Independent Sets on Trees: Consecutive Log-Concavity Failures and an Exact Covariance Threshold

Abstract

This preprint combines three investigations of independent sets on finite trees. It proves arbitrarily long consecutive blocks of strict log-concavity failures in a fixed family of unweighted subcubic trees. For every fixed positive epsilon less than one, a suitable fixed terminal path gives at least a constant multiple of n to the power one minus epsilon consecutive failures along that family. The coefficients in the resulting blocks are nevertheless decreasing. The paper also develops a message-based covariance bound and then proves its exact activity threshold: (500 + 225 sqrt(5))/64, approximately 15.6736764834. The threshold is the same for uniform activity and for positive vertex activities bounded by a common ceiling. At the critical value, the worst normalized covariance norm on trees with at most n vertices grows on the order of the square of log n. Full proofs, counterexamples to specific proof shortcuts, and exact finite, rational-interval and symbolic checks are included. These results do not prove or disprove the general tree independence-sequence unimodality conjecture. The universal assertions rest on the written arguments, not the finite tests. Prior constructions and external results are credited; global priority, independent peer review and proof-assistant certification are not claimed. Author: Seth Christopher, Independent Researcher. Version 1.0.0. Manuscript source and evidence accompany the paper; verification software is supplied separately. CC BY 4.0 applies only to author-held rights; third-party rights remain unchanged. AI assistance: ChatGPT substantially assisted with mathematical development, proof drafting, literature work, verification software, testing and manuscript preparation. Seth Christopher directed the research and is the sole listed author responsible for the content.