SilverSight/docs/research/CHIRAL_BATCH_ENCODING.md

Chiral Batch Encoding: Hundreds of Configurations per Run

Status: REFINEMENT — connects chiral braid chirality to batch Sidon filtering Date: 2026-07-04 Depends on: DUAL_QUATERNION_SIDON_FILTER.md, braid_group_action.md, TOROIDAL_POLOIDAL_REFINEMENT.md, weird_machine_conservation_law.md Key insight: Chirality (over/under = ±1 per crossing) means a braid word of length k encodes 2^k configurations. Batch-encode hundreds, Sidon-filter in one pass.

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1. The Chiral Braid Structure

1.1 Chirality = Handedness = ±1 per Crossing

In the braid group B_n, each generator σ_i has two chiral forms: σ_i⁺¹ = over-crossing (right-handed) σ_i⁻¹ = under-crossing (left-handed)

A braid word of length k has 2^k possible chiral configurations: w = σ_{i₁}^{ε₁} σ_{i₂}^{ε₂} ... σ_{iₖ}^{εₖ} where εⱼ ∈ {+1, -1}

1.2 Chirality in the CRT Embedding

The CRT embedding already has chirality built in: Identity axis: a mod L₀ = poloidal (no reflection = "straight through") Reflection axes: S-a mod Lᵢ = toroidal (reflection = "flipped")

The S-a reflection IS the chiral operation: S-a = "over" (positive chirality) a-S = "under" (negative chirality, equivalent to -(S-a))

Each reflection axis Lᵢ contributes one chiral bit. With k reflection axes, there are 2^k chiral configurations per identity axis choice.

1.3 Chirality in Dual Quaternions

Dual quaternions have natural chirality: q = q_r + ε q_d (standard) q* = q_r - ε q_d (conjugate = opposite chirality)

The conjugate reverses the translation direction (toroidal flip) while preserving the rotation (poloidal). This is exactly the S-a ↔ a-S flip.

A dual quaternion pair (q_i, q_j) has 4 chiral configurations: (q_i, q_j) — both standard (q_i*, q_j) — i flipped (q_i, q_j*) — j flipped (q_i*, q_j*) — both flipped

With n boundary points, there are 4^(n choose 2) chiral configurations of the full pairwise product set. We don't test all of these — we batch-encode a representative sample and Sidon-filter.

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2. Batch Encoding: How It Works

2.1 The Problem with Sequential Testing

Current approach (v2/v3): test one (shape, n, q) configuration per run.

• 5 shapes × 3 n-values × 5 q-values = 75 configurations
• 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.

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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

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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.

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5. Implementation Plan

Phase 1: Chiral Batch CRT (Python, exact arithmetic)

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?

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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).

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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?