From Axioms to the Path Sum

You have seen gravity derived from records (AP08). You have seen quantum mechanics derived from the pre-state (AP09). Now watch what happens when you let them talk to each other. The path sum — the object that captures every possible way the break could have unfolded — falls out of structures you already have. No imports. No new axioms. Just the completeness of what the axioms already built.

2.1 — What a record is

A record is one irreversible act of writing. Axiom R: the monoid (ℳ, ·) has no non-identity inverses. Once a record is written, it is written. The accumulated record is the environment (AP13 §2.1), the classical world (AP13 §4.1), the curvature of spacetime (AP08).

Each record has a minimum action: ℏ. Axiom B — the break has a minimum size. One record = one ε = one unit of action ℏ (AP12 §1). You cannot write half a record. You cannot write a tenth of a record. The record is discrete. The monoid is countable. Every measurement you have ever made was one of these records.

2.2 — The propagator

AP09 derives three results this paper requires. (i) The Hilbert space ℋ exists (AP09 §3): the pre-state has the structure of a complex vector space with inner product. (ii) Between record-writing events, the pre-state evolves unitarily: |ψ(t)⟩ = Û(t)|ψ(0)⟩, with Û(t) = exp(−iĤt/ℏ) (AP09 §7.3). (iii) The Born rule P = |ψ|² = ψψ* = Light × Dark (AP09 §6) extracts probabilities.

From (i) and (ii), the transition amplitude between any two states is the matrix element of the evolution operator: K(ψ_f, t_f | ψ_i, t_i) = ⟨ψ_f | Û(t_f − t_i) | ψ_i⟩. Not imported from quantum field theory — the definition of a matrix element on the Hilbert space AP09 derives. Given ℋ exists and Û exists, K exists.

2.3 — The path sum

The derivation requires one result from the Born rule: the measurement outcomes — the monoid states — form a complete basis for ℋ. The Born rule’s normalisation gives this directly: if Σ_m P(m|ψ) = 1 for all |ψ⟩, then Σ_m |m⟩⟨m| = 𝟙. Each measurement writes a record to the monoid (AP09 §5), so the monoid states resolve the identity. Not an assumption — a theorem of the Born rule and the measurement axiom.

Partition the interval [t_i, t_f] into N steps of width Δt and insert a complete set of monoid states at each partition point. The amplitude becomes a sum of products of matrix elements over all sequences of intermediate states — exact for any N, an identity from completeness, not an approximation.

In standard quantum field theory the intermediate states form a continuum, N → ∞, and the result is the Feynman path integral. Three axioms prevent this. Axiom R: the monoid is countable — the sum at each step is over a countable set. Axiom B: each record has minimum action ℏ — choose Δt = ℏ/E_max so at most one record is written per step; a finer partition adds no new physics because sub-record processes do not exist. Axiom C: propagation is bounded by c — the causal diamond has finite 4-volume, so the number of Planck cells is V/ℓ_P⁴.

Result: the transition amplitude is a finite sum over finite sequences of discrete intermediate states. K(f|i) = Σ_paths W(path), where each path is a sequence of record-writing events (or non-events) and the sum has finitely many terms. If you can count them, they are finite. The path sum. Not the path integral. Derived from the Hilbert space (AP09) via completeness of monoid states, constrained by Axioms B, R, C. Nothing imported.

2.4 — Virtual records

The path sum contains many intermediate sequences. Some steps involve record-writing: the monoid changes from m_k to m_k · r_k. If all such records appear in the final state m_f, they are committed — real, irreversible, permanent (Axiom R). But the path sum includes all admissible intermediate sequences, including those where a record is considered — its effect on the amplitude computed — yet does not appear in m_f.

Definition. A virtual record is an intermediate record-writing event that appears in the path sum but whose record does not appear in the final monoid state m_f. It is a record the structure considers but does not commit.

Not imposed — it falls out of the path-sum structure of §2.3. And it does not violate Axiom R. Axiom R states: once a record IS written (committed to the monoid), it cannot be erased. A virtual record is never committed. It exists in the amplitude, not in the monoid. It lives in the white space, not on the black curve. The monoid contains only committed records; the Hilbert space, through the path sum, considers all possibilities including the virtual ones. The Born rule extracts probability from the full amplitude. The amplitude includes the virtual records. The monoid does not.

2.5 — The natural decomposition

The path sum decomposes by counting virtual records per path. Let K₀ be the sum over paths with zero virtual records — the classical amplitude. Let K_n be the sum over paths with exactly n virtual records. Then K(f|i) = K₀ + K₁ + K₂ + … + K_{N_max}.

The series terminates. In a region of 4-volume V, the maximum number of virtual records is V/ℓ_P⁴ (Axioms B + C), so K_n = 0 for n > V/ℓ_P⁴. The sum is finite. Not perturbation theory imported from quantum field theory — a counting decomposition on a finite sum, partitioned by the number of virtual records per path. At one loop the dimensional-uniqueness theorem (§3.4) shows K₁ is suppressed relative to K₀ by ℓ_P²/L². At observation scales L ≫ ℓ_P the series converges rapidly with K₁ dominant; at L ~ ℓ_P all terms contribute equally. The expansion parameter emerges from axiom-derived scales — it is not imposed.

2.6 — How virtual records affect measured geometry

Geometry is the accumulated record (AP08) — exact and unmodified by anything in this paper. A virtual record, by definition (§2.4), does not appear in the accumulated record. It does not directly alter geometry. What virtual records alter is which geometry is realised.

The Born rule gives P(m_f) = |K(m_f | m_i)|². The amplitude K includes virtual-record contributions; different configurations produce different amplitudes for different final states. The probability distribution over final geometries is shifted. The effective coupling at scale L is ⟨G_eff(L)⟩ = Σ_{m_f} P(m_f) × G(m_f; L), and the quantum correction is δG(L) = ⟨G_eff⟩_full − ⟨G_eff⟩_classical, where “classical” means K₀ only. At leading order the dominant correction comes from K₁ — one virtual record.

No semiclassical gravity is invoked. No ⟨T̂_μν⟩ source term. No geometry treated as a classical background with quantum perturbation. The derivation uses only: geometry = record (AP08, unmodified); the path sum includes virtual records (§2.3, derived); the Born rule gives probabilities (AP09, unmodified); effective geometry = probability-weighted geometry (definition). The virtual record does not source geometry. It shifts the distribution over geometries. Everything you needed was already in the path sum.