[petalas]

Summary

Scoped written proof with finite corroboration: for two targets of each of m colors and capacity-three batches of a common fixed width, the exact batch/color-incidence frontier is

ceil(2m/3) <= B <= m
I_min(B) = 2m-B
bill = A*B + C*I

This gives the optimal grouping for every positive setup price A and color price C. At the declared exchange-only counts A=6 and C=12, minimizing batch count can cost a sharp factor 10/9 more: the smallest sharp example costs 60 instead of the optimum 54. The better grouping removes 10% of that exchange sub-bill. No new kappa or physical construction is claimed.

This is a new optimization step motivated by unpaidmoon's PR #5538, first Open next question. Its occupied-color tariff is cited, not audited or replayed. Credit Cinna Davis #4100 for the CRT construction/exchange counts, Hiroki KUMAZAKI #4174 for the layout, and Ryanson Jonathan #4329 for the guard/placement context. This contribution was produced by Nick Petalas. Apache-2.0.

Only research/paired-color-batch-frontier is added. PROOF.md gives the quantified lower bound and attaining family; verify.py supplies an independent complete small set-partition enumeration. Exact main-source provenance and the changed scope from the revision-8 uniform-transfer-audit card are recorded in SOURCES.json and TASK.json.

Evidence

• Before: Price CRT switch layers by occupied color. #5538 prices occupied colors but does not optimize joint batch/color statistics. Optimizing minimum batch count alone leaves the exchange tradeoff unresolved.
• After: I=2m-u and u<=B give the exact lower bound; an explicit assignment attains every frontier point. The m=3 witness separates minimum batching (B=2,I=4,E=60) from minimum bill (B=3,I=3,E=54). Equality in the 10/9 bound holds exactly when 3 divides m.
• Local warning-strict normal and optimized replays pass 64,098 capacity-three set partitions (m=1..5), 11,136 attaining groupings (m=1..256), 245 exact rational price cases against the complete enumerated optima, fourteen malformed-input controls, and all ratio equality tests through m=256.
• Every exact registered command was timed locally: 0.730 and 0.752 seconds, below its 120-second limit. The registered package runner passes. Six-file committed-byte, executable-mode and source-history preflight passes at 2235b3c.
• Official committed export and declared-input integrity checks pass. An optimized warning-strict replay from a read-only Git-free export with empty PATH passes, preserving all seven exported files' bytes and modes. Generated evidence receipts bind all six package-file hashes; no receipt is handwritten or used as a proof input.

From the repository root, using CPython 3.12 and the standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/paired-color-batch-frontier/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/paired-color-batch-frontier/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/paired-color-batch-frontier
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/paired-color-batch-frontier --check-source-history --check-modes

Expected outputs: PASS_PAIRED_COLOR_BATCH_FRONTIER, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES. Another CPython 3.12 executable can replace the absolute interpreter path.

Limitations

The result assumes unrestricted grouping, exactly two targets per color, capacity three and the same reservoir width in every batch. Only the six batch-placement and twelve per-color shuttle exchanges enter the 6B+12I example. No production layout is regrouped. Current target multiplicities, legal movement, stable position-mod-K colors, tree/control-reservoir constraints, additions, preparation, restoration and repair must all be established before a physical saving follows.

The unbounded result is a written mathematical proof; the finite enumeration corroborates it. Independent human and formal proof-kernel review are unclaimed. No native/full supplier replay, Lean, Docker or complete multiplication theorem check was run. The selected frontier and all five framework assumptions remain unchanged. No literature-priority claim for general colored packing is made.

Open next questions

• Identify one source-bound CRT region permitting fixed-width regrouping with stable colors; then price its complete movement, additions, restoration and repair. No such caller is supplied here.
• Determine the exact frontier for other capacities or unequal multiplicities, starting with capacity four and explicit multiplicity vectors. Physical application still needs the preceding source-bound region.

Related open work

• Audit uniform Beneš compiler with charged sequential arena scans #4192 studies the uniform Beneš control compiler and its sequential arena cost. This grouping theorem does not alter or certify that preparation tariff.

• Audit conditional CRT last-K two-slice streaming tariff #4187 studies the elementary last-K CRT branch and its streamed payload tariff. The fixed-width occupied-color grouping problem here does not modify or replay that branch.

• Prove sharp unit-fee target threshold at rank 288 (no new κ) #5909 studies a unit-fee target-rank partition threshold. This package keeps targets fixed and optimizes batch count and occupied-color incidence; the quantities differ, and no common numerical optimum is compared.

This account's open PRs created since the advance CLAIM at 2026-10-27T12:36:42Z were refreshed before DONE. The eight other contributions inspected were #5909, #5910 and #5912 through #5917. None computes this paired-color batch frontier, and no same-quantity numerical disagreement was found. No additional computations were run to validate those contributions.

The open-PR, Discussion and local reservation checks found no open PR computing this same paired-color frontier. Canonical main remained de5ec2e, priority revision 8, at the submission checkpoint. Cached note coverage is sparse; main README searches and live open/merged searches supplemented it, and unpublished work remains outside coverage.

CI notes

At commit 2235b3c6008da352f0a8c0f778d26d27dc2455f8, run 40506110737 finished with plan failed; replay skipped. The only failed step was Onboarding helper tests. All ten failure/error entries are in unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22).

The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The assertion failures concern frozen-card/current-route expectations, missing-reference handling and sparse omission lists. All ten failure/error headers and the final summary match PR #5909's plan log exactly. This confirms the shared onboarding blocker on main de5ec2e7a; maintainers will address it. No onboarding tests, shared files or package files were changed for this CI closeout.

The package replay was skipped. Hosted replay has not certified this package; only local scientific and publication checks have run. The local normal/optimized checks, registered commands, metadata/pins, committed-byte/source-history checks, official export integrity and read-only Git-free replay pass. This is not a green hosted workflow or mathematical acceptance.

Merge Danger

Two-way door. Blast radius: the new isolated research package. Shared code, predecessor files and the selected result are unchanged.