Mathematical Foundations of the Danieletto Axiom (Paper 0 of the Danieletto Program)

Axiomatic foundations of the Danieletto Program. From four structural axioms of a critical elastic (Cosserat) medium -- existence of the medium, a stored-energy/constraint functional pair, elastic stability, and the marginal no-scale variational principle dT/dF=0 -- supplemented by two explicitly flagged inputs (a biaxial-chiral constitutive class and a geometric-quantization postulate), eight theorems derive the field content of the Standard Model together with gravity as the emergent low-energy physics of the medium: relaxed existence of the critical vacuum, a forced no-scale attractor, the topological classification and selection of the stable defect, canonical quantization with spin-1/2 and Fermi statistics, Einstein gravity, the gauge group (SU(3)xSU(2)xU(1))/Z6 with forced hypercharge, exactly three fermion generations, and the Born rule as quantum equilibrium. Every statement carries an explicit status tag and a master debt ledger isolates the two structural inputs. This is the foundational reference volume for Papers I-IV.