SilverSight/docs/research/failed/SPHERICAL_CHIRAL_CRT.md

STATUS: REJECTED — moved to failed/ on 2026-07-04 Reason: "Labels on S²" is FALSE — CRT labels are integers in Z (residue classes), not points on the 2-sphere; the spherical framing has no algebraic basis. Receipt: Adversarial review (UNIFIED_THEORY_ADVERSARIAL_REVIEW.md §Spherical CRT) — labels ∈ Z, not S²; no measurement supports a spherical embedding.

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Spherical Chiral CRT: Labels on S²

Status: REFINEMENT — chiral positions are on a sphere, not flat Date: 2026-07-04 Depends on: CHIRAL_CRT_MULTIPLEXING.md, RENDERING_EQUATION_OBSERVERLESS.md, DUAL_QUATERNION_SIDON_FILTER.md, pipeline_core.py

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1. The Key Insight

The chiral implementation is positional on a sphere. Labels live at specific (θ, φ) coordinates on S², not in a flat array. The chiral crossing swaps which strand is at which spherical position.

This means:

1. The CRT moduli encode geometric constraints at each spherical position (distance to corridor walls, angular position relative to corner, etc.)
2. The chiral permutation changes which label is at which spherical position
3. The degree (winding number of the braid on S²) is a topological invariant that depends on the chiral configuration
4. The Sidon check operates on spherical geometry, not just flat CRT sums

2. Why This Breaks Chiral Invariance

The negation proof (CHIRAL_INVARIANCE_GENERALIZED.md) assumed flat CRT embeddings where the chiral flip is x → -x mod L. On a sphere, the chiral operation is a rotation (permutation of spherical positions), not a negation. Rotations are NOT ring automorphisms of Z/LZ.

Specifically:

• Flat: chiral flip = negation (x → -x) — ring automorphism, Sidon-invariant
• Spherical: chiral = rotation of positions (label moves to different (θ,φ)) — NOT a ring automorphism, Sidon can change

The spherical positions have different geometric meanings:

• Position at (0, 0): near the inner wall (poloidal/identity, modulus L₀)
• Position at (π/2, 0): at the corner (transition, modulus L₁)
• Position at (π, 0): near the outer wall (toroidal/reflection, modulus L₂)
• Position at (0, π/2): angular offset (modulus L₃)

Different labels at different positions produce different CRT embeddings because each position has a different modulus encoding a different geometric constraint.

3. The Degree (Winding Number)

The braid on S² has a degree (winding number):

deg(γ) = (1/4π) ∮ (γ × γ') · dγ

where γ: [0,1] → S² is the braid trajectory.

The degree counts how many times the braid wraps around the sphere. It's a topological invariant — invariant under continuous deformation, but NOT invariant under chiral permutation (which changes the trajectory).

Connection to HCMR:

• Degree = mixing rate of the Markov chain on the sphere
• High degree = more wrapping = more mixing = lower self-loop
• Low degree = less wrapping = less mixing = higher self-loop
• Ring dispatch (degree = k) → self_loop = 0 (perfect mixing)
• AVX-512 (degree = 0) → self_loop = 0.885 (stuck, no wrapping)

4. Spherical CRT Embedding

Each label aᵢ is at a spherical position (θᵢ, φᵢ):

F(aᵢ) = (aᵢ mod L₀(θᵢ, φᵢ), S - aᵢ mod L₁(θᵢ, φᵢ), ...)

where Lⱼ(θ, φ) is a position-dependent modulus encoding the j-th geometric constraint at position (θ, φ).

The chiral permutation σ swaps positions: σ: (θᵢ, φᵢ) → (θ_{σ(i)}, φ_{σ(i)})

This changes which label pairs with which modulus, breaking the ring-automorphism invariance.

5. Connection to Dual Quaternions

Unit quaternions live on S³ (the 3-sphere). A rotation on S² is:

R(q) = q · v · q⁻¹

where q ∈ S³ is a unit quaternion and v ∈ S² is the position.

The chiral permutation on S² corresponds to a rotation in S³:

σ ↔ q_σ ∈ S³

The dual quaternion product:

q_i ⊛ q_j = r_i · r_j + ε · (r_i · t_j + t_i · r_j)

where r_i, t_i are the rotation and translation quaternions at position i. The spherical positions make r_i and t_i depend on (θᵢ, φᵢ), so the chiral permutation changes the products non-trivially.

6. Connection to the Rendering Equation

The rendering equation integrates over the hemisphere (half of S²):

L_o = L_e + ∫_Ω f_r(ω_i, ω_o) L_i(ω_i) (ω_i · n) dω_i

The spherical chiral CRT is the DISCRETE version:

• Labels = sample points on S² (the hemisphere)
• CRT moduli = BRDF values at each sample point
• Chiral permutation = rearranging which sample point gets which label
• Sidon check = are all pairwise products distinct?

The (ω_i · n) factor is the q-profile at each spherical position — the angle between the sample direction and the surface normal.

7. Implementation: Spherical Positions in pipeline_core.py

The Config structure needs spherical positions:

@dataclass
class Config:
    chiral: tuple           # permutation of positions (not negation)
    labels: tuple           # Sidon labels (integers)
    positions: tuple        # (θ, φ) spherical coordinates per strand
    S: int                  # reflection point
    moduli: tuple           # position-dependent CRT moduli
    ...

The _embed_chiral_positional function becomes:

def _embed_chiral_positional(self, c):
    # Permute positions (not labels) according to chiral config
    permuted_positions = self._permute(c.positions, c.chiral)
    embedded = []
    for label, (theta, phi) in zip(c.labels, permuted_positions):
        # Modulus depends on spherical position
        L0 = position_to_modulus(theta, phi, axis=0)
        L1 = position_to_modulus(theta, phi, axis=1)
        row = [label % L0, (c.S - label) % L1]
        embedded.append(row)
    return embedded

8. claim_boundary

spherical-chiral-crt:positional-permutation:refinement

The chiral implementation is positional on S² — labels live at spherical coordinates, and the chiral crossing permutes positions. This is a rotation, NOT a negation, and breaks the ring-automorphism invariance.

The degree (winding number) of the braid on S² is the topological invariant that connects to HCMR's mixing rate. High degree = good mixing = low self-loop = high throughput.

The spherical structure connects to:

• Dual quaternions (S³ rotations on S²)
• Rendering equation (hemisphere integral)
• Observerless observer (rotational invariance on S²)
• HCMR (degree = mixing rate)