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