# Matching counts: literal schedule exclusion and bounded fragility reference

Family 113; selected manuscript/construction and formal-scope addendum. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`.

The FPRAS main statement and selected introduction, quantitative sampling guarantee, annealing/counting parameters, amplification and prescribed-degree corollary were read. The literal nonempty routine has `S=ceil(10 sigma^-2(n+K+1))`, with `sigma<=1/(10^7D)` and large enlarged-graph parameters. At two vertices and error `1/10`, confidence failure `1/20`, its general schedule requests about `4.52*10^48` fresh runs per stage. This is an exact declared-parameter calculation, not a runtime observation or impossibility for a different method. Empty/infeasible branches and practical special cases remain separate. The prescribed-degree/fixed-size sampler promises total-variation approximation, not exact or pointwise almost-uniform sampling.

The companion entropy introduction (lines 1–240) and selected deterministic algorithm (lines 1–155) were read. Entropy bounds require a feasible perfect-matching marginal vector, with general-graph odd cuts. The deterministic count guarantee uses factor `512^n`, not fixed relative-error accuracy, and explicitly warns its polynomial degree is large. The selected family scope covers the FPRAS, entropy/count bounds and triangle-expansion relation, but excludes the sharp global face-dimension bound. No independent formal check was run.

Defer the literal production counter. [Dossier 006](../../../../opportunities/006-matching-flexibility-analyzer/2026-10-09-v2.md) now describes a conventional bounded exact subset-DP reference, rank replay and single-edge failure counts. Its 93 finite checks do not verify the source FPRAS or buyer value. Full proof, sampler implementation and deletion reduction remain pending.

Exact source files read:

- `preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex` (lines 1–250, 2350–2725, 2820–2845, 3080–3140).
- `preprints/Entropy-and-Face-Dimension-of-the-Perfect-Matching-Polytope-September-23-2026/build/sections/01-introduction.tex` (1–240).
- `preprints/Entropy-and-Face-Dimension-of-the-Perfect-Matching-Polytope-September-23-2026/build/sections/09-algorithm.tex` (1–155).
- `lean/docs/113.md`.

See [source metadata](source.json), [parameter calculations](../../../../snapshots/2026-10-08-baseline/matching-count-parameters-2026-10-09-v1.json), and the [exact utility](../../../../tools/perfect_matching_reference.py).