Hadamard Powers on Bipartite Cones and Block Decompositions

This preprint treats Hadamard powers on cones over bipartite graphs and on the stated bipartite/chordal block decompositions. It proves the sharp exponent threshold, including singular cases and the specified signed-apex extension, using factor-width and gluing arguments. The technical supplement preserves energy formulations, sufficient regions and rigorously identified limitations of stronger auxiliary methods. The factor-width power theorem is prior work of Yaroslav Shitov and is explicitly attributed. The arbitrary-graph GKR universal-vertex problem remains unresolved. Consolidated preprint edition 1.0, dated 5 November 2026. The main PDF and its source/certificate archive form one article. The accompanying technical supplement belongs to this 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.