Cubic Median Inequalities and the Power Conjecture of Liu (2024)

This consolidated preprint proves Liu's cubic median inequality with optimal coefficient 1/72 and classifies the universal positive-exponent set as a closed interval with endpoints approximately 0.0589422833021817898 and 4.6683652379684839567. The positive branch concerns all nondegenerate triangles. The reversed negative-exponent branch holds on nondegenerate nonobtuse triangles. Equality cases, counterexamples, exact coefficient certificates and outward interval certificates accompany the computer-assisted proofs. The original conjectures and earlier median inequalities are credited to Jian Liu. Consolidated preprint edition 1.0, dated 5 November 2026. The main PDF and its source/certificate archive form one article. Conceptual origin and AI assistance. The initial idea and conceptual starting point of this investigation are based on the framework of the original geometric paper by Christos Panorias and Kostandina Theodhori, Canonical Parametric Representation of Plane Euclidean Geometry (DOI: 10.5281/zenodo.23837735). This attribution concerns the conceptual origin and methodological organization of the research; every mathematical premise required by the present proofs is stated explicitly here. OpenAI ChatGPT/Codex assisted verification of calculations, editorial polishing and selected, delimited arguments during proof development, including substantive mathematical reasoning in those arguments. Christos Panorias is the sole named author and retains responsibility for reviewing, adopting and submitting the mathematical claims and final text. Research preprint for scholarly scrutiny. Independent peer review, proof-assistant formalization and worldwide priority are not certified.