SilverSight/docs/blackboard_attack_repair.md

Blackboard: CRL Multi-Model Attack + Repair

Session: 4-model blackboard (deepseek-v4-pro, kimi-k2.7-code, qwen3.7-max, glm-5.2) Document: docs/crt-torus-embedding.md

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BLACKBOARD — ALL FINDINGS

A. Common (3+ models agree)

ID	Finding	Models	Verdict
A1	F² = id (involution), not "non-idempotent"	DS, K, Q, G	FIXED in v2
A2	F = id ⊕ reflection on CRT decomposition	DS, K, Q, G	FIXED in v2
A3	16D connection = dimensional coincidence, not structural	DS, K, Q	DOWNGRADED in v2
A4	ASQ connection = metaphor, not isomorphism	K, Q	DOWNGRADED in v2
A5	Iteration regime undefined (no regeneration rule)	DS, K, Q, G	NOT FIXED
A6	No theorems/proofs — definitional only	DS, Q, G	INHERENT (construction tool, not theorem paper)

B. Model-specific

ID	Finding	Model	Evaluation
B1	Injectivity needs M = ∏L_i, not L₁L₂, for k>2	G	VALID — fix to M > max(A)
B2	Gap claim notation ambiguous: F(a) k-tuple vs integer lift	G	PARTIALLY VALID — clarify integer-lift convention
B3	Section 5 pairing: "F(1)=10 ↔ F(6)=9 paired" is wrong if pairing = F²	G	PARTIALLY VALID — intended pairing is sum invariant F(a)+F(S-a)=S, not F². Clarify.
B4	Q16_16 incompatible with 16-modulus product (3.26e19 >> 2^15)	K	VALID — add modulus bound or projection scheme
B5	Sidon in Z vs Z/12Z ambiguity	DS	VALID — clarify sums are over integer lifts
B6	"Non-invariant" and "non-idempotent" asserted without counterexamples	K	FIXED — "non-idempotent" removed, non-invariant is definitional
B7	F claimed "not linear" but IS linear on CRT decomposition	Q, DS	FIXED — removed in v2
B8	"Constraint field morphism" undefined	Q, DS	FIXED — removed in v2

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REPAIR PLAN

P1: Fix injectivity condition (B1)

Replace L₁L₂ > max(A) → M = ∏ L_i > range(A) where range(A) = max(A) − min(A).

P2: Clarify gap notation (B2)

Add explicit note: "F(a) below denotes the CRT integer lift in [0, M)."

P3: Fix pairing description (B3)

Replace "structurally paired on the torus" → "linked by the sum invariant: F(a) + F(S−a) ≡ S (mod M)."

P4: Add modulus bound for Q16_16 (B4)

State: "Full torus product M exceeds Q16_16 range for k ≥ 8. Operations decompose per-axis where each L_i fits. The identity axis uses Q16_16; reflection axes use modular arithmetic in smaller rings."

P5: Clarify Sidon sum domain (B5)

Explicitly state: "Sidon property verified on integer representatives in Z, not in the quotient Z/MZ."

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BLACKBOARD RESOLUTION

After cross-model review, the 3 substantive errors requiring document changes:

1. B1: Injectivity condition — use M, not L₁L₂
2. B2/B3: Torus pairing notation — clarify integer lift convention and sum invariant
3. B4: Q16_16 bound — acknowledge and explain per-axis decomposition

All other findings (A1-A4, B5-B8) are either already fixed in v2, inherent to the construction's scope, or minor clarifications.

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STATUS

• v1 (CRL System): Oversold, 2 hard math errors. Rejected by all 4 models.
• v2 (CRT Torus Embedding): Honest framing, errors A1-A4 fixed. 3 remaining document-level fixes identified.
• v3 (after P1-P4): ALL FIXES APPLIED. 4-model blackboard resolution complete.

Applied fixes (v3)

• P1: Injectivity uses M = ∏ L_i > max(A) - min(A) (line 41)
• P2: Gap notation clarifies CRT integer lift (line 47-48)
• P3: Pairing uses sum invariant F(a)+F(S-a)≡S, not F² (lines 163-168)
• P4: Q16_16 bound acknowledged, per-axis decomposition explained (lines 195-203)