docs(research): chiral invariance — CRT Sidon check is chiral-invariant

20,000 random trials found no boundary case. The chiral flip
(S-a vs a-S mod L) is a ring automorphism (negation) that preserves
the Sidon property. All chiral configs give the same Sidon result.

Proof: (a-S) mod L = -(S-a) mod L. The negation x→-x preserves
collision structure (x≡-x iff 2x≡0, same condition for both).

Implication: the chiral filter is trivial for CRT sums. It matters
for dual quaternion products (which involve multiplication, not
just addition). Next step: implement dual quaternion Sidon filter.

docs/research/CHIRAL_INVARIANCE_FINDING.md

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+# 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.