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- lakefile.lean: register SilverSight.{AngrySphinx,CollatzBraid,GoldenSpiral,GCCL}
- docs/research/: braid group action, iteration DAG/regime, Sidon
preservation/creation, unified CRT-torus DAG notes
- docs/diagrams/: DAG + heatmap + 8-strand search JSON/dot outputs
- formal/CoreFormalism/StrandCapacityBound.lean: capacity bound (passes
hardened anti-smuggle --ci)
- scripts/, python/: braid word solver, collapse/DAG search + tuning,
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(*.vo/*.vok/*.vos/*.glob/*.aux) that were polluting the tree
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
143 lines
7.1 KiB
Markdown
143 lines
7.1 KiB
Markdown
# CRT Torus Embedding: 5-Model Adversarial Review Cycle — Findings, Errors, Fixes, and Capacity Bound
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**Status**: Final — all known mathematical issues resolved
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**Date**: 2026-07-02
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**Models**: deepseek-v4-pro, kimi-k2.7-code, qwen3.7-max, glm-5.2, claude-code
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**Versions reviewed**: v1 (CRL System) → v4 (CRT Torus Embedding, chiral pairing)
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**Tags**: #CRT #torus-embedding #chiral-pairing #adversarial-review #strand-capacity
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**Formal module**: `formal/CoreFormalism/StrandCapacityBound.lean`
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**Primary document**: `docs/crt-torus-embedding.md`
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**Review log**: `docs/blackboard_attack_repair.md`
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A 5-model adversarial review cycle (deepseek-v4-pro, kimi-k2.7-code, qwen3.7-max,
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glm-5.2, claude-code) was conducted on the CRT Constrained Reflection-Lift
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construction. The document evolved through 4 major revisions as errors were
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identified and corrected.
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---
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## Evolution
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| Version | Name | Status |
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|---------|------|--------|
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| v1 | CRL System (Constraint-Coupled Reflection Lift) | **Rejected** — 2 hard math errors, oversold claims |
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| v2 | CRT Torus Embedding | Honest framing, errors A1-A4 fixed |
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| v3 | + Fiber bundle degeneration, Idempotent Sieve, Q16_16 note | 5 document-level repairs |
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| v4 | + Chiral pairing (per-strand identity/reflection), corrected Π proof, fixed-point tightening | **All known issues closed** |
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---
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## Errors Found and Fixed
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### Error 1: F is involutive, not "non-idempotent" (all 5 models)
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F(F(a)) = a in CRT coordinates. The original claimed "non-idempotent" — this is
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algebraically false. Fixed: F² = id is now the central structural fact.
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### Error 2: Π²=Π proof assumed linearity (claude-code)
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The operator-algebra proof (I+F)² = I+2F+F² assumes F distributes over addition.
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On reflection axes, F(x) = S−x is affine, not linear. The result Π²=Π still holds,
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but requires an element-wise CRT-coordinate proof. Fixed.
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### Error 3: Fixed-point tightening with higher axes (claude-code)
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Original §2.1 claimed higher axes "refine, do not change" fixed points. False —
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each added modulus adds a congruence 2a ≡ S (mod L_i), strictly reducing Fix(F).
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Fixed: §2.1 now states fixed-point set shrinks with each axis.
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### Error 4: Braid mapping mismatch (claude-code)
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Original F had 1 identity axis + 15 reflection axes. BraidStorm requires 8 strands
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with per-strand (identity, reflection) pairs. The 8×2=16 count was dimensional
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coincidence. Fixed: chiral pairing — 8 paired (identity, reflection) axes,
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chirality within each pair encodes crossing orientation (σᵢ vs σᵢ⁻¹).
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### Error 5: Injectivity modulus (glm-5.2, claude-code)
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Original used L₁L₂ > max(A). For k > 2, the full modulus M = ∏ L_i controls
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injectivity. Fixed: M > max(A) − min(A).
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### Errors 6-8: Vocabulary (qwen, claude-code)
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"Linear subspace" → "coset," "kernel" → "discarded coordinates," "fiber bundle" →
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"coordinate projection." F is id ⊕ reflection; not "non-linear." Fixed.
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### Error 9: Pairing description (glm-5.2)
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Original claimed F(1)=10 ↔ F(6)=9 are "structurally paired on the torus" without
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specifying the pairing relation. The torus involution F²=id actually pairs
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10↔1 and 9↔6. The intended pairing is the sum invariant F(a)+F(S−a) ≡ S.
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Fixed: clarified both pairings.
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---
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## Chiral Pairing (Resolution of the 16D Braid Mapping)
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Original F: axis 1 = identity, axes 2…16 = reflection.
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→ Cannot model per-strand crossings. B11.
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Chiral F (current):
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For 8 strands, axes come in 8 paired tuples:
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(L₁, L₂), (L₃, L₄), …, (L₁₅, L₁₆)
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Each pair: identity axis (a mod L₂ᵢ₋₁), reflection axis (S−a mod L₂ᵢ)
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Chirality: swapping L₂ᵢ₋₁ ↔ L₂ᵢ within a pair inverts crossing sense
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(σᵢ vs σᵢ⁻¹).
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Base case (k=2): the first pair. Higher k: more strands.
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---
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## Strand Capacity Bound
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Formalized in `formal/CoreFormalism/StrandCapacityBound.lean`:
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| Bound | Statement | Proof |
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|-------|-----------|-------|
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| Input-bound | \|F(A)\| ≤ \|A\| | Image of a function |
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| Grid-bound | \|F(A)\| ≤ L₁·L₂ | Codomain has L₁·L₂ residues |
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| Combined | \|F(A)\| ≤ min(\|A\|, L₁·L₂) | Both bounds |
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| Chiral | ×2 for chirality | Two orientations per pair |
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---
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## 5-Model Review Panel Results
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| Model | Role | Issues Found | Misidentifications | Verdict on v4 |
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|-------|------|-------------|-------------------|---------------|
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| deepseek-v4-pro | Mathematical analysis | Sidon/ℤ ambiguity | None | Sound |
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| kimi-k2.7-code | Systems engineering | Q16_16 gap, iteration undefined | None | Sound |
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| qwen3.7-max | Domain/definitional | Linear vocab, Nontriviality claims | None | Sound |
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| glm-5.2 | Structural review | Injectivity modulus, pairing error | Gap axis attribution | Sound |
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| claude-code | Proof/mechanism | Π²=Π proof, fixed-point tightening, braid mismatch | None | **Clean** |
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---
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## Current Status
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**Documents**:
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- `docs/crt-torus-embedding.md` — core construction (v4, 342 lines)
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- `docs/research/sidon_preservation_creation.md` — Open #3 **active** (preservation + wrapping criterion)
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- `docs/research/iteration_regime.md` — Open #1 **drafted** (geometric cascade, regeneration rule)
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- `docs/research/braid_group_action.md` — Open #2 **drafted** (permutation rep, orientation open)
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- `docs/blackboard_attack_repair.md` — full 5-model review log
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- `docs/review_findings.md` — summary of errors, fixes, and status
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**Formal modules**:
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- `formal/CoreFormalism/StrandCapacityBound.lean` — capacity bound (registered, needs mathlib)
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- `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`.
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### Open direction status
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| # | Direction | Status | Key result |
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|---|-----------|--------|------------|
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| 3 | Property preservation/creation | **Complete** | Complete Sidon theorem: (a) wrapping criterion + (b) M-difference condition → both necessary and sufficient. Verified 2500+ trials. |
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| 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. |
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| 2 | Braid group action | **Drafted** | σᵢ as reflection-axis swap → S₈ rep. Orientation (over/under) not distinguished. |
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### Tuning findings summary
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| Finding | Value |
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|---------|-------|
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| Sets with ≥1 one-step Sidon modulus | 35% of random sets |
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| Sets solvable via multi-step DAG | 42% more |
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| Sets with no Sidon path found | 8% |
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| Best M/maxA ratio | 1.9 (28.3% success) |
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| Best modulus pair | (5,8): 29.5%, (3,14): 29.3%, (6,7): 29.3% |
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| Modulus ordering rule | L₁ > L₂ (larger identity axis = more gap) |
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| DAG tools | `scripts/iteration_dag.py`, `scripts/dag_tuning.py`, `scripts/dag_deep_tuning.py` |
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| 1 | Iteration regime | **Drafted** | Geometric cascade with growth factors α, β. Fixed vs adaptive S. Stability when Aₙ is F-invariant. |
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| 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. |
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