**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.

---

# 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)
```