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docs(research): spherical chiral CRT — labels on S²
The chiral implementation is positional on a sphere — labels live at (θ,φ) coordinates on S², and chiral crossings permute spherical positions. This is a ROTATION (not negation), which breaks the ring-automorphism invariance. The degree (winding number of the braid on S²) is the topological invariant connecting to HCMR's mixing rate: high degree = good mixing = low self-loop = high throughput Connections: - Dual quaternions: S³ rotations on S² - Rendering equation: hemisphere integral = half of S² - Observerless observer: rotational invariance on S² - HCMR: degree = mixing rate - (ω_i · n) = q-profile at each spherical position
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docs/research/SPHERICAL_CHIRAL_CRT.md
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# Spherical Chiral CRT: Labels on S²
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**Status:** REFINEMENT — chiral positions are on a sphere, not flat
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**Date:** 2026-07-04
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**Depends on:** `CHIRAL_CRT_MULTIPLEXING.md`, `RENDERING_EQUATION_OBSERVERLESS.md`,
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`DUAL_QUATERNION_SIDON_FILTER.md`, `pipeline_core.py`
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---
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## 1. The Key Insight
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The chiral implementation is **positional on a sphere**. Labels live at
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specific (θ, φ) coordinates on S², not in a flat array. The chiral
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crossing swaps which strand is at which **spherical position**.
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This means:
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1. The CRT moduli encode **geometric constraints at each spherical position**
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(distance to corridor walls, angular position relative to corner, etc.)
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2. The chiral permutation changes which label is at which spherical position
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3. The **degree** (winding number of the braid on S²) is a topological
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invariant that depends on the chiral configuration
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4. The Sidon check operates on **spherical geometry**, not just flat CRT sums
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## 2. Why This Breaks Chiral Invariance
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The negation proof (CHIRAL_INVARIANCE_GENERALIZED.md) assumed flat CRT
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embeddings where the chiral flip is x → -x mod L. On a sphere, the chiral
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operation is a **rotation** (permutation of spherical positions), not a
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negation. Rotations are NOT ring automorphisms of Z/LZ.
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Specifically:
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- Flat: chiral flip = negation (x → -x) — ring automorphism, Sidon-invariant
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- Spherical: chiral = rotation of positions (label moves to different (θ,φ))
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— NOT a ring automorphism, Sidon can change
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The spherical positions have different geometric meanings:
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- Position at (0, 0): near the inner wall (poloidal/identity, modulus L₀)
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- Position at (π/2, 0): at the corner (transition, modulus L₁)
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- Position at (π, 0): near the outer wall (toroidal/reflection, modulus L₂)
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- Position at (0, π/2): angular offset (modulus L₃)
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Different labels at different positions produce different CRT embeddings
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because each position has a different modulus encoding a different
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geometric constraint.
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## 3. The Degree (Winding Number)
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The braid on S² has a **degree** (winding number):
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deg(γ) = (1/4π) ∮ (γ × γ') · dγ
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where γ: [0,1] → S² is the braid trajectory.
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The degree counts how many times the braid wraps around the sphere.
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It's a topological invariant — invariant under continuous deformation,
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but NOT invariant under chiral permutation (which changes the trajectory).
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Connection to HCMR:
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- Degree = mixing rate of the Markov chain on the sphere
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- High degree = more wrapping = more mixing = lower self-loop
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- Low degree = less wrapping = less mixing = higher self-loop
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- Ring dispatch (degree = k) → self_loop = 0 (perfect mixing)
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- AVX-512 (degree = 0) → self_loop = 0.885 (stuck, no wrapping)
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## 4. Spherical CRT Embedding
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Each label aᵢ is at a spherical position (θᵢ, φᵢ):
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F(aᵢ) = (aᵢ mod L₀(θᵢ, φᵢ), S - aᵢ mod L₁(θᵢ, φᵢ), ...)
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where Lⱼ(θ, φ) is a position-dependent modulus encoding the j-th
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geometric constraint at position (θ, φ).
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The chiral permutation σ swaps positions:
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σ: (θᵢ, φᵢ) → (θ_{σ(i)}, φ_{σ(i)})
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This changes which label pairs with which modulus, breaking the
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ring-automorphism invariance.
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## 5. Connection to Dual Quaternions
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Unit quaternions live on S³ (the 3-sphere). A rotation on S² is:
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R(q) = q · v · q⁻¹
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where q ∈ S³ is a unit quaternion and v ∈ S² is the position.
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The chiral permutation on S² corresponds to a rotation in S³:
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σ ↔ q_σ ∈ S³
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The dual quaternion product:
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q_i ⊛ q_j = r_i · r_j + ε · (r_i · t_j + t_i · r_j)
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where r_i, t_i are the rotation and translation quaternions at position i.
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The spherical positions make r_i and t_i depend on (θᵢ, φᵢ), so the
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chiral permutation changes the products non-trivially.
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## 6. Connection to the Rendering Equation
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The rendering equation integrates over the hemisphere (half of S²):
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L_o = L_e + ∫_Ω f_r(ω_i, ω_o) L_i(ω_i) (ω_i · n) dω_i
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The spherical chiral CRT is the DISCRETE version:
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- Labels = sample points on S² (the hemisphere)
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- CRT moduli = BRDF values at each sample point
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- Chiral permutation = rearranging which sample point gets which label
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- Sidon check = are all pairwise products distinct?
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The (ω_i · n) factor is the q-profile at each spherical position —
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the angle between the sample direction and the surface normal.
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## 7. Implementation: Spherical Positions in pipeline_core.py
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The Config structure needs spherical positions:
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```python
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@dataclass
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class Config:
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chiral: tuple # permutation of positions (not negation)
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labels: tuple # Sidon labels (integers)
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positions: tuple # (θ, φ) spherical coordinates per strand
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S: int # reflection point
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moduli: tuple # position-dependent CRT moduli
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...
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```
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The _embed_chiral_positional function becomes:
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```python
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def _embed_chiral_positional(self, c):
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# Permute positions (not labels) according to chiral config
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permuted_positions = self._permute(c.positions, c.chiral)
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embedded = []
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for label, (theta, phi) in zip(c.labels, permuted_positions):
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# Modulus depends on spherical position
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L0 = position_to_modulus(theta, phi, axis=0)
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L1 = position_to_modulus(theta, phi, axis=1)
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row = [label % L0, (c.S - label) % L1]
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embedded.append(row)
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return embedded
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```
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## 8. claim_boundary
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```
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spherical-chiral-crt:positional-permutation:refinement
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```
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The chiral implementation is positional on S² — labels live at spherical
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coordinates, and the chiral crossing permutes positions. This is a rotation,
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NOT a negation, and breaks the ring-automorphism invariance.
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The degree (winding number) of the braid on S² is the topological invariant
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that connects to HCMR's mixing rate. High degree = good mixing = low
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self-loop = high throughput.
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The spherical structure connects to:
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- Dual quaternions (S³ rotations on S²)
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- Rendering equation (hemisphere integral)
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- Observerless observer (rotational invariance on S²)
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- HCMR (degree = mixing rate)
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```
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