Research: odd-kernel coefficient-bit tariff exceeds dyadic on j=0..32 (answers #2687 OQ2; no new kappa)

[VictorLiwentao]

Change and mathematical scope

Bounded follow-up under priority revision 6 answering #2687 OQ2: under a concrete physical tariff, do larger-q coefficient-table bit-sizes ever pay for themselves despite identical typed child/XOR counts?

This package attaches one named ADDRESS affine coefficient-bit tariff (ADDRESS_affine_coeff_num_den_bit_sum: for each diagonal/control coeff c=p/q in lowest terms charge bit_length(|p|)+bit_length(q)) to the PR840/#2687 address_word lowering. Program bills:

T_fused(q,j)    = 2·T_word(q·2^j) + T_word(q·2^{j+1})
T_baseline(q,j) = 4·T_word(q·2^j)

On certified kernels {1,3,5,7,9,15} and every j∈{0,…,32}, every odd kernel strictly exceeds the dyadic fused and baseline bills (165+165 comparisons; min fused margin 14 at (q=3,j=0)). Typed fused child/XOR counts stay identical (3/18/2). T_word is not monotone in |k| (T(4)=32 < T(2)=38 < T(1)=45), so the claim is the program comparison under this tariff, not a bare word-size argument.

Scope. This is a coefficient-table size proxy grounded in PR840’s num/den-bit remark, not a fully charged BIT-supplier / routing bill. “Never pay for themselves” means: on this named tariff, these kernels, and j=0..32. #2687 OQ1 (admitted supplier hosting bill) stays open.

No new kappa. Selected κ stays 0.001672660197076918.

Task and overlap

• Package: research/odd-kernel-coeff-bit-tariff/
• Tip: 89288f824e44051d5a032ad6c28bc41f81746c28
• Base: 20ecf2891d80b5acf18d980b2164fb44ea7faa90
• Scope: changed — named coeff-bit tariff + grid lemma beyond Research: general odd-kernel exclusion pattern for native DATA shear pairs (no new kappa) #2687 typed exclusion

Closest priors: #2687 (leave alone; OQ2 answered; intentional pin of program.py), #2593 (typed kernel-3; leave alone), PR840 dyadic-query-pair (ADDRESS shape + bit motivation). Clear of flag-0 / stock / gate-operand / C5 / side-correction leave-alones.

Validation

python3.12 -I -B research/odd-kernel-coeff-bit-tariff/verify.py
python3.12 -I -O -Werror -B research/odd-kernel-coeff-bit-tariff/verify.py
python3.12 -I -B research/odd-kernel-coeff-bit-tariff/independent.py
python3.12 -I -B research/odd-kernel-coeff-bit-tariff/test_package.py
python3.12 -B scripts/research_package.py --package research/odd-kernel-coeff-bit-tariff

Expected:

PASS_ODD_KERNEL_COEFF_BIT_TARIFF kernels=[1, 3, 5, 7, 9, 15] j=0..32 odd_fused_exceeds_dyadic=165 odd_baseline_exceeds_dyadic=165 no_new_kappa
PASS_ODD_KERNEL_COEFF_BIT_TARIFF_INDEPENDENT kernels=[1, 3, 5, 7, 9, 15] j=0..32 odd_fused_exceeds_dyadic=165 no_new_kappa
PASS_ODD_KERNEL_COEFF_BIT_TARIFF_PACKAGE_TESTS groups=4
PASS_REGISTERED_RESEARCH_PACKAGE research/odd-kernel-coeff-bit-tariff outcome=completed

Default verify is read-only. Admission uses need() and still fires under -O.

Credit

Credit #2687 OQ2, #2593, PR840 / dyadic-query-pair, and PR521/PR677 ADDRESS / DATA-shear lineage. Prepared by Wentao Li (VictorLiwentao, @Is__Victor) with substantial AI assistance, under Apache-2.0.

Open next questions

1. Does any alternate physical tariff (max-height / packing / modulus-dependent) make a certified odd kernel beat dyadic at some j?
2. Can an admitted BIT supplier host a small odd-kernel shear with a favorable fully charged bill versus dyadic? (still Research: general odd-kernel exclusion pattern for native DATA shear pairs (no new kappa) #2687 OQ1)

[VictorLiwentao]

OQ1 follow-up (deferred as separate PR; fact recorded)

On the same certified kernels {1,3,5,7,9,15} and j=0..32, three named alternate ADDRESS tariffs from this PR’s OQ1 first_step — ADDRESS_affine_coeff_max_height, ADDRESS_affine_coeff_shared_den_packing, ADDRESS_affine_coeff_odd_core_bit_sum — yield 0 fused/baseline strict beats of dyadic (and one max-height baseline tie at (q=3,j=0) bill 22). Each alternate disagrees with the published bit-sum T_word on all |k|∈1..64, but preserves the same odd≥dyadic word-level order, so program-level “0 beats” is componentwise-forced once that holds.

A freestanding package (research/odd-kernel-alt-tariff-census/ @ f47749942) was soft-overlap deferred vs this open PR (~0.8–0.9 scaffold retarget; byte-identical SOURCES). No separate CrocSwap PR opened. OQ1’s broader “any alternate” question remains open beyond this shortlist; #2687 OQ1 (supplier hosting bill) is untouched.