Density Mathematics: Numbers as Emergents of Density Clustering — Static Core (EN)

Abstract

Static core of Density Mathematics: numbers as emergents of density clustering. Objects are density points P=m·p (mass × shape); operations are merge ⊕ (mass conservation) and resonance ⊗ (mass = inner product; Bayesian contraction). Axioms A1-A4 with model-based independence arguments; full proofs of core theorems: T1 δ-alignment equivalence (standard arithmetic = the exact image of the δ-aligned subspace), T3 reduced semiring (no nilpotents), T9 Grothendieck completion (the unique generation path of subtraction/negatives, with the universal property). Main volume: discrete grid model (exact rational arithmetic, fully enumerable); continuous version conditional. Machine verification 24/24 (companion script, exact fractions). Includes one correction log (machine verification refuted an earlier independence argument, §4.R1). Static closure of the number-system chain (natural → integer → rational → real → complex).