# General matching: literal backend prerequisite prevents a practical speedup claim

Family 120; construction addendum, 9 October 2026 Australia/Brisbane. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`.

The introduction and uniform program were read, together with selected engine-sizing definitions and the backend parameter/guard block. The input is a simple undirected unweighted graph with isolates charged in `p=n+m`. One finite program interleaves parameter branches with a deterministic exact fallback and outputs explicit matching edges. The common exponent envelope is defined by a supremum over actual instruction counts and is not a numerical practical crossover.

For accelerated branch `t>=1`, `a=1000+t`. The backend requires `P_star<=b`, with `Lambda=2^(1000a^2)(1+u)`, `P_star=Lambda^(400a^2)`, `b=M^(1/a^2)` and `M=2^u<=S^2`. Therefore `u>=400000a^6` and `log2(S)>=200000a^6`. At `a=1001`, this is an astronomical necessary master-size bound, not a sufficient threshold. Structural easy cases and the fallback may still terminate. The bound neither contradicts the theorem nor excludes a redesigned algorithm.

Defer the literal accelerated backend for commercial speedups. An [independent finite matching audit](../../../../tools/audit_matching_certificate.py) now checks candidate feasibility and a supplied odd-component upper bound, certifying maximum cardinality when attained. It passed tests on all 1,024 five-vertex graphs against brute-force optima. A loose bound leaves optimality unknown. This is a classical component, not the source's new solver or a certificate constructor. Weighted/fairness/side-constraint claims remain outside the model.

Exact source sections:

- `preprints/Almost-Linear-Time-Maximum-Cardinality-Matching-in-Sparse-General-Graphs-September-24-2026/build/sections/01-introduction.tex`
- `preprints/Almost-Linear-Time-Maximum-Cardinality-Matching-in-Sparse-General-Graphs-September-24-2026/build/sections/05-engine.tex` (selected size/degree-reduction parameters)
- `preprints/Almost-Linear-Time-Maximum-Cardinality-Matching-in-Sparse-General-Graphs-September-24-2026/build/sections/07-backend-quality.tex` (parameter and guard block)
- `preprints/Almost-Linear-Time-Maximum-Cardinality-Matching-in-Sparse-General-Graphs-September-24-2026/build/sections/09-uniformization.tex`

The full backend proof/update machinery and prescribed-degree factor construction have not been independently reproduced. No family formal scope is listed. See [source metadata](source.json), [revised dossier 007](../../../../opportunities/007-general-graph-allocation/2026-10-09-v3.md) and [feasibility derivation](../../../../FEASIBILITY-2026-10-09-v2.md). No profitability or demand is established.

## Existing-solver integration addendum

The [bounded adapter](../../../../tools/solve_and_audit_matching.py) now uses NetworkX 3.4.2 to propose matching edges and, when needed, a subset from vertex-deletion solves. A separate integer graph audit accepts only a feasible matching attaining a valid upper bound. The public 34-vertex karate-club graph, a generated 41-vertex star and an 80-vertex regular graph all have saved attaining certificates. The adapter passed 72 additional checks against brute-force four-vertex optima, malformed-input/cap tests and these fixtures. It rejects weighted inputs. No source accelerated backend, formal program verification, customer workload or business value is demonstrated. [Saved report](../../../../snapshots/2026-10-08-baseline/matching-integration-2026-10-09-v1.json).