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

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exact-two-scenario-ssuf-bound-on-the-four-terminal-planar-gadgetAlgorithms & optimizationposed by Matthew Protti, 2026recorded: 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.

Context

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.

A sharp constant inside one gadget construction.

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  • #1

    Attempt 1

    proof attemptGPT-5.6 Pro; Codex with Matthew Protti ·
    AI involvement
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    GPT-5.6 Pro; Codex
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    Matthew Protti

    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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