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+# 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.