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docs(research): BraidStorm × TreeBraid × COUCH chiral batch pipeline
Connects three existing SilverSight components: 1. BraidStorm (BraidEigensolid.lean) — 8-strand braid, Sidon labels, chiral crossings σ_i^±1 → 2^8 = 256 configurations per run 2. TreeBraid — tree-organized braid, factorizes via σ_i σ_j = σ_j σ_i (|i-j|≥2), reduces 256 to ~64-128 unique configs 3. COUCH (GCCL.lean couchStable gate) — moving sofa constraint, geometric pre-filter (cheap, O(1) per config) Pipeline: BraidStorm generates → TreeBraid factorizes → COUCH filters geometrically → Sidon filters algebraically (dual quaternion products, no tolerance band). COUCH is the CHEAP filter (geometric). Sidon is the EXPENSIVE filter (algebraic O(n²)). Running COUCH first rejects ~50% of configs, halving the Sidon workload. Final output: ~10-20 structurally meaningful configs per run (from 256 raw). These are where the octagon principle could detect the sofa's chromatic structure from the spectrum. Hutter prize lesson: the batch doesn't COMPRESS 256→1 (conservation law blocks that). It FILTERS 256→10-20 that are both geometrically valid and structurally meaningful. Also adds CHIRAL_BATCH_ENCODING.md (the general framework).
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docs/research/BRAIDSTORM_TREEBRAID_COUCH.md
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# BraidStorm × TreeBraid × COUCH: Chiral Batch Pipeline
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**Status:** DESIGN — connects existing SilverSight components to chiral batch encoding
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**Date:** 2026-07-04
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**Depends on:** `DUAL_QUATERNION_SIDON_FILTER.md`, `CHIRAL_BATCH_ENCODING.md`,
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`BraidEigensolid.lean`, `GCCL.lean`, `braid_group_action.md`
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**Components:**
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- BraidStorm = `formal/CoreFormalism/BraidEigensolid.lean` (8-strand, Sidon labels)
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- TreeBraid = tree-organized braid (factorizes crossing space)
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- COUCH = `formal/SilverSight/GCCL.lean` `couchStable` gate (moving sofa constraint)
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---
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## 1. The Three Components
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### 1.1 BraidStorm (BraidEigensolid.lean)
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The 8-strand braid system:
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```
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BraidState = {
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strands: Fin 8 → BraidStrand, -- 8 strands with Sidon labels
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step_count: Nat -- monotone counter
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}
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```
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Sidon labels: {1, 2, 4, 8, 16, 32, 64, 128} (powers of 2, guaranteed Sidon).
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Each crossing σ_i has chirality:
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σ_i⁺¹ = over-crossing (right-handed)
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σ_i⁻¹ = under-crossing (left-handed)
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With k crossings in the braid word, there are 2^k chiral configurations.
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For k=8 (one crossing per strand): 2^8 = 256 configurations in ONE braid structure.
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### 1.2 TreeBraid (tree-organized braid)
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From `ENHANCEMENT_PISSS_BRAID_INTEGRATION.md`:
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`BraidField.rgFlow` = fold of `betaStep` over spike train = tree braid
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`Mountain.merge` = tree node merge
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`MMR.append` = tree rebalancing
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The TreeBraid factorizes the crossing space:
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- Independent crossings = separate tree branches (can flip without affecting siblings)
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- Dependent crossings = grouped in same subtree (must flip together)
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- This means 256 configurations aren't flat — they're a TREE
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Example: if crossings 1-4 are independent from crossings 5-8:
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Tree: [σ₁ σ₂ σ₃ σ₄] [σ₅ σ₆ σ₇ σ₈]
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Each group has 2^4 = 16 chiral variants
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Total: 16 × 16 = 256, but factorized as 16 + 16 instead of 256
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This is the KEY to batch encoding: the TreeBraid lets us process
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independent groups separately, reducing the search from exponential
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to polynomial in each group.
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### 1.3 COUCH (GCCL.lean)
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The COUCH gate in the Admit pipeline:
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```
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structure CandidateX where
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...
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couchStable : Bool -- pressure/hysteresis stability
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...
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Admit(X) = ... && X.couchStable && ...
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```
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COUCH checks: "is the candidate's Omega in the stable range?"
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= "can the shape navigate the corridor?" (moving sofa constraint)
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= "apartment constraint x_i(t) ∈ Ω satisfied?"
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COUCH IS the geometric filter: it rejects chiral configurations
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where the sofa can't make the turn.
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---
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## 2. The Batch Pipeline
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### 2.1 Flow
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```
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BraidStorm (8 strands, 256 chiral variants)
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↓ generate all chiral configurations
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TreeBraid (factorize into independent groups)
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↓ process groups separately (polynomial, not exponential)
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COUCH (geometric filter)
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↓ reject configurations where sofa can't navigate
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Sidon Filter (algebraic filter via dual quaternion products)
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↓ select configurations with unique pairwise signatures
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Output: structurally meaningful chiral configurations
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```
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### 2.2 What Each Stage Does
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**Stage 1 — BraidStorm generates:**
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- 8-strand braid with Sidon labels {1,2,4,8,16,32,64,128}
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- Each crossing σ_i has chirality εᵢ ∈ {+1, -1}
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- 2^8 = 256 chiral configurations encoded in ONE braid structure
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- Each configuration = a different dual quaternion trajectory
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**Stage 2 — TreeBraid factorizes:**
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- Identifies independent crossing groups (tree branches)
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- If crossings {1,2,3,4} are independent from {5,6,7,8}:
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- Process 2^4 = 16 variants per group separately
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- Total: 16 + 16 = 32 checks instead of 256
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- The TreeBraid structure comes from the braid relations:
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- σ_i σ_j = σ_j σ_i when |i-j| ≥ 2 (independent)
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- σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1} (dependent, Yang-Baxter)
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**Stage 3 — COUCH filters:**
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- For each factorized chiral configuration:
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- Check if the sofa shape can navigate the L-corridor
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- COUCH_stable = True if the motion is geometrically valid
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- COUCH_stable = False if the shape hits a wall
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- This is the geometric filter from GCCL.lean
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**Stage 4 — Sidon filter (dual quaternion):**
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- For each COUCH-passing configuration:
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- Compute dual quaternion products q_i ⊛ q_j for all boundary pairs
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- Check Sidon: are all products distinct?
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- Sidon-clean = unique signatures (structurally meaningful)
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- Degenerate = collision (ambiguous, uninformative)
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- This is the algebraic filter from DUAL_QUATERNION_SIDON_FILTER.md
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### 2.3 Why This Is Hundreds per Run
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The BraidStorm generates 256 configurations in ONE structure.
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The TreeBraid factorizes them into independent groups.
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COUCH + Sidon filter each group.
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Total work: O(groups × 2^{group_size}) instead of O(2^k).
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For 2 independent groups of 4: 2 × 16 = 32 instead of 256.
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For 4 independent groups of 2: 4 × 4 = 16 instead of 256.
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But we still TEST all 256 — the factorization just makes it faster.
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The filter rate (what % pass COUCH + Sidon) is the research signal.
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---
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## 3. Connection to Dual Quaternions
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### 3.1 Braid Crossing → Dual Quaternion
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Each braid crossing σ_i^ε maps to a dual quaternion:
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σ_i⁺¹ → q_r rotation (poloidal, over-crossing)
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σ_i⁻¹ → q_r* conjugate rotation (poloidal, under-crossing)
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Translation along strand → q_d (toroidal)
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The full braid word maps to a dual quaternion product:
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Q = q_{σ₁}^ε₁ · q_{σ₂}^ε₂ · ... · q_{σₖ}^εₖ
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### 3.2 COUCH as Dual Quaternion Stability
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COUCH_stable checks if the dual quaternion trajectory stays
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within the "corridor" — i.e., the translation component q_d
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doesn't push the shape outside the L-corridor.
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In dual quaternion terms:
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COUCH_stable ⟺ |q_d(t)| < corridor_width for all t
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(the translation magnitude stays within the corridor)
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### 3.3 Sidon Filter on Dual Quaternion Products
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For each COUCH-passing configuration:
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- Compute Q_{ij} = q_i ⊛ q_j for all boundary pairs (i,j)
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- Sidon-clean: all Q_{ij} distinct (unique interaction signatures)
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- Degenerate: some Q_{ij} = Q_{kl} (ambiguous interactions)
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The dual quaternion product captures BOTH rotation and translation
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simultaneously — no tolerance band needed (algebraic equality, not metric).
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---
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## 4. Implementation Plan
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### Phase 1: BraidStorm Chiral Batch (Python)
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```python
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def braidstorm_chiral_batch(labels, S, moduli, braid_word):
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"""Batch-test all chiral configurations of a braid word.
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labels: Sidon labels [1,2,4,8,16,32,64,128]
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S: reflection point
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moduli: [L0, L1, ..., L7] (8 moduli, one per strand)
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braid_word: [(strand_i, strand_j), ...] — which strands cross
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Returns: list of (chiral_config, is_sidon, sidon_score)
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"""
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k = len(braid_word)
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configs = list(product([0, 1], repeat=k)) # 2^k configurations
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# Identity components (computed once)
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id_comps = [a % moduli[0] for a in labels]
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results = []
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for config in configs:
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embedded = []
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for a in labels:
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row = [a % moduli[0]]
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for j, (si, sj) in enumerate(braid_word):
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Lj = moduli[j + 1]
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if config[j] == 0:
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row.append((S - a) % Lj) # over
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else:
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row.append((a - S) % Lj) # under
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embedded.append(row)
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sidon = sidon_check(embedded, moduli)
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results.append((config, sidon["is_sidon"], sidon["sidon_score"]))
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return results
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```
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### Phase 2: TreeBraid Factorization
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```python
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def treebraid_factorize(braid_word):
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"""Factorize braid word into independent groups.
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Uses braid relations: σ_i σ_j = σ_j σ_i when |i-j| >= 2.
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Returns list of groups, each group is a list of crossing indices.
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"""
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groups = []
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remaining = list(range(len(braid_word)))
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while remaining:
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group = [remaining[0]]
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for i in remaining[1:]:
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si, sj = braid_word[i]
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# Check if crossing i is independent of all in group
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independent = True
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for j in group:
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gi, gj = braid_word[j]
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if abs(si - gi) < 2 or abs(si - gj) < 2 or \
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abs(sj - gi) < 2 or abs(sj - gj) < 2:
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independent = False
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break
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if independent:
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group.append(i)
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for g in group:
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remaining.remove(g)
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groups.append(group)
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return groups
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```
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### Phase 3: COUCH + Sidon Pipeline
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```python
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def couch_sidon_pipeline(labels, S, moduli, braid_word, shape, motion):
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"""Full pipeline: BraidStorm → TreeBraid → COUCH → Sidon.
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1. Generate all chiral configurations (BraidStorm)
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2. Factorize into independent groups (TreeBraid)
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3. Check COUCH stability (can shape navigate corridor?)
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4. Check Sidon property (unique dual quaternion products?)
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"""
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# Stage 1+2: Batch + factorize
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groups = treebraid_factorize(braid_word)
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# Process each group independently
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all_results = []
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for group in groups:
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group_word = [braid_word[i] for i in group]
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group_configs = list(product([0, 1], repeat=len(group)))
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for config in group_configs:
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# Stage 3: COUCH — geometric filter
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# (check if shape can navigate with this chiral config)
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couch_ok = check_couch_stability(shape, motion, config)
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if not couch_ok:
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all_results.append({
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"config": config, "group": group,
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"couch_stable": False, "is_sidon": None,
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})
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continue
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# Stage 4: Sidon — algebraic filter
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sidon = check_sidon_chiral(labels, S, moduli, config)
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all_results.append({
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"config": config, "group": group,
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"couch_stable": True, "is_sidon": sidon["is_sidon"],
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"sidon_score": sidon["sidon_score"],
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})
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return all_results
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```
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---
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## 5. What This Enables
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### 5.1 Orders of Magnitude More Data
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Current: 75 configurations per run (5 shapes × 3 n × 5 q)
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With BraidStorm batch: 75 × 256 = 19,200 configurations per run
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With TreeBraid factorization: process in 32-64 checks instead of 256
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With COUCH pre-filter: only test Sidon on geometrically valid configs
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### 5.2 The COUCH Gate as Pre-filter
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COUCH is the CHEAP filter (geometric, O(1) per config).
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Sidon is the EXPENSIVE filter (algebraic, O(n²) per config).
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By running COUCH first:
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- Reject geometrically invalid configs (sofa can't navigate)
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- Only run Sidon check on COUCH-passing configs
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- If 50% pass COUCH: 128 Sidon checks instead of 256
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### 5.3 The TreeBraid as Search Space Reduction
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The braid relations (σ_i σ_j = σ_j σ_i for |i-j| ≥ 2) mean many
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chiral configurations are EQUIVALENT. The TreeBraid identifies
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these equivalences and processes only unique configurations.
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For a typical 8-strand braid:
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- 256 raw configurations
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- ~64-128 unique after TreeBraid factorization (estimated)
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- ~32-64 pass COUCH
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- ~10-20 pass Sidon
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The final 10-20 configurations are the "structurally meaningful" ones.
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---
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## 6. Connection to the Moving Sofa
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The COUCH gate's "apartment constraint" IS the moving sofa:
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x_i(t) ∈ Ω (shape stays in corridor)
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The braid word describes the boundary point worldlines through the corner.
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The chiral configurations describe different ways the boundary points
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can cross (over/under) during the motion.
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COUCH filters: which chiral configurations correspond to physically
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realizable sofa motions (shape doesn't hit walls).
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Sidon filters: which of those motions have unique boundary interactions
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(no two pairs of boundary points produce the same dual quaternion product).
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The COMBINED filter (COUCH ∧ Sidon) selects motions that are BOTH
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geometrically valid AND structurally meaningful — these are the
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configurations where the octagon principle could detect the sofa's
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chromatic structure from the spectrum.
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---
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## 7. claim_boundary
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```
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braidstorm-treebraid-couch:batch-pipeline:design
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```
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This document connects three existing SilverSight components:
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1. BraidStorm (8-strand, Sidon labels, chiral crossings) — generates 256 configs
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2. TreeBraid (tree-organized, factorizes via braid relations) — reduces search
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3. COUCH (GCCL gate, moving sofa constraint) — geometric pre-filter
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Combined with the dual quaternion Sidon filter, this pipeline batch-processes
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hundreds of chiral configurations per run, with COUCH as the cheap geometric
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pre-filter and Sidon as the expensive algebraic filter.
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The Hutter prize lesson applies: the batch doesn't COMPRESS 256 configs into 1
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(conservation law blocks that). It FILTERS 256 configs down to the ~10-20
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that are both geometrically valid and structurally meaningful.
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316
docs/research/CHIRAL_BATCH_ENCODING.md
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docs/research/CHIRAL_BATCH_ENCODING.md
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# Chiral Batch Encoding: Hundreds of Configurations per Run
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**Status:** REFINEMENT — connects chiral braid chirality to batch Sidon filtering
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**Date:** 2026-07-04
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**Depends on:** `DUAL_QUATERNION_SIDON_FILTER.md`, `braid_group_action.md`,
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`TOROIDAL_POLOIDAL_REFINEMENT.md`, `weird_machine_conservation_law.md`
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**Key insight:** Chirality (over/under = ±1 per crossing) means a braid word of
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length k encodes 2^k configurations. Batch-encode hundreds, Sidon-filter in one pass.
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---
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## 1. The Chiral Braid Structure
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### 1.1 Chirality = Handedness = ±1 per Crossing
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In the braid group B_n, each generator σ_i has two chiral forms:
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σ_i⁺¹ = over-crossing (right-handed)
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σ_i⁻¹ = under-crossing (left-handed)
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A braid word of length k has 2^k possible chiral configurations:
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w = σ_{i₁}^{ε₁} σ_{i₂}^{ε₂} ... σ_{iₖ}^{εₖ} where εⱼ ∈ {+1, -1}
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### 1.2 Chirality in the CRT Embedding
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The CRT embedding already has chirality built in:
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Identity axis: a mod L₀ = poloidal (no reflection = "straight through")
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Reflection axes: S-a mod Lᵢ = toroidal (reflection = "flipped")
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The S-a reflection IS the chiral operation:
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S-a = "over" (positive chirality)
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a-S = "under" (negative chirality, equivalent to -(S-a))
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Each reflection axis Lᵢ contributes one chiral bit. With k reflection
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axes, there are 2^k chiral configurations per identity axis choice.
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### 1.3 Chirality in Dual Quaternions
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Dual quaternions have natural chirality:
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q = q_r + ε q_d (standard)
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q* = q_r - ε q_d (conjugate = opposite chirality)
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The conjugate reverses the translation direction (toroidal flip) while
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preserving the rotation (poloidal). This is exactly the S-a ↔ a-S flip.
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A dual quaternion pair (q_i, q_j) has 4 chiral configurations:
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(q_i, q_j) — both standard
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(q_i*, q_j) — i flipped
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(q_i, q_j*) — j flipped
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(q_i*, q_j*) — both flipped
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With n boundary points, there are 4^(n choose 2) chiral configurations
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of the full pairwise product set. We don't test all of these — we
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batch-encode a representative sample and Sidon-filter.
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---
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## 2. Batch Encoding: How It Works
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### 2.1 The Problem with Sequential Testing
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Current approach (v2/v3): test one (shape, n, q) configuration per run.
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- 5 shapes × 3 n-values × 5 q-values = 75 configurations
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- Each takes ~10s = 12.5 minutes total
|
||||
- Each is a separate Sidon check
|
||||
|
||||
This is slow and doesn't exploit the braid structure.
|
||||
|
||||
### 2.2 Chiral Batch Encoding
|
||||
|
||||
The chiral braid allows encoding MANY configurations into a SINGLE run:
|
||||
|
||||
1. Choose a base braid word w = σ₁ σ₂ σ₃ ... (the "spine")
|
||||
2. For each crossing, choose chirality εᵢ ∈ {+1, -1}
|
||||
3. A batch of B configurations = B different chirality assignments
|
||||
ε¹ = (+1, +1, +1, ...), ε² = (+1, +1, -1, ...), etc.
|
||||
4. All B configurations share the same braid SPINE (which strands cross)
|
||||
but differ in CHIRALITY (how they cross)
|
||||
|
||||
5. For each configuration, compute the CRT embedding with the chiral
|
||||
reflection choices:
|
||||
- εᵢ = +1 → S-a mod Lᵢ (standard reflection)
|
||||
- εᵢ = -1 → a-S mod Lᵢ (flipped reflection)
|
||||
|
||||
6. Apply the Sidon filter ONCE to the entire batch:
|
||||
- For each configuration, check if the CRT-reconstructed sums are Sidon
|
||||
- The filter selects which chiral configurations produce unique
|
||||
pairwise signatures
|
||||
|
||||
### 2.3 Why This Is Hundreds per Run
|
||||
|
||||
With k reflection axes:
|
||||
- 2^k chiral configurations per (identity, label_set) pair
|
||||
- For k=8 (our standard 8-strand braid): 2^8 = 256 configurations
|
||||
- For k=10: 2^10 = 1024 configurations
|
||||
|
||||
Each run can batch-test ALL 256 (or 1024) chiral configurations with
|
||||
a SINGLE CRT reconstruction pass — the identity axis is computed once,
|
||||
and each chiral variant only changes the reflection components.
|
||||
|
||||
The Sidon filter then selects which of the 256 configurations are
|
||||
structurally meaningful (Sidon-clean) vs degenerate (collision).
|
||||
|
||||
### 2.4 Connection to the Hutter Prize Filtering
|
||||
|
||||
The Hutter prize lesson: compression is dead, filtering works.
|
||||
|
||||
Batch encoding is the APPLICATION of this lesson:
|
||||
- Don't compress 256 configurations into 1 (impossible — conservation law)
|
||||
- Don't test 256 configurations sequentially (slow)
|
||||
- DO: batch-encode all 256, then FILTER to the Sidon-clean ones
|
||||
|
||||
The filter selects which chiral configurations have unique pairwise
|
||||
signatures. The rest are noise (degenerate, collision). This is the
|
||||
same filtering mechanism from the weird machine conservation law:
|
||||
program (chiral configuration) + residual (dropped configs) ≥ K(data)
|
||||
But we don't care about the residual — we care about which configurations
|
||||
the filter KEEPS.
|
||||
|
||||
---
|
||||
|
||||
## 3. The Chiral Sidon Filter (Concrete)
|
||||
|
||||
### 3.1 Algorithm
|
||||
|
||||
```
|
||||
Input:
|
||||
- Label set A = {a₁, ..., aₙ} (Sidon in ℤ)
|
||||
- Identity modulus L₀
|
||||
- Reflection moduli L₁, ..., Lₖ (pairwise coprime)
|
||||
- Reflection point S
|
||||
- Batch size B (number of chiral configurations)
|
||||
|
||||
Output:
|
||||
- For each of B chiral configurations: is_sidon (bool), sidon_score
|
||||
|
||||
Algorithm:
|
||||
1. Compute identity component once: id_i = a_i mod L₀ for all i
|
||||
2. For each chiral configuration c ∈ {0, 1}^k (binary vector):
|
||||
a. For each reflection axis j ∈ {1, ..., k}:
|
||||
- If c[j] = 0: ref_i_j = (S - a_i) mod Lⱼ (standard)
|
||||
- If c[j] = 1: ref_i_j = (a_i - S) mod Lⱼ (flipped)
|
||||
b. CRT reconstruct: val_i = CRT(id_i, ref_i_1, ..., ref_i_k)
|
||||
c. Check Sidon: all pairwise sums val_i + val_j distinct mod M?
|
||||
3. Return filter results for all B configurations
|
||||
```
|
||||
|
||||
### 3.2 Computational Cost
|
||||
|
||||
- Identity component: O(n) — computed ONCE
|
||||
- Per configuration: O(n·k) for reflection + O(n²) for Sidon check
|
||||
- Total: O(n) + B × O(n·k + n²)
|
||||
- For n=21, k=8, B=256: 21 + 256 × (168 + 441) = 21 + 155,904 ≈ 156K ops
|
||||
- vs sequential: 256 × (21 + 168 + 441) = 256 × 630 = 161K ops
|
||||
|
||||
The speedup is modest for small k, but the REAL advantage is:
|
||||
1. The identity component is shared (not recomputed)
|
||||
2. The Sidon check can be parallelized across configurations
|
||||
3. The filter selects which configurations are worth deeper analysis
|
||||
|
||||
### 3.3 What the Filter Selects
|
||||
|
||||
The chiral Sidon filter selects configurations where:
|
||||
- The chiral choices (which axes are flipped) produce unique pairwise sums
|
||||
- This means the chiral pattern is "informative" — it breaks symmetries
|
||||
that would otherwise cause collisions
|
||||
|
||||
Configurations that FAIL the filter:
|
||||
- Have chiral choices that create sum collisions (degenerate)
|
||||
- The chiral pattern doesn't break existing symmetries
|
||||
- These are "uninformative" — the chirality doesn't help
|
||||
|
||||
The filter rate (fraction of configurations that pass) measures how
|
||||
much chiral information the braid structure carries:
|
||||
- High pass rate (>50%): chirality doesn't matter much (symmetric problem)
|
||||
- Low pass rate (<10%): chirality is critical (most configs degenerate)
|
||||
- Medium pass rate (~30%): chirality selects a specific structural class
|
||||
|
||||
---
|
||||
|
||||
## 4. Connection to q-Profile
|
||||
|
||||
### 4.1 q-Profile as Chiral Ratio
|
||||
|
||||
The q-profile = L₁/L₀ (reflection/identity = toroidal/poloidal).
|
||||
|
||||
In the chiral batch:
|
||||
- q < 1: L₁ < L₀ → reflection axis smaller → chiral flip has less impact
|
||||
- q > 1: L₁ > L₀ → reflection axis larger → chiral flip has more impact
|
||||
- q = 1: L₁ = L₀ → chiral flip is symmetric → degenerate
|
||||
|
||||
The q-profile sweep showed q > 1 has 100% Sidon rate. In chiral terms:
|
||||
larger reflection axis → chiral flips create more diverse products →
|
||||
fewer collisions → higher Sidon rate.
|
||||
|
||||
### 4.2 Chiral q-Sweep
|
||||
|
||||
Instead of sweeping q across fixed chiral configurations:
|
||||
1. Fix q at the optimal value (q > 1, e.g. q = 3/2)
|
||||
2. Sweep chiral configurations (256 variants)
|
||||
3. Measure: which chiral patterns have highest Sidon score?
|
||||
|
||||
This separates the q-effect (axis ratio) from the chirality effect
|
||||
(which axes are flipped). The q-profile sweep couldn't do this —
|
||||
it tested one chirality per q value.
|
||||
|
||||
---
|
||||
|
||||
## 5. Implementation Plan
|
||||
|
||||
### Phase 1: Chiral Batch CRT (Python, exact arithmetic)
|
||||
|
||||
```python
|
||||
def chiral_batch_sidon(labels, S, L0, Ls, batch_configs=None):
|
||||
"""Batch-test chiral configurations for Sidon property.
|
||||
|
||||
Ls = [L1, ..., Lk] reflection moduli
|
||||
batch_configs = list of binary tuples (length k), each specifying
|
||||
which axes are flipped (1 = flipped, 0 = standard)
|
||||
If None, test ALL 2^k configurations.
|
||||
"""
|
||||
k = len(Ls)
|
||||
if batch_configs is None:
|
||||
batch_configs = list(product([0, 1], repeat=k))
|
||||
|
||||
# Identity component (computed once)
|
||||
id_comp = [a % L0 for a in labels]
|
||||
|
||||
results = []
|
||||
for config in batch_configs:
|
||||
# Reflection components with chiral choices
|
||||
embedded = []
|
||||
for a in labels:
|
||||
row = [a % L0] # identity
|
||||
for j, Lj in enumerate(Ls):
|
||||
if config[j] == 0:
|
||||
row.append((S - a) % Lj) # standard
|
||||
else:
|
||||
row.append((a - S) % Lj) # flipped
|
||||
embedded.append(row)
|
||||
|
||||
# CRT reconstruct + Sidon check
|
||||
sidon = sidon_check(embedded, [L0] + Ls)
|
||||
results.append({
|
||||
"config": config,
|
||||
"is_sidon": sidon["is_sidon"],
|
||||
"sidon_score": sidon["sidon_score"],
|
||||
"collisions": sidon["collisions"],
|
||||
})
|
||||
|
||||
return results
|
||||
```
|
||||
|
||||
### Phase 2: Chiral q-Sweep
|
||||
|
||||
1. Fix L₀ = 7 (optimal from capacity envelope)
|
||||
2. For each q ∈ {3/2, 2, 5/2, 3}:
|
||||
- Set L₁ = L₀ × q
|
||||
- Batch-test all 2^k chiral configurations
|
||||
- Measure: pass rate, best config, worst config
|
||||
3. Compare to sequential q-sweep results
|
||||
|
||||
### Phase 3: Dual Quaternion Chiral Filter
|
||||
|
||||
1. For each chiral configuration, compute dual quaternion products
|
||||
2. Apply Sidon filter to dual quaternion products (not CRT sums)
|
||||
3. Compare: does the dual quaternion filter select different configs
|
||||
than the CRT filter?
|
||||
|
||||
---
|
||||
|
||||
## 6. What This Enables
|
||||
|
||||
### 6.1 Orders of Magnitude More Configurations
|
||||
|
||||
Current: 75 configurations per run (5 shapes × 3 n × 5 q)
|
||||
With chiral batch: 75 × 256 = 19,200 configurations per run
|
||||
With k=10: 75 × 1024 = 76,800 configurations per run
|
||||
|
||||
### 6.2 Statistical Power
|
||||
|
||||
With 256+ configurations per (shape, n, q):
|
||||
- Can compute Sidon pass rate with statistical confidence
|
||||
- Can identify which chiral patterns are optimal
|
||||
- Can detect phase transitions (where pass rate drops sharply)
|
||||
|
||||
### 6.3 Connection to the Moving Sofa
|
||||
|
||||
The sofa motion through the L-corridor IS a braid:
|
||||
- Boundary point worldlines = braid strands
|
||||
- Corner navigation = braid crossings
|
||||
- Each crossing has chirality (over/under = which strand is in front)
|
||||
|
||||
Batch-encoding chiral braid configurations = batch-encoding different
|
||||
sofa motion variants. The Sidon filter selects which motions have
|
||||
unique boundary interactions (structurally meaningful) vs degenerate
|
||||
(symmetric, uninformative).
|
||||
|
||||
---
|
||||
|
||||
## 7. claim_boundary
|
||||
|
||||
```
|
||||
chiral-batch-encoding:efficiency-multiplier:conceptual
|
||||
```
|
||||
|
||||
The chiral braid structure allows batch-encoding 2^k configurations
|
||||
per run (k = number of reflection axes). The Sidon filter then selects
|
||||
which configurations are structurally meaningful. This is the Hutter
|
||||
prize lesson applied: filtering works, compression doesn't, and
|
||||
batching makes filtering efficient.
|
||||
|
||||
**MEASURED:** q > 1 has 100% Sidon rate (from q-profile sweep)
|
||||
**PREDICTED:** chiral batch will show ~30-50% pass rate per q value,
|
||||
with specific chiral patterns being optimal
|
||||
**OPEN:** does the chiral filter select different configs than the
|
||||
CRT sum filter?
|
||||
Loading…
Add table
Reference in a new issue