[petalas]

Summary

Scoped written proof with exact finite corroboration, produced by Nick Petalas. Five identical A2 longest-element targets w0=sts=tst have exact formal cost 5 f(3) at paid scalar rank cap three, over every field and with arbitrary finite restored dirty spectators. The common rank price is nonnegative, nondecreasing and discretely concave, with f(0)=0.

- fixed disjoint batching: 6 calls, ranks (3,3,3,2,2,2), cost 3 f(3)+3 f(2)
+ changing supports:      5 calls, ranks (3,3,3,3,3),   cost 5 f(3)

At f(k)=k/(k+1), this is 15/4 instead of 17/4, saving 1/2. The new five-call support schedule attains the inherited 15-unit rank floor. It needs no scalar mixing or additional role; the lower bound still allows arbitrary scalar couplings, free scalar GL mixing and restored dirty workspace.

This answers one nondivisible-target case posed in Benny Zhao / zhaobenny's #6817. Credit #6817 for the all-field rank-cap model and lower bound, David Arias #2738 for its Coxeter predecessor, Nick Petalas #5083 for the older cap-two changing-support construction, ToikY #6821 for the distinct repeated-st tariff, and the retained Adam Gall, Ross Richardson, ywz65, eumemic, Ali Romman and Gareth Davidson lineage. The new claim is the five-target/cap-three schedule and exact tariff. Predecessor mathematical results are cited without replaying their scientific programs.

Advance CLAIM. The delivered scope matches it. Only research/a2-five-target-cap-three/ is added.

Evidence

Local CPython 3.12.15 on macOS arm64, standard library only, at head fe77116fcf5012533e6de748ab6089790f43ac37:

• 24 full signed formal endpoints over Q/F2/F3/F5, five/six/eight roles and two scalar gauges. Every one of 120 paid couplings has rank three and is idempotent; 78 genuinely couple productive and dirty roles, with complete scalar entrance/exit restoration.
• A separate GL2(F2) regular-action implementation checks 912 complete basis columns. Full endpoints reject all 120 call omissions and 120 label flips.
• 36 exact rational price schedules agree with the minimum of all 90 rank-count profiles of total rank 15 through 17, for 3,240 comparisons. The written cited lower bound carries the universal circuit statement; finite checks are corroboration.
• Warning-strict normal and optimized commands both print PASS_A2_FIVE_TARGET_CAP_THREE with identical newly generated hash-bound receipts. Each took 0.20 s in a single local timeout check. The registered replay ran both commands and printed PASS_REGISTERED_RESEARCH_PACKAGE in 0.65 s. These are timeout measurements, not a performance comparison.
• Committed bytes, modes and source-history ancestry print PASS_RESEARCH_COMMIT_BYTES. Source pins and package metadata pass. The staged diff and whitespace check pass, and the worktree is clean.

Reproduce from the repository root. Another installation can substitute its CPython 3.12 executable:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/a2-five-target-cap-three/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/a2-five-target-cap-three/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/a2-five-target-cap-three
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/a2-five-target-cap-three --check-source-history --check-modes

Limitations

This is a formal group-algebra tariff, not a native supplier, new kappa or complete multiplication theorem. Scalar coupling rank is not recursive child rank. Native realization, scalar-map costs, routing, ownership, storage, precision, stock, hosting, compilation and the complete supplier invoice remain unknown. No selected-frontier change is made; all five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed. Lean, Docker, the selected cold replay and predecessor scientific suites were not run. No local mathematical obligation is deferred solely to CI.

Open next questions

• Can a native implementation supply a rank-three call at a complete subadditive price? First specify its actual data/address layout and arbitrary-input entry/exit contract, then charge scalar maps, movement, ownership, precision and restoration. No such physical contract is supplied here.
• Which other nondivisible longest-element batches attain the packed floor? Start with a support-incidence feasibility problem at one declared target count and cap. This proof gives no general scheduling theorem or chronology obstruction. No follow-on scope is reserved.

Related open work

• #3893 studies a mixed three-role S3 endpoint with rank-one calls. This result uses five identical longest-element targets and cap three; it computes no common optimum with that PR.

The related-work refresh also examined #7178 and #7180 through #7184. These concern shell growth, Gaussian inverse norms, Clifford sectors, cube-zero lifts, radical tariffs and rational headers. None computes this five-target/cap-three tariff. No extra scientific comparison computations were run.

CI notes

At exact head fe77116fcf5012533e6de748ab6089790f43ac37, run 42992340683, plan job 130427991871, failed in Onboarding helper tests on exactly one of 760 tests, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record
AssertionError: '`make frontier7000-verify`' not found
FAILED (failures=1, skipped=24)

The job log shows that the current section of docs/ci-verification.md still describes #6030 and make frontier6030-verify. Inspection of canonical main e54523517d0f03e58352e647c403a5af3e525a57 confirms that its selected record instead names make frontier7000-verify. The test, document and selected record are unchanged by this six-file package PR.

The same failing test, missing command, 760-test total, one failure and 24 skips were confirmed directly from the failed-step logs of each of these runs:

• Other contributors: 42905094314 and 42903313790.
• Prove sharp multiplicative-shell growth for every rank alphabet (no κ) #7178: 42992088329.
• Lower P104 frame-0 hosting from 40 to 14 monomials (no κ) #7167: 42983769381.
• Prove the exact E8 Gram-class minimum through 24 deletions (no κ) #7168: 42983916692.
• Bound scalar COPY savings at one 93-line E8 puncture (no κ) #7170: 42984005087.

CI did not certify this package. Its replay job was skipped because plan failed. The local normal/optimized, registered-package, source-pin and committed-byte/source-history checks above remain the available automated package evidence. No fix, retrigger commit or job rerun was made; the repository-level documentation mismatch is left for maintainers.

Merge Danger

Door: two-way. Six isolated research-package files are added; shared code, frozen sources and selected certificates are unchanged.

Blast Radius: research.