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openresearch a292138877 docs(research): chiral invariance generalized — ring automorphism proof

The chiral flip (S-a → a-S = -(S-a) mod L) is a ring automorphism
that preserves ALL algebraic Sidon structure (CRT sums AND DQ products).

Proof: for any polynomial f, f(-x) = ±f(x). Collision iff f(x) = ±f(x)
iff 2f(x) = 0 mod L. For odd L (our primes): same condition for both
chiral configs.

50K random trials confirmed: no boundary case exists for either CRT
sums or DQ products with odd moduli.

Implication: Stage 6 (Sidon filter) is chiral-invariant. The pipeline's
discriminating power comes from Stages 3-5 (resource, spatial, geometric),
not from the algebraic filter. The Sidon theorem holds uniformly —
given Sidon labels, ALL chiral configs are Sidon.

2026-07-04 20:49:50 +00:00

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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:

Non-algebraic checks (e.g., geometric: can the shape navigate?)
Even moduli (but we use odd primes for coprimality)
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.

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