# AP21 — DIVERGED, no source of record

**Checked 7 September 2026** by the paragraph-level test of record. The closest
candidate is `AP21_The_Web_FINAL.docx` (found in Formatted), and it is **not** the published
text: 5 paragraph(s) in the source are absent from the
published PDF, and 1 published sentence(s) are absent from
the source.

**Until a matching source is produced, the published PDF's text is the source for
AP21, 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 | `/AP21_The_Web.pdf` |
| sha256 | `97bc105cf77ae0009e439c4d27c1a0bdc17557fa8ac4adf4415a8bfe87803a1a` |
| Closest candidate | `AP21_The_Web_FINAL.docx` (Formatted) — NOT of record |

## In the source, absent from the published paper

- 5.1 (Stretching energy). DERIVED. E = Tl from Axioms B and R, AP18 Lemma 1, AP05/AP08 (Noether), AP15, and AP20.
- AP14 (The Correction). The quantum-gravity correction δG/G = γlp2/L2 is the micro-scale face of the same tension (section 4).
- The stretching energy of the vacuum is therefore Etot = TΣli, summed over all field lines. The vacuum configuration that minimises Etot is the one that minimises the total length of all field lines, subject to the constraint that every line must close.
- Proof. By Proposition 1, the system minimises Etot = Tltot. This is equivalent to minimising the total length ltot of all field lines. The manifold is a metric space (AP20). The problem of finding the minimum total length of connections among N points in a metric space is the Euclidean Steiner Tree Problem. The solution is a well-known mathematical result:
- Proposition 1 (Lemma + AP08 + AP15 + AP20) → E = Tl, T = λ → stretching energy proportional to length.

## In the published paper, absent from the source

- energy stored in the line is E = T , where T = λ (substrate stiffness from AP15).

*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.*