Research: sharp invocation tariff for rank-bounded UT3 block calls

[Arham-Tahir64]

The rank-one UT3 central-target floor does not determine the fewest invocations when a paid call may have scalar rank up to k. This package proves the sharp count 2 max(2,ceil(2w/k)) for every target count w and arbitrary finite restored dirty workspace, with a cyclic batching construction that simultaneously attains each letter's 2w rank units.

For w=3,k=2 the complete formal boundary uses six rank-two calls, versus twelve rank-one calls or eight disjoint-block calls. All charge twelve total rank units. The sharp nonnegative affine per-letter tariff is t(alpha_a+alpha_b)+2w(beta_a+beta_b), t=max(2,ceil(2w/k)): only invocation overhead can improve under rank-linear pricing. Scalar rank here is not an admitted supplier's recursive child rank.

Only research/ut3-rank-bounded-block-calls/ is added: written uniform proof, deterministic receipt, full formal and faithful regular checkers, source/task pins and an explicit replay hook. The changed hypothesis is paid rank-bounded couplings in the two-letter alphabet. No central letter, clean allocation, free address maps, physical supplier or new kappa is claimed.

Validation on Windows / CPython 3.12.14, standard library:

• Registered optimized replay and two CLI/evidence-integrity test groups pass, including normal/optimized source export, read-only checks, receipt regeneration and four certificate mutations.
• 102 complete forward/inverse dirty fixtures, 16,160 regular basis columns, 27,931 arbitrary short words, 900 formal/regular two-factor comparisons and 1,060 same-label decompositions pass.
• 62 omitted-call and 10 model/boundary controls reject, including altered dirt, undercharged rank and a nonzero formal perturbation hidden by the lower-bound quotient. 306 rational tariffs pass.
• Package metadata/Git provenance, selected baseline bindings and whitespace checks pass. The restricted Windows temp path used an explicit writable scratch directory; the earlier flag-placement test mistake was fixed before the final replay.

Reproduce:

python scripts/research_package.py --package research/ut3-rank-bounded-block-calls

In a restricted Windows sandbox set UT3_TEST_TMPDIR to an existing writable scratch directory. The theorem is the written proof; finite fixtures corroborate it. Hosted CI is separate from local validation. Full selected cold replay, physical/compiler/common-label/hosting/precision interfaces, nonlinear recursive tariff, outer transfer, Lean and independent human review remain outside scope.

Base current main 0b1cccf86e2f5132ed286a26ef0a681a495740ea; chooser nonce c79b0a54a0d8afe483e69e03ae03fa5b, revision 3, refined bit-new-register-model. Claim and handoff. Final overlap snapshot: 556 open PRs including drafts and 536 Discussion body/comment/reply records, fully paginated at 13:41–13:42 UTC; local notes, UT3 and D8 aliases checked. No novelty guarantee.

Credit steph4n-gh #1573 (0b8c5a69297c794f2de22c04494c83e41300ae36), Arham Tahir #1589 (010ec8c01c0edbbb08b541be236ab78d096f2ca1), Hanson Sun #1607 (9d6117918025cee6a0c5b7ef30ff8d936ca0621f) and Josie Elliston #1792 (cc3a4adbae23fb07687803d1361bb7a1bcd449b0) for the inherited rank/coefficient method and rank-linear floor. The new result is the invocation optimum and cyclic batching; rank-volume preservation is credited corroboration. woklerx #1841 (0c305488a4e27cd96edd61460bb5429ed5e8abf0) treats the distinct added-central-label premise. The README retains the broader predecessor lineage and source licenses.

Next step: realize and completely price the six-call rank-two boundary against twelve rank-one calls. No physical realization is supplied. Prepared by Arham Tahir with substantial OpenAI Codex assistance in proof, implementation, verification and writing; Apache-2.0; no subagents or independent human review.

[Mister-Meeseeks]

Batch review at f5610483cab0b5547be778a4e8983faf78aefa69.

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

Reviewed scope: All-register coefficient-span proof and cyclic block construction attain2max(2,ceil(2w/k)) calls with4w scalar rank mass.27931 short words,102 complete formal/faithful fixtures and306 rational tariff cases pass. Invocation savings assume a rank-capped block primitive; no recursive-rank or supplier saving follows.

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.