[petalas]

Result

No κ claim, no word and no supplier change.

For a simple graph Γ on n points, G_Γ is the star-configuration graph of #7114: points and edges of Γ as vertices, a point adjacent to the pairs containing it, two pairs adjacent iff disjoint. A k-block pattern has invertible diagonal blocks and zero blocks on non-adjacent pairs; H_f(G;F) is the infimum of rank/k.

Theorem 1. For Γ the wheel W_4 (n = 5) and Γ = K_{3,3} (n = 6), every k-block pattern on G_Γ has rank at least (n+1)k, for every k and every field. With the Gram upper bound of #7114, H_f(G_{W_4};F) = 6 and H_f(G_{K_{3,3}};F) = 7 over every field. Both graphs have minimum degree 3 and contain no K_4.

Corollary 2. In a star configuration (K, Γ, (T_e)) with |K| = m, if Γ has a subgraph isomorphic to W_4 or K_{3,3}, every k-block pattern has rank at least (m+1)k; in the notation of #7114 Theorem 3', j >= k. #7114 proves this when every non-isolated vertex of Γ has degree at least 4.

Corollary 3. All 26 labelled graphs on five points with minimum degree at least 3 contain W_4, so Corollary 2 covers each of them.

Remark 4. H_f(G_{C_n};F) = n for every cycle, so the floor n+1 can fail at minimum degree 2.

Claim types: Theorem 1 is a finite certificate (two exact rational duals) combined with a scoped written proof that such a dual bounds every block pattern. Corollary 2 and Remark 4 are scoped written proofs. Corollary 3 is an exhaustive count. No statement is made for all graphs of minimum degree 3.

Evidence

• certificates.json: 86 dual entries for W_4 and 197 for K_{3,3}. Each entry is one instance of "a vertex adds at most one" or of elementary submodularity, with a positive rational multiplier.
• verify.py (standard library, exact Fraction arithmetic) averages every entry over the automorphisms of Γ (8 and 72), applies the decoding closure, and checks that the combination is exactly f(empty) >= n+1. It rebuilds the Gram labels as an integer matrix and checks unit diagonal and zeros on non-adjacent pairs. It enumerates the 27 labelled graphs of minimum degree at least 3 on four and five points, builds rank-n patterns for cycles of length 3 to 6, and runs 14 rejecting controls (damaged certificates, a deleted edge, a moved edge, a removed adjacency).
• README section 2 gives the rank argument: for a k-block pattern with row space W, f(S) = dim(W + E_S)/k satisfies normalization, slope, submodularity and decoding. This is the argument of Tuafo's Certify P111/P112 block-rank lower bound nine and exact binary broadcast rate #1744, restated for a general graph.

Reproduce

From the repository root:

~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/star-pattern-degree-three-floor
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/star-pattern-degree-three-floor --check-source-history
~/git/integer-mult-bounds/.venv/bin/python3.12 -I -B -W error -X dev research/star-pattern-degree-three-floor/verify.py

Any CPython 3.12 works in place of that interpreter path. The verifier ends with

PASS_STAR_PATTERN_DEGREE_THREE_FLOOR graphs=2 controls=14 cycles=4 census=27 no_new_kappa

Local results on commit a8ae38561deb40edbd41b9f8790fab6ac949c0a3: both registered commands pass (normal and -O), about 0.4 s each; PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed; PASS_RESEARCH_COMMIT_BYTES with 8 files.

derive_certificates.py shows how the duals were found. It needs numpy and scipy, is not registered, and the verifier never imports it.

Limitations and unrun obligations

• Two graphs only. The general degree-3 statement is not proved.
• The prism is not certified: the floating-point optimum of its program is 7.0, but the rounded dual was rejected by the exact check. No bound is claimed for it.
• The linear program gives lower bounds only; it meets the Gram upper bound on these two graphs.
• The model is the whole-block mask of Port windows are a fractional Haemers bound: w >= 53/6 on P104 (no new κ) #6933 and P104 port window is exactly 9 in characteristic 2 (no κ) #7114. Sub-block masks and port-dependent block sizes are not covered.
• The orthogonal 8-sets of P104, P112 and P120 have minimum degree 5, 6 and 7, so their floor 9 does not need this result.
• P104 port window is exactly 9 in characteristic 2 (no κ) #7114 is open and unreviewed. Only its definitions and Gram labels are used, and the labels are re-checked here.
• The Petersen case (Γ = K_4) is not re-proved; its value 5 is cited from Blasiak, Kleinberg and Lubetzky and from Certify P111/P112 block-rank lower bound nine and exact binary broadcast rate #1744.
• No human or proof-kernel review is claimed.

Open next questions

1. Is j >= k forced for every star configuration whose Γ has minimum degree at least 3? First test: the cube Q_3, which contains none of K_4, W_4, K_{3,3} (20 vertices in G_Γ).
2. Does the prism have value 7? It needs an exact dual; the float support has 2,133 orbit variables.
3. What are the values at minimum degree 2 beyond cycles? A local, unshipped run of the same exact pipeline gave lower bounds 9/2 for K_4 minus an edge and 11/2 for K_{2,3}; no upper bound below n+1 was sought.

Related open work

• Prove the E8 ports-per-dimension law N_e = 2^(e-2) + 4 in closed form (no κ) #7127: the same port family (P120 and P104) but a different quantity, the number of ports in a subspace of each dimension. No number is shared with this PR.
• P104 port window is exactly 9 in characteristic 2 (no κ) #7114 (DreamingOfClouds): source of the question. It proves j >= k at minimum degree 4 and H_f(Petersen;F) >= 9/2, and lists the Petersen value as open. This PR computes neither number. The Petersen value 5 is already on main through Certify P111/P112 block-rank lower bound nine and exact binary broadcast rate #1744 and the cited literature.

Credit

DreamingOfClouds (#6933, #7114): block patterns, star configurations, G_Γ, Theorem 3', the Gram upper bound and the open question. Tuafo (#1744): the all-field rank argument and the exact Petersen dual. Blasiak, Kleinberg and Lubetzky: the linear program. Bukh and Cox: H_f. The two certificates, the corollaries, the cycle remark and the verifier were produced by Nick Petalas.

CI notes

CI did not certify this package. At head a8ae38561deb40edbd41b9f8790fab6ac949c0a3, run 42908142215 (job 130151433953) failed in plan, and replay was skipped because plan failed. The evidence for this PR is therefore the local runs listed under Reproduce.

Cause, read from the job log: the "Onboarding helper tests" step ran 760 tests (24 skipped) and exactly one failed,

FAIL: test_current_ci_section_matches_selected_record (test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record)
AssertionError: '`make frontier7000-verify`' not found in '# CI verification\n\n## Current selected-family replay: #6030 ...

On main at e54523517d0f03e58352e647c403a5af3e525a57, certificates/selected-result.json has "reproduce": "make frontier7000-verify", while docs/ci-verification.md still opens with "Current selected-family replay: #6030". The test compares those two files, and this PR changes neither; it adds only research/star-pattern-degree-three-floor/.

I checked the failed logs of six other runs from today. Each shows the same single failing test, 760 tests run and 24 skipped: 42905094314 and 42903313790 (other contributors' branches), and 42905202459, 42905330981, 42905806694, 42906716667 (#7125, #7126, #7127, #7128).

No fix, retrigger or re-run was made from this PR.