# AP14 — DIVERGED, no source of record

**Checked 7 September 2026** by the paragraph-level test of record. The closest
candidate is `AP14_The_Correction_v2.docx` (found in Formatted), and it is **not** the published
text: 21 paragraph(s) in the source are absent from the
published PDF, and 11 published sentence(s) are absent from
the source.

**Until a matching source is produced, the published PDF's text is the source for
AP14, and the AP30 republish procedure is not available to it.** No file is
copied into this folder, so nothing here can be harvested by mistake.

| | |
|---|---|
| Published PDF | `/AP14_The_Correction.pdf` |
| sha256 | `faa73eafd907c83b6f36b4d3259e1d5467cc56dcea6c738f33f50af2982cd163` |
| Closest candidate | `AP14_The_Correction_v2.docx` (Formatted) — NOT of record |

## In the source, absent from the published paper

- 6 - The Planck Scale: Where the Two Faces Meet · lP as the meeting of the quantum face (ħ) and the gravitational face (G), bounded by c; behaviour above, at, and below the scale. 22
- lP is the Planck length √(ħG/c3). L is the observation scale (the external parameter). γ is the dimensionless one-loop combinatoric coefficient - of order unity, form-fixed, value open (Gap 1, KS-26). ħ is the minimum action per record (Axiom B, AP12). G = 2κ/mε2 is the gravitational coupling (AP08). c is the causal bound (Axiom C). Λ is the cosmological constant (Lovelock, AP08 §9).
- R̂ (R-hat) is the record-writing operator; its explicit matrix elements are Gap 1. K is the transition amplitude; K0, K1, Kn are the sub-sums by virtual-record count (§2.5). ε is the minimum break (Axiom B), identified with the electron (The Lock); mε is its mass. EH is the Embedding Hypothesis, proved in AP20.
- Gap 1 - the explicit form of the record-writing operator R̂. The one-loop sum requires its matrix elements; AP09 gives existence and properties but not the closed form. Closing Gap 1 converts the scaling result into an exact γ.
- Logarithmic refinement. Dimensional uniqueness fixes the power-law lP2/L2 but cannot exclude corrections lP2/L2 × f(ln(L/lP)). Whether the discrete path sum produces or excludes logarithmic running depends on Gap 1. Λ-dependent (cosmological-scale) corrections are an IR question tied to the cosmological constant problem.
- Result 2. The correction has the form Geff = G(1 + γ lP2/L2), where lP is the Planck length, L is the observation scale, and γ is a dimensionless constant determined by the monoid structure.
- 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).
- The maximum number of Planck cells is Nmax = V/lP4, 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 = Σk 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 amplitu
- δG/G = γ × ħG/(c3L2) = γ × lP2/L2, where γ is a dimensionless constant determined by the monoid combinatorics.
- The two sectors L and P are related by the involution σ. The Born rule P = ψψ = Light × Dark (AP09 §6) gives the probability of each final state from the total amplitude. Both sectors must agree, so the measure is fixed uniquely. Without Axiom S the extraction of probabilities would be ambiguous - computed from which sector? With Axiom S there is no ambiguity: the probability is the product of bot
- The Planck length is lP = √(ħG/c3). Not the collision of two independent theories - the meeting point of two faces of one break. ħ is the quantum face (Axiom B: minimum action per record). G = 2κ/mε2 is the gravitational face (AP08 §5). c is the propagation limit (Axiom C). The Planck length is where the minimum record (ħ) and the curvature scale (G) meet, bounded by c. At this scale, one ε writin
- Above the Planck scale (L ≫ lP): many records, smooth geometry, classical limit (AP13). The gravitational sector dominates; quantum corrections suppressed by (lP/L)2. The world you observe. At the Planck scale (L ~ lP): δG/G ~ γ, of order unity. The pre-state and the record are equally present; the geometry is not smooth; the monoid is sparse; the discrete structure is exposed. Below the Planck sc
- Result: Geff(L) = G(1 + γ lP2/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).
- Status. LIVE - EMPIRICAL. Each higher loop is conjectured to add one lP2/L2 factor and the geometric construction (§4.5) is argued to exclude new operators, but the combinatorics grow at each order and the formal inheritance is outstanding (D7).
- Test. Measure or derive the commutator at scales approaching lP. A deviation from iħ at the predicted order tests the claim.
- §3.4 - Dimensional uniqueness. - δG/G = γ lP2/L2, exponents locked. DERIVED (form); γ OPEN (Gap 1).
- All-orders finiteness. - Each loop adds one lP2/L2; no new operators. STRUCTURAL CONJECTURE (D7).
- Gap 1 (explicit record-writing operator R̂ - fixes γ). Gap 2 (explicit form of the virtual measure - reduces to Gap 1). Gap 3 (EH as an explicit large-N convergence theorem - EH itself supplied by AP20). Logarithmic and Λ-dependent refinements of the correction.
- lP - The Planck length √(ħG/c3). lP4 is the Planck 4-volume - the spacetime occupied by one minimum record.
- H (Hilbert) - The complex Hilbert space of the pre-state (AP09). Context distinguishes from ħ the action quantum.
- σ - The involution relating the two sectors L and P (Axiom S). σ = complex conjugation in the Born rule.

## In the published paper, absent from the source

- Geff = G(1 + γ P2/L2), where P is the Planck length, L the observation
- The one-loop result Geff(L) = G(1 + γ P2/L2) with exponents locked (§3.4).
- virtual records is V/ P4 (Axioms B + C), so Kn = 0 for n > V/ P4.
- The effective gravitational coupling at scale L is Geff(L) = G × (1 + γ P2/L2).
- V/ P4 terms, each bounded by 1 (unitarity), the index set finite.
- amplitude is finite; the induced correction scales as P2/L2 (§3.4).
- + γ P2/L2), where P = √(ħG/c3) is the Planck length, L is the observation
- §3.4 - Dimensional uniqueness. - δG/G = γ P2/L2, exponents locked.
- All-orders finiteness. - Each loop adds one P2/L2; no new operators.
- - The Planck length √(ħG/c3). P4 is the Planck 4-volume - the spacetime
- reaching Geff ~ G(1 + const × P2/L2) by an independent route.

*Each item is either a pre-publication draft difference, which is nothing, or a paragraph G edited after publication, which becomes a dated note. That is G's reading to make.*