The One-Loop Correction

The derivation narrows. Everything above built the path sum and identified virtual records. Now: what is the leading correction? One virtual record. One loop. And you will see the functional form is not a guess — it is the only form the axioms allow.

3.1 — Setup

Consider a region of M with spacetime 4-volume V. The leading correction comes from the K₁ contribution — paths with exactly one virtual record (§2.5). The manifold is 4-dimensional (AP10: three spatial from {C, S, B}, one temporal from R). Under EH (proved in AP20), the discrete monoid embeds faithfully into M. The minimum record occupies the Planck 4-volume ℓ_P⁴ = (ℏG/c³)², the intersection of the break’s quantum face (ℏ) and gravitational face (G), bounded by c. Not imposed — it follows from the axioms.

3.2 — The sum

The maximum number of Planck cells is N_max = V/ℓ_P⁴, finite: Axiom B sets a minimum 4-volume per record, Axiom C bounds the spatial extent, Axiom R makes the set discrete. The K₁ amplitude is the sum over all positions where a single virtual record could be written: K₁ = Σ_k a(k), where k runs over Planck cells and a(k) is the amplitude contribution at cell k. The Born rule acts on the total amplitude, not on individual terms.

3.3 — The amplitude contribution

At cell k, the contribution from one virtual record is the matrix element a(k) = ⟨m_k · r | Û(Δt) | m_k⟩ — a complex number, an amplitude, describing one record-writing event. The record r has action ℏ (Axiom B), gravitational coupling G (AP08), and propagation bounded by c (Axiom C). These are the only scales that enter, because the matrix element describes one record interacting with geometry under the causal bound. The sum K₁ = Σ_k a(k) is over a finite set, so it converges. The exact value of each a(k) requires the explicit matrix elements of ℛ̂ (Gap 1).

3.4 — The result: dimensional uniqueness

The leading correction δG/G is dimensionless and first-order in ℏ (one virtual record). The axiom-derived dimensionful constants are ℏ (Axiom B), G (AP08), c (Axiom C), and Λ (Lovelock, AP08 §9). The observation scale L is the only external parameter. At sub-cosmological scales ΛL² ≪ 1, so Λ-dependent corrections are subleading.

Theorem (dimensional uniqueness at one loop). Let δG/G = ℏ¹ G^b c^d L^e be dimensionless. With [ℏ] = M L² T⁻¹, [G] = M⁻¹ L³ T⁻², [c] = L T⁻¹, [L] = L, the three constraints give b = 1, d = −3, e = −2 — a unique solution.

δG/G = γ × ℏG/(c³L²) = γ × ℓ_P²/L², where γ is a dimensionless constant determined by the monoid combinatorics.

The exponents are locked. You cannot change them without breaking the dimensional analysis. The same logical structure as Lovelock’s theorem: given the axiom-derived inputs, the functional form is unique up to one undetermined constant. Lovelock leaves Λ undetermined; the one-loop theorem leaves γ undetermined. Finiteness of K₁ (§3.2) guarantees the one-loop amplitude is finite — at most V/ℓ_P⁴ terms, each a(k) bounded by 1 (matrix elements of a unitary operator), the sum over a countable set. γ’s finiteness is structurally motivated; its exact value requires the explicit monoid coupling (Gap 1; KS-26).

The effective gravitational coupling at scale L is G_eff(L) = G × (1 + γ ℓ_P²/L²). At the Planck scale L = ℓ_P the correction is of order unity — quantum and gravitational sectors of equal weight. At laboratory scales L ~ 1 m the correction is of order 10⁻⁷⁰. Unmeasurable. But finite. The quantum correction to gravity is not infinite. It is not zero. It is ℓ_P²/L² — and the axioms forced that form with no freedom in the exponents.

[Note on refinements. Dimensional uniqueness fixes the power-law but cannot exclude logarithmic corrections ℓ_P²/L² × f(ln(L/ℓ_P)), which arise in the standard effective-field-theory treatment (Donoghue 1994); whether the discrete path sum produces them depends on Gap 1. Λ-dependent corrections are subleading at sub-cosmological scales and tied to the cosmological constant problem. Donoghue’s independent route reaches the same G_eff ~ G(1 + const × ℓ_P²/L²); the contribution here is the structural finiteness argument and the axiom-level derivation of the path sum.]