The Draken Framework v5.0: A Hodge-Theoretic Coherence Theory for Multi-Scale Systems

Version 5.0 of the Draken monograph rebuilds the framework on one mathematical spine: the discrete Hodge decomposition of cochains of a cellular sheaf on a finite 2-complex. Relational data split orthogonally into a gradient part (removable by re-gauging local states), a harmonic part (the Draken invariant, a class in H^1) and a curl part (curvature on declared faces). Local states are scored by a resolvent coherence, the normalised cost of the cheapest coherent repair, reported with a declared stiffness and reference. Original to this version: the anti-totalisation theorem and admissibility rule (deleting relations always raises unanchored coherence, so coherence gains are admissible only on a fixed base, while anchored compliance under a declared load is punished by the same deletion); the principle that only relational data carry H^1; face-filling reclassification of obstruction; the extension of the invariant to arbitrary relational data and the identification of lifting obstructions with integration obstructions; and a measurement discipline (reporting standard, debiased estimator, noise floors, and the result that coherence is not soundness). Standard results are reproved in this notation and classified in Appendix G (23 standard, 13 elementary, 9 original). Graded bridges connect the framework to clock synchronisation and the Sagnac gap, Berry and Aharonov-Bohm holonomy, CSS and Tanner codes, Kuramoto synchrony, structural balance, Gaussian Markov random fields and score-based diffusion, trophic levels and Leontief planning, and Rayleigh monotonicity. Thirteen definitional forks in the corpus are settled and thirteen errors in v4.4 are corrected. Files: the monograph as PDF and DOCX, and a source archive with the Markdown source, the reproducibility script that recomputes every worked example and figure, the audit script, and the site rendering. Revised after two external reviews. Supersedes v4.4. Companion corpus: draken.info.