[petalas]

Result

Scoped proof, every hub count h ≥ 1. No new κ. This answers Open next question 1 of unpaidmoon's #4145 (research/p112-third-hub-extremal). The model is unchanged: the relaxed literal actual-reader model on the h-hub star, over any field of characteristic zero. In it, the complete extremal table has 2(3h(h+1) + 2^(h+1)) rows, and the minimum charge is

16h(h+1) + 2^(h+3),

so the excess over the one-unit baseline is exactly 4h(h+1) for every h. The charges are 48, 128, 256, 448 and 736 for h = 1..5. #4145 proved the lower bound for every h, but equality only for h ≤ 3, by subset census.

What is new:

• Readers. Eliminating each hub coordinate V_j turns every minimal reader into a circuit through E in one fixed seven-vector configuration {E, G, E+F, E+F+G, E+G, F+G, E−F} of a three-dimensional space, with distinct hubs assigned to the last three vectors. That configuration has exactly nine such circuits, which gives the Record the three-hub extremal actual-reader charge. #4145 reader list, 3h(h+1) per mark, for every h.
• Sources. Every maximal source is the zero set of a functional φ with φ(E)=1, φ(F)=0 and φ(G) ∈ {0, −1}. Each hub then picks one of two bundles, giving 2^(h+1) per mark.
• Costs. Outside families A–C, the evaluator with φ(G) = 0 when g is present (else −1) and every φ(V_j) = 0 leaves exactly one query nonzero. Every source has a one-hub reader of cost 1.

The cost-two lower bound for families A–C is cited from #4145, not reproved.

Evidence and reproduction

Package: research/p112-hub-excess-all-h (PROOF.md, verify.py, test_controls.py). Interpreter: CPython 3.12 (~/git/integer-mult-bounds/.venv/bin/python3.12), run from the repository root:

python3.12 -B scripts/research_package.py --package research/p112-hub-excess-all-h
python3.12 -B scripts/check_research_commit.py --package research/p112-hub-excess-all-h --check-source-history
python3.12 -B research/p112-hub-excess-all-h/verify.py                                  # ~9 s
python3.12 -B research/p112-hub-excess-all-h/verify.py --hmax 5 --closed-form-hmax 3    # ~40 s
python3.12 -B research/p112-hub-excess-all-h/test_controls.py                           # <1 s

Local results: PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed (about 52 s for all three registered checks), PASS_RESEARCH_COMMIT_BYTES, PASS_HUB_EXCESS_ALL_H ... h4:rows=184,charge=448,excess=80 ... h5:rows=308,charge=736,excess=120 and PASS_HUB_EXCESS_ALL_H_CONTROLS mutations_rejected=3.

verify.py checks the finite core of the proof exactly over Q: the nine reduced circuits (all 63 subsets), the excluded three-query hub, the per-hub source cases, and every cost-one witness on the closed-form table for h ≤ 12. As a control, it enumerates every query subset for h ≤ 4 (h = 5 is a separate registered check), compares the census with the closed-form table, and recomputes every row cost by exhaustive paid-set search. The census is a control. The proof does not depend on it.

Limitations

• This is a statement about one finite relaxed row model. It is not a physical tariff, a supplier invoice or a κ change.
• The argument needs characteristic zero and does not cover characteristic two, where E−F = E+F.
• The cost-two lower bound for families A–C comes from Record the three-hub extremal actual-reader charge. #4145. This package only recomputes those costs for h ≤ 5.
• Not run: Lean, Docker, any selected cold replay. No independent human or proof-kernel review is claimed.

Open next questions

1. In characteristic two, E−F = d, so the reduced configuration gains a coincidence. Does the extremal table or the excess change? First step: rerun the 63-subset reduced-circuit check over GF(2) and GF(4).
2. For which core triples and hub vectors does the hub excess stay quadratic in h? First step: parametrize the seven reduced vectors and enumerate their circuits through E.
3. Does the quadratic excess transfer to a physical actual-reader tariff? This needs a bound caller, not more census.

Related open work

No open PR studies this star, its extremal table or its hub-count excess. A search of open PR titles for hub, extremal, reader and 4h turned up only #5721 (a Git blob reader, unrelated).

• Prove P112 quadratic extension threshold is exactly 7 (no new κ) #5957 (P112 quadratic extension threshold): also named P112, but it studies line/frame quadratic extension under deletions, not the hub star or its row model. No quantity is shared, so no numbers are compared.

Credit

Predecessors: unpaidmoon's #4145 (the table through three hubs, families A–C, the lower bound), and Nick Petalas's #3573/#3583/#3698 (star, second hub, row model), which carry Philippe Van Velzen's expanded identity, ctoth's coverage question and the actual-reader lineage. Base 19fdb086ea96af9a9f87936a2b888e10af4ed810, priority revision 8. Claim: #1166 (comment)

CI notes

All checks passed for head a74c52264aef8540d128485e5442da7493deb1bd: plan (4m35s) and replay (research/p112-hub-excess-all-h) (5m29s), run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40530725484.