Show a restored third bank does not add an unpaid coordinate.

[unpaidmoon]

Change and mathematical scope

Scoped obstruction for the third-bank cycle on the coverage-law card. No new κ. The selected frontier stays

κ = 836330098538459/500000000000000000 (0.001672660197076918).

On the nested binary frames of the inherited C_q contract, the unpaid count equals the initial rank and the acquired count equals the rank difference. The selected family starts at one coordinate and acquires 72. A restored dirty bank acquires nothing, and copying the datum into a second bank acquires one paid direction, so the family unpaid count stays 1. One or two extra rank-1 children per live family, still at unpaid 1, put the ideal idle relaxation strictly above 1 at the selected saving. The unpaid-2 ceiling remains the separate 3.77× lever.

This is a dimension count plus a ceiling call. It is not a Clifford replay, a plane-port decoder, or a replacement coverage identity.

Task and overlap

Base commit bb4bccbf42844d2801487450727c2f63e10d1d0f. Task coverage-law-change, scope refined: the card asks whether any template can leave two unpaid coordinates; this package answers only the cycle X → Y → Z left open in research/multipair-schedule-bound/README.md. A two-dimensional initial frame is the plane-port contract and is not repriced.

The two-unpaid lever is the largest ideal packing multiplier in research/complex-local-word-ceiling/levers.py. It applies when the initial data frame has dimension two. A third bank that does not enlarge that frame leaves the deficit at 320/9.

Nearest neighbours, and what this adds:

• PR Cap line-port replicas at one unpaid coordinate per family #3561 caps a line-port replica offset at one unpaid coordinate per family (896). This package does not rerun that offset.
• PR Record the two-unpaid idle mass #3757 records that two-unpaid idle mass 13918/9 is not an integer histogram on one call. The same idle mass appears here as the one-step excursion, with the deficit kept at 320/9, so the ceilings move in opposite directions.
• research/plane-port-copy-budgets prices the actual unpaid-2 contract at about 160–164 copy charge per call. PR Price the plane-port u=2 screen against the rank-8 cell's 16-child sl… #4156 prices a plane-port u=2 screen against rank-8 slack. Neither is repeated.
• Centre-path, connector-alphabet, and shear packages change other levers.

Live inputs are pinned in research/three-bank-restored-unpaid/SOURCES.json. The domain is four explicit frames on F_2^73 plus the selected word at grid 1/10^12.

Validation

Python 3.12, standard library only, from the repository root:

python3 -B research/three-bank-restored-unpaid/verify.py
python3.12 scripts/research_package.py --package research/three-bank-restored-unpaid

verify.py prints

PASS_THREE_BANK_RESTORED_UNPAID restored_adds_unpaid=false one_step_exceeds_selected=true no_new_kappa

and requires RESULT.json to match a fresh run. Rejection controls: a restored bank with positive acquired rank, a copied bank counted as unpaid 2, and a one-step excursion that still meets the selected saving.

Certified high endpoints: selected 169005411/100000000000, one-step 1659158299/1000000000000, two-step 1629372687/1000000000000, unpaid-2 6377984109/1000000000000 (ratio 708664901/187783790).

Not run: Clifford phase replay, the 946 local frames, a native tape, and any Lean or Docker suite.

Credit

• themilkman216, research/multipair-schedule-bound, for the third-bank question.
• Douglas Colkitt, research/complex-local-word-ceiling, with Claude (Anthropic) assistance, for the ceiling and the 3.77× lever. Apache-2.0.
• The nested Fourier-rank rule and the C_q endpoints are research/split-centres-global/proofs/INHERITED-COMPLEX-GEOMETRY.txt; predecessor attribution stays in that file.
• jwr404, PRs Cap line-port replicas at one unpaid coordinate per family #3561 and Record the two-unpaid idle mass #3757, for the line-port cap and the 13918/9 idle integrality fact.
• Plane-port copy budgets and PR Price the plane-port u=2 screen against the rank-8 cell's 16-child sl… #4156 remain separate.

Prepared by unpaidmoon with substantial Cursor Grok assistance for the proof check, the ceiling calls, and the package text. New work is Apache-2.0.

Open next questions

See research/three-bank-restored-unpaid/README.md.

1. Can a non-nested erasure and rewrite leave two unpaid coordinates without becoming a plane port? First step: one explicit three-coordinate schedule and dim(initial ∩ final) after every paid climb. Blocker: the cleared-source example here has intersection zero, and no other candidate is supplied.
2. If the third bank is a second initial direction, does any scalar endpoint pass the paired-plane obstruction inside the copy budget near 160 per call? Blocker: this package has no scalar endpoint and no native copy program.

Workflow feedback (optional)

The multipair README already names this cycle as the 3.77× lever, which made the scope check short. The useful split was keeping the bank's own rank separate from the family's initial frame.