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

docs/research/SPHERICAL_CHIRAL_CRT.md

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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`
+
+---
+
+## 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:
+
+```python
+@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:
+
+```python
+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)
+```