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Scenario-Cover Geometry for a Four-Terminal Planar Unsplittable-Flow Gadget

Matthew Protti

For the released fixed four-terminal planar acyclic gadget with coordinatewise positive cost-difference scenarios, the normalized additive upper arc-deviation supremum is (299−41√41)/32 with one scenario, 17/8 with two, 3 with three, and 4 with at least four. The two-scenario value is not attained under scenario-wise cost non-increase.

Sources for this record

What was already known

An earlier planar SSUF counterexample is distinct. The project first established a one-scenario lower bound, then a fixed-gadget two-scenario 17/8 result.

AI involvement AI-centered

Protti chose targets, scope and corrections. GPT-5.6 Pro supplied substantive searches, symbolic derivations and analytic proofs, with other model sessions implementing checks.

The provenance record identifies substantive model-generated constructions and proofs retained in a human-directed research program.

Importance Modest

Determines the exact scenario-count ladder for this fixed gadget.

Limitations

This does not settle the unrestricted planar SSUF constant. Exact finite checks and an external review cover only parts of the analytic ladder; the involved SC-006 and high-heterogeneity arguments lack a full external reconstruction.

Technical verification Partial

The repository reports exact finite enumeration and certificates plus a scope-limited external human review. Neither covers every analytic step of the full v0.3.0 ladder; complete external reconstruction or proof-assistant verification is explicitly absent.

Independent checking

Finite fixed-gadget cells, witnesses, and scenario-cover calculations supporting the ladder. exact arithmetic, exhaustive enumeration, and regression checks

Checker
Project verification scripts and exact atlas
Scope
Finite fixed-gadget cells, witnesses, and scenario-cover calculations supporting the ladder
Basis
Reported by the linked checker

Duality and short three-/many-scenario arguments, excluding unpacked SC-006 and high-heterogeneity proofs. partial proof reconstruction

Checker
Unnamed external human reviewer, independence undocumented
Scope
Duality and short three-/many-scenario arguments, excluding unpacked SC-006 and high-heterogeneity proofs
Basis
Reported by the linked checker
Evidence source

Human understanding Partial

Protti's correction and scope decisions and the external reconstruction document understanding of the fixed-gadget approach, while the most involved SC-006 and tail arguments lack an external human reconstruction.

Correction and publication history