Derivation Chain

AP09 (complex Hilbert space) + AP05 (Lorentzian spacetime) + AP08 (EFEs) + AP21 (Energy–Measure Bridge) + Axiom C (Constraint) → Derived QFT satisfies conditions for modular theory (Stage 1).

AP17 (1-pole/0-pole) + Axiom C (Constraint) + AP08 (horizons exist) → σ-boundary = event horizon (Part A).

Axiom R (Record — irreversibility) + Axiom S (two sectors) → Two-sector topology: no Regions III/IV (Proposition 0). Commutant of 𝒶_ℒ is 𝒶_𝒫 (Corollary).

Conditions for modular theory + Killing horizon (AP08) + two-sector topology + Sewell / Kay–Wald → Modular conjugation J maps 𝒶_ℒ → 𝒶_𝒫 and J = CPT (Stage 2).

Axiom S (Symmetry — σ is involution) + Part A (σ-boundary = horizon) + Tomita–Takesaki structure → σ̂ satisfies defining properties of J (Stage 3).

Tomita–Takesaki uniqueness (polar decomposition) → σ̂ = J = CPT (Stage 4). Theorem 1 proved.

Theorem 1 + AP17 (Eye topology) + AP21 (filament fragmentation) → Segregation: matter outward, antimatter inward (§5.1).

Axiom B (Break — ε has no σ-image) → Uncancellable entry (Proposition 1, unconditional).

Theorem 1 + Proposition 1 → Global quantum number balance Q_total = Q(ε) (Proposition 2).