[VictorLiwentao]

Change and scope

Completes open question 2 of #2593 (research/triadic-query-pair, kernel-3 specialization of #840): for every positive odd kernel q, the same three-query fusion at a=±q·2^j attains the typed invoice below on every odd modulus coprime to q. Even moduli and odd moduli sharing a prime with q are excluded. Typed costs do not improve over the dyadic (q=1) case.

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

Certificate kernels: {1,3,5,7,9,15} (dyadic recovery, triadic cross-check, and primary new {5,7,9,15}). Pattern: admitted ⇔ odd and gcd(modulus,q)=1.

Research construction / recursive interface; no new kappa. Leave-alone through #2686 (including #2677, #2643, #2593) was not reopened.

Result

Package: research/odd-kernel-query-pair/.
Tip: c319093b5b96211e89165f4ad9dd0d20d5039414.

Exact invoice children 4→3; affine 24→18; XOR 4→2 uniformly across certified odd kernels; no typed gain over dyadic.

Validation

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

Expected:

PASS_ODD_KERNEL_QUERY_PAIR kernels=[1, 3, 5, 7, 9, 15] primary_new=[5, 7, 9, 15] children=4->3 affine=24->18 xor=4->2 no_typed_gain_over_dyadic ... no_new_kappa
PASS_INDEPENDENT_ODD_KERNEL_QUERY_PAIR kernels=[1, 3, 5, 7, 9, 15] ... no_new_kappa
PASS_ODD_KERNEL_QUERY_PAIR_PACKAGE_TESTS groups=6
PASS_REGISTERED_RESEARCH_PACKAGE research/odd-kernel-query-pair outcome=completed

Default verify is read-only. Admission uses need(); -O also passes.

Overlap and credit

Closest priors: #840 dyadic; #2593 kernel-3 (OQ2). Freestanding generalization; does not edit those packages. No competing open PR on this multi-kernel pattern at refresh.

Credit Wentao Li / PR840 / #2593 lineage. Prepared by Wentao Li (VictorLiwentao, @Is__Victor) with substantial AI assistance, under Apache-2.0.

[VictorLiwentao]

OQ1 follow-up attempt (deferred; OQ1 still open)

A candidate package research/odd-kernel-source-return-host/ @ db3deecf6 named an immutable-source / exclusive-target “admitted-screen” host bill on top of the #3702 coeff-bit T_word: I=0 recovers zero odd fused beats on j=0..32; I=1 yields a unique beat at (q=3,j=0) (674<677) with critical I*=14/17.

Soft-overlap / weak-need deferred: the odd-vs-dyadic comparison is ΔT + I·ΔA, so fixed units C,X,D,R_SR cancel from that order; R_SR never binds the proved inequalities; screen ODD_MODULI_3_99 / unit routing I are author-chosen (near-minimal upper bound with I*<1). This is not a grounded admitted-BIT-supplier hosting lemma with a pinned source-return boundary that charges routing/precision/dispatch non-cancellably.

No separate CrocSwap PR opened. OQ1 remains open.

[VictorLiwentao]

Odd-kernel run-batching attempt (deferred; OQ1 still open)

A candidate package research/odd-kernel-query-runs/ @ 327262ebe on branch cursor/odd-kernel-query-runs-60b9 lifts the PR953 dyadic-run schedule through this PR’s signed q-dyadic admission: for certified kernels and all 63 nonempty J⊆{0,…,5}, the batch walk attains children 2r+c / affine 6(2r+c) / XOR 2r with no typed gain over dyadic.

Soft-overlap / weak-need deferred: soft-overlap ≈0.90–0.95 vs PR953 (same schedule/bound/seed-55, powers×q) and ≈0.55–0.70 vs this PR’s admission census; combined compose-and-retarget ≈0.9. The 2r+c invoice is forced once the dyadic-run walk is scaled into the odd-kernel alphabet; machine “lower bound” equals the constructed program. Does not answer OQ1 (admitted BIT-supplier / non-cancellable source-return bill).

No separate CrocSwap PR opened. Prefer a grounded OQ1 hosting lemma or a disjoint non-odd-kernel-run card.