Exact SSUF scenario-count ladder on the four-terminal planar gadget
openconfidence 70%
VibeMathed reports this item as open. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Precise statement
For the released four-terminal planar acyclic single-source unsplittable-flow gadget, require one unsplittable routing to be no more expensive than a prescribed fractional routing under each of $m$ strictly positive full-demand route-cost-difference scenarios. What is the worst-case normalized additive upper-arc deviation? The v0.3.0 fixed-gadget scenario-count ladder proves
$$\beta_G^{(m,+)}=
\begin{cases}
\dfrac{299-41\sqrt{41}}{32},&m=1,\\
\dfrac{17}{8},&m=2,\\
3,&m=3,\\
4,&m\ge4.
\end{cases}$$
The $m\ge2$ values are non-attained suprema; no attainment or non-attainment assertion is made for $m=1$. The one-scenario value also holds for legally realizable signed and zero route-cost differences; no such multi-scenario extension is claimed.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
GPT-5.6 Pro; Codex
GPT-5.6 Pro generated and developed substantial portions of the construction and parameter searches, symbolic derivations, analytic proofs, exact verifiers, adversarial analyses, and manuscript drafts. Codex sessions implemented or reconstructed code paths, performed repository-level checks, and attacked proposed claims. Matthew Protti selected and framed the targets, required exact arithmetic, exhaustive finite checks, mutation tests and hostile review, identified and forced correction of material errors and overclaims, chose the final theorem scope and public wording, reviewed the work, and authorized the release. Role-separated model review reduced some shared-context risk but was not independent human verification.
The unrefereed v0.3.0 release contains complete analytic proof sources, exact certificates, deterministic replay, claim ledgers, and review evidence for the fixed-gadget ladder. The $m=2$ rung was first released on 28 July 2026 in immutable v0.2.1; Matthew Protti's no-error review covered that corpus only. v0.3.0 adds the exact $m=1$, $m=3$, and $m\ge4$ branches while preserving the earlier source. Exact-arithmetic checks, exhaustive finite enumeration, mutation tests, claim-scoped role-separated AI reviews, a blind independent reconstruction of the fixed finite atlas, and one scope-limited human-directed AI-assisted adversarial report are included. The latter did not inspect every theorem component. No Lean proof, full independent expert verification, or journal peer review is documented, so the catalog status remains Unreviewed. The analytic proofs are authoritative; computation is corroborative.
Fixed four-terminal planar DAG; positive route-cost differences for m≥2, with non-attained suprema; signed/zero only for m=1, whose attainment is unstated. No topology-wide/unrestricted-planar claim.
Known method families
argument (source-reported)
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
What remains uncertain
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
VibeMath has not independently verified the mathematical claim.
AI-attempt independence and training-data exposure are unknown unless explicitly documented.
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.