[petalas]

Result

Claim type: scoped written proof with exact finite corroboration. No κ change; the selected record is untouched.

A paid fusion replaces paid ranks b_1,...,b_m by their sum and leaves the productive multiset T fixed. Write M_H(c) = Σ_{h∈H} h^c. For every c ∈ (0,1):

1. Strict subadditivity. (a+b)^c < a^c + b^c for all integers a,b ≥ 1, with equality at c = 1 and the inequality reversed for c > 1. So every paid fusion strictly lowers M_P(c) on the whole sub-unit range, and every productive split strictly raises M_T(c) there.
2. Sharp single-fusion cap. D(a,b) = a^c+b^c-(a+b)^c satisfies 0 < D(a,b) < min(a,b)^c. The constant one is sharp: D(a,b)/a^c → 1 as b/a → ∞ with a fixed, and is never attained. At a fixed total paid rank r = a+b, D(a,r-a) is instead maximized by the balanced split.
3. Total fusion budget. Replacing any partition of a paid multiset P by its block sums gives M_{P'}(c) ≥ (Σ_{p∈P} p)^c, hence
   M_P(c) − M_{P'}(c) ≤ G(c) = Σ_{p∈P} p^c − (Σ_{p∈P} p)^c, with equality only for the one-block collapse. G(c) is shared by every paid-only fusion scheme.
4. Crossing consequence. A rewrite that only fuses paid jobs leaves the crossing exponent at most the root of M_T(c) = (Σ_{p∈P} p)^c. No choice of which jobs to fuse, and no number of fusions, passes that root.

Point 4 is the usable form: one scalar root decides a whole fusion family, with no word replay. It also shows why fusion pays at all here — only because the crossing exponent is below one.

Evidence and reproduction

The package is research/paid-fusion-moment-ceiling/. verify.py encloses every a^c in an exact rational interval (CPython decimal at 90 digits, correctly rounded per its documentation, widened by one unit in the last place; all later arithmetic exact rational) and then compares exactly. It certifies 12,631 comparisons with zero inconclusive cases:

• points 1 and 2 for all 1 ≤ a ≤ b ≤ 40 at c = 1/1000, 1/279, 363005817/100000000000, 1/2, 3/4;
• the m-block form for m = 2,3,4 on five block shapes per exponent;
• 0.99 < D(1,b) < 1 for b = 10^4..10^12 at c = 1/2, with exact Newton enclosures of both square roots;
• balanced-split optimality at fixed total rank against every other split for 3 ≤ r ≤ 60;
• the total budget on five explicit multisets and every partition shape, with full collapse attaining the bound, plus a non-fusion control where a paid split strictly raises the moment;
• rejection controls at c = 1, 2, 3/2 and for non-positive ranks or c = 0.

RESULT.json stores the SHA-256 of the canonical summary and of PROOF.md; the verifier recomputes both and fails on any mismatch.

From the repository root with CPython 3.12:

~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/paid-fusion-moment-ceiling
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/paid-fusion-moment-ceiling --check-source-history

Both pass locally at e077c646db493ac3dedebf35fda6e2950f35d7d9. The registered replay is {python} -B {package}/verify.py and runs in a few seconds.

Limitations

• The ceiling is a statement about the moment model the signed invoice evaluates. No word, caller, compiler, precision tape, ordinary-work budget or outer row is certified here.
• Only the paid side is bounded. The productive-side balanced-split optimum and halving test for a concave rank tariff is Prove the exact uniform-fee target refinement optimum and halving test (no κ) #6651; it is cited, not re-derived or replayed.
• No invoice instance is evaluated. Points 3 and 4 are a template that needs the exact T and P the gate consumes, and that rank convention is not yet pinned.
• x ↦ x^c concavity is classical. The new content is the quantitative cap in 2, the total budget in 3 and the crossing consequence in 4.

Open next questions

1. Apply the ceiling to one live fusion family: take the exact invoice histogram the interval gate consumes, collapse P, bisect the root of M_T(c) = (Σp)^c, and report whether the family's target is above it. First step: read the gate's own audit input rather than the invoice summary block of research/shared-wire-split-composition/FULL-RESULT.json, which is not interchangeable with it. Blocked by pinning that histogram's rank convention.
2. Does the budget stay separable when a rewrite fuses paid jobs and also adds productive credit? First step: state the joint model and look for a counterexample to separability.

Related open work

• Bound P104 public MOVE fusion contacts (0% κ change) #7312 bounds P104 public MOVE fusion contacts on an exact word; this package bounds the paid-moment effect of any such fusion, with no number compared.
• Bound BIT sender fusion at owned recipients (0% κ change) #7159 bounds BIT sender fusion at owned recipients; same relation, no number compared.
• Bound timed input support for serial MOVE fusions (0% κ change) #7171 bounds timed input support for serial MOVE fusions; same relation, no number compared.
• Bound fixed-output complex child consolidation (0% κ change) #7245 bounds fixed-output complex child consolidation; a consolidation is a paid fusion only if it replaces paid ranks by their sum, so the ceiling applies to that case and not to changed outputs.
• Exclude two simultaneous Type B phase transfers (0% κ change) #7323 and Exclude literal P104 helper entrance raises (0% κ change) #7319 exclude specific P104 transfers and entrance raises; those are local rewrites, not paid fusions, and are outside this bound.
• Bound signed invoice classes by a polarized extreme (no κ) #7325 bounds signed invoice classes by a polarized extreme; both packages bound invoice classes, but that one polarizes an extremal functional while this one bounds paid fusion, so no number is compared.

No common numerical quantity is computed in more than one of these packages, so no agreement or disagreement is asserted.

CI notes

Hosted CI did not certify this package. At head e077c646db493ac3dedebf35fda6e2950f35d7d9, run 43107374986, plan job 130814108478, failed in Onboarding helper tests on exactly one test of 760 (24 skipped):

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record

with AssertionError: 'make frontier7000-verify' not found in '# CI verification ...'. The replay job was skipped because plan failed, so nothing in this package was certified by CI.

The cause is on base main e54523517, not in this branch. I read both base files: certificates/selected-result.json names make frontier7000-verify, while docs/ci-verification.md still describes #6030. Neither file is touched here. The same sole failing test, the same assertion and the same 760 tests / one failure / 24 skipped appear in runs 43105694304, 43107053327 and 42905094314 from other heads. No fix, retrigger commit, workflow rerun or out-of-package edit was made; the repair belongs to the maintainers.

Local evidence at this head is unaffected: package verification, the registered replay, the committed-byte/mode/source-history preflight and the receipt-hash checks pass, with 12,631 certified comparisons and zero inconclusive cases.