M-Compatible Gauge Classification for Banach Presheaf Coboundaries: The ℓ¹ Maximal-Symmetry Endpoint

Abstract

Title. M-Compatible Gauge Classification for Banach Presheaf Coboundaries: The ℓ¹ Maximal-Symmetry Endpoint Abstract. For Banach-presheaf coboundaries over a finite poset, any faithful edge-additive diagnostic that is compatible with all bounded linear endomorphisms of each stalk is forced to a weighted ℓ¹ form over native stalk norms. The unweighted global ℓ¹ form follows only under edge-uniformity, single-orbit, or explicit normalization. The result characterizes one maximal-symmetry observer class; weaker monoids admit larger gauge families (M-compatible taxonomy). CPTP monotonicity alone is treated as a compatibility condition, not as a theorem selecting the trace norm. Why a researcher should care. Presheaf gluing failures, constraint residuals, projection defects, and distributed consistency errors all choose a diagnostic norm. That choice is not free: under maximal local linear symmetry plus edge-additive aggregation, weighted ℓ¹ is selected. The paper separates that endpoint from non-claims about physics or universal norm uniqueness. What this does not claim. No finite-structural universal ℓ¹; no CPTP→trace-norm uniqueness alone; no collapse of weaker monoids to native stalk norms; no unweighted global scalar without uniformity; no projection short-time quantum bound (PO-000); no governance authority. Key closed forms (replayable). Weighted edge sum of native stalk norms under P1–P3; uniform collapse under extra hypotheses; monoid counterexamples in the Python suite (EX-class gauges). Reproduction. python -m pytest code/tests -q -p no:debugging Version / DOI chain. - Concept DOI: `10.5281/zenodo.18897080`- Latest published (v4.6): `10.5281/zenodo.21763610`- Historical (v4.2, often cited): `10.5281/zenodo.20564214`- This deposit: **new version** (suggested label v4.7) — write back DOI after mint