# AP15 — DIVERGED, no source of record

**Checked 7 September 2026** by the paragraph-level test of record. The closest
candidate is `AP15_The_Connection_v2.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
AP15, 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 | `/AP15_The_Connection.pdf` |
| sha256 | `37e75ffe4a1db1a864b7a89a293197ec00555b908fe20dcf6629705d52d114c1` |
| Closest candidate | `AP15_The_Connection_v2.docx` (Formatted) — NOT of record |

## In the source, absent from the published paper

- AP09 (Quantum Mechanics). The complex Hilbert space H, complex amplitudes, the Born rule (P = ψψ), the Schrödinger equation, the path sum, and the involution σ identified with complex conjugation (AP09 section 3.2). Load-bearing for the U(1) phase symmetry (section 1) and the action principle (Theorem 3).
- Axiom S (two sectors / involution). The two-sector structure of H gives the involution σ = complex conjugation. σ acts as ψ → ψ. The Born rule P = ψψ is the structural fingerprint of σ. The Born rule's invariance under ψ → e^{iθ}ψ is the U(1) phase symmetry that this paper derives electromagnetism from.
- AP09 (Quantum Mechanics). AP09 derives H, complex amplitudes, the Born rule, the Schrödinger equation, and the path sum. AP15 reads the Born rule's phase symmetry as U(1) and identifies the path sum as the containing structure for the action principle (Theorem 3).
- AP09 section 3.2 identifies σ with complex conjugation on H: σ(ψ) = ψ. For ψ = |ψ|e^{iφ}, complex conjugation reverses the phase: σ(ψ) = |ψ|e^{-iφ}. In the U(1) representation where a state with charge q transforms as ψ → e^{iqθ}ψ, its conjugate transforms as ψ → e^{-iqθ}ψ, i.e. with charge -q.
- (iii) The bundle structure group is U(1), not its universal cover R (section 3: the fiber at each point is S1, the phase circle inherited from H). Matter fields coupled to this bundle must transform in genuine representations of U(1), not representations of R. The charge of any field is therefore an integer: q ∈ Z. This is where the architecture's construction differs from the standard Lie-algebra
- ψ - State vector in H. ψ = |ψ|e^{iφ} carries magnitude and phase.

## 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.*