Structural Irreversibility: The Second Law of Thermodynamics as a Consequence of Axiom D3 of Structuristics

The second law of thermodynamics is currently postulated rather than derived. Boltzmann's H-theorem required the Stosszahlansatz, an assumption about molecular correlations before each collision, which Loschmidt showed cannot follow from reversible mechanics. This paper closes the gap using a single axiom from the Structuristics framework (Moura, 2026): D3, the non-injectivity of the decomposition operator D. Three theorems are derived in sequence, all by pure algebra with no physical hypotheses about states or initial conditions. Theorem 1 shows that a non-injective D necessarily produces structural loss, with a lower bound proportional to the distance between states that D collapses. Theorem 2 shows that iterated applications of D produce strictly nested equivalence classes, and that the best Decompositional Integrity any reconstruction operator can guarantee is monotonically non-increasing. Theorem 3 shows that the information lost at each step is irrecoverable by any operator, however powerful, and that accumulated irrecoverable loss is monotonically non-decreasing. A structural entropy H_D is defined as the negative logarithm of the optimal Decompositional Integrity and shown to be monotonically non-decreasing: this is the second law in structural form. The Boltzmann Bridge (Proposition 8.4) connects this structural result to thermodynamic entropy via a measure postulate (M1) and an isoperimetric bound, with explicit regime of validity. A central result is the regime tripartition (Proposition 8.5): the three structural regimes of Structuristics: Homeostatic, Corrigible, and Reconstitutable, constitute the natural validity domains of the Bridge. The Corrigible regime corresponds exactly to classical Boltzmann-Gibbs thermodynamics. The Reconstitutable regime is where generalised entropies (Tsallis, Renyi) are empirically required. A falsifiable prediction follows: in the Reconstitutable regime, the Tsallis parameter q equals the ratio d_H/N between the Hausdorff dimension of the indistinguishability cell and the ambient dimension. The two minimal physical inputs, that every macroscopically realizable D is non-injective (partial trace theorem, Nielsen and Chuang, 2000) and that a compatible Borel measure exists on the structural domain (Liouville or Hilbert-Schmidt measure) are stated as separate propositions entirely outside the mathematical derivation. Loschmidt's paradox is resolved algebraically: it presupposes the existence of the inverse of D, which Theorem 3B proves cannot exist for any non-injective D. The document is produced under CAMAF CS2 standards (Moura, 2026), with full epistemic labelling, declared limitations, and explicit falsification conditions.