Graph sampling, all-cut trees and spectral construction
This edition extends the earlier direct-backend exclusions. New source readings clarify three graph-product boundaries. Finite reference checks do not verify the source proofs or establish buyer value.
Sampling protocol changes the law
Family 131's bound applies to the half-hold, uniform-four-set, uniform-ordered-pair chain on the complete host. Invalid proposals hold and count as steps. Its quarter-TV bound is 2n^8; exactness uses an ordinary run of 24n^2*binom(n,4)*(B+2D+2) steps and a rare exhaustive residual correction, with B=binom(n,2) and D=4B+10. The exact algorithm has expected bit cost, exponential correction work and unbounded rejection-draw tails. The selected formal scope excludes that exact sampler.
The bounded audit enumerates up to six vertices and calculates the precise rational law from one start. On degree one with a six-cycle host, its two feasible states are isolated; TV remains 1/2. For degree two on six vertices, sixty six-cycles have twelve valid neighbors and ten two-triangle graphs have eighteen. Selecting uniformly among valid neighbors gives stationary triangle-event mass 1/5, versus uniform mass 1/7, and TV 2/35. This generic accepted-event chain is separate from an exhaustive external-library audit. NetworkX's swap API counts performed swaps and has a separate attempt cap; its interface cannot inherit the paper's step bound without a kernel/time correspondence. NetworkX documentation.
Saved exact laws, counterexamples and schedules, 69 finite checks, source kernel, scope.
Fundamental cuts do not certify all cuts
For a path tree in K4, fundamental tree-cut ratios are at most 1/3, while an alternating-vertex cut crosses three tree edges out of four original edges: actual thinness 3/4. The exact all-cut audit finds this witness. It handles positive binary multiplicities without expanding copies and computes exact connectivity and achieved k*alpha. Its 16-vertex cap and five-million edge-examination budget support exponential finite checking. Partial scanning grants no all-cut certificate.
Family 174's constructive signing estimate takes Cs=2^20, giving declared cut factor 1/2+3Cs*r^(-1/8). That conservative factor certifies strict contraction only for r>(6Cs)^8, a 55-digit quantity. This is not a final C, numerical input-size/runtime lower bound or impossibility for another method. The iteration cutoff and final constant remain numerically unselected. The fractional corollary promises simultaneous cut/cost rounding for nonnegative rational inputs with an all-cut mass promise; it is not separately listed in the selected family scope.
Saved ratio/constant calculations, 30 checks, iteration source, signing source.
Spectral acceptance is separate from graph construction
Family 178 fixes a=1/10000, so its largest-primary auxiliary group has ceil(10/(0.03a))=3,333,334 points and uses both signs: 6,666,668 signed auxiliary points during early/cleanup phases. These are parameter points, not a moment order, complexity exponent or required simultaneous-memory allocation. Repair enumerates P^2 affine seeds for a prime of order n^Cprime, with the sufficiently-large fixed exponent not numerically selected here; search can stop at the first passing candidate. The fixed-degree threshold and polynomial exponent remain uninstantiated.
The theorem outputs a full adjacency list at sufficiently large even orders for fixed degree. It does not supply joint-polynomial dependence on degree and size or local adjacency queries. The existing rational contrast test remains a useful exact finite acceptance component. A heuristic generator plus a passing test does not implement the source's deterministic polynomial constructor.
Exact parameter record, setup, repair, existing checker.
The revised graph businesses are method calibration, finite cut assurance and exact spectral acceptance, with full constructors retained as separate R&D objectives. Their usefulness, repeatability and pricing still need buyer validation. Dossier 005, dossier 024, dossier 036.