From bd330074408c777999a37f25337222c9beb5bfbe Mon Sep 17 00:00:00 2001
From: openresearch <bot@openresearch.sh>
Date: Sat, 4 Jul 2026 20:35:35 +0000
Subject: [PATCH] =?UTF-8?q?docs(research):=20chiral=20invariance=20?=
 =?UTF-8?q?=E2=80=94=20CRT=20Sidon=20check=20is=20chiral-invariant?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

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 | 45 ++++++++++++++++++++++
 1 file changed, 45 insertions(+)
 create mode 100644 docs/research/CHIRAL_INVARIANCE_FINDING.md

diff --git a/docs/research/CHIRAL_INVARIANCE_FINDING.md b/docs/research/CHIRAL_INVARIANCE_FINDING.md
new file mode 100644
index 00000000..3e68f8ad
--- /dev/null
+++ b/docs/research/CHIRAL_INVARIANCE_FINDING.md
@@ -0,0 +1,45 @@
+# 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.