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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.
45 lines
1.8 KiB
Markdown
45 lines
1.8 KiB
Markdown
# Chiral Invariance of CRT Sidon Check
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**Status:** MEASURED — chiral flip doesn't affect Sidon property
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**Date:** 2026-07-04
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**Method:** 20,000 random trials (10K with 2 moduli, 10K with 3 moduli)
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## Finding
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No boundary case found in 20,000 trials: the chiral flip (S-a vs a-S)
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does NOT change the Sidon property. All chiral configurations give the
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same Sidon result (either all Sidon or all non-Sidon).
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## Proof
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The chiral flip negates the reflection component:
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Standard: (S - a) mod L
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Flipped: (a - S) mod L = -(S - a) mod L
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The pairwise sum of two reflection components:
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Standard: (S-a) + (S-b) = 2S - (a+b) mod L
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Flipped: (a-S) + (b-S) = (a+b) - 2S mod L = -(2S - (a+b)) mod L
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These are negatives of each other. Two values x and -x mod L collide
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(x ≡ -x) iff 2x ≡ 0 mod L. For odd L (all our moduli are odd primes),
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this requires x ≡ 0 mod L — the same condition for both standard and
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flipped. Therefore the collision structure is identical.
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**The negation map x → -x is a ring automorphism that preserves the
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Sidon property.** The chiral flip is exactly this automorphism applied
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to the reflection component. It cannot create or break Sidon collisions.
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## Implication
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The chiral filter is **trivial for CRT sum-based Sidon checks**. All
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2^k chiral configurations give the same Sidon result.
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The chiral structure matters for **dual quaternion products** (which
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involve quaternion multiplication, not just addition). The dual quaternion
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product q_i ⊛ q_j includes both rotation (multiplication of real parts)
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and translation (cross terms), so the negation affects the product
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non-trivially.
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**Next step:** Implement the dual quaternion Sidon filter (not the CRT
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sum filter) to test whether chiral configurations produce different
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dual quaternion product collision structures.
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