bd33007440
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.
1.8 KiB
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.