Mean-payoff source procedure, finite certificates and variant boundaries

9 October 2026. The independently written deterministic reference implements paper Procedure 4.1 with exact integers/rational masses and work/depth caps. It is the 36th research utility. 1,097 new controls pass, bringing saved totals to 21,714 in 54 reports. Finite generated evidence does not accept source proofs or establish buyer value.

Models and measured costs

The ordinary deterministic paper returns winning states and adds value/strategy reductions. The randomized main claim has a 7/8 whole-output guarantee and every-tape time bound; its checked-potential retry variant has expected runtime. The stochastic claim thresholds expected pathwise liminf. The parity conjunction returns a winning set and may require infinite-memory Max strategies. Variant and business dossier.

All 324 complete two-vertex games with four ownership assignments and weights -1/0/1 complete the reference and match enumeration. Of thirty hash-derived three-vertex games, eleven complete and nineteen return unknown. Every completed answer matches. Fourteen selected controls follow. Timings are sequential single local samples; the tiny oracle additionally recovers values/strategies, while the source reference returns threshold labels only.

The all-negative recursive two-vertex game completes after 237,877 calls; its all-positive counterpart exhausts two million selected charges. The source call expression is an upper bound, never an observed cost or lower bound. The 2^64 one-vertex control shows large reward magnitude alone need not make this deterministic recursion slow. No caching or heuristic simplification was applied.

Independent finite certificates

For all 368 generated games, the exhaustive comparator supplies threshold regions and positional choices accepted by a separate closure/cycle checker. It uses original edges and retains distance potentials. For Max it excludes negative original cycles; for Min it excludes positive transformed cycles t(e)=(n+1)w(e)+1, forcing original simple-cycle sums <= -1. Closure and cycle decomposition give guarantees against arbitrary histories. This evidence can pass when the source reference is unknown.

Eight initial adversarial controls and two explicit opponent-escape tests check false regions, bad/missing choices, zero/positive cycles and closure. A suboptimal zero-loop strategy passes a threshold certificate even though its start has value one: threshold-winning is not value-optimality. Schema, models, weight limits, work and depth also have controls. No universal correctness or formal code proof is inferred.

Schedules and semantic examples

For a one-vertex 2^64 positive loop, the randomized algorithm's first Ask would perform J=2854495385411919762116571938898990272765493249 trials literally. A zero-loop input instead selects Basic before any wrapper. The deterministic algorithm has no such J. Exact schedules.

The stochastic fair-split plan has H=122880, N=483183820800 and 2,847 outer width updates. The width schedule is label-independent and was counted arithmetically; no stochastic recursion ran. Two encoded 1/2 probabilities give Q=4. A binary denominator is not an iteration count. Parameter plan.

A fair split into +1/-1 absorbing rewards has expected limiting reward zero but nonnegative-path probability 1/2. In the two-state parity example, R increasing rounds give reward -R over R(R+1)/2+2R edges: mean -2/(R+5) tends to zero while priority zero recurs infinitely. Every deterministic finite-memory play is ultimately periodic and cannot simultaneously achieve that mean threshold and parity. A forced first reward -100 followed by reward-one looping wins mean payoff but violates zero-credit energy. These are model distinctions, not deployed controller tests.

Replay and preserved evidence

python3 research/tools/mean_payoff_label_reference.py research/prototypes/mean-payoff-comparison-2026-10-09-v1/one_high_binary-input.json
python3 research/tools/mean_payoff_label_reference.py research/prototypes/mean-payoff-comparison-2026-10-09-v1/all_negative_recursive-input.json --work-limit 2000000

The importable certify_regions function takes separate edge-index strategies; the CLI exposes labels. Caps are 12 vertices, 96 edges, 256-bit signed weights, two MiB CLI input, depth 128 and two million selected charges maximum. Bit arithmetic, parsing, preprocessing, allocation and time are outside the counter. Unknown emits no partial winning set. Unsupported stochastic/parity/energy declarations are refused.

1,077 initial controls, 20 targeted controls, 324 tiny inputs/results, 30 hash-derived inputs/results, main comparison/source/runtime record, recursive extension. Python 3.10.19 was reused with standard-library modules only; no installation was performed. Archived tools remain inert evidence.

PRISM-games documentation, reviewed 9 October 2026, distinguishes expected/almost-sure objectives and restricts expected long-run/ratio queries to controllable-multichain games. It was not executed. Semantic translation and a compatible production baseline remain open. Commercial confidence stays low; facility-planning assurance retains priority.