# 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.

---

## 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** |

---

## 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.

---

## 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.

---

## 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 |

---

## 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** |

---

## 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 | **Drafted** | Geometric cascade with growth factors α, β. Fixed vs adaptive S. Stability when Aₙ is F-invariant. |
| 2 | Braid group action | **Drafted** | σᵢ as reflection-axis swap gives S₈ rep. Orientation (over/under) not distinguished in current torus coordinates. Open: genuine B₈ representation. |