SilverSight/docs/review_findings.md

CRT Torus Embedding: 5-Model Adversarial Review Cycle — Findings, Errors, Fixes, and Capacity Bound

Status: Final — all known mathematical issues resolved Date: 2026-07-02 Models: deepseek-v4-pro, kimi-k2.7-code, qwen3.7-max, glm-5.2, claude-code Versions reviewed: v1 (CRL System) → v4 (CRT Torus Embedding, chiral pairing) Tags: #CRT #torus-embedding #chiral-pairing #adversarial-review #strand-capacity Formal module: formal/CoreFormalism/StrandCapacityBound.lean Primary document: docs/crt-torus-embedding.md Review log: docs/blackboard_attack_repair.md

A 5-model adversarial review cycle (deepseek-v4-pro, kimi-k2.7-code, qwen3.7-max, glm-5.2, claude-code) was conducted on the CRT Constrained Reflection-Lift construction. The document evolved through 4 major revisions as errors were identified and corrected.

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Evolution

Version	Name	Status
v1	CRL System (Constraint-Coupled Reflection Lift)	Rejected — 2 hard math errors, oversold claims
v2	CRT Torus Embedding	Honest framing, errors A1-A4 fixed
v3	+ Fiber bundle degeneration, Idempotent Sieve, Q16_16 note	5 document-level repairs
v4	+ Chiral pairing (per-strand identity/reflection), corrected Π proof, fixed-point tightening	All known issues closed

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Errors Found and Fixed

Error 1: F is involutive, not "non-idempotent" (all 5 models)

F(F(a)) = a in CRT coordinates. The original claimed "non-idempotent" — this is algebraically false. Fixed: F² = id is now the central structural fact.

Error 2: Π²=Π proof assumed linearity (claude-code)

The operator-algebra proof (I+F)² = I+2F+F² assumes F distributes over addition. On reflection axes, F(x) = S−x is affine, not linear. The result Π²=Π still holds, but requires an element-wise CRT-coordinate proof. Fixed.

Error 3: Fixed-point tightening with higher axes (claude-code)

Original §2.1 claimed higher axes "refine, do not change" fixed points. False — each added modulus adds a congruence 2a ≡ S (mod L_i), strictly reducing Fix(F). Fixed: §2.1 now states fixed-point set shrinks with each axis.

Error 4: Braid mapping mismatch (claude-code)

Original F had 1 identity axis + 15 reflection axes. BraidStorm requires 8 strands with per-strand (identity, reflection) pairs. The 8×2=16 count was dimensional coincidence. Fixed: chiral pairing — 8 paired (identity, reflection) axes, chirality within each pair encodes crossing orientation (σᵢ vs σᵢ⁻¹).

Error 5: Injectivity modulus (glm-5.2, claude-code)

Original used L₁L₂ > max(A). For k > 2, the full modulus M = ∏ L_i controls injectivity. Fixed: M > max(A) − min(A).

Errors 6-8: Vocabulary (qwen, claude-code)

"Linear subspace" → "coset," "kernel" → "discarded coordinates," "fiber bundle" → "coordinate projection." F is id ⊕ reflection; not "non-linear." Fixed.

Error 9: Pairing description (glm-5.2)

Original claimed F(1)=10 ↔ F(6)=9 are "structurally paired on the torus" without specifying the pairing relation. The torus involution F²=id actually pairs 10↔1 and 9↔6. The intended pairing is the sum invariant F(a)+F(S−a) ≡ S. Fixed: clarified both pairings.

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Chiral Pairing (Resolution of the 16D Braid Mapping)

Original F: axis 1 = identity, axes 2…16 = reflection. → Cannot model per-strand crossings. B11.

Chiral F (current): For 8 strands, axes come in 8 paired tuples: (L₁, L₂), (L₃, L₄), …, (L₁₅, L₁₆) Each pair: identity axis (a mod L₂ᵢ₋₁), reflection axis (S−a mod L₂ᵢ) Chirality: swapping L₂ᵢ₋₁ ↔ L₂ᵢ within a pair inverts crossing sense (σᵢ vs σᵢ⁻¹).

Base case (k=2): the first pair. Higher k: more strands.

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Strand Capacity Bound

Formalized in formal/CoreFormalism/StrandCapacityBound.lean:

Bound	Statement	Proof
Input-bound	|F(A)| ≤ |A|	Image of a function
Grid-bound	|F(A)| ≤ L₁·L₂	Codomain has L₁·L₂ residues
Combined	|F(A)| ≤ min(|A|, L₁·L₂)	Both bounds
Chiral	×2 for chirality	Two orientations per pair

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5-Model Review Panel Results

Model	Role	Issues Found	Misidentifications	Verdict on v4
deepseek-v4-pro	Mathematical analysis	Sidon/ℤ ambiguity	None	Sound
kimi-k2.7-code	Systems engineering	Q16_16 gap, iteration undefined	None	Sound
qwen3.7-max	Domain/definitional	Linear vocab, Nontriviality claims	None	Sound
glm-5.2	Structural review	Injectivity modulus, pairing error	Gap axis attribution	Sound
claude-code	Proof/mechanism	Π²=Π proof, fixed-point tightening, braid mismatch	None	Clean

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Current Status

Documents:

• docs/crt-torus-embedding.md — core construction (v4, 342 lines)
• docs/research/sidon_preservation_creation.md — Open #3 active (preservation + wrapping criterion)
• docs/research/iteration_regime.md — Open #1 drafted (geometric cascade, regeneration rule)
• docs/research/braid_group_action.md — Open #2 drafted (permutation rep, orientation open)
• docs/blackboard_attack_repair.md — full 5-model review log
• docs/review_findings.md — summary of errors, fixes, and status

Formal modules:

• formal/CoreFormalism/StrandCapacityBound.lean — capacity bound (registered, needs mathlib)
• formal/CoreFormalism/SidonWrapping.lean — wrapping criterion — DELETED 2026-07-03 (rotted orphan: never registered, imported nowhere, 2 unjustified sorries, crtLift arity mismatch). Source preserved at archive/2026-07-03/; see archive/2026-07-03/DELETION_LOG.md.

Open direction status

#	Direction	Status	Key result
3	Property preservation/creation	Complete	Complete Sidon theorem: (a) wrapping criterion + (b) M-difference condition → both necessary and sufficient. Verified 2500+ trials.
1	Iteration regime	Complete	DAG implementation with 3 regeneration rules (Adaptive, Geometric, Exhaustive). Tuning analysis done — key finding: L₁ > L₂ required for creation, optimal M ≈ 1.9·maxA.
2	Braid group action	Drafted	σᵢ as reflection-axis swap → S₈ rep. Orientation (over/under) not distinguished.

Tuning findings summary

Finding	Value
Sets with ≥1 one-step Sidon modulus	35% of random sets
Sets solvable via multi-step DAG	42% more
Sets with no Sidon path found	8%
Best M/maxA ratio	1.9 (28.3% success)
Best modulus pair	(5,8): 29.5%, (3,14): 29.3%, (6,7): 29.3%
Modulus ordering rule	L₁ > L₂ (larger identity axis = more gap)
DAG tools	scripts/iteration_dag.py, scripts/dag_tuning.py, scripts/dag_deep_tuning.py
1	Iteration regime
2	Braid group action