[unpaidmoon]

Change and mathematical scope

Scoped obstruction. No new κ. The selected conditional exponent stays 605457647/200000000000 (0.003027288235).

A one-dimensional birth cannot leave two unpaid coordinates under a non-nested erasure and rewrite. For any finite sequence of subspaces A_0, ..., A_t, both dim(A_0 ∩ ... ∩ A_t) and dim(A_0 ∩ A_t) are at most dim(A_0). The selected C_q line therefore never becomes a plane port by rewriting a cleared direction.

On F_2^73 the explicit schedule ⟨e_0⟩, then ⟨e_1⟩, then the whole space has surviving-birth count 0, naive endpoint intersection 1, and dimension charge 2 + 72 = 74. The nested climb ⟨e_0⟩ → F_2^73 still pays 72 and keeps unpaid count 1. That is the 72W term of the coverage identity. A plane birth ⟨e_0, e_1⟩ does keep unpaid count 2; that is the separate plane-port contract and is not repriced.

rank(A, B) = dim(A+B) - dim(A∩B) agrees with the nested Fourier-rank sentence when A ⊆ B. Off that hypothesis it is only the dimension charge used in the paid-rank induction. A cheaper physical step would be a different acquisition cost. It still could not raise either intersection above the birth dimension.

This is a finite certificate of that ledger plus the general induction in PROOF.md. It is not a Clifford replay, a compiler theorem for non-nested steps, or a multiplication theorem.

Task and overlap

Base: 32890bc5de034ab8fc93ac88ead7279a6a416d04 (upstream CrocSwap/integer-mult-bounds main). Priority revision 7. Card coverage-law-change, scope refined.

Ordered eligible list for session r45, item 13. Order: the priority table in docs/research/current-priorities.md, then catalogue tasks that are not already in that table. A row stays while revision 7 still states its next decision as open. Work this account already delivered is not repeated; a later row may still be taken when the piece is distinct.

1. uniform-transfer-audit
2. supplier-boundary-audit
3. joint-query-controller
4. r437-carrier-integration
5. p104-side-correction
6. independent-moment
7. formal-interface-lemma
8. bit-helper-contract
9. bit-new-register-model
10. complex-rank-shape-decoder
11. complex-new-helper-inventory
12. centre-path-sharing
13. This package. coverage-law-change

The nested third-bank cycle on this card is merged research/three-bank-restored-unpaid (upstream CrocSwap#4164). Its open question 1 asks whether a non-nested erasure can leave two unpaid coordinates without becoming a plane port. This package answers that question. It does not rerun the nested frames, the excursion ceilings, or research/plane-port-copy-budgets.

No open PR states this surviving-birth bound. Fork branch cursor/three-bank-unpaid-obstruction-cc55 is the nested predecessor, not this erasure.

Validation

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

python3 -B research/coverage-birth-erasure/verify.py
python3 scripts/research_package.py --package research/coverage-birth-erasure --check-only
python3.12 -B scripts/research_package.py --package research/coverage-birth-erasure
python3 scripts/check_research_commit.py --package research/coverage-birth-erasure --check-source-history

Observed:

PASS_COVERAGE_BIRTH_ERASURE line_unpaid_at_most_1=true erasure_creates_second_unpaid=false no_new_kappa
PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
PASS_REGISTERED_RESEARCH_PACKAGE research/coverage-birth-erasure outcome=negative-result
PASS_RESEARCH_COMMIT_BYTES

verify.py checks 4,096 length-3 histories on F_2^3, 67,335 line-born length-3 histories on F_2^4, the step lemma (1,056 and 34,371 instances), the explicit width-73 erasure, and 400 random length-6 histories on F_2^8. An optimized rerun passed. The induction is in PROOF.md. Not run: Clifford phase replay, the 946 local frames, a native tape, Lean, or Docker.

Credit

The erasure question is open question 1 of research/three-bank-restored-unpaid. The nested Fourier-rank sentence and m = 73 are research/split-centres-global/proofs/INHERITED-COMPLEX-GEOMETRY.txt. The third-bank question was left by themilkman216 in research/multipair-schedule-bound. Plane-port budgets are not recomputed.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0. No independent human review or proof-assistant acceptance is claimed.

Open next questions

1. Does any checked non-nested compiler step have Fourier rank strictly below dim(A+B) - dim(A∩B)? The unpaid bound does not need that formula. The paid-rank lower bound does. First step: compare one of the geometry note's 3,120 used transitions with this dimension count. Blocked because clifford_exact.py is not replayed here.
2. At the current selected κ, does the existing plane-port copy budget still fail the complex word? That birth is the remaining way to obtain unpaid count 2 inside this ledger. First step: rerun the pinned plane-port budget at 605457647/200000000000. Blocked because that package pins an older selected record, and this package does not reprice ceilings.

Workflow feedback (optional)

The merged third-bank README already named the erasure question with a first step and a blocker. Answering that question was a smaller and more reviewable piece than reopening the two-unpaid lever.