Direct-candidate feasibility: schedules, laws and solver certificates

This edition adds explicit construction findings for families 115 and 120 to the earlier parameter exclusions. The source results remain claims at the pinned revision; no independent proof check was run. The calculations below are derived from stated schedules and guards, not observed runtimes or a proof that no practical variant could exist.

Fixed-margin tables: the literal schedule is a complexity result

The sampler sets d=10+(m+1)(n+1), L=d^12, U=d^20, C=N+dL+U+2 and b=d+ceil(log2(C+2)). Its bounded outer routine chooses T=d^200(k+b)^2 transitions per trial and caps trials at d^4(k+1). The exact version chooses D=d^6 b^2, accuracy k=2D, and a correction branch with probability 2^-D. That branch exhaustively tabulates the implemented law; its extremely small probability controls expected cost. It does not provide a deadline bound on every exact-sampling execution.

For the two-by-two margins [2,2] and [2,2], the literal outer schedule already has hundreds of decimal digits in its transition count, before charging a transition's dense-completion work. The general sampler has no cell bounds; the separate cell-bounded paper provides approximate counting and explicitly emphasizes complexity over practical runtime. No production crossover has been demonstrated.

Parameter definitions, sampling and exact correction, cell-bounded scales, selected scope.

Uniform tables and conditional independence are different laws

SciPy's random_table is designed for the fixed-margin conditional independence distribution and offers Boyett/Patefield generation. Its probability implementation uses product(row!)*product(column!)/(total!*product(cell!)). It does not give each integer table equal probability. SciPy documentation, versioned probability implementation.

With two-by-two margins [2,2], there are three tables, indexed by the top-left cell 0,1,2. Uniform-table probabilities are each 1/3. Conditional-independence probabilities are 1/6, 2/3, 1/6. The exact total-variation distance is 1/3; the probability of the two extreme tables is 2/3 under the uniform law and 1/3 under conditional independence. This difference is a model choice, not a flaw in the existing statistical sampler. A uniform-table engine must not silently replace an independence-test sampler.

The saved law comparison uses exact fractions. The new finite reference counts every feasible table once and un-ranks a supplied integer into a table, also respecting explicit cell bounds. Uniform rank selection maps to uniform tables. It is a conventional bounded dynamic program, capped at dimension six, total 128 and work/state budgets; it does not implement the new polynomial-bit backend. Budget exhaustion returns unknown rather than zero. Recorded ranks permit exact replay of generated examples.

General matching: an astronomical prerequisite for the literal backend

The uniform program uses accelerated branches indexed by t>=1, with a=1000+t, and a deterministic fallback. In the cycle backend, M=2^u, b=M^(1/a^2), Lambda=2^(1000a^2)(1+u), and P_star=Lambda^(400a^2). The mandatory check P_star<=b, together with M<=S^2, implies

u/a^2 >= log2(P_star) >= 400000 a^4
u >= 400000 a^6
log2(S) >= u/2 >= 200000 a^6.

At the first accelerated branch, a=1001. This is an astronomical necessary master-size condition before that backend can pass its first guard. It is not a sufficient threshold; other checks are stronger. Early structural completions, local easy cases and the deterministic fallback may still terminate. The derivation does not disprove the asymptotic theorem, establish an input-size threshold from an unspecified big-O constant, or exclude a substantially redesigned practical variant.

The source's common exponent envelope is defined through a supremum over actual instruction counts, rather than supplied as a numerical crossover estimate. A literal general-purpose implementation therefore lacks a defensible commercial speedup claim. Backend parameters and guards, uniform branches and schedule.

What can be implemented now for allocation workflows

The matching certificate audit checks an unweighted candidate and an explicit vertex subset S. If q is the number of odd components after removing S, at least q-|S| vertices must remain unmatched: each odd component needs an outside match or an unmatched vertex, and at most |S| components can consume vertices in S. Hence every matching has at most (n-q+|S|)/2 edges, also at most floor(n/2).

A feasible matching attaining either checked bound is maximum cardinality. A loose bound is inconclusive, so the utility preserves unknown optimality instead of rejecting a potentially maximum matching. This is a classical certificate component, not the paper's new solver or a barrier constructor. Weighted objectives, fairness and other allocation constraints need their own model and evidence. NetworkX already offers a blossom-based matcher with a maximum-cardinality option; it is a baseline rather than proof of a new market gap. NetworkX API.

Saved evidence and earning consequence

Exact schedule/guard calculations record formulas and source locations. Contingency validation has 325 checks against closed-form and independent Cartesian references. Matching validation covers all 1,024 five-vertex graphs, compares 32,768 subset bounds against independent brute-force optima, and checks malformed/invalid candidates. It is finite component validation, not an almost-linear solver benchmark.

Defer literal research backends for dossiers 004 and 007. Retain constrained-data law auditing and existing-solver allocation certificates as more plausible integration hypotheses. Demand, delivery cost and recurring value still need validation; small reference scripts alone do not establish profitable products.