[petalas]

Result

Scoped proof and finite certificate in the PR1898 abstract tariff; no new κ. Package: research/partitioned-walk-pair-sum-floor/.

This answers open question 2 of Wentao Li's #2779 (research/independent-walks-7q-obstruction): repeated labels inside a partitioned walk never cover at Q=7. Covering needs Σ(n_i−1) ≥ 5, and at Q=7 this is tight, so each walk must have independent labels. PR #2779's R=112 is therefore the exact minimum over all label patterns.

For any labels (repeats and interior zero revisits allowed) and any number of walks, the cost-2 masks are pair sums of one walk. This gives

R ≥ max(62, 122 − (Q−2)(Q−3)).

The floor is attained at Q=8 (R=92, walks (0,10), (0,22,5,31,24,20)) and Q=9 (R=80, walks (0,3), (0,18,1,5,11,20,27)). The tariff floor 38Q+2R is at least 470 at every Q ≥ 7. No partitioned independent dual undercuts the six-query endpoint tariff 436 at any query count.

Evidence and reproduction

verify.py imports PR #2779's model.py (pinned live input). It checks:

• the table lemma (finite iff in span; cost 2 iff pair sum) on all 924,482 restored walks of length 2–5;
• the Q=7 rank census: 624,960 of 894,660 length-5 walks and 26,040 of 28,861 length-4 walks have full rank, all with distinct nonzero labels;
• 4,000 seeded random duals with Q from 7 to 11, of which 2,456 cover, against both floors and against Research: partitioned independent-walks 7q readout obstruction (no new kappa) #2779's two-port dual;
• the witnesses and the tariff floors.

~/git/integer-mult-bounds/.venv/bin/python3.12 -B research/partitioned-walk-pair-sum-floor/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/partitioned-walk-pair-sum-floor
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/partitioned-walk-pair-sum-floor --check-source-history

All three passed locally (Python 3.12, about 25 s for the verifier).

Limitations

• The result is stated in the abstract tariff only. It is not a physical supplier and not a multiplication theorem.
• The k ≥ 3 walk cost is the natural min-plus extension of the two-port dual. No earlier package defines it.
• Same-walk, transfer-coupled and shared-dirt schedules are out of scope.
• The random controls are sanity checks; the floor rests on the written proof. Independent human and formal review are unclaimed.

Open next questions

1. What are the exact minima at Q=10, 11? The floor gives 66 and 62; a seeded search found 72 and 66.
2. Can a pair-sum count with the shared prefix charged once bound transfer-coupled duals at every Q?

Related open work

• Research: transfer-coupled (5,1,1) 7q readout obstruction (no new kappa) #2817 (transfer-coupled (5,1,1) at Q=7) studies the same readout in the transfer-coupled interface. That interface is outside this package's scope, so no number is compared.

CI notes

At head 1cf36dd5f4d484a6c764f2b73f4f1e21c3eb3108, both hosted checks passed, plan and replay (research/partitioned-walk-pair-sum-floor), in run 40608745282.