# AP18 — DIVERGED, no source of record

**Checked 7 September 2026** by the paragraph-level test of record. The closest
candidate is `AP18_The_Floor_FINAL.docx` (found in Formatted), and it is **not** the published
text: 6 paragraph(s) in the source are absent from the
published PDF, and 0 published sentence(s) are absent from
the source.

**Until a matching source is produced, the published PDF's text is the source for
AP18, 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 | `/AP18_The_Floor.pdf` |
| sha256 | `d89f7d8d76b1aa828debb7664ea879adc6c386f165dddf9521621b332d3c7c1f` |
| Closest candidate | `AP18_The_Floor_FINAL.docx` (Formatted) — NOT of record |

## In the source, absent from the published paper

- The two sectors L and P are connected by the involution σ (Axiom S). The tension field of ε between 0 and 1 has field lines that leave 1 (propagation, matter, the visible) and return to 0 (fold, collapse, the dark). Every field line must close.
- The monoid (M, ·) accumulates records (Axiom R). Records are irreversible; the monoid grows monotonically. Under EH (proved in AP20: AS = manifold identity), the monoid admits embedding into a smooth manifold M. The manifold is the accumulated record. Cosmologically, the manifold is expanding; the expansion rate is the Hubble parameter H0.
- Lemma 1 (Measure homomorphism). The monoid (M, ·) admits a measure μ: M → R+ satisfying μ(m1 · m2) = μ(m1) + μ(m2) for disjoint records.
- Proof. (i) The monoid (M, ·) accumulates records (Axiom R). Records are irreversible and each record is a distinct actualisation event - written once at a specific spacetime location. Distinct records occupy disjoint regions of the accumulated structure; you cannot double-count them. (ii) Under AP20 (AS = manifold), the monoid embeds into a smooth Riemannian manifold M. The embedding is injective:
- Proof. The coherent arc of length l = Rh is centred on the apex: the Z2 involution σ (Axiom S) maps the two half-arcs onto each other, so the apex is the unique midpoint. The integration limits are therefore -Rh/2 to +Rh/2, or equivalently (by symmetry) 0 to Rh/2 doubled. From Proposition 1(c), α = (2/Rh) ∫0^{Rh/2} [T(s)/T0] ds. Substituting T(s)/T0 = sec(s/Rh) and u = s/Rh: α = 2 ∫0^{1/2} sec(u) 
- M, ·, μ - the record monoid (M, ·) and its additive measure μ: M → R+ (Lemma 1).

## In the published paper, absent from the source

*(none)*

*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.*