Paper 1. Fifteen Cards by Hand: the Histogram Route

Osman, Mohamed A.

Paper 1. Fifteen Cards by Hand: the Histogram Route

A hand proof of g(6) >= 16: degree bounds, a lemma on edge covers of K7, and thirteen degree histograms

Mohamed A. Osman · ORCID 0009-0004-5912-999X · independent researcher · 2026-10-19

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Abstract. Symbol degrees are forced into {2,3,4}; a finite lemma about edge covers of K7, proved by hand and also checked exhaustively, supplies the two hardest degree exclusions; a weighted counting inequality cuts the possible degree histograms to thirteen by a short count; and the thirteen are eliminated by incidence arguments. Hence g(6) is at least 16, with every step a hand argument.

No priority is claimed for any result, and this is not presented as a settled answer. The lower bound is computer-assisted and has not been independently reviewed by a human or by a proof assistant. AI tools were used throughout the work and its review; every computational claim is regenerated by a script or certified by a proof file in code/ and results/ of the repository. Licence: CC BY 4.0 (text), MIT (code).