[petalas]

Summary

Scoped uniform proof with exact finite corroboration. For the binary signed two-port code with vertex stars and the empty star, every prescribed target graph with exactly one cycle in total, of odd length, has the exact deletion budget

delta_+(T) = 2p(p-1) - r - floor((2p-r+1)^2/4),  r=|T|, p>=3.

- PR669: general nonzero-constant lower bound and exact triangle case
+ Uniform attainment for every odd-unicyclic target set

Attached trees, other tree components and isolated vertices are allowed. A balanced root construction attains the bound despite the removed perfect matching, and explicit vector coefficients give every target its unit decoder. A five-cycle on six vertices needs exactly six deletions. The triangle specialization remains Sam Sung's credited result.

Only research/odd-unicyclic-star-budgets/ is added. This contribution was produced by Nick Petalas. Credit Sam Sung / the-missionary-company #669 for the model, graph-target reductions and inherited lower bound; Martinez-Bernal, Valencia-Bucio and Villarreal for the graph-incidence bridge; and the retained Joel Pulikkan / GamingPuzzled, icekylinx and eumemic lineage.

Evidence

Before, #669 supplies a general partition optimization and prescribed forest/triple results. After, PROOF.md gives a closed form and an attaining construction for this entire remaining feasible cyclic class.

• Exhaustive independent row-space oracle over all 4,096 retained six-vertex graphs. All 864 eligible prescribed target sets agree with the formula and literal constructor. Their counts at r=3,4,5,6 are 8,72,288,496; 128 require the root replacement.
• 2,704 larger fixed fixtures check 101,231 retained input columns, with odd cycle lengths 3 through 31. These are fixture cases with repetition allowed, not an exhaustive larger-graph census.
• Five invalid-input controls, three decoder mutations and all twelve single-edge restorations of the p=4 witness are rejected. The concentration inequality passes 1,600 integer pairs.
• CPython 3.12.15, macOS arm64, standard library only. Warning-strict normal and optimized commands pass with identical generated source-bound receipts. They take 0.311 and 0.315 seconds locally; these are timeout observations, not a performance claim. Both registered commands pass within their 120-second limits.
• Metadata/source pins, committed bytes, file modes and source ancestry pass at 4fece6ddc4dff91ea1bbfd820f6c447c9a38387c. Official export integrity and declared-input coverage pass. Both isolated normal and optimized Git-free replays pass from read-only exported files with an empty executable search path; all eight exported files remain byte-identical.

From the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/odd-unicyclic-star-budgets/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/odd-unicyclic-star-budgets/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/odd-unicyclic-star-budgets
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/odd-unicyclic-star-budgets --check-source-history --check-modes

Another CPython 3.12 executable may replace the local interpreter path. Expected markers are PASS_ODD_UNICYCLIC_STAR_BUDGETS, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Canonical base 721c03822c421b7f7fcbb2598993b947c4d88419 was refreshed unchanged before publication. The card's changed scope to an algebraic side result is explicit. Main, merged work, live open PRs, recent account contributions and Discussion claims were searched before claiming and committing. The cached remote search covered 407 open PRs and 5,828 Discussion records with complete pagination; local note coverage was sparse and was supplemented with Git searches over main. Search absence is not a novelty certificate.

Limitations and unrun obligations

The proof is uniform for all p>=3; the executable constructor's fixture interface is bounded to p<=20. The finite tests corroborate the written proof and do not establish its quantifiers by enumeration. Independent human and formal proof-kernel review are unclaimed.

This is an algebraic observation threshold. Physical helper acquisition, native frames, dirty restoration, precision, tape movement, recursive pricing and supplier admission remain unproved here. No selected supplier cold replay, Lean, Docker or complete multiplication theorem was run or claimed. No new kappa or selected-frontier change.

Open next questions

What is the exact C+ budget for a prescribed forest with r>p? Start with one p>=6 forest above the source's uniform no-benefit range and compare its C- matching optimum with the nonzero-constant bipartite bound. A forest does not force a nonzero empty-star coefficient, so both decoder types must be considered. This follow-up is left unclaimed.

Related open work

• #732 proves the uniform one-target proper-star distance. This contribution studies prescribed cyclic multi-target sets at k=2 with the empty star allowed, building on Research: exact star-decoder hierarchies, target budgets, and COPY bounds #669's graph model; it does not re-check Research note: sharp off-target proper-star puncture threshold #732's distance or implementation.

The refreshed related-work check found no additional open PR on this prescribed odd-unicyclic star quantity or a competing same-instance value. No extra numerical comparisons were run.

CI notes

Uncertified by CI. At head 4fece6ddc4dff91ea1bbfd820f6c447c9a38387c, run 40726363879 failed plan. The package job replay (research/odd-unicyclic-star-budgets) was skipped and did not run in CI. This hosted run certified none of the package's scientific checks. A maintainer re-run is needed after main is fixed.

The failed-job log confirms exactly two onboarding helper test failures, ending FAILED (failures=2, skipped=23):

• test_research_current_status.CurrentStatusTests.test_current_section_matches_selected_record expects the literal **0.003125804** in docs/research/current-status.md. The file instead bolds the decimal and fraction together as 0.003125804 = 781451/250000000, so the exact substring assertion fails.
• test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record expects make frontier6030-verify in docs/ci.md, which still describes PR5599 and make frontier5599-verify.

These failures are inherited from main commit 721c03822c421b7f7fcbb2598993b947c4d88419, Admit conditional PR6030 kappa after independent application audit [skip ci]. That admission changed the selected record without updating the onboarding documents to satisfy these checks. Neither failing document is changed by this isolated package.

PR #6120's run 40691776027 was checked directly and has the same two failures and exact FAILED (failures=2, skipped=23) summary. Open PR #6050 targets the documentation synchronization. Local scientific and publication checks remain passes. No documentation fix, new commit or push was made for this blocker.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Six new package files. Shared implementations, predecessor sources, upstream archives and the selected record are unchanged.