Research: three-query odd-kernel-3 native DATA shear pair (no new kappa)

[VictorLiwentao]

Change and scope

Complements #840 (research/dyadic-query-pair), which fuses two completed DATA shears for dyadic powers a=±2^j with an all-odd-radix lowering and leaves non-dyadic powers to introduce prime exclusions. This freestanding package fixes the odd kernel to 3, takes a=±3·2^j, and prices the same three-query fusion with every excluded modulus named.

x <- h^-a x
x ^= y
x <- h^(2a) x
x ^= y
x <- h^-a x

Complete cost item	Separate pair	Fused odd-kernel pair
Ordinary recursive children	4	3
Ordered affine events	24	18
Aligned XOR scans	4	2
Extra full private rows	0	0

The lowering specializes to every odd modulus not divisible by 3. Multiples of 3 and even rings are rejected. These are exact interface counts, not measured runtime.

Research construction / recursive interface; no new kappa. Leave-alone tooling through #2487 and pending mutable-source-789q-invoice were not reopened.

Result

Package: research/triadic-query-pair/.
Tip: 77041f2d139276115ffa32ab876455d3a58ef2cb.

Exact invoice children 4→3; affine 24→18; XOR 4→2 with named exclusions (excluded_moduli=11 in the verify receipt) and independent matrix checks.

Validation

python3.12 -I -B research/triadic-query-pair/verify.py
python3.12 -I -O -B research/triadic-query-pair/verify.py
python3.12 -I -B research/triadic-query-pair/independent.py
python3.12 -I -B research/triadic-query-pair/test_package.py
python3.12 -B scripts/research_package.py --package research/triadic-query-pair

Expected:

PASS_TRIADIC_QUERY_PAIR children 4->3; affine 24->18; XOR 4->2; kernel=3 ... no_new_kappa
PASS_INDEPENDENT_TRIADIC_QUERY_PAIR children=4->3 xor=4->2 kernel=3 ...
PASS_TRIADIC_QUERY_PAIR_PACKAGE_TESTS groups=5
PASS_REGISTERED_RESEARCH_PACKAGE research/triadic-query-pair outcome=completed

Default verify is read-only. Admission uses need() (not assert); -O also passes.

Overlap and credit

Closest prior: #840 dyadic all-odd-radix case (does not price kernel-3 non-dyadic lowering). Leave-alone through #2487. No competing open PR on this freestanding package at refresh.

Credit Wentao Li / PR840 and the retained PR521/PR677 lineage. Prepared by Wentao Li (VictorLiwentao, @Is__Victor) with substantial AI assistance, under Apache-2.0.