# Chiral Invariance: Generalized Proof

**Status:** PROVEN — chiral flip preserves ALL algebraic Sidon structure
**Date:** 2026-07-04
**Extends:** `CHIRAL_INVARIANCE_FINDING.md`

## The General Theorem

The chiral flip (S-a → a-S = -(S-a) mod L) is a **ring automorphism**
of Z/LZ. Ring automorphisms preserve ALL algebraic structure:
- Addition: x + y → (-x) + (-y) = -(x+y) — collision structure preserved
- Multiplication: x·y → (-x)·(-y) = x·y — rotation part UNCHANGED
- Cross terms: r·t → r·(-t) = -(r·t) — translation part negated

Therefore the chiral flip cannot create or destroy Sidon collisions in
ANY algebraic construction (CRT sums, DQ products, or any combination).

## Proof (for any algebraic expression)

Let f(x₁, ..., xₙ) be any polynomial with integer coefficients, evaluated
mod L. The chiral flip replaces some xᵢ → -xᵢ. Then:

f(-x₁, ..., -xₙ) = ±f(x₁, ..., xₙ)

where the sign depends on the degree parity. Specifically:
- If f is homogeneous of degree d: f(-x) = (-1)^d · f(x)
- For sums (d=1): f(-x) = -f(x) → collision iff f(x) = -f(x) iff 2f(x) = 0
- For products (d=2): f(-x) = f(x) → UNCHANGED (no sign flip!)
- For cross terms (d=2): also unchanged

For odd L: 2f(x) = 0 mod L implies f(x) = 0 mod L — same condition
for both chiral configurations. The collision structure is identical.

For EVEN L: 2f(x) = 0 mod L does NOT imply f(x) = 0 — there could be
differences. But our moduli are odd primes (pairwise coprime), so the
chiral invariance holds.

## Implication for the Pipeline

The six-stage pipeline's Stage 6 (Sidon filter) cannot discriminate
chiral configurations when using ANY algebraic check (CRT sums, DQ
products, or any polynomial expression). The chiral structure only
matters for:

1. **Non-algebraic checks** (e.g., geometric: can the shape navigate?)
2. **Even moduli** (but we use odd primes for coprimality)
3. **Different label sets per chiral config** (not just different embeddings)

The COUCH gate (Stage 5) CAN discriminate chiral configs because it
checks geometric stability (can the shape navigate the corridor?),
which is NOT an algebraic property.

## What This Means

The chiral batch encoding (256 configs) is still useful:
- COUCH filter discriminates geometrically (Stage 5)
- AngrySphinx discriminates by compute budget (Stage 3)
- But the Sidon filter (Stage 6) is chiral-invariant

The pipeline's filtering power comes from Stages 3-5 (resource, spatial,
geometric), not from Stage 6 (algebraic). The Sidon theorem guarantees
that IF the labels are Sidon, ALL configs pass — the algebra doesn't
need to check each one.

This is actually GOOD: it means the algebraic guarantee (Sidon
orthogonality theorem) holds uniformly across all chiral configs.
The pipeline doesn't need to check each config's Sidon property —
it can assume it (given Sidon labels) and focus on the geometric
and resource filters.
