INDEPENDENT PHASE-8 AUDIT AND MANUSCRIPT CONNECTIONS 10 October 2026 VERDICT All 38 delivered local modules compile independently with Lean 4.26.0 on this Mac. The 34 earlier modules are unchanged from the phase-7 audit. All delivered hashes match. All 1,588 theorem/lemma declarations have matching axiom reports; only propext, Classical.choice and Quot.sound occur. No sorry, admit, custom axiom, unsafe, native_decide, extern or implemented_by was found by the comment/string-aware scan. Fresh logs agree with delivered logs after normalising only the leading ./ in warning locations. No invalid theorem was found in the semantic interfaces examined. Kernel checking validates the formal statements; their correspondence to prose is a separate review, recorded below and in the map. The three outstanding mathematical tasks from phase 7 are substantially complete: real-number sample complexity and minimum sample count; Fourier basis/tail and specified delta extension; general weak mastery and strategy connections. The existing winning-game certificates for all three bypasses remain valid. It would still be inaccurate to say that the entire paper, including observations, background literature and informal learning hypotheses, is a Lean theorem. EXTENT 38 local Lean modules; 1,588 theorem/lemma declarations; 632 def declarations. 24,493 source lines; 21,813 nonblank source lines; 17,689 code lines after comment/string removal and excluding #print axioms commands. Counts exclude Mathlib, Lean/Std, dependencies, logs and audit copies. Auxiliary results count. Phase 8 adds 4 modules, 261 declarations, 3,096 source lines and 2,075 code lines. The real-number files use Mathlib v4.26.0, pinned to 2df2f0150c275ad53cb3c90f7c98ec15a56a1a67. Qualitatively this checks an interconnected construction: game semantics, parity reduction, probability, executable move selectors, resource-bounded circuit families, network compilation, controllers and whole-game guarantees. It is much more extensive than checking a selection of algebraic identities. REPRODUCTION rebuild.py rebuilt the 36 core modules from copied sources into new oleans. rebuild_real.py rebuilt RealNoise and RealFourier against the pinned official mathlib cache in this audit's isolated .lake directory. The source delivery and its dependencies were not modified. reproduction_results.json records exits and times; input_manifest.json records sources/hashes; audit_results.json records scans, counts, hashes and axiom coverage. scan_audit.py reproduces the checks. The official cache was fetched only for the required imports (1,008 dependency files); these dependencies are trusted in the usual Lean/Mathlib sense, rather than claimed to have been independently rebuilt from source in this audit. SEMANTIC FINDINGS 1. RealNoise uses genuine Mathlib real arithmetic. Its adapters verify that the generic field operations are the intended operations. mUp = ceil(4/a^2) is a natural sample count, ach_exists establishes existence, and mMin_spec proves attainment/minimality. For 0=2 its sign is not a signed character; N=1 is explicitly degree two. This is now the exact claim in the paper. Claude also proves uniqueness of the degree-2N character agreeing on ALL single-change pairs, including increases and unchanged boards. That theorem must not be advertised as a legal-decreases-only result. The paper does not make that stronger assertion. 3. WeakMastery formalises roles, histories, deterministic strategies, legal opponents, forfeits, terminating play and winning guarantees. The equivalence to moving to a P-position on all policy-consistent decision histories is substantive. Initially losing roles give vacuous weak mastery. Pairing has a compiled selector and a genuine second-player guarantee; multi-head and two-frame strategies are instantiated. Strong mastery uses arbitrary legal histories, and the (3)->(0) example separates it from weak mastery. The manuscript now specifies these domains and gives that simple example. 4. The latest paper contained stale implementation language. BoardStore already checks one-pass storage (NB+B+r bits, N+1 updates plus reset) as well as the replay alternative. Phase 7 already checks the outer early-exit and all-head schedules, including actual rollout execution, reset and head switching. The updated paper distinguishes classification from move selection and counts the read-only original board separately from controller registers. It records M+2B+2 component evaluations for all heads plus final readout, not just M simulated legal moves. The circuits and networks linked to these semantics have depth <=34 and layers <=70 per controller tick. These are sufficient resource bounds, not hardware timings or optimised implementations. REMAINING BOUNDARIES AND OPTIONAL FORMAL WORK A. Classical results are intentionally external. HastadNegl is a theorem parameter, not a hidden Lean axiom. The main paper-specific application to AC0 readouts, finite probabilities and the negligible final error is checked. The analytic passage from a published exponential correlation estimate to that named negligible-family theorem remains at the accepted classical boundary. General Sprague-Grundy, Ajtai and other cited background are also literature results, not claimed as newly formalised here. B. Fourier.TailPremise is not instantiated against one precise LMN/Tal theorem with reconciled size/depth conventions. The tail-to-correlation implication is checked. Completing that adapter is useful final boundary work, but is not needed for the paper's principal search bound, whose correlation route is separate. The revised verification paragraph discloses this explicitly. C. WholeGame.ReplayWins is not instantiated in the newer Mastery.Hist model. This is an optional unification bridge: the replay game and network guarantee already exist in their original semantics. Do not describe it as an absent recurrent winning proof. The same distinction applies to storage semantics. D. Strategies in the new general game framework are deterministic; probability over testing rules is formalised separately. No randomized game-policy or learning-convergence theorem is claimed. A universal adaptive-FSM compiler is not claimed; the two concrete schedules are checked. E. Training curves, checkpoint observations, plateau causation and empirical implementation are not kernel-certified. This audit did not rerun training. Mathematical baseline counts are checked; they do not establish why SGD remains near 0.54 or when it improves. The paper already states this carefully. PRESENTATION AND CORRESPONDENCE DELIVERED The canonical candidate source has been edited in place, preserving its title, article format, narrative, experiments and principal results. A byte-for-byte pre-edit source is saved as paper_before_phase8.tex. Changes clarify mastery, noise independence, the Fourier connection, the unknown-index extension, controller/storage costs and the scope of verification. A short probability appendix supplies the previously deferred minimum-sample definition and proof. No Lean source was changed. paper_proof_correspondence.json/.tsv map 51 stable LaTeX labels, including every mathematical label, to exact checked declarations, source locations, hashes, hypotheses and external/empirical boundaries. declaration_index.json indexes all 1,588 checked declarations. make_correspondence.py validates label and declaration existence. The map records the manuscript hash. It is a reviewable correspondence record, not an automated proof that English and Lean mean the same thing. No interactive site has been built; the canonical text and stable labels are prepared for that later layer. Final public URLs remain placeholders as requested. PUBLICATION STATUS The paper builds as a 40-page article. Its abstract is 187 whitespace-separated words. The local TeX build resolves references/citations; the built-in single-file compiler cannot load the external figure files. Use the verified project PDF. Final journal formatting, actual public links, author declarations and the public proof/experiment release remain pre-submission tasks. No submission or publication was performed. The source bundle is a review candidate, not a claim that these final author/publication tasks are complete. FINAL VALIDATION: all 40 pages rendered and visually reviewed in overviews; changed verification, Fourier, resource-table and noise-proof pages also inspected individually. Final build has no warnings, overfull boxes or undefined references. The isolated arXiv source snapshot also compiles successfully. All 1,588 declaration index entries have source line locations. The canonical source hash is recorded in both publication_validation.json and paper_proof_correspondence.json.