Open Gaps and Debts Owed

The structural argument for one-loop finiteness is complete. Three gaps remain for the explicit calculation, and one debt is owed for the all-orders claim.

Gap 1 — the explicit record-writing operator ℛ̂. The one-loop sum requires the matrix elements ⟨m₂|ℛ̂|m₁⟩. AP09 derives the existence and properties of measurements but not the operator in closed form. Closing this gap converts G_eff = G(1 + γℓ_P²/L²) into an exact result with computed γ.

Gap 2 — the measure on virtual records (PARTIALLY CLOSED). The Born rule operates on the amplitude, which already contains the virtual records; there is no separate virtual measure to define. What remains open is whether the path-sum weights are well-defined for virtual records in the gravitational sector — the monoid is discrete (Axiom R) and the sum finite (Axioms B + C), guaranteeing convergence, but the specific coupling structure requires Gap 1. Measure existence is established; its explicit form is open.

Gap 3 — EH as an explicit theorem (Open Problem 7). The source manuscript framed the entire derivation as conditional on EH and wished for a large-N convergence proof of the discrete monoid to a smooth manifold. AP20 proves EH and QRA, so AP14 is unconditional on EH in the current corpus. Gap 3 is retained here only as the standing wish for a fully explicit convergence construction, not as an open conditionality.

D7 — virtual-record sum and higher-derivative terms. Section 4.5 argues the record algebra cannot generate higher-order curvature terms. A formal proof that the virtual-record sum inherits the algebra’s geometric constraints — that summing over non-committed records cannot generate higher-derivative contributions to an effective action — is owed. The principal debt for the all-orders finiteness claim. Without it, the one-loop result stands but the all-orders extension is a structural conjecture, not a proof.

Gap 1 affects the value of the correction. Gap 2 reduces to Gap 1. D7 affects the all-orders extension. None affects the structure of the argument: B + C + R + S + Lovelock(N = 3+1) → finiteness.