 .. k - 1]
+                        absorbRows k s
                         eliminate (k - 1)
 
-        -- visits holds x(0) .. x(k-1) in order.
-        substitute k visits
-            | k >= n = pure visits
+        substitute visits k
+            | k >= n = pure ()
             | otherwise = do
-                terms <-
-                    mapM
-                        (\(i, x) -> (x *) <$> readArray a (index i k))
-                        (zip [0 ..] visits)
-                substitute (k + 1) (visits ++ [sum terms])
+                value <- substitutionTerm visits k
+                Mutable.unsafeWrite visits k value
+                substitute visits (k + 1)
 
+        substitutionTerm visits k = accumulate 0 0
+          where
+            accumulate i total
+                | i >= k = pure total
+                | otherwise = do
+                    visitsAtI <- Mutable.unsafeRead visits i
+                    factor <- Mutable.unsafeRead a (index i k)
+                    let next = total + visitsAtI * factor
+                    next `seq` accumulate (i + 1) next
+
     feasible <- eliminate (n - 1)
     if feasible
-        then Just <$> substitute 1 [1]
+        then do
+            visits <- Mutable.replicate n 0
+            Mutable.unsafeWrite visits 0 1
+            substitute visits 1
+            Just <$> Storable.freeze visits
         else pure Nothing
 
 {- | Compute the extremal stationary distributions, one per recurrent class
@@ -240,9 +270,13 @@
   where
     dim = stateCardinalityInt @state
     matrix = unTransitionMatrix p
+    graph = tmSupport p
     closedClasses =
-        [classMembers c | c <- communicatingClasses p, classClosed c]
-    distributionOn members = do
+        [ (map toState indices, indices)
+        | indices@(first : _) <- G.components graph
+        , G.inClosedComponent graph first
+        ]
+    distributionOn (members, indices) = do
         solution <- stationaryOfBlock (subMatrix indices indices matrix)
         let placed :: Unboxed.UArray Int Double
             placed =
@@ -255,5 +289,3 @@
             ( members
             , DistributionVector (LA.fromList [placed Unboxed.! i | i <- [0 .. dim - 1]])
             )
-      where
-        indices = map toIndex members
diff --git a/src/Dtmc/Analysis/VisitCount.hs b/src/Dtmc/Analysis/VisitCount.hs
--- a/src/Dtmc/Analysis/VisitCount.hs
+++ b/src/Dtmc/Analysis/VisitCount.hs
@@ -58,7 +58,7 @@
     communicatingClasses,
     recurrentState,
  )
-import Dtmc.Analysis.Classification.Internal (
+import Dtmc.Analysis.Classification.Reachability.Internal (
     backwardReachable,
  )
 import Dtmc.Analysis.Event (
@@ -68,6 +68,10 @@
 import Dtmc.Analysis.Expectation (
     Expectation (..),
  )
+import Dtmc.Analysis.FiniteHorizon.Internal (
+    branchingDenseBackend,
+    denseBoundedVisitExpectation,
+ )
 import Dtmc.Analysis.HittingTime qualified as Hit
 import Dtmc.Analysis.Initial.Internal (
     expectationUnderEither,
@@ -377,7 +381,12 @@
 support size, and @w@, @e@, and @u@ are per-step upper bounds on stored source
 states, traversed transition edges, and accumulated destination states.
 
-Complexity: excluding the initial 'distributionWeights' call,
+For a 'TransitionMatrix' with a branching row in the initial support, an
+internal adaptive path uses dense matrix-vector multiplication in
+@O(n log(s + 1) + k n^2)@ time and @O(n)@ temporary space. Deterministic
+initial rows retain the sparse recurrence.
+
+Generic complexity: excluding the initial 'distributionWeights' call,
 'transitionLaw', and predicate evaluation,
 @O(s log(s + 1) + k (w + e log(u + 1) + u))@ time, @O(w + u)@ temporary
 space, and @O(1)@ result space. At @k = 0@ the function takes @O(1)@ time and
@@ -394,9 +403,23 @@
     transition ->
     (TransitionState transition -> Bool) ->
     Double
+boundedExpectation 0 _ _ _ = 0
 boundedExpectation bound initial transition isVisited =
-    go bound (Map.fromList (distributionWeights initial)) 0
+    case optimizedExpectation of
+        Just expectation -> expectation
+        Nothing -> go bound initialWeights 0
   where
+    initialWeights = Map.fromList (distributionWeights initial)
+    optimizedExpectation = do
+        backend <- branchingDenseBackend transition (Map.keys initialWeights)
+        pure
+            ( denseBoundedVisitExpectation
+                bound
+                initialWeights
+                backend
+                isVisited
+            )
+
     go 0 _ expectation = expectation
     go remaining weights expectation =
         let visitProbability =
diff --git a/src/Dtmc/Dynamics.hs b/src/Dtmc/Dynamics.hs
--- a/src/Dtmc/Dynamics.hs
+++ b/src/Dtmc/Dynamics.hs
@@ -63,16 +63,17 @@
 evolveVector (DistributionVector v) p =
     DistributionVector (LA.tr (unTransitionMatrix p) LA.#> v)
 
-{- | Compute the distribution after @k@ transitions as
-@evolveVector mu (power k p)@. Exponent zero is the original distribution
-mathematically.
+{- | Compute the distribution after @k@ transitions. A cost model chooses
+between repeated matrix-vector multiplication and powering the matrix, so a
+moderate number of steps does not construct a full matrix power unnecessarily.
+Exponent zero returns the original distribution.
 
-This powers the matrix rather than iterating 'evolveVector', so the two
-calculations may differ by floating-point rounding. The result is not
-revalidated.
+The two strategies are mathematically equivalent but may differ by ordinary
+floating-point rounding. The result is not revalidated.
 
-Complexity: @O(n^2 + n^3 log(k + 1))@ time, @O(n^2)@ temporary space, and
-@O(n)@ result space.
+Complexity: @O(k n^2)@ time when @k <= n@ and
+@O(n^2 + n^3 log(k + 1))@ otherwise. Temporary space is @O(n)@ in the
+iterated case and @O(n^2)@ in the powered case; result space is @O(n)@.
 -}
 evolveVectorN ::
     (FiniteState state) =>
@@ -80,8 +81,25 @@
     DistributionVector state ->
     TransitionMatrix state ->
     DistributionVector state
-evolveVectorN k mu p =
-    evolveVector mu (power k p)
+evolveVectorN k mu@(DistributionVector initial) p
+    | k == 0 = mu
+    | useIteration = DistributionVector (iterateVector k initial)
+    | otherwise = evolveVector mu (power k p)
+  where
+    matrix = unTransitionMatrix p
+    transposed = LA.tr matrix
+    dimension = LA.rows matrix
+
+    -- A deliberately conservative threshold retains the highly tuned matrix
+    -- power path for long runs on small matrices. Focused benchmarks show the
+    -- repeated matrix-vector path winning once the matrix dimension reaches
+    -- the requested step count.
+    useIteration = toInteger k <= toInteger dimension
+
+    iterateVector 0 vector = vector
+    iterateVector remaining vector =
+        let next = transposed LA.#> vector
+         in next `seq` iterateVector (remaining - 1) next
 
 {- | Push any finite-support 'Distribution' through one locally finite kernel
 step. The result uses t'DistributionMap' because a general kernel does not
diff --git a/src/Dtmc/Simulation.hs b/src/Dtmc/Simulation.hs
--- a/src/Dtmc/Simulation.hs
+++ b/src/Dtmc/Simulation.hs
@@ -13,12 +13,16 @@
     sample,
     step,
     simulate,
+    simulateMatrix,
 ) where
 
+import Control.Monad (when)
 import Control.Monad.Primitive (
     PrimMonad,
     PrimState,
  )
+import Data.Array qualified as Array
+import Data.Array.Unboxed qualified as Unboxed
 import Data.List qualified as List
 import Dtmc.Distribution (
     Distribution (..),
@@ -26,9 +30,20 @@
 import Dtmc.Simplex.Internal (
     simplexTolerance,
  )
+import Dtmc.State (
+    FiniteState,
+ )
+import Dtmc.State.Internal (
+    stateFromInt,
+    stateIndexInt,
+ )
 import Dtmc.Transition (
     Transition (..),
  )
+import Dtmc.Transition.Matrix.Internal (
+    TransitionMatrix,
+    unTransitionMatrix,
+ )
 import Numeric.LinearAlgebra qualified as LA
 import Numeric.Natural (
     Natural,
@@ -155,3 +170,109 @@
         case result of
             Left problem -> pure (Left problem)
             Right next -> go (remaining - 1) next (next : reversed)
+
+-- | A validated cumulative row used by 'simulateMatrix'. The final positive
+-- index is retained as a defensive fallback if the random generator returns
+-- the upper endpoint of its requested floating-point interval.
+data PreparedMatrixRow = PreparedMatrixRow
+    !(Unboxed.UArray Int Int)
+    !(LA.Vector Double)
+    !Double
+    !Int
+
+{- | Simulate a finite transition matrix without converting every visited row
+to a map-backed distribution. Each distinct row is validated and converted to
+a shared cumulative vector on first use; sampling that row thereafter uses a
+binary search.
+
+This has the same validation and generator-advancement contract as 'simulate':
+an invalid visited row is reported before drawing a random number, unvisited
+rows are not inspected, and a zero-step simulation inspects neither the matrix
+nor the generator.
+
+For matrix dimension @n@, @k@ transitions, and @r@ distinct visited rows,
+complexity is @O(r n + k log n)@ time, @O(r n + n + k)@ temporary space, and
+@O(k)@ result space. The row cache lives only for this simulation call.
+-}
+simulateMatrix ::
+    forall state m.
+    (FiniteState state, PrimMonad m) =>
+    Natural ->
+    TransitionMatrix state ->
+    state ->
+    MWC.Gen (PrimState m) ->
+    m (Either SimulationError [state])
+simulateMatrix 0 _ initial _ = pure (Right [initial])
+simulateMatrix transitions matrix initial generator =
+    go transitions initial [initial]
+  where
+    stored = unTransitionMatrix matrix
+    dimension = LA.rows stored
+
+    -- 'Array' is lazy in its elements, so only rows reached by the trajectory
+    -- are converted and validated. Forced entries are then shared.
+    preparedRows :: Array.Array Int (Either SimulationError PreparedMatrixRow)
+    preparedRows =
+        Array.listArray
+            (0, dimension - 1)
+            [prepareMatrixRow (LA.toList row) | row <- LA.toRows stored]
+
+    go 0 _ reversed = pure (Right (reverse reversed))
+    go remaining current reversed =
+        case preparedRows Array.! stateIndexInt current of
+            Left problem -> pure (Left problem)
+            Right prepared -> do
+                index <- samplePreparedMatrixRow prepared generator
+                case stateFromInt index of
+                    Nothing ->
+                        pure
+                            (Left (SampleIndexOutOfBounds index dimension))
+                    Just next ->
+                        go (remaining - 1) next (next : reversed)
+
+prepareMatrixRow :: [Double] -> Either SimulationError PreparedMatrixRow
+prepareMatrixRow weights = do
+    let entries = filter ((/= 0) . snd) (zip [0 ..] weights)
+    when (null entries) (Left EmptySupport)
+    repaired <- traverse repairWeight (zip [0 ..] (map snd entries))
+    let cumulative = drop 1 (List.scanl' (+) 0 repaired)
+        total = List.foldl' (+) 0 repaired
+    validateTotal total
+    let lastPositive =
+            List.foldl'
+                (\latest (index, weight) -> if weight > 0 then index else latest)
+                0
+                (zip [0 ..] repaired)
+        outcomes =
+            Unboxed.listArray
+                (0, length entries - 1)
+                (map fst entries)
+    pure
+        ( PreparedMatrixRow
+            outcomes
+            (LA.fromList cumulative)
+            total
+            lastPositive
+        )
+
+samplePreparedMatrixRow ::
+    (PrimMonad m) =>
+    PreparedMatrixRow ->
+    MWC.Gen (PrimState m) ->
+    m Int
+samplePreparedMatrixRow
+    (PreparedMatrixRow outcomes cumulative total lastPositive)
+    generator = do
+    target <- MWC.uniformR (0, total) generator
+    pure (outcomes Unboxed.! firstGreater target cumulative lastPositive)
+
+firstGreater :: Double -> LA.Vector Double -> Int -> Int
+firstGreater target cumulative fallback = search 0 (LA.size cumulative)
+  where
+    search lower upper
+        | lower >= upper =
+            if lower < LA.size cumulative then lower else fallback
+        | cumulative `LA.atIndex` middle > target = search lower middle
+        | otherwise = search (middle + 1) upper
+      where
+        middle = lower + (upper - lower) `quot` 2
diff --git a/src/Dtmc/Transition.hs b/src/Dtmc/Transition.hs
--- a/src/Dtmc/Transition.hs
+++ b/src/Dtmc/Transition.hs
@@ -8,28 +8,9 @@
 "Dtmc.Transition.Matrix" and "Dtmc.Transition.Kernel".
 -}
 module Dtmc.Transition (
-    Transition (..),
+    Transition (TransitionState, transitionLaw),
 ) where
 
-import Dtmc.Distribution.Map (DistributionMap)
-
-{- | A time-homogeneous transition rule whose law from any supplied state has
-finite support. The complete state space may be finite or infinite.
-
-This capability is sufficient for exact finite-horizon map-backed algorithms.
-It does not imply that states can be enumerated, so it cannot by itself support
-generic classification, stationary, eventual-hitting, or expectation
-algorithms.
--}
-class Transition transition where
-    -- | State type governed by this transition representation.
-    type TransitionState transition
-
-    {- | Return the validated finite-support law of the next state.
-
-    Complexity: implementation-dependent.
-    -}
-    transitionLaw ::
-        transition ->
-        TransitionState transition ->
-        DistributionMap (TransitionState transition)
+import Dtmc.Transition.Internal (
+    Transition (TransitionState, transitionLaw),
+ )
diff --git a/src/Dtmc/Transition/Internal.hs b/src/Dtmc/Transition/Internal.hs
new file mode 100644
--- /dev/null
+++ b/src/Dtmc/Transition/Internal.hs
@@ -0,0 +1,89 @@
+{-# OPTIONS_HADDOCK not-home #-}
+
+{- |
+Module      : Dtmc.Transition.Internal
+Description : Transition class and optional dense finite-state backend.
+
+The public "Dtmc.Transition" module exposes only 'TransitionState' and
+'transitionLaw'. The remaining method exposes an optional representation
+capability, kept internal so optimized library analyses do not enlarge the
+public API or burden third-party instances.
+-}
+module Dtmc.Transition.Internal (
+    DenseTransitionBackend (..),
+    denseRowBranches,
+    Transition (..),
+) where
+
+import Data.Vector.Storable qualified as Storable
+import Dtmc.Distribution.Map (
+    DistributionMap,
+ )
+import Numeric.LinearAlgebra qualified as LA
+
+{- | Dense finite-state data shared by representation-specific analysis
+implementations. State order, indexing, and the matrix must describe the same
+canonical coordinate system.
+
+This type is internal: public transition instances need only implement
+'transitionLaw'.
+-}
+data DenseTransitionBackend state = DenseTransitionBackend
+    { denseTransitionMatrix :: LA.Matrix Double
+    , denseTransitionStates :: [state]
+    , denseTransitionIndex :: state -> Int
+    }
+
+{- | Test whether one stored row has more than one positive transition. The
+scan stops after its second positive entry.
+-}
+denseRowBranches :: DenseTransitionBackend state -> state -> Bool
+denseRowBranches backend state = go 0 0
+  where
+    stored = denseTransitionMatrix backend
+    rowIndex = denseTransitionIndex backend state
+    entries =
+        LA.flatten
+            ( LA.subMatrix
+                (rowIndex, 0)
+                (1, LA.cols stored)
+                stored
+            )
+    entryCount = Storable.length entries
+
+    go :: Int -> Int -> Bool
+    go offset positiveCount
+        | positiveCount > 1 = True
+        | offset == entryCount = False
+        | entries Storable.! offset > 0 = go (offset + 1) (positiveCount + 1)
+        | otherwise = go (offset + 1) positiveCount
+
+{- | A time-homogeneous transition rule whose law from any supplied state has
+finite support. The complete state space may be finite or infinite.
+
+This capability is sufficient for exact finite-horizon map-backed algorithms.
+It does not imply that states can be enumerated, so it cannot by itself support
+generic classification, stationary, eventual-hitting, or expectation
+algorithms.
+-}
+class Transition transition where
+    -- | State type governed by this transition representation.
+    type TransitionState transition
+
+    {- | Return the validated finite-support law of the next state.
+
+    Complexity: implementation-dependent.
+    -}
+    transitionLaw ::
+        transition ->
+        TransitionState transition ->
+        DistributionMap (TransitionState transition)
+
+    -- Optional representation capability. Existing and third-party instances
+    -- inherit the locally finite, representation-independent fallback.
+    transitionDenseBackend ::
+        transition ->
+        Maybe (DenseTransitionBackend (TransitionState transition))
+    transitionDenseBackend _ = Nothing
+
+    {-# MINIMAL transitionLaw #-}
diff --git a/src/Dtmc/Transition/Matrix.hs b/src/Dtmc/Transition/Matrix.hs
--- a/src/Dtmc/Transition/Matrix.hs
+++ b/src/Dtmc/Transition/Matrix.hs
@@ -85,8 +85,8 @@
 {- | Construct a row-stochastic matrix from a grid of weights in canonical
 state order, stopping at the first problem. Within each accepted row,
 tolerated coordinate error is clamped to @[0, 1]@ and the repaired row is
-normalised. The support graph remains lazy, and the empty @0 x 0@ matrix is
-accepted.
+normalised. The support graph and classification cache remain lazy, and the
+empty @0 x 0@ matrix is accepted.
 
 This inverts 'toRows' up to that repair and needs no @hmatrix@ value: the
 shape is checked here and reported as 'WrongRowCount' or 'WrongRowWidth'
@@ -117,8 +117,8 @@
 {- | Materialise a finite-state kernel as a dense transition matrix. Kernel
 rows are already validated 'Dtmc.Distribution.Map.DistributionMap' values, so
 this conversion is total and performs no additional clamping or
-renormalisation. Missing coordinates become exact zeros. The support graph
-remains lazy, and the empty @0 x 0@ matrix is accepted.
+renormalisation. Missing coordinates become exact zeros. The support graph and
+classification cache remain lazy, and the empty @0 x 0@ matrix is accepted.
 
 Complexity: excluding evaluation of 'finiteStates' and the kernel laws,
 @O(n^2)@ time and @O(n^2)@ temporary and result space.
@@ -155,7 +155,7 @@
 multiplication mathematically, but floating-point rounding can accumulate.
 
 Complexity: @O(n^3)@ worst-case time and @O(n^2)@ temporary and result space.
-The support graph is built lazily.
+The support graph and classification cache are built lazily.
 -}
 compose ::
     TransitionMatrix state ->
diff --git a/src/Dtmc/Transition/Matrix/Internal.hs b/src/Dtmc/Transition/Matrix/Internal.hs
--- a/src/Dtmc/Transition/Matrix/Internal.hs
+++ b/src/Dtmc/Transition/Matrix/Internal.hs
@@ -1,11 +1,13 @@
+{-# LANGUAGE GADTs #-}
+
 {- |
 Module      : Dtmc.Transition.Matrix.Internal
 Description : Raw carrier for transition matrices (unsafe underbelly).
 
 Raw carrier behind t'Dtmc.Transition.Matrix.TransitionMatrix': an hmatrix
-matrix paired with its lazy support graph. The public smart constructor
-validates its square shape and canonicalises rows; this internal module
-exposes unchecked construction.
+matrix paired with its lazy support graph and complete classification. The
+public smart constructor validates its square shape and canonicalises rows;
+this internal module exposes unchecked construction.
 
 The constructor is positional so the public matrix projection cannot act as a
 record-update setter and desynchronise the matrix from its cached graph.
@@ -14,10 +16,17 @@
     TransitionMatrix (TransitionMatrix),
     unTransitionMatrix,
     tmSupport,
+    tmClassification,
     unsafeTransitionMatrix,
     matrixRowAt,
 ) where
 
+import Data.Maybe (fromMaybe)
+import Data.Vector.Storable qualified as Storable
+import Dtmc.Analysis.Classification.Internal (
+    Classification,
+    classificationFromGraph,
+ )
 import Dtmc.Distribution.Map (
     fromDistribution,
  )
@@ -26,12 +35,15 @@
  )
 import Dtmc.State (
     FiniteState,
+    finiteStates,
  )
 import Dtmc.State.Internal (
     stateCardinalityInt,
+    stateFromInt,
     stateIndexInt,
  )
-import Dtmc.Transition (
+import Dtmc.Transition.Internal (
+    DenseTransitionBackend (..),
     Transition (..),
  )
 import Dtmc.Transition.Matrix.Internal.Graph (
@@ -47,14 +59,19 @@
 tolerant row validation and canonicalisation. The internal constructor and
 arithmetic instances do not revalidate.
 
-Each value also carries its support graph as a /lazy/ second argument, so any
-graph-based analyses on the same value share one build. Construct internal
-values with @unsafeTransitionMatrix@ rather than pairing a matrix and graph
-directly.
+Each value also carries its support graph and complete typed classification as
+/lazy/ arguments, so graph-based analyses on the same value share both the
+graph build and public classification results. Construct internal values with
+@unsafeTransitionMatrix@ rather than pairing these fields directly.
 -}
-data TransitionMatrix state
-    = -- | Unchecked matrix/cache pair; the graph must match the matrix.
-      TransitionMatrix (LA.Matrix Double) Graph
+data TransitionMatrix state where
+    -- | Unchecked matrix/cache triple; both caches must match the matrix.
+    TransitionMatrix ::
+        (FiniteState state) =>
+        LA.Matrix Double ->
+        Graph ->
+        Classification state ->
+        TransitionMatrix state
 
 -- Nominal role prevents coercion between distinct state types, including
 -- state types with the same cardinality.
@@ -67,7 +84,7 @@
 unTransitionMatrix ::
     TransitionMatrix state ->
     LA.Matrix Double
-unTransitionMatrix (TransitionMatrix matrix _) = matrix
+unTransitionMatrix (TransitionMatrix matrix _ _) = matrix
 
 {- | Return the lazy support graph, with edge @i -> j@ exactly when the stored
 entry is strictly positive. No tolerance is applied: a tiny positive rounding
@@ -80,8 +97,21 @@
 space; the resulting graph occupies @O(n + E)@ space for @E@ support edges.
 -}
 tmSupport :: TransitionMatrix state -> Graph
-tmSupport (TransitionMatrix _ support) = support
+tmSupport (TransitionMatrix _ support _) = support
 
+{- | Return the lazy complete classification associated with the matrix.
+Whole-chain classification queries on the same matrix therefore share their
+typed classes and state lists as well as the underlying graph facts.
+
+Complexity: @O(1)@ projection time and space. The first full evaluation takes
+the classification cost documented by
+'Dtmc.Analysis.Classification.communicatingClasses'; later projections reuse
+the retained @O(n)@ classification.
+-}
+tmClassification :: TransitionMatrix state -> Classification state
+tmClassification (TransitionMatrix _ _ classification) = classification
+{-# INLINE tmClassification #-}
+
 -- Manual 'Show': 'Graph' has no 'Show', and the derived cache should not
 -- appear in the rendering.
 instance Show (TransitionMatrix state) where
@@ -90,19 +120,28 @@
             showString "TransitionMatrix "
                 . showsPrec 11 (unTransitionMatrix p)
 
-{- | Pair a raw matrix with its lazy support graph. This performs no
-row-stochastic, finiteness, or simplex validation; internal callers must
-establish the required invariant.
+{- | Pair a raw matrix with its lazy support graph and classification. This
+performs no row-stochastic, finiteness, or simplex validation; internal
+callers must establish the required invariant.
 
 Complexity: @O(1)@ construction time and @O(1)@ construction space. Forcing
 the support graph takes @O(n^2)@ time and @O(n^2)@ temporary space; the graph
 occupies @O(n + E)@ space for @E@ support edges.
 -}
 unsafeTransitionMatrix ::
+    (FiniteState state) =>
     LA.Matrix Double ->
     TransitionMatrix state
 unsafeTransitionMatrix matrix =
-    TransitionMatrix matrix (supportGraphOf matrix)
+    TransitionMatrix matrix support classification
+  where
+    support = supportGraphOf matrix
+    classification = classificationFromGraph toState support
+    toState index =
+        fromMaybe
+            (error "Dtmc.Transition.Matrix.Internal: graph vertex out of bounds")
+            (stateFromInt index)
+{-# INLINE unsafeTransitionMatrix #-}
 
 {- | Wrap one stored matrix row as a distribution vector without revalidation.
 The finite-state index makes the lookup total.
@@ -132,21 +171,29 @@
     transitionLaw matrix =
         fromDistribution . matrixRowAt matrix
 
+    transitionDenseBackend matrix =
+        Just
+            DenseTransitionBackend
+                { denseTransitionMatrix = unTransitionMatrix matrix
+                , denseTransitionStates = finiteStates
+                , denseTransitionIndex = stateIndexInt
+                }
+
 -- Use strict positivity without tolerance so graph queries reflect the stored
 -- matrix exactly; keep construction here so the cache cannot become stale.
 supportGraphOf ::
     LA.Matrix Double ->
     Graph
 supportGraphOf matrix =
-    fromAdjacency
-        dim
-        [ ((i, j), entry > 0)
-        | (i, row) <- zip [0 ..] rows
-        , (j, entry) <- zip [0 ..] row
-        ]
+    fromAdjacency dim associations
   where
-    rows = LA.toLists matrix
-    dim = length rows
+    dim = LA.rows matrix
+    associations = Storable.ifoldr associationFor [] (LA.flatten matrix)
+    associationFor offset probability rest =
+        let (row, column) = offset `quotRem` dim
+         in row `seq`
+                column `seq`
+                    ((row, column), probability > 0) : rest
 
 {- | Matrix multiplication as transition composition: @p '<>' q@ takes a @p@
 step followed by a @q@ step. Exact products preserve row-stochasticity and
@@ -158,7 +205,8 @@
         TransitionMatrix state ->
         TransitionMatrix state ->
         TransitionMatrix state
-    p <> q = unsafeTransitionMatrix (unTransitionMatrix p LA.<> unTransitionMatrix q)
+    p@TransitionMatrix{} <> q =
+        unsafeTransitionMatrix (unTransitionMatrix p LA.<> unTransitionMatrix q)
 
 {- | The identity matrix represents zero transitions and is the unit of the
 transition-composition monoid.
diff --git a/test/Dtmc/Analysis/AbsorptionSpec.hs b/test/Dtmc/Analysis/AbsorptionSpec.hs
--- a/test/Dtmc/Analysis/AbsorptionSpec.hs
+++ b/test/Dtmc/Analysis/AbsorptionSpec.hs
@@ -1,7 +1,6 @@
 {-# LANGUAGE DataKinds #-}
 {-# LANGUAGE DeriveAnyClass #-}
 {-# LANGUAGE DeriveGeneric #-}
-{-# LANGUAGE TypeApplications #-}
 
 module Dtmc.Analysis.AbsorptionSpec (
     spec,
@@ -72,8 +71,35 @@
     either (error . show) id $
         fromRows (chunksOf 2 [0, 1, 1, 0])
 
+illConditionedTransientBlock :: TransitionMatrix (Finite 4)
+illConditionedTransientBlock =
+    either (error . show) id $
+        fromRows
+            ( chunksOf
+                4
+                [ 1 - epsilon
+                , 0
+                , epsilon
+                , 0
+                , 0
+                , 0
+                , 0
+                , 1
+                , 0
+                , 0
+                , 1
+                , 0
+                , 0
+                , 0
+                , 0
+                , 1
+                ]
+            )
+  where
+    epsilon = 1e-14
+
 closeTo :: Double -> Double -> Bool
-closeTo expected actual = approxEq testTolerance expected actual
+closeTo = approxEq testTolerance
 
 finiteCloseTo :: Double -> E.Expectation -> Bool
 finiteCloseTo expected (E.FiniteExpectation x) = closeTo expected x
@@ -116,6 +142,12 @@
             Absorption.probabilityGivenInitialState chain C C `shouldBe` Right 1
             Absorption.probabilityGivenInitialState chain D C `shouldBe` Right 0
 
+        it "does not force an ill-conditioned transient solve at a recurrent state" $ do
+            Absorption.probabilityGivenInitialState illConditionedTransientBlock 2 2
+                `shouldBe` Right 1
+            Absorption.probabilityGivenInitialState illConditionedTransientBlock 2 3
+                `shouldBe` Right 0
+
         it "is exactly zero for a transient target" $
             Absorption.probabilityGivenInitialState chain A B `shouldBe` Right 0
 
@@ -131,6 +163,35 @@
                     Hitting.eventualProbabilityGivenInitialState chain [C, D] A
                         `shouldSatisfy` rightCloseTo (toC + toD)
                 other -> expectationFailure ("solve failed: " <> show other)
+
+    describe "probabilityMatrix" $ do
+        it "returns the complete closed-form absorption table" $
+            case Absorption.probabilityMatrix chain of
+                Left err -> expectationFailure ("solve failed: " <> show err)
+                Right (transient, recurrent, rows) -> do
+                    transient `shouldBe` [A, B]
+                    recurrent `shouldBe` [C, D]
+                    concat rows
+                        `shouldSatisfy` ( and
+                                            . zipWith
+                                                closeTo
+                                                [17 / 20, 3 / 20, 11 / 20, 9 / 20]
+                                        )
+
+        it "returns empty rows when no state is transient" $
+            Absorption.probabilityMatrix twoCycle
+                `shouldBe` Right ([], [False, True], [])
+
+        it "agrees with every scalar absorption probability" $
+            case Absorption.probabilityMatrix chain of
+                Left err -> expectationFailure ("solve failed: " <> show err)
+                Right (transient, recurrent, rows) ->
+                    sequence_
+                        [ Absorption.probabilityGivenInitialState chain target initial
+                            `shouldSatisfy` rightCloseTo expected
+                        | (initial, row) <- zip transient rows
+                        , (target, expected) <- zip recurrent row
+                        ]
 
     describe "expectationByState" $ do
         it "matches the closed form of the notes" $
diff --git a/test/Dtmc/Analysis/CanonicalDifferentialSpec.hs b/test/Dtmc/Analysis/CanonicalDifferentialSpec.hs
--- a/test/Dtmc/Analysis/CanonicalDifferentialSpec.hs
+++ b/test/Dtmc/Analysis/CanonicalDifferentialSpec.hs
@@ -1,5 +1,4 @@
 {-# LANGUAGE DataKinds #-}
-{-# LANGUAGE TypeApplications #-}
 
 module Dtmc.Analysis.CanonicalDifferentialSpec (
     spec,
@@ -161,7 +160,7 @@
         and
             [ let law = Oracle.hittingLaw 4 matrix target source
                   exact = known (Oracle.lawProbability (EqualTo time) law)
-                  dense = (hitProbabilityByState (EqualTo time) matrix [2])
+                  dense = hitProbabilityByState (EqualTo time) matrix [2]
                in close (Hit.probabilityGivenInitialState (EqualTo time) matrix target source) exact
                     && close (dense !! fromIntegral source) exact
             | source <- finites
@@ -170,7 +169,7 @@
             && and
                 [ let law = Oracle.hittingLaw 4 matrix target source
                       bounded = known (Oracle.lawProbability (LessThan bound) law)
-                      dense = (hitProbabilityByState (LessThan bound) matrix [2])
+                      dense = hitProbabilityByState (LessThan bound) matrix [2]
                    in close
                         (Hit.probabilityGivenInitialState (LessThan bound) matrix target source)
                         bounded
@@ -182,7 +181,7 @@
         and
             [ let law = Oracle.returnLaw 4 matrix source
                   exact = known (Oracle.lawProbability (EqualTo time) law)
-                  dense = (returnProbabilityByState (EqualTo time) matrix)
+                  dense = returnProbabilityByState (EqualTo time) matrix
                in close (Return.probabilityGivenInitialState (EqualTo time) matrix source) exact
                     && close (dense !! fromIntegral source) exact
             | source <- finites
@@ -191,7 +190,7 @@
             && and
                 [ let law = Oracle.returnLaw 4 matrix source
                       bounded = known (Oracle.lawProbability (LessThan bound) law)
-                      dense = (returnProbabilityByState (LessThan bound) matrix)
+                      dense = returnProbabilityByState (LessThan bound) matrix
                    in close
                         (Return.probabilityGivenInitialState (LessThan bound) matrix source)
                         bounded
@@ -263,7 +262,7 @@
         case hitEventualProbabilityByState terminalChain [1] of
             Left _ -> False
             Right dense ->
-                and (zipWith close (dense) hitValues)
+                and (zipWith close dense hitValues)
                     && and
                         [ rightClose expected (Hit.eventualProbabilityGivenInitialState terminalChain [1] state)
                         | (state, expected) <- zip states hitValues
@@ -281,7 +280,7 @@
         case hitRaceProbabilityByState terminalChain [1] [2] of
             Left _ -> False
             Right dense ->
-                and (zipWith close (dense) raceValues)
+                and (zipWith close dense raceValues)
                     && and
                         [ rightClose
                             expected
@@ -305,7 +304,7 @@
         case returnEventualProbabilityByState terminalChain of
             Left _ -> False
             Right dense ->
-                and (zipWith close (dense) returnValues)
+                and (zipWith close dense returnValues)
                     && and
                         [ rightClose expected (Return.eventualProbabilityGivenInitialState terminalChain state)
                         | (state, expected) <- zip states returnValues
@@ -373,5 +372,6 @@
                     Right matrix -> property (finiteAndBoundedChecks matrix)
 
     describe "canonical infinite-horizon differential baseline" $ do
-        it "all eventual, race, expectation, and total-visit queries match a completed path law" $
+        it
+            "all eventual, race, expectation, and total-visit queries match a completed path law"
             terminalChecks
diff --git a/test/Dtmc/Analysis/ClassificationSpec.hs b/test/Dtmc/Analysis/ClassificationSpec.hs
--- a/test/Dtmc/Analysis/ClassificationSpec.hs
+++ b/test/Dtmc/Analysis/ClassificationSpec.hs
@@ -256,7 +256,7 @@
     powers = take bound (drop 1 (iterate (boolMul s) (boolIdentity dim)))
     returns = [k | (k, m) <- zip [1 :: Int ..] powers, (m !! i) !! i]
 
-classesAsInts :: (KnownNat n) => TransitionMatrix (Finite n) -> [[Integer]]
+classesAsInts :: TransitionMatrix (Finite n) -> [[Integer]]
 classesAsInts = map (map getFinite . classMembers) . communicatingClasses
 
 cyclicClassesAsInts :: (KnownNat n) => TransitionMatrix (Finite n) -> Maybe [[Integer]]
diff --git a/test/Dtmc/Analysis/EventSpec.hs b/test/Dtmc/Analysis/EventSpec.hs
--- a/test/Dtmc/Analysis/EventSpec.hs
+++ b/test/Dtmc/Analysis/EventSpec.hs
@@ -66,12 +66,10 @@
                 let threshold = asNatural rawThreshold
                     value = asNatural rawValue
                  in property $
-                        and
-                            [ matches (GreaterThan threshold) value
-                                /= matches (AtMost threshold) value
-                            , matches (AtLeast threshold) value
+                        matches (GreaterThan threshold) value
+                            /= matches (AtMost threshold) value
+                            && matches (AtLeast threshold) value
                                 /= matches (LessThan threshold) value
-                            ]
 
     describe "includesInfiniteOutcome" $ do
         it "includes infinity exactly in upper-tail events" $ do
diff --git a/test/Dtmc/Analysis/FiniteTimeCanonicalSpec.hs b/test/Dtmc/Analysis/FiniteTimeCanonicalSpec.hs
--- a/test/Dtmc/Analysis/FiniteTimeCanonicalSpec.hs
+++ b/test/Dtmc/Analysis/FiniteTimeCanonicalSpec.hs
@@ -1,5 +1,4 @@
 {-# LANGUAGE DataKinds #-}
-{-# LANGUAGE TypeApplications #-}
 
 module Dtmc.Analysis.FiniteTimeCanonicalSpec (
     spec,
diff --git a/test/Dtmc/Analysis/HittingTimeCanonicalSpec.hs b/test/Dtmc/Analysis/HittingTimeCanonicalSpec.hs
--- a/test/Dtmc/Analysis/HittingTimeCanonicalSpec.hs
+++ b/test/Dtmc/Analysis/HittingTimeCanonicalSpec.hs
@@ -1,5 +1,4 @@
 {-# LANGUAGE DataKinds #-}
-{-# LANGUAGE TypeApplications #-}
 
 module Dtmc.Analysis.HittingTimeCanonicalSpec (
     spec,
@@ -115,7 +114,7 @@
         [ let law = Oracle.hittingLaw 4 matrix isTarget initial
               oracle = known (Oracle.lawProbability event law)
               scalar = Hit.probabilityGivenInitialState event matrix isTarget initial
-              dense = (hitProbabilityByState event matrix [2])
+              dense = hitProbabilityByState event matrix [2]
            in close scalar oracle
                 && close (dense !! fromIntegral initial) oracle
         | initial <- finites
@@ -138,19 +137,19 @@
             Hit.probabilityGivenInitialState (AtLeast 0) terminalChain target 0 `shouldBe` 1
             Hit.probabilityGivenInitialState (AtLeast 1) terminalChain target 0 `shouldBe` 1
             Hit.probabilityGivenInitialState (AtLeast 2) terminalChain target 0 `shouldBe` 0.5
-            (hitProbabilityByState (GreaterThan 1) terminalChain [1])
+            hitProbabilityByState (GreaterThan 1) terminalChain [1]
                 `shouldBe` [0.5, 0, 1]
-            (hitProbabilityByState (AtMost 1) terminalChain [1])
+            hitProbabilityByState (AtMost 1) terminalChain [1]
                 `shouldBe` [0.5, 1, 0]
 
         it "keeps empty-target and time-zero boundaries structural" $ do
-            (hitProbabilityByState (EqualTo 3) terminalChain [])
+            hitProbabilityByState (EqualTo 3) terminalChain []
                 `shouldBe` [0, 0, 0]
-            (hitProbabilityByState (AtMost 3) terminalChain [])
+            hitProbabilityByState (AtMost 3) terminalChain []
                 `shouldBe` [0, 0, 0]
-            (hitProbabilityByState (GreaterThan 3) terminalChain [])
+            hitProbabilityByState (GreaterThan 3) terminalChain []
                 `shouldBe` [1, 1, 1]
-            (hitProbabilityByState (AtLeast 0) terminalChain [1])
+            hitProbabilityByState (AtLeast 0) terminalChain [1]
                 `shouldBe` [1, 1, 1]
             Hit.probabilityGivenInitialState (EqualTo 0) terminalChain (== 1) 1 `shouldBe` 1
             Hit.probabilityGivenInitialState (GreaterThan 0) terminalChain (== 1) 1 `shouldBe` 0
diff --git a/test/Dtmc/Analysis/NamespaceCompileSpec.hs b/test/Dtmc/Analysis/NamespaceCompileSpec.hs
--- a/test/Dtmc/Analysis/NamespaceCompileSpec.hs
+++ b/test/Dtmc/Analysis/NamespaceCompileSpec.hs
@@ -131,5 +131,7 @@
                 `shouldBe` Right (FiniteExpectation 0.5)
             Absorption.probability matrix 1 mixedInitial
                 `shouldBe` Right 1
+            Absorption.probabilityMatrix matrix
+                `shouldBe` Right ([0], [1], [[1]])
             Absorption.expectation matrix mixedInitial
                 `shouldBe` Right (FiniteExpectation 0.5)
diff --git a/test/Dtmc/Analysis/ProbabilityOracle.hs b/test/Dtmc/Analysis/ProbabilityOracle.hs
--- a/test/Dtmc/Analysis/ProbabilityOracle.hs
+++ b/test/Dtmc/Analysis/ProbabilityOracle.hs
@@ -17,6 +17,7 @@
     getFinite,
  )
 import Data.List (
+    elemIndex,
     findIndex,
  )
 import Data.Map.Strict (
@@ -178,7 +179,7 @@
   where
     paths = weightedTrajectories horizon [(initial, 1)] matrix
     firstReturn path =
-        fromIntegral . (+ 1) <$> findIndex (== initial) (drop 1 path)
+        fromIntegral . (+ 1) <$> elemIndex initial (drop 1 path)
 
 visitLawBefore ::
     (FiniteState state) =>
diff --git a/test/Dtmc/Analysis/ReturnTimeCanonicalSpec.hs b/test/Dtmc/Analysis/ReturnTimeCanonicalSpec.hs
--- a/test/Dtmc/Analysis/ReturnTimeCanonicalSpec.hs
+++ b/test/Dtmc/Analysis/ReturnTimeCanonicalSpec.hs
@@ -1,5 +1,4 @@
 {-# LANGUAGE DataKinds #-}
-{-# LANGUAGE TypeApplications #-}
 
 module Dtmc.Analysis.ReturnTimeCanonicalSpec (
     spec,
@@ -112,7 +111,7 @@
         [ let law = Oracle.returnLaw 4 matrix initial
               oracle = known (Oracle.lawProbability event law)
               scalar = Return.probabilityGivenInitialState event matrix initial
-              dense = (returnProbabilityByState event matrix)
+              dense = returnProbabilityByState event matrix
            in close scalar oracle
                 && close (dense !! fromIntegral initial) oracle
         | initial <- finites
@@ -123,28 +122,28 @@
 spec = do
     describe "canonical return probability" $ do
         it "enforces the time-zero exclusion exactly" $ do
-            (returnProbabilityByState (EqualTo 0) terminalChain)
+            returnProbabilityByState (EqualTo 0) terminalChain
                 `shouldBe` [0, 0, 0]
-            (returnProbabilityByState (LessThan 1) terminalChain)
+            returnProbabilityByState (LessThan 1) terminalChain
                 `shouldBe` [0, 0, 0]
-            (returnProbabilityByState (AtMost 0) terminalChain)
+            returnProbabilityByState (AtMost 0) terminalChain
                 `shouldBe` [0, 0, 0]
-            (returnProbabilityByState (GreaterThan 0) terminalChain)
+            returnProbabilityByState (GreaterThan 0) terminalChain
                 `shouldBe` [1, 1, 1]
-            (returnProbabilityByState (AtLeast 0) terminalChain)
+            returnProbabilityByState (AtLeast 0) terminalChain
                 `shouldBe` [1, 1, 1]
-            (returnProbabilityByState (AtLeast 1) terminalChain)
+            returnProbabilityByState (AtLeast 1) terminalChain
                 `shouldBe` [1, 1, 1]
 
         it "implements every relation and carries non-return mass in upper tails" $ do
             Return.probabilityGivenInitialState (EqualTo 1) terminalChain 2 `shouldBe` 1
             Return.probabilityGivenInitialState (AtMost 1) terminalChain 2 `shouldBe` 1
             Return.probabilityGivenInitialState (GreaterThan 1) terminalChain 2 `shouldBe` 0
-            (returnProbabilityByState (AtMost 1) terminalChain)
+            returnProbabilityByState (AtMost 1) terminalChain
                 `shouldBe` [0, 0, 1]
-            (returnProbabilityByState (GreaterThan 1) terminalChain)
+            returnProbabilityByState (GreaterThan 1) terminalChain
                 `shouldBe` [1, 1, 0]
-            (returnProbabilityByState (AtLeast 2) terminalChain)
+            returnProbabilityByState (AtLeast 2) terminalChain
                 `shouldBe` [1, 1, 0]
 
         it "preserves locally finite kernels and tiny survivor mass directly" $ do
diff --git a/test/Dtmc/Analysis/TimeSpecSupport.hs b/test/Dtmc/Analysis/TimeSpecSupport.hs
--- a/test/Dtmc/Analysis/TimeSpecSupport.hs
+++ b/test/Dtmc/Analysis/TimeSpecSupport.hs
@@ -320,7 +320,7 @@
                 Right result ->
                     sequence_
                         [ x `shouldSatisfy` closeTo (1 - fromIntegral i / 4)
-                        | (i, x) <- zip [0 :: Int ..] (result)
+                        | (i, x) <- zip [0 :: Int ..] result
                         ]
 
         it "solves the oscillator race to a single absorbing state" $ do
@@ -329,7 +329,7 @@
                 Right result ->
                     sequence_
                         [ x `shouldSatisfy` closeTo v
-                        | (x, v) <- zip (result) [2 / 3, 1 / 3, 1, 0]
+                        | (x, v) <- zip result [2 / 3, 1 / 3, 1, 0]
                         ]
 
         it "is all zero for an empty target" $
@@ -355,7 +355,7 @@
                                                 (x >= -testTolerance && x <= 1 + testTolerance)
                                         | otherwise -> x === 0
                                 | (i, x) <-
-                                    zip (finites :: [Finite 4]) (result)
+                                    zip (finites :: [Finite 4]) result
                                 ]
 
         prop "satisfies the first-step equations off the target (random @4)" $
@@ -371,42 +371,42 @@
                              in conjoin
                                     [ property (closeTo hi pi_)
                                     | (i, hi, pi_) <-
-                                        zip3 (finites :: [Finite 4]) (h) pushed
+                                        zip3 (finites :: [Finite 4]) h pushed
                                     , i /= 0
                                     ]
 
     describe "bounded hitting times" $ do
         it "returns an empty result for the empty chain" $
-            ((hitProbabilityByState . LessThan) 3 (identity @(Finite 0)) [])
+            (hitProbabilityByState . LessThan) 3 (identity @(Finite 0)) []
                 `shouldBe` []
 
         it "places all time-zero mass on the target" $
-            ((hitProbabilityByState . EqualTo) 0 oscillator [2])
+            (hitProbabilityByState . EqualTo) 0 oscillator [2]
                 `shouldBe` [0, 0, 1, 0]
 
         it "gives zero exact-time mass for an empty target" $
-            ((hitProbabilityByState . EqualTo) 5 oscillator [])
+            (hitProbabilityByState . EqualTo) 5 oscillator []
                 `shouldBe` [0, 0, 0, 0]
 
         it "matches a one-step gambler's-ruin hit" $
-            ((hitProbabilityByState . EqualTo) 1 (gambler 0.5) [0])
+            (hitProbabilityByState . EqualTo) 1 (gambler 0.5) [0]
                 `shouldBe` [0, 0.5, 0, 0, 0]
 
         it "uses a strict time bound" $ do
-            ((hitProbabilityByState . LessThan) 0 oscillator [2])
+            (hitProbabilityByState . LessThan) 0 oscillator [2]
                 `shouldBe` [0, 0, 0, 0]
-            ((hitProbabilityByState . LessThan) 1 oscillator [2])
+            (hitProbabilityByState . LessThan) 1 oscillator [2]
                 `shouldBe` [0, 0, 1, 0]
             (Hit.probabilityGivenInitialState . LessThan) 2 (gambler 0.5) (== 0) 1
                 `shouldSatisfy` closeTo 0.5
 
         it "ignores duplicate and reordered targets" $
-            ((hitProbabilityByState . LessThan) 4 oscillator [2, 3, 2])
-                `shouldBe` ((hitProbabilityByState . LessThan) 4 oscillator [3, 2])
+            (hitProbabilityByState . LessThan) 4 oscillator [2, 3, 2]
+                `shouldBe` (hitProbabilityByState . LessThan) 4 oscillator [3, 2]
 
         it "single-state queries look up the all-state results" $ do
-            let exact = ((hitProbabilityByState . EqualTo) 3 oscillator [2])
-                bounded = ((hitProbabilityByState . LessThan) 4 oscillator [2])
+            let exact = (hitProbabilityByState . EqualTo) 3 oscillator [2]
+                bounded = (hitProbabilityByState . LessThan) 4 oscillator [2]
             sequence_
                 [ (Hit.probabilityGivenInitialState . EqualTo) 3 oscillator (== 2) i
                     `shouldSatisfy` closeTo exactAt
@@ -464,11 +464,10 @@
                 `shouldBe` Right 0
 
         it "gives all zeros for identical successful and competing sets" $
-            ( hitRaceProbabilityByState
+            hitRaceProbabilityByState
                 oscillator
                 [2, 3]
                 [2, 3]
-            )
                 `shouldBe` Right (replicate 4 0)
 
         it "gives all zeros for an empty successful set" $
@@ -487,7 +486,7 @@
                 (Right before, Right plain) ->
                     sequence_
                         [ x `shouldSatisfy` closeTo y
-                        | (x, y) <- zip (before) (plain)
+                        | (x, y) <- zip before plain
                         ]
 
         it "is exactly zero when the successful set is unreachable" $
@@ -514,7 +513,7 @@
                 (Right withDuplicates, Right once) ->
                     sequence_
                         [ x `shouldSatisfy` closeTo y
-                        | (x, y) <- zip (withDuplicates) (once)
+                        | (x, y) <- zip withDuplicates once
                         ]
 
         it "ignores target order" $ do
@@ -532,7 +531,7 @@
                 (Right reordered, Right ordered) ->
                     sequence_
                         [ x `shouldSatisfy` closeTo y
-                        | (x, y) <- zip (reordered) (ordered)
+                        | (x, y) <- zip reordered ordered
                         ]
 
         it "single-state lookups match the all-state vector" $
@@ -550,7 +549,7 @@
                             i
                             `shouldSatisfy` either (const False) (closeTo x)
                         | (i, x) <-
-                            zip (finites :: [Finite 4]) (result)
+                            zip (finites :: [Finite 4]) result
                         ]
 
         it "solves the oscillator race against a competing absorber" $ do
@@ -559,7 +558,7 @@
                 Right result ->
                     sequence_
                         [ x `shouldSatisfy` closeTo v
-                        | (x, v) <- zip (result) [2 / 3, 1 / 3, 1, 0]
+                        | (x, v) <- zip result [2 / 3, 1 / 3, 1, 0]
                         ]
 
         it "matches a hand-computed symmetric race (gambler p = 0.5)" $ do
@@ -568,7 +567,7 @@
                 Right result ->
                     sequence_
                         [ x `shouldSatisfy` closeTo (fromIntegral i / 4)
-                        | (i, x) <- zip [0 :: Int ..] (result)
+                        | (i, x) <- zip [0 :: Int ..] result
                         ]
 
         it "disjoint races sum to one when the union is hit almost surely" $
@@ -581,7 +580,7 @@
                     (Right wins, Right losses) ->
                         sequence_
                             [ (x + y) `shouldSatisfy` closeTo 1
-                            | (x, y) <- zip (wins) (losses)
+                            | (x, y) <- zip wins losses
                             ]
                 | pp <- [0.3, 0.5, 0.7]
                 , let g = gambler pp
@@ -676,38 +675,38 @@
 returnTimeSpec = do
     describe "bounded first-return times" $ do
         it "returns an empty result for the empty chain" $
-            ((returnProbabilityByState . LessThan) 3 (identity @(Finite 0)))
+            (returnProbabilityByState . LessThan) 3 (identity @(Finite 0))
                 `shouldBe` []
 
         it "has no return mass at time zero" $
-            ((returnProbabilityByState . EqualTo) 0 oscillator)
+            (returnProbabilityByState . EqualTo) 0 oscillator
                 `shouldBe` [0, 0, 0, 0]
 
         it "uses the transition diagonal at time one" $
-            ((returnProbabilityByState . EqualTo) 1 nonUniformRecurrent)
+            (returnProbabilityByState . EqualTo) 1 nonUniformRecurrent
                 `shouldBe` [0.9, 0.6]
 
         it "counts only the first return" $ do
-            ((returnProbabilityByState . EqualTo) 1 oscillator)
+            (returnProbabilityByState . EqualTo) 1 oscillator
                 `shouldBe` [0, 0, 1, 1]
-            ((returnProbabilityByState . EqualTo) 2 oscillator)
+            (returnProbabilityByState . EqualTo) 2 oscillator
                 `shouldBe` [0.25, 0.25, 0, 0]
-            ((returnProbabilityByState . EqualTo) 2 twoCycle)
+            (returnProbabilityByState . EqualTo) 2 twoCycle
                 `shouldBe` [1, 1]
 
         it "uses a strict time bound" $ do
-            ((returnProbabilityByState . LessThan) 0 oscillator)
+            (returnProbabilityByState . LessThan) 0 oscillator
                 `shouldBe` [0, 0, 0, 0]
-            ((returnProbabilityByState . LessThan) 1 oscillator)
+            (returnProbabilityByState . LessThan) 1 oscillator
                 `shouldBe` [0, 0, 0, 0]
-            ((returnProbabilityByState . LessThan) 2 oscillator)
+            (returnProbabilityByState . LessThan) 2 oscillator
                 `shouldBe` [0, 0, 1, 1]
-            ((returnProbabilityByState . LessThan) 3 twoCycle)
+            (returnProbabilityByState . LessThan) 3 twoCycle
                 `shouldBe` [1, 1]
 
         it "single-state queries look up the all-state results" $ do
-            let exact = ((returnProbabilityByState . EqualTo) 3 oscillator)
-                bounded = ((returnProbabilityByState . LessThan) 4 oscillator)
+            let exact = (returnProbabilityByState . EqualTo) 3 oscillator
+                bounded = (returnProbabilityByState . LessThan) 4 oscillator
             sequence_
                 [ (Return.probabilityGivenInitialState . EqualTo) 3 oscillator i
                     `shouldSatisfy` closeTo exactAt
@@ -788,7 +787,7 @@
                                                         ( zipWith
                                                             (*)
                                                             row
-                                                            (hits)
+                                                            hits
                                                         )
                                              in counterexample
                                                     (show (i, f, firstStep))
@@ -797,7 +796,7 @@
                                         zip3
                                             (finites :: [Finite 4])
                                             rows
-                                            (returns)
+                                            returns
                                     ]
 
         it "is one for an absorbing state" $
@@ -934,7 +933,7 @@
                 `shouldSatisfy` either (const False) (closeTo 0.5)
 
         it "solves bounded hitting queries in named state order" $
-            ((hitProbabilityByState . LessThan) 3 namedGambler [Won])
+            (hitProbabilityByState . LessThan) 3 namedGambler [Won]
                 `shouldBe` [0, 0, 0.25, 0.5, 1]
 
         it "solves named expected hitting and return times" $ do
diff --git a/test/Dtmc/Analysis/VisitCountCanonicalSpec.hs b/test/Dtmc/Analysis/VisitCountCanonicalSpec.hs
--- a/test/Dtmc/Analysis/VisitCountCanonicalSpec.hs
+++ b/test/Dtmc/Analysis/VisitCountCanonicalSpec.hs
@@ -1,5 +1,3 @@
-{-# LANGUAGE TypeApplications #-}
-
 module Dtmc.Analysis.VisitCountCanonicalSpec (
     spec,
 ) where
@@ -143,7 +141,7 @@
 generatedTotalChecks matrix =
     and
         [ let scalar = checked (Visit.totalProbabilityGivenInitialState event matrix 0 initial)
-              dense = (checked (visitTotalProbabilityByState event matrix 0))
+              dense = checked (visitTotalProbabilityByState event matrix 0)
            in close (dense !! fromIntegral initial) scalar
                 && scalar >= negate testTolerance
                 && scalar <= 1 + testTolerance
diff --git a/test/Dtmc/Analysis/VisitCountSpec.hs b/test/Dtmc/Analysis/VisitCountSpec.hs
--- a/test/Dtmc/Analysis/VisitCountSpec.hs
+++ b/test/Dtmc/Analysis/VisitCountSpec.hs
@@ -1,5 +1,3 @@
-{-# LANGUAGE TypeApplications #-}
-
 module Dtmc.Analysis.VisitCountSpec (
     spec,
 ) where
@@ -147,7 +145,7 @@
     describe "totalProbabilityByState" $ do
         it "matches the geometric law for a transient target" $ do
             let probabilities count =
-                    (checked ((visitTotalProbabilityByState . EqualTo) count transientVisitChain 0))
+                    checked ((visitTotalProbabilityByState . EqualTo) count transientVisitChain 0)
             sequence_
                 [ actual `shouldSatisfy` closeTo expected
                 | (actual, expected) <-
@@ -163,7 +161,7 @@
                 | (actual, expected) <-
                     zip (probabilities 3) [3 / 64, 3 / 128, 0]
                 ]
-            (checked (visitInfiniteProbabilityByState transientVisitChain 0))
+            checked (visitInfiniteProbabilityByState transientVisitChain 0)
                 `shouldBe` [0, 0, 0]
 
         it "puts all positive recurrent-target mass at infinity" $ do
@@ -173,14 +171,14 @@
                 [ actual `shouldSatisfy` closeTo expected
                 | (actual, expected) <-
                     zip
-                        ((checked (visitInfiniteProbabilityByState recurrentVisitChain 2)))
+                        (checked (visitInfiniteProbabilityByState recurrentVisitChain 2))
                         [2 / 3, 1 / 3, 1, 0]
                 ]
             sequence_
                 [ actual `shouldSatisfy` closeTo expected
                 | (actual, expected) <-
                     zip
-                        ((checked ((visitTotalProbabilityByState . EqualTo) 0 recurrentVisitChain 2)))
+                        (checked ((visitTotalProbabilityByState . EqualTo) 0 recurrentVisitChain 2))
                         [1 / 3, 2 / 3, 0, 1]
                 ]
 
@@ -205,7 +203,7 @@
                                             [ (Visit.totalProbabilityGivenInitialState . EqualTo) count matrix 0 initial
                                                 === Right probability
                                             | (initial, probability) <-
-                                                zip (finites :: [Finite 3]) (probabilities)
+                                                zip (finites :: [Finite 3]) probabilities
                                             ]
                                 | count <- [0, 1, 3]
                                 ]
@@ -216,7 +214,7 @@
                                         [ Visit.infiniteProbabilityGivenInitialState matrix 0 initial
                                             === Right probability
                                         | (initial, probability) <-
-                                            zip (finites :: [Finite 3]) (probabilities)
+                                            zip (finites :: [Finite 3]) probabilities
                                         ]
                             ]
 
diff --git a/test/Dtmc/SimulationSpec.hs b/test/Dtmc/SimulationSpec.hs
--- a/test/Dtmc/SimulationSpec.hs
+++ b/test/Dtmc/SimulationSpec.hs
@@ -18,14 +18,15 @@
     Distribution (..),
  )
 import Dtmc.Distribution.Map (
-    pointMass,
     fromDistribution,
+    pointMass,
  )
 import Dtmc.Distribution.Vector qualified as Vector
 import Dtmc.Simulation (
     SimulationError (..),
     sample,
     simulate,
+    simulateMatrix,
     step,
  )
 import Dtmc.State (FiniteState)
@@ -70,9 +71,7 @@
     distributionWeights (UncheckedDistribution entries) = entries
 
 checkedSimulation :: (Monad m) => m (Either SimulationError value) -> m value
-checkedSimulation action = do
-    result <- action
-    pure (either (error . show) id result)
+checkedSimulation action = either (error . show) id <$> action
 
 cyclicThree :: TransitionMatrix (Finite 3)
 cyclicThree =
@@ -98,6 +97,15 @@
         fromRows @NamedSample
             (chunksOf 3 [0, 1, 0, 0, 0, 1, 1, 0, 0])
 
+iidThree :: TransitionMatrix (Finite 3)
+iidThree =
+    either (error . show) id $
+        fromRows
+            [ [0.2, 0.3, 0.5]
+            , [0.2, 0.3, 0.5]
+            , [0.2, 0.3, 0.5]
+            ]
+
 absorbingTwo :: TransitionMatrix (Finite 2)
 absorbingTwo =
     either (error . show) id $
@@ -181,6 +189,21 @@
     after <- MWC.save generator
     pure (result, before == after)
 
+zeroStepMatrixAndGeneratorState :: (Either SimulationError [Finite 3], Bool)
+zeroStepMatrixAndGeneratorState = runST $ do
+    generator <- MWC.create
+    before <- MWC.save generator
+    result <-
+        simulateMatrix
+            0
+            ( error "zero-step matrix simulation evaluated its matrix" ::
+                TransitionMatrix (Finite 3)
+            )
+            0
+            generator
+    after <- MWC.save generator
+    pure (result, before == after)
+
 emptyKernel :: Kernel.TransitionKernel Int
 emptyKernel =
     Kernel.fromLaws
@@ -285,3 +308,41 @@
                     simulate 3 emptyKernel 0 generator
                 )
                 `shouldBe` Left EmptySupport
+
+    describe "simulateMatrix" $ do
+        it "follows a deterministic matrix cycle" $
+            let trajectory = runST $ do
+                    generator <- MWC.create
+                    checkedSimulation (simulateMatrix 4 cyclicThree 0 generator)
+             in trajectory `shouldBe` [0, 1, 2, 0, 1]
+
+        it "supports named finite states" $
+            let trajectory = runST $ do
+                    generator <- MWC.create
+                    checkedSimulation
+                        (simulateMatrix 4 namedCyclicThree FirstSample generator)
+             in trajectory
+                    `shouldBe` [ FirstSample
+                               , SecondSample
+                               , ThirdSample
+                               , FirstSample
+                               , SecondSample
+                               ]
+
+        it "samples a dense row with the expected frequencies" $
+            let trajectory = runST $ do
+                    generator <- MWC.create
+                    checkedSimulation (simulateMatrix 20000 iidThree 0 generator)
+                observed :: Finite 3 -> Double
+                observed state =
+                    fromIntegral (length (filter (== state) (drop 1 trajectory)))
+                        / 20000
+             in do
+                    (abs (observed 0 - 0.2) < 0.02) `shouldBe` True
+                    (abs (observed 1 - 0.3) < 0.02) `shouldBe` True
+                    (abs (observed 2 - 0.5) < 0.02) `shouldBe` True
+
+        it "does not inspect the matrix or advance the generator at zero steps" $ do
+            let (result, unchanged) = zeroStepMatrixAndGeneratorState
+            result `shouldBe` Right [0]
+            unchanged `shouldBe` True
diff --git a/test/Dtmc/TestSupport.hs b/test/Dtmc/TestSupport.hs
--- a/test/Dtmc/TestSupport.hs
+++ b/test/Dtmc/TestSupport.hs
@@ -1,5 +1,4 @@
 {-# LANGUAGE ScopedTypeVariables #-}
-{-# LANGUAGE TypeApplications #-}
 
 module Dtmc.TestSupport (
     testTolerance,
diff --git a/test/Dtmc/Transition/InterfaceSpec.hs b/test/Dtmc/Transition/InterfaceSpec.hs
--- a/test/Dtmc/Transition/InterfaceSpec.hs
+++ b/test/Dtmc/Transition/InterfaceSpec.hs
@@ -1,5 +1,3 @@
-{-# LANGUAGE TypeApplications #-}
-
 module Dtmc.Transition.InterfaceSpec (
     spec,
 ) where
@@ -18,6 +16,9 @@
     chunksOf,
     genTransitionRows,
  )
+import Dtmc.Transition (
+    Transition (..),
+ )
 import Dtmc.Transition qualified as Transition
 import Dtmc.Transition.Kernel qualified as Kernel
 import Dtmc.Transition.Matrix (
@@ -43,6 +44,14 @@
 checked :: (Show error) => Either error value -> value
 checked = either (error . show) id
 
+data PublicCustomTransition = PublicCustomTransition
+
+-- A downstream instance needs to implement only the documented public
+-- method; internal optimization hooks keep their library defaults.
+instance Transition PublicCustomTransition where
+    type TransitionState PublicCustomTransition = Int
+    transitionLaw _ = DistributionMap.pointMass . (+ 1)
+
 finiteChain :: TransitionMatrix (Finite 3)
 finiteChain =
     checked $
@@ -75,6 +84,11 @@
 spec :: Spec
 spec =
     describe "Transition interface" $ do
+        it "keeps custom transition instances source-compatible" $
+            Distribution.distributionWeights
+                (transitionLaw PublicCustomTransition 4)
+                `shouldBe` [(5, 1)]
+
         it "exposes a matrix row as a finite-support transition law" $
             Distribution.distributionWeights (Transition.transitionLaw finiteChain 1)
                 `shouldBe` [(1, 0.2), (2, 0.8)]
