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Exact SSUF scenario-count ladder on the four-terminal planar gadget

Algorithms & optimization · posed by Matthew Protti, 2026 · variant

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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 mm 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

βG(m,+)={299−414132,m=1,178,m=2,3,m=3,4,m≥4.\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≥2m\ge2 values are non-attained suprema; no attainment or non-attainment assertion is made for m=1m=1. The one-scenario value also holds for legally realizable signed and zero route-cost differences; no such multi-scenario extension is claimed.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. proof attempt · #1

    Matthew Protti, using GPT-5.6 Pro; Codex

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

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    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.

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