# Chiral Invariance of CRT Sidon Check

**Status:** MEASURED — chiral flip doesn't affect Sidon property
**Date:** 2026-07-04
**Method:** 20,000 random trials (10K with 2 moduli, 10K with 3 moduli)

## Finding

No boundary case found in 20,000 trials: the chiral flip (S-a vs a-S)
does NOT change the Sidon property. All chiral configurations give the
same Sidon result (either all Sidon or all non-Sidon).

## Proof

The chiral flip negates the reflection component:
  Standard: (S - a) mod L
  Flipped:  (a - S) mod L = -(S - a) mod L

The pairwise sum of two reflection components:
  Standard: (S-a) + (S-b) = 2S - (a+b)  mod L
  Flipped:  (a-S) + (b-S) = (a+b) - 2S  mod L = -(2S - (a+b)) mod L

These are negatives of each other. Two values x and -x mod L collide
(x ≡ -x) iff 2x ≡ 0 mod L. For odd L (all our moduli are odd primes),
this requires x ≡ 0 mod L — the same condition for both standard and
flipped. Therefore the collision structure is identical.

**The negation map x → -x is a ring automorphism that preserves the
Sidon property.** The chiral flip is exactly this automorphism applied
to the reflection component. It cannot create or break Sidon collisions.

## Implication

The chiral filter is **trivial for CRT sum-based Sidon checks**. All
2^k chiral configurations give the same Sidon result.

The chiral structure matters for **dual quaternion products** (which
involve quaternion multiplication, not just addition). The dual quaternion
product q_i ⊛ q_j includes both rotation (multiplication of real parts)
and translation (cross terms), so the negation affects the product
non-trivially.

**Next step:** Implement the dual quaternion Sidon filter (not the CRT
sum filter) to test whether chiral configurations produce different
dual quaternion product collision structures.
