Research: one-repeat partitioned (2,6) 8q priced obstruction (no new kappa)

[VictorLiwentao]

Change and scope

Freestanding #840-style priced obstruction under #1898 at (b,a,x)=(38,2,2). Names the one-repeat companion of the partitioned (2,6) Hamilton layer at Q=8. Answers open question 1 of #2851: do non-distinct (one-repeat) label layers inside a partitioned Q=8 walk improve on distinct-label R=92 at (2,6)?

Dual schedule	Covering pairs	Exact min R	Tariff floor (X≥0)	Undercuts 436?	Beats distinct 92?
Distinct-label (2,6) (#2851)	3916416	92	≥488	no	—
One-repeat long (2,6) (this)	59996160	112	≥528	no	no

Every covering one-repeat dual has exact R=112 (spectrum {112}). Only rank-4 longs cover, always with Σe=40 and σ outside the long span (16 choices), so R=2·40+32=112. Abstract tariff 38·8+2·112=528>436. Distinct-label floor 92 is not improved. Length-1 components are not restored. Rank<4 longs are dual-scanned with covering_rank_lt4=0.

Scoped obstruction / priced alphabet; no new kappa. Leave-alone through #2860 / #2858 / #2851 / #2846 was not reopened.

Result

Package: research/independent-walks-onerepeat-8q/.
Tip: b07bc1e16752a1980c7dab098d6a4b98295d73c0.

Exact: one-repeat min R 112 (tariff floor 528); no undercut of 436; does not beat distinct 92.

Validation

python3.12 -I -B research/independent-walks-onerepeat-8q/verify.py
python3.12 -I -O -B research/independent-walks-onerepeat-8q/verify.py
python3.12 -I -B research/independent-walks-onerepeat-8q/independent.py
python3.12 -I -B research/independent-walks-onerepeat-8q/test_package.py
python3.12 -B scripts/research_package.py --package research/independent-walks-onerepeat-8q

Expected:

PASS_INDEPENDENT_WALKS_ONEREPEAT_8Q_OBSTRUCTION one_repeat_min_R=112 ... tariff_floor=528 ... answers_independent_walks_8q_oq1_onerepeat no_new_kappa
PASS_INDEPENDENT_WALKS_ONEREPEAT_8Q covering=59996160 ... one_repeat_min_R=112 tariff_floor=528 ... no_new_kappa
PASS_INDEPENDENT_WALKS_ONEREPEAT_8Q_PACKAGE_TESTS groups=6
PASS_REGISTERED_RESEARCH_PACKAGE research/independent-walks-onerepeat-8q outcome=completed

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

Overlap and credit

Closest prior: #2851 (distinct partitioned Q=8), #2858 (one-repeat (3,5)/(4,4)), #2860 (two-repeat), #2846, #2779. Freestanding one-repeat layer on partitioned (2,6); no competing open PR at refresh.

Credit the mutable-source / #2851 / #2858 / #2860 / #2846 / #2779 / PR1898 lineage. Prepared by Wentao Li (VictorLiwentao, @Is__Victor) with substantial AI assistance, under Apache-2.0.