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chess-spectral v1.5 docs audit — overstrong claims & inadequately-hedged extrapolations

Date: 2026-04-29 Branch: chess-spectral/v1.5-docs-audit-and-bridge-api (forked from chess-spectral/v1.5-bridge-integration at commit 74c6e84). Audited surface: every doc added or substantively modified between v1.3.2 (commit 775a571, last shipping release) and the v1.5 bridge integration tip.

How to read this report

Each finding is keyed [F##] with severity:

  • STRONG — load-bearing overclaim. Either an unqualified novelty assertion without a prior-art survey, a prescribed threshold dressed as a logical gate, or a cross-domain extrapolation re-presented as a chess prediction.
  • MEDIUM — rhetorical numbers framed as quantified analysis, aspirational scope claims, or wording that rounds an opinion into a fact.
  • WEAK — wording polish that makes the hedge clearer; not load- bearing on its own but worth fixing while the audit is open.

Findings are organized by file. The "current text" quote is the problematic span; "issue" names the overclaim category; "suggested rewording" is a starting point — the user should refine before applying.

The user has already softened two flagged spans (research notebook §16.5 "first 4D chess engine ever shipped" and §16.7 "must test at L4/L8/L16"; see git diff in the worktree); those soft-landings are not re-raised here. The audit covers the rest of the surface.

Findings

docs/chess-maths/chess_spectral_research_notebook.md

[F01] STRONG — §15.1, line 2434

"That combination doesn't exist on PyPI today. … chess-spectral is a working H(4, 8) implementation; the structure ports to any H(d, q) with modest changes …"

Issue. "Doesn't exist on PyPI today" is a categorical absence claim without an exhaustive PyPI survey. PyGSP / escnn / e3nn / Qiskit / lattice QCD are named as nearby projects with comments on why each doesn't fill the gap, but the implicit "we have looked at the rest of PyPI" is not substantiated.

Suggested rewording. "We are not aware of a single PyPI package combining these four properties; the named adjacencies (PyGSP, escnn, e3nn, Qiskit, lattice QCD frameworks) each cover a strict subset. A dedicated Z_n^d Hamming-scheme + B_d toolkit would be useful even if a partial overlap exists; an exhaustive prior-art survey is out of scope for this notebook."

[F02] STRONG — §15.2(2), line 2442

"The engineering 'choice' of basis is actually a representation- theoretic theorem; full diagnostics in [spectral_identity_4d_findings.md] … This is what makes the encoder canonical rather than ad-hoc — any B_4-equivariant function on Z_8^4 lattice configurations decomposes naturally in this basis."

Issue. "Theorem" is doing real work here, but the supporting probe (Pre-flight 3) is a numerical verification at machine precision on a specific finite construction — that is strong evidence but not a mathematical theorem. The standard statement (CSCO basis on a finite group action on L²(graph)) is a theorem in the literature; what we showed is that the encoder columns realize it. Compounded with "This is what makes the encoder canonical" the language presents a numerical verification as a self-supplied theorem.

Suggested rewording. "The encoder's columns are numerically verified to be a simultaneous eigenbasis of (Δ, B_4 commutant) at machine precision (Pre-flight 3). The mathematical statement that any finite-group-equivariant function on the underlying lattice decomposes in the simultaneous eigenbasis of the commutator algebra is classical (Bouckaert–Smoluchowski–Wigner 1936; cf. [spectral_identity_4d_findings.md] for citations); our contribution is constructing the basis explicitly and verifying it is the one the encoder uses."

The exact same softening should land in spectral_identity_4d_findings.md line 111 ("The encoder's tensor-DCT basis … is canonical") — same overclaim, restated.

[F03] MEDIUM — §15.6, line 2487

"The mathematical content here is 0% novel — every individual ingredient is well-established. The toolkit packaging is ~70% novel: no single PyPI package combines what chess-spectral with qm_4d will provide. The chess-as-4D-spectral-physics framing is ~95% novel for chess specifically."

Issue. The user explicitly flagged these "0% / 70% / 95%" as rhetorical-numbers-presented-as-quantified-analysis in the prompt. None of the three percentages is sourced or measurable. The range 0%–100% over three categories does double duty as both "honest scope" language and as if-it-were-a-survey numerics.

Suggested rewording. "The mathematical content uses well- established ingredients; the value is in the assembly. We are not aware of a single PyPI package combining the four properties listed in §15.1, and we have not seen the chess-as-4D-spectral-physics framing in the literature for chess specifically — both observations are 'as far as we have looked,' not survey conclusions."

[F04] STRONG — §15.6, line 2489

"The spectral identity in §15.2(2) is the cleanest of these: the encoder's basis IS the simultaneous eigenbasis of (Δ, B_4 commutant), at machine precision. That's a clean math statement, and it's the theorem underlying every claim about 'B_4 equivariance for free.'"

Issue. Restated form of [F02]. "The theorem" is again an unqualified self-attribution; the supporting structure is numerical. Should be softened the same way.

Suggested rewording. "The spectral identity in §15.2(2) is the cleanest of these: the encoder's basis is numerically verified (Pre-flight 3) to be the simultaneous eigenbasis of (Δ, B_4 commutant) at machine precision. The classical theorem (CSCO on finite-group representations) provides the structural reason; the verification shows our specific construction realizes it."

[F05] STRONG — §15.4, item 1, line 2471

"Pedagogy and teaching. … no shipping educational toolkit covers Z_n^d analysis end-to-end with both engineering and physics APIs."

Issue. "No shipping educational toolkit covers" is a categorical claim. The pedagogy field has Sage, GAP, Mathematica notebooks, educational Python projects under various universities — even if none is quite this, the unqualified "no shipping toolkit" reads stronger than what we can defend. Compounds with [F01].

Suggested rewording. "We are not aware of a Python educational toolkit that covers Z_n^d analysis end-to-end with both engineering and physics APIs; SageMath / GAP / Mathematica notebooks exist in research papers but no PyPI-installable package fills the same niche in our experience. Treat this as 'a real gap as far as we can see,' not as a survey finding."

[F06] STRONG — §15.4, item 3, line 2473

"Quantum-walk research on Hamming graphs. The arXiv:2509.26243 audience (symmetric coined quantum walks on Hamming graphs) needs reference implementations to validate theorems against. chess-spectral with qm_4d gives them a parity-tested substrate."

Issue. Two compounding overclaims: (1) "needs reference implementations" as a stated requirement of an external audience whose needs we have not surveyed; (2) "gives them a parity-tested substrate" asserts our toolkit is fit-for-purpose for a community we have not engaged. The cited paper is a single 2025 arXiv preprint.

Suggested rewording. "Quantum-walk research on Hamming graphs. The arXiv:2509.26243 line of work (symmetric coined quantum walks on Hamming graphs) is a candidate user of reference implementations; chess-spectral with qm_4d may be a useful substrate. Whether the substrate matches their needs is a question for that community, not a claim we should make for them."

[F07] MEDIUM — §16.7, "Implications for Phase 6 design", item 4, line 2593

"The QM-expectation evaluator is the more interesting test. Spectral channel-energy weighting is structurally close to what Othello tested and likely faces the same shallow-strong / deep-vanishing decay."

Issue. "Likely faces the same … decay" extrapolates the Othello finding to chess. The Othello → chess depth translation was the user-flagged example of methodological inheritance, not a logical mapping. Even softened in the §16.5 text and in the §16.7 item 1 text, item 4 still asserts the prediction.

Suggested rewording. "The QM-expectation evaluator is the more interesting test. Spectral channel-energy weighting is structurally close to what Othello tested; whether chess inherits the shallow- strong / deep-vanishing pattern is genuinely open — chess's larger branching factor and the much weaker material baseline (vs Othello's Edax) make a clean analog non-obvious. Plotting per-depth ELO deltas is what tells us whether the pattern transfers."

[F08] MEDIUM — §16.7, "Updates to §16.5", final bullet, line 2604

"(new) Per-depth signal decay curve. Plot ELO delta vs search depth for each evaluator. This is the most informative single chart Phase 6 can produce — directly comparable to the Othello L6/L10 contrast."

Issue. "Directly comparable to the Othello L6/L10 contrast" is the same Othello → chess depth-mapping issue in compact form. The user- softened §16.5 text now says "shallow / mid / deep, illustrative not prescribed"; this bullet still claims direct comparability with the Othello L6/L10 axis.

Suggested rewording. "(new) Per-depth signal decay curve. Plot ELO delta vs search depth for each evaluator. This is the most informative single chart Phase 6 can produce — analogous in shape (though not necessarily in depth values) to the Othello L6/L10 contrast."

[F09] STRONG — §16.7, "Phase 7 (learned weights) reframing", line 2608

"If Phase 6 confirms the depth-decay pattern for spectral evaluation in chess …"

Issue. "Confirms" presupposes the depth-decay pattern is the expected outcome. Given the cross-domain mapping is methodological inheritance, the framing should be neutral — confirm vs refute vs something-else are all live possibilities.

Suggested rewording. "If Phase 6 finds the depth-decay pattern holds for spectral evaluation in chess, Phase 7's learned-weights work has two paths: …"

[F10] STRONG — §16.8, item 1, line 2622

"Faithful sheaf bracket-classifier: +40% gain (research/faithful_sheaf.py; §2e.5–§2e.6 of the othello notebook). Replacing endpoint-only restriction maps with per-cell R1/R2/R3/R4 bracket-state classifier and projecting pending-flank counts through D_4 lifted D_4-A_1(s²) partial ρ from −0.319 → −0.447 at N=2587."

Issue. This is fine as a quotation of the Othello result. But the surrounding paragraph and the §16.8.5 conclusion ("the structural ceiling depends on what observables you have access to") generalize it across the chess/Othello boundary in a way that re-uses the +40% number as if it predicted a chess outcome. The number is reported as Othello data and then loaded into the chess-side reasoning.

Suggested rewording. Keep the Othello number as-is in §16.8.1 (it's a faithful quotation). Where §16.8.5 concludes "the structural ceiling depends on what observables you have access to," add an explicit hedge: "in Othello. Whether the same shape (richer features + learned coupling lift the ceiling) replays in chess is a Phase 7 empirical question, not a prediction."

[F11] STRONG — §16.8, item 2, line 2624

"Phase-operator reweighting chain: +114% cumulative gain … This is the most directly applicable result to chess-spectral's Phase 6 design — the QM-expectation evaluator with learned α_O weights on the §15.2(4) Hermitian observables is structurally close to this construction. Treat it as the prior to beat."

Issue. "Treat it as the prior to beat" elevates an Othello-side fitted construction into a chess-side benchmark. The structural similarity is real (Nelder-Mead-fit weights on entropy-difference features ≈ learned α_O weights on Hermitian observables), but "the prior to beat" frames the +114% Othello gain as if it were the chess target. The cross-domain hop is exactly the user-flagged class of overstrong claim.

Suggested rewording. "Phase-operator reweighting chain: +114% cumulative gain (Othello). The structural analog for chess-spectral Phase 7 is the QM-expectation evaluator with learned α_O weights on the §15.2(4) Hermitian observables. The Othello gain doesn't predict the chess gain (different game, different baseline, different feature basis); use it as a methodology pointer (Nelder-Mead-fit on entropy- diff features works well in this class of problem) rather than a target number."

[F12] MEDIUM — §16.8.4, item 4, line 2677

"Three-regime channel decomposition. Phase 6's per-channel ablation should classify chess channels into the (monotone / trajectory-driven / turn-order-coupled) regimes."

Issue. "Should classify chess channels" prescribes a classification scheme inherited from Othello. The three regimes were identified in Othello data; whether chess channels separate into the same three regimes is empirically open.

Suggested rewording. "Three-regime channel decomposition. Phase 6's per-channel ablation can test whether chess channels exhibit the same (monotone / trajectory-driven / turn-order-coupled) regime structure the Othello data showed. Treating the Othello categorization as a hypothesis to test, not a prescriptive decomposition."

[F13] MEDIUM — §16.8.5, "Combined working hypothesis", item 1, line 2686

"Phase 6 ships naive linear spectral weights (analogous to zen-pike's Architecture A) and likely confirms the depth-decay pattern."

Issue. Same prediction-from-Othello shape as [F07] and [F09]. "Likely confirms" again presupposes the chess outcome will replay the Othello pattern.

Suggested rewording. "Phase 6 ships naive linear spectral weights (analogous to zen-pike's Architecture A); whether they replay the Othello depth-decay or break the pattern (e.g., because chess's material baseline is far weaker than Edax) is the load-bearing empirical question."

[F14] STRONG — §16.8.5, "Combined working hypothesis", item 3, line 2688

"Validation target should be game outcomes, not Stockfish eval at fixed depth — Edax d=20 didn't bridge the perfect-play gap on Othello, and there's no reason Stockfish at any heuristic depth would do better on chess."

Issue. "There's no reason Stockfish at any heuristic depth would do better on chess" extrapolates the Othello/Edax finding to chess without engaging the differences (Stockfish is a much-deeper search engine than typical Edax leaves; chess-side game-theoretic structure differs). The Othello finding (Edax d=20 ≠ perfect-play correlation target) is a real result; the chess prediction it implies is much weaker than the framing suggests.

Suggested rewording. "Validation target should preferentially be game outcomes, not engine evals at fixed depth — Edax d=20 didn't bridge the perfect-play gap on Othello, which suggests engine-eval- as-truth has limitations even with deep heuristic search. Whether Stockfish at any depth bridges the gap on chess is empirically open; using game outcomes as primary truth and engine evals as a regularizer is conservative."

[F15] STRONG — §17.6, "The line is bright", line 2818

"chess-spectral exposes capabilities; chess4D-OC chooses how to use them."

Issue. Stylistically clean but rhetorically overstrong. There isn't actually a bright line — getDrawStatus() taking has_legal_moves: bool from the consumer (per §17.5 narrative) is exactly the kind of capability/policy intermixing the slogan claims isn't happening. The line is reasonably bright most places; saying "bright" unconditionally papers over the gray edges.

Suggested rewording. "The intended split: chess-spectral exposes capabilities; chess4D-OC chooses how to use them. Edge cases like getDrawStatus's has_legal_moves parameter (a capability that takes a policy bit from the consumer) sit on the boundary — those are documented per-method rather than hidden under the slogan."

[F16] STRONG — §18.2, line 2840

"The Phase 1 spectral-identity result + the Phase 3.5 phase- distinguishability result together establish that chess-spectral's QM extension is more than notation — the channels are an irreducibly quantum decomposition (irrep-typed eigenbasis of (Δ, B_4 commutant) per Pre-flight 3) AND moves act as channel-distinguishable unitaries (Probe 1)."

Issue. "Establish that chess-spectral's QM extension is more than notation" is doing a lot of work — Probe 1 showed 73.6% strong- distinguishability on a specific 9,500-pair sample with the channel- phase formulas in ADR-001. That's evidence the Aaronson-trap escape valve is open, which is exactly how §18.1 phrases it. The §18.2 reframing pushes from "escape valve open" to "irreducibly quantum decomposition," which is a much stronger claim (and "irreducibly quantum" is a term of art with a specific meaning in QM literature that doesn't quite line up).

Suggested rewording. "The Phase 1 spectral-identity result + the Phase 3.5 phase-distinguishability result together show that chess- spectral's QM extension is not a tautological re-notation — the channels carry an irrep-typed decomposition (per Pre-flight 3) AND moves act as channel-distinguishable unitaries with the ADR-001 phase convention (Probe 1's 73.6% strong-distinguishability rate). Whether this rises to 'irreducibly quantum content' in the technical sense is a stronger claim we don't make; what we have shown is the Aaronson- trap escape valve is open."

[F17] STRONG — §18.2, second paragraph, line 2842

"… it does close the door on the strongest version of the Aaronson critique: at minimum, our channel-as-PVM measurements distinguish moves, which is more than a basis-aligned PVM on basis-aligned states could ever do."

Issue. "Closes the door on the strongest version of the Aaronson critique" overstates what Probe 1 showed. Probe 1 showed channel- distinct phases produce non-trivial inner-product overlaps between post-move states; that's the Aaronson escape valve being open on this specific construction, not "the strongest version of the critique closed." Aaronson's critique survives all sorts of post-hoc constructions; what we have is empirical evidence that this particular construction doesn't reduce to it.

Suggested rewording. "… it shows the strongest version of the Aaronson critique — that this is just classical permutations in QM notation — does not apply to the ADR-001 channel-phase construction on the Probe 1 sample. Whether weaker variants of the critique survive (e.g., 'the channel-distinct phases don't carry chess- relevant information') is the §16 / Phase 6 empirical question."

docs/chess-maths/chess_spectral_4d_notebook.md

[F18] STRONG — qm_4d Pre-flight Findings → "Pre-flight 3" interpretation, line 775

"The 'engineering choice' of basis is a representation-theoretic theorem. Any B_4-equivariant function on Z_8^4 configurations decomposes naturally in the encoder's basis."

Issue. Same as [F02] / [F04]. The encoder's basis being the simultaneous eigenbasis of (Δ, B_4 commutant) was verified at machine precision; the underlying decomposition theorem (for any finite-group action on L² of a graph respecting the operator) is classical, not a result of this work. Calling our verification "a representation- theoretic theorem" elevates it.

Suggested rewording. "The encoder's choice of basis is verified to realize the simultaneous eigenbasis of (Δ, B_4 commutant) at machine precision, putting it in correspondence with the classical representation-theoretic decomposition. Any B_4-equivariant function on Z_8^4 configurations therefore decomposes naturally in this basis."

[F19] MEDIUM — Phase 6 plan section, line 781 (lightly already softened)

The user already softened the "first 4D chess engine ever shipped" language. The current text reads:

"… we don't have an exhaustive survey of prior 4D chess search implementations, so we make no 'first ever' claim."

Status. Already softened — no further action. Recorded for completeness.

docs/chess-maths/chess-spectral/docs/adr/qm_4d/ADR-001-phase-convention-for-unitary-moves.md

[F20] STRONG — §3.2 "Aaronson escape", line 115

"A superposition input ψ spanning multiple channels accumulates different phase factors per channel, so |<phi|U_move ψ>|^2 is no longer just the classical permutation overlap — interference arises."

Issue. "Interference arises" is presented as a structural consequence; Probe 1 retroactively showed this empirically (73.6% strong distinguishability, 100% below-threshold), but at the time ADR-001 was written this was a structural argument. Post-Probe-1, the ADR claim survives empirically — but the wording still presents the prediction as a guarantee. Probe 1's amendment record (PHASE_3_5_PROBE_RESULTS) calls the result "PASS — ship as-is," which is correct for the specific phase formulas in §3.1, but doesn't generalize to "any ADR-001-shaped construction produces interference."

Suggested rewording. "A superposition input ψ spanning multiple channels accumulates different phase factors per channel, so for states with non-trivial channel support |<phi|U_move ψ>|^2 will generically differ from the classical permutation overlap (i.e., interference is structurally available). Whether the interference is quantitatively meaningful for chess-relevant ψ states was tested in Phase 3.5 Probe-1 (PASS at 73.6% strong distinguishability)."

[F21] MEDIUM — §3.3 "Why these specific phase values" (pi/4, 2pi/3, etc.), lines 137–157

"The factor pi/4 is chosen so a single-step move accumulates pi/4 (smallest nonzero phase that survives to displacement = 8 without wrapping). … FIB_SYM channels (cube roots of unity scaled by d_path): The three fiber-SVD directions span a 3D rank space; phases 2pi/3, 4pi/3, 2pi encode the three irrep components as 3rd roots of unity."

Issue. The §3.3 derivations are post-hoc rationalizations — the phases are design choices that produce empirically distinguishable moves, but the framing reads as if there's a unique principled derivation. Section 5.4 (Option D variants) and the user's own "Krawtchouk values rejected because opaque" reasoning shows the choice is closer to "readable rationals that work" than "uniquely principled."

Suggested rewording. Keep the table; reframe the prose as "These specific values are design choices that satisfy the constraints in §3.2 while staying readable; they are not uniquely determined by representation theory. Krawtchouk-based variants (rejected in §5.4) or other rational choices would also satisfy the constraints. The load-bearing property is that channels accumulate distinct phases, not that the specific values are canonical."

[F22] STRONG — §4.2 "What this forecloses", line 188

"No path-history dependence on the phase. The phase depends only on origin, destination, piece type, and channel — not on how the piece's ψ-amplitude got to origin. This is intentional (keeps U_move Markov on the QM state) but rules out 'true' path-integral phase that integrates over all paths the piece *might have taken."*

Issue. "True path-integral phase" presents Berry-phase-from- path-integration as the canonical version, when in fact path integrals in QM live on continuous configuration spaces; on a discrete graph there is no "true" path integral, only various discrete approximations (Berry-phase analogs, sum-over-paths on the graph, etc.). The framing imports continuum-QM language unhelpfully.

Suggested rewording. "No path-history dependence on the phase. The phase depends only on origin, destination, piece type, and channel — not on how the piece's ψ-amplitude reached origin. On a discrete graph this is the simplest phase convention; richer alternatives (graph-Berry-phase along the path, sum-over-paths on the graph) are documented in Option B (§5.2) and deferred to v1.7+."

docs/chess-maths/chess-spectral/docs/adr/qm_4d/ADR-002-time-evolution-semantics.md

[F23] MEDIUM — §4.1 "Captures collapse to a single rescale", line 217

"A reviewer asking 'how does QM chess handle captures?' gets a one-line answer: 'we project and renormalize — the textbook Born- rule operation.'"

Issue. Captures aren't textbook Born-rule projections in any standard sense — Born-rule projection is "measure observable, collapse to eigenspace of measured eigenvalue, renormalize." Capture-as-rescale is a partial-isometry composed with renormalization, which is related but not the textbook Born picture. Calling it "the textbook Born-rule operation" misnames it for rhetorical convenience.

Suggested rewording. "Captures collapse to a single rescale of ψ. The non-unitary structure is contained in N — a partial-isometry followed by norm restoration. This is structurally similar to a Born-rule measurement projection (project + renormalize) but is specifically the projection-then-renormalize step of a measurement without the eigenvalue-sampling step."

docs/chess-maths/chess-spectral/docs/adr/qm_4d/ADR-003-per-channel-move-transformation.md

[F24] STRONG — §3.1 channel 0 "Verification", line 132

"Pi_0 @ encode_4d_A1(pos_pre) = encode_4d_A1(pos_post) exactly, provided sig values for non-o, non-d squares are unchanged."

Status. This claim was empirically falsified by PR #76 and explicitly amended by ADR-003-AMENDMENT-orbit-restriction.md. The amendment file exists and PHASE_3_5_PROBE_RESULTS.md flags the amendment. However, the original §3.1 prose still reads as written — a downstream reader landing on §3.1 first will read a verification claim that is no longer supported. The amendment file fixes this in narrative but doesn't update the §3.1 text itself.

Suggested action. Add a 2-line "Amended" callout box at the top of §3.1 pointing at the amendment file ("This claim is restricted to same-orbit non-capture moves per ADR-003 amendment; cross-orbit moves fall back to measurement-only re-encode"). The amendment file already contains the precise wording.

[F25] MEDIUM — §4.2 "What this forecloses", line 363

"Full-strict-QM Track B in v1.5. Channels 5-7 and 10 are not strictly evolved unitarily."

Issue. The post-Phase-3.5 picture is more restrictive than the ADR's original framing: A_1, STD4_X/Y/Z/W are now only same-orbit strict-unitary (per the AMENDMENT); FIB_SYM are measurement-only; FD_DIAG is rank-1 + renorm. So "Channels 5-7 and 10 are not strictly evolved" understates the actual scope-narrowing. The "Aggregate v1.5 unitarity" table in §3 is also pre-amendment.

Suggested rewording. Add a "Post-Phase-3.5 / post-PR-#76 status" row that names the actual v1.5 strict-unitary tier (A_1 same-orbit; STD4 same-orbit; FA_PAWN axis-flip non-capture); FIB_SYM and FD_DIAG were already best-effort; cross-orbit / capture cases route through measurement-only. The ADR's original analysis remains valid as the design rationale; the table should explicitly note "amended — see PHASE_3_5_PROBE_RESULTS and ADR-003-AMENDMENT for current status."

docs/chess-maths/chess-spectral/docs/adr/qm_4d/ADR-004-z2-superselection-structure.md

[F26] STRONG — §3.4, line 162

"This means U_move @ J_op = −J_op @ U_move (U_move *anti- commutes with J_op, not commutes). Equivalently: U_move is a Z_2-graded operator of odd parity (it changes sector). … This is a strong constraint on U_move. Verification: for each of the strict-unitary channel constructions in ADR-003, the per-channel Pi_c must satisfy Pi_c @ J_c = −J_c @ Pi_c …"*

Status. This claim was empirically falsified by Phase 3.5 Probe-4 (anti-commutator residual = 128, target 1e-10 — fails by 30 orders of magnitude). PHASE_3_5_PROBE_RESULTS.md amends this section to "the parity sector change is mediated by state_to_psi's sign multiplier; per-channel anti-commutation is not required." The probe results file says §3.4 should be replaced.

However, the §3.4 text still reads as written. A reader landing on ADR-004 directly will read a constraint that is empirically known to fail. As with [F24], the amendment lives in a separate file but doesn't update the original ADR text.

Suggested action. Add the same "Amended — see PHASE_3_5_PROBE_RESULTS.md" callout at the top of §3.4 with the specific replacement wording from the probe-results file.

[F27] WEAK — §4.1 "Pre-flight 2 commutation cross-check is trivial", line 257

"Pre-flight 2 commutation cross-check is trivial. The 5 H_piece_4 observables commute with J_op by construction (graph- adjacency matrices of B_4-symmetric reach graphs commute with B_4- central elements). The cross-check is a closed-form test, not a deep theorem."

Issue. Minor — "by construction" is correct if the lift uses B_4-symmetric reach predicates, which it does, but the test is still worth running (and the section says ~80 LOC of test). "Trivial" is true given the construction; calling out why it's trivial in a sentence rather than asserting it would be cleaner.

Suggested rewording. "Pre-flight 2's commutation cross-check is trivial given the construction (graph-adjacency matrices of B_4-symmetric reach graphs automatically commute with B_4-central elements), but the test is still asserted at construction time as an invariant."

docs/chess-maths/chess-spectral/docs/adr/qm_4d/ADR-005-pawn-pseudo-hermitian-eta-metric.md

[F28] STRONG — §3.3.1 "M_pawn_w_white^T = M_pawn_w_black", lines 199–200

"M_pawn_w_white^T = M_pawn_w_black (push directions reverse)"

Status. Phase 3.5 Probe-3 found this holds only in the no-double-push variant (residual 0.0); the full operator including double-push has residual 32.0. PHASE_3_5_PROBE_RESULTS.md amends to the M_single + M_double decomposition. Same shape as [F24] and [F26]: the original ADR text still asserts the un-amended identity.

Suggested action. Add an "Amended" callout at the top of §3.3.1 with the M_single / M_double decomposition wording from the probe- results file.

[F29] MEDIUM — §5.5 "which η operator to use" / §3.3 "the unique η", lines 216, 451

"The η operator that makes M_pawn_w_white pseudo-Hermitian is: η_pawn_w := P_w … Within Option δ, the choice of η is constrained by the directed-push structure. The §3.3 derivation η = P_w (axis parity flip) is the unique η-operator candidate that …"

Issue. "The unique η-operator candidate" overclaims — uniqueness is up to scale, and up to the no-double-push idealization. Probe-3 showed J_op also satisfies the no-double-push pseudo-Hermiticity at zero residual (per the findings table). So η is not unique even among the candidates the probe checked.

Suggested rewording. "Within Option δ, P_w is the natural η- operator candidate constrained by the directed-push structure (axis parity ↔ flip the pawn's push direction). Probe-3 showed both P_w and J_op satisfy the no-double-push pseudo-Hermiticity at zero residual; we pick P_w for its physical interpretability. Other non-trivial candidates (e.g., scaled variants) exist by the up-to- positive-scalar non-uniqueness in the pseudo-Hermitian framework."

docs/chess-maths/chess-spectral/python/research/spectral_identity_4d_findings.md

[F30] STRONG — "The encoder's tensor-DCT basis is canonical", line 111

"The encoder's tensor-DCT basis e_i (x) e_j (x) e_k (x) e_l is canonical: it diagonalizes Delta (Kronecker-sum eigenvalues) AND every P_lambda commutes with every pi(g) (B_4-action). The choice of basis within each multidimensional eigenspace is therefore not arbitrary engineering — it is determined by the simultaneous spectral decomposition of the operator algebra < Delta, pi(B_4) >."

Issue. Same shape as [F02] / [F04] / [F18]. The basis is canonical up to a CSCO completion — within each multidim eigenspace the simultaneous-eigenbasis-of-Δ-and-B_4-commutant constraint leaves freedom (there's a finite group acting; pick the irrep-decomposition basis vs other bases). The paragraph correctly says "sub-decomposition within a B_4-stable subspace, not a re-choice of basis," but the "canonical" framing rounds over the within-eigenspace freedom.

Suggested rewording. "The encoder's tensor-DCT basis e_i ⊗ e_j ⊗ e_k ⊗ e_l simultaneously diagonalizes Δ and commutes with all pi(g) — the necessary structure for any B_4-equivariant analysis. Within each multidimensional eigenspace, further refinement into B_4-irrep components is a choice; the simultaneous-eigenbasis constraint fixes the eigenspace decomposition but not the basis within. The 'canonical' label refers to the eigenspace-level property, not to a unique within-eigenspace choice."

docs/chess-maths/chess-spectral/python/research/track_b_pawn_pt_symmetry_findings.md

[F31] MEDIUM — Recommendation §, line 69

"The spectrum is fully real because M is nilpotent (all eigenvalues = 0); pawn pushes are non-iterable on a finite board. ADR-005's PT-realness gate is trivially satisfied."

Issue. "Trivially satisfied" — true in the literal sense (zero spectrum is real), but the framing reads as if the gate was passed meaningfully. A nilpotent operator with all-zero eigenvalues satisfies any spectrum-realness gate trivially; the gate doesn't distinguish meaningful PT-symmetry from triviality. The PT-symmetry question for pawns is essentially undetermined by the eigenvalue test.

Suggested rewording. "The spectrum is fully real because M is nilpotent (all eigenvalues = 0; pawn pushes don't iterate on a finite board). ADR-005's PT-realness gate is satisfied trivially — but because the eigenvalues are all zero, the gate doesn't actually distinguish a meaningfully PT-symmetric operator from a generically nilpotent one. The pseudo-Hermiticity check (which probes the operator's structure beyond its spectrum) is the more discriminating test, and it's what the M_single / M_double decomposition resolves."

docs/chess-maths/chess-spectral/docs/adr/qm_4d/ADR-003-AMENDMENT-orbit-restriction.md

[F32] MEDIUM — §3.3, "Predicted to fail cross-orbit by the same mechanism", line 96

"STD4_X/Y/Z/W (B3a): Pi_a = D_a @ swap @ D_a^{-1} (rescaled swap; ADR-003 §3.1 channels 1-4). The rescaling is diagonal and commutes with the orbit projector iff the swap does — i.e., the same orbit dichotomy applies. Predicted to fail cross-orbit by the same mechanism; B3a's tests will quantify."

Issue. "Predicted to fail cross-orbit by the same mechanism" is a structural prediction that was subsequently verified in B3a (per the dynamics module status post-B5). The wording is honest at the time of writing — "to verify in B3a" — but a downstream reader after B5 should be told the prediction was confirmed. As with [F24]/[F26]/[F28] the issue is that the doc isn't updated post-empirical-result, so the "prediction" wording survives past when it became "confirmed."

Suggested action. Add a post-B3a status update: "Post-B3a: this prediction was confirmed empirically; STD4_X/Y/Z/W exhibit the same same-orbit / cross-orbit dichotomy as A_1."

Internal-consistency check (cross-doc contradictions)

Walking the docs in parallel surfaced these:

[F33] STRONG — Channel count consistency

The 4D notebook line 7 (architecture summary), 2D notebook §15.2(3), ADR-003 §3, ADR-001 §3.1 (table), and qm_4d_bridge.py _SHIPPED_CHANNELS all consistently report 11 channels. ✅ No drift.

The 2D notebook §16.1 item 2 says "11 channels (4D) or 10 channels (2D)" — that's the 2D vs 4D difference (2D ships the 10-channel encoder, 4D ships 11). Cross-checked against the python README's 2D vs 4D split: ✅ consistent. No drift.

(Recorded as [F33] because the user explicitly asked about this class of contradiction; nothing to fix, but the audit checked.)

[F34] STRONG — ADR text vs amendment file divergence

Three ADRs (ADR-003 §3.1, ADR-004 §3.4, ADR-005 §3.3.1) carry text that was empirically falsified by Phase 3.5 probes (see [F24], [F26], [F28]). The amendments live in separate documents (PHASE_3_5_PROBE_RESULTS.md, ADR-003-AMENDMENT-orbit-restriction.md) that explicitly note the original text "is preserved as the design record at the time of the decision." This is a deliberate convention per the README ("ADRs themselves are preserved … this doc captures the empirical reality").

However, that convention only works if every ADR has a forward- pointing callout to its amendment. Currently: - ADR-003-AMENDMENT-orbit-restriction.md is forward-linked from README.md and ADR-003-per-channel-move-transformation.md is not updated to point to its amendment. - ADR-004 §3.4 has no callout pointing at the Probe 4 amendment. - ADR-005 §3.3.1 has no callout pointing at the Probe 3 amendment.

Suggested action (cross-cutting). Add a 2-line "Amended — see PHASE_3_5_PROBE_RESULTS.md" callout at the top of each amended section. This preserves the "ADR is the historical record" convention while giving downstream readers a forward link to current status. The README.md already has the table; each ADR just needs the section-level pointer.

[F35] MEDIUM — The "qm_4d.py" module's H_piece spectrum claims

Pre-flight 2 quoted spectrum bounds for rook ([-4, 28] integer) but also notes "non-rook pieces have larger non-integer spectra." 4D notebook lines 749–754 give the empirical full sweep. ADR-004 §3.3 asserts "real-symmetric matrices have real spectrum (Pre-flight 2 already confirmed)" which is true, but elsewhere ([F27]'s context) the claim "trivial" is overstated.

No contradiction — but the per-piece bounds are documented in three places (4D notebook qm_4d Pre-flight section; phase_operators_4d audit; 2D notebook §15.2(4)) and any future per-piece spectrum adjustment must update all three. Not an error today; flagged for future hygiene.

Summary

Total findings: 35 (33 substantive + 2 internal-consistency checks).

Severity Count
STRONG 18
MEDIUM 12
WEAK 1
Already softened (recorded for completeness) 1 (F19)
Internal consistency / no-drift confirmation 1 (F33)
Cross-cutting hygiene (ADR ↔ amendment) 1 (F34)
Future-hygiene only 1 (F35)

Top 5 highest-priority items (most egregious overclaims):

  1. [F02] / [F04] / [F18] / [F30] — "encoder basis is the theorem" overclaim. Four restatements across the 2D notebook (§15.2, §15.6), 4D notebook (Pre-flight 3 interpretation), and spectral_identity_4d_findings.md. Numerical verification is represented as a self-supplied theorem in each restatement. Single fix recommended: pick one canonical phrasing ("numerically verified to realize the classical CSCO basis") and use it consistently in all four locations.

  2. [F24] / [F26] / [F28] — three ADRs carry empirically- falsified text without in-section amendment callouts. ADR-003 §3.1's verification claim, ADR-004 §3.4's anti-commutation requirement, and ADR-005 §3.3.1's transpose identity are all amended in external files but the original ADR sections remain untouched. Risk: a reader landing on ADR-004 §3.4 reads a constraint that is known to fail. The cross-cutting fix is [F34].

  3. [F03] — §15.6's "0% / 70% / 95% novel" rhetorical percentages. User explicitly flagged this in the audit prompt; the percentages are not sourced.

  4. [F11] / [F14] / [F09] / [F13] — Othello → chess extrapolation overclaims. Even after the user's softening of §16.5 / §16.7 item 1, downstream items in §16.7, §16.8.4, and §16.8.5 still translate Othello numbers into chess predictions ("the prior to beat", "no reason Stockfish would do better", "likely confirms the depth-decay pattern").

  5. [F01] / [F05] / [F06] — categorical "no PyPI package / no shipping toolkit / community needs this" claims without exhaustive surveys. Three near-restatements; consistent fix is to soften "doesn't exist" → "we are not aware of."

Aspirational-vs-accomplishment items (less urgent but worth addressing): [F15] (the bright-line rhetoric), [F17] (closing the door on Aaronson), [F23] (textbook Born-rule misnaming).

Out-of-section text adjustments: [F25], [F32] — the two predict-then-verify spans where the doc never circled back to mark the prediction as verified after it landed empirically.

Internal inconsistencies spotted

  • F33 (no drift): Channel count is consistent across docs and code at 11 (4D) / 10 (2D).
  • F34 (real divergence): ADR text vs amendment file. Three ADR sections carry text that was empirically falsified, but the amendments live in separate files without in-section forward links. This is the only structural inconsistency in the audited surface, and the fix is mechanical (add 2-line callouts).
  • F35 (non-issue today): Per-piece spectrum bounds duplicated in three places; no drift now, but future updates need to keep all three in sync.

No "all 11 channels ship vs all 10 channels ship" type contradiction was found — the bridge code, ADR-001 / ADR-003 channel tables, and both notebooks agree on 11 channels for 4D throughout.