Summary of Derivation

Axiom B → minimum record ε, action-scale ℏ, discrete monoid → finite number of virtual records per volume → UV finite.

Axiom C → finite propagation c, bounded causal diamond, Λ sets maximum volume → IR finite.

Axiom R → monoid countable, no inverses; path sum from completeness of monoid states (§2.3); Born rule gives canonical measure → well-defined sum.

Axiom S → involution σ = complex conjugation, P = ψψ*, two sectors fix the measure → canonical measure.

AP10 (N = 3 spatial) → D = 4 → Lovelock’s theorem → G_μν + Λg_μν is the unique geometry → no higher-order counterterms (§4.5; formal proof outstanding, D7).

AP09 §4.4 (the loop) → closing IS opening, defragmentation at saturation → singularity resolved.

Result: G_eff(L) = G(1 + γ ℓ_P²/L²). Finite one-loop correction. No divergences at one loop. No counterterms within the algebra’s geometric construction for committed-record geometry; inheritance by virtual-record sums remains a formal debt (D7).