Scoped negative result: one frame relation cannot save paid rank at any register count

[bbMisk]

Scoped negative result; no new kappa.

This extends Ali Romman’s PR #468 at-most-one-relation argument from three scalar registers to arbitrary register counts in the abelian rational-address bit model. If the k paid rank-one frames have rational span at least k−1, every diagonal monomial target is a sum of a disjoint subset of the paid frames. Thus the sum of target ranks is at most k, including any extra registers restored on arbitrary dirty inputs. Targets need not be invertible or triangular.

For example, four rank-four targets cannot use 15 paid calls in this class. A 16-call attaining fixture is included. Two or more frame relations and different operation alphabets remain outside the result. The proof handles cyclic projector patterns with incoming-only or outgoing-only extra roles; a zero projector diagonal does not make such a role inactive.

The package includes a uniform written proof, exact source pins, and two separate implementations: packed coefficient matrices versus per-entry Laurent exponent sets, with different catalogue and rational-rank algorithms. They agree on all 155 projectors at sizes 1–4, 577 conjugations, 13 rational formal-word fixtures and nine hypothesis/scope controls. These finite checks support the proof; they are not a formal verification of its arbitrary-register statement.

Reproduce with Python 3.12 and the standard library:

python3.12 scripts/research_package.py --package research/bit-one-relation-seed04

Expected: PASS_REGISTERED_RESEARCH_PACKAGE research/bit-one-relation-seed04 outcome=negative-result.

Local validation passed: registered Python 3.12 replay, isolated optimized Python 3.14 replay, three corruption/relocation tests, 20 research-workflow tests and five selected-record tests. The selected supplier, frontier and upstream files are unchanged. Native/hosted supplier admission and a full multiplication theorem are not claimed.

Exact base: cd91ac8e2652bfee6bf09da3b8724253bcf78f5f; OS assignment seed 4. TASK.json records scope selection, overlap coverage and open next questions. SOURCES.json pins the predecessor and provenance. Only the new research directory is added.

Credit Ali Romman / PR #468 for the proof strategy and adapted eight-call scope fixture, eumemic / PR #288 for the model, and huxint / PR #203 for earlier common-flag work. Prepared for Hanson Sun (@bbMisk) with substantial OpenAI Codex assistance in derivation, implementation, checks and writing; Apache-2.0. Separate implementations do not imply independent human review.

Proof and finite certificate.

[Mister-Meeseeks]

100-PR batch review at 2971637b4a291c355353742f3f187ad1b1e4c58d.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Arbitrary-register one-relation rational-frame rank bound, using the Laurent specialization argument and exact finite controls. Noncommuting and multiple-relation families are outside scope; parallel PR1456 credit retained.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.