# AP09 — DIVERGED, no source of record

**Checked 7 September 2026** by the paragraph-level test of record. The closest
candidate is `AP09_The_Break_Empty_Set_FINAL.docx` (found in Formatted), and it is **not** the published
text: 11 paragraph(s) in the source are absent from the
published PDF, and 6 published sentence(s) are absent from
the source.

**Until a matching source is produced, the published PDF's text is the source for
AP09, 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 | `/AP09_The_Break_Empty_Set.pdf` |
| sha256 | `2f3f21d2a198556d961569c3da7d168f82aecedc3b666569a036007d50455f35` |
| Closest candidate | `AP09_The_Break_Empty_Set_FINAL.docx` (Formatted) — NOT of record |

## In the source, absent from the published paper

- AP11 (The Spin): The Z2 of Axiom S read on the rotation group. Builds on this paper's Hilbert space construction and σ ↔ time-reversal identification.
- Axiom S - symmetry. Two sectors, L and D, with an involution σ mapping one to the other. One sector reads the state as 0. The other reads the same state as 1. The involution flips between them. They are perfectly symmetric. They are one.
- Axiom B - one element ε ∈ L with no σ-image. The tiniest asymmetry the structure can tolerate while still being readable. The first record. The crack in the mirror.
- Complex numbers form a field, C, with addition, multiplication, and inverses for both.
- The set of such functions - all maps from {r1, ..., rn} to C - is Cn, a complex vector space by the standard construction.
- Any construction from a finite monoid acting on a set with C-valued weights naturally produces a C-module, and every finitely generated C-module is a vector space.
- Kill switch KS-Q.7. If a construction from R + C + EH that does not produce a vector space can be equally motivated from the axioms, this step fails. LIVE - HARD.
- If the resulting inner-product space is not complete, let H be its Hilbert completion under ‖·‖ - standard functional-analytic completion, structural bookkeeping, not new physics.
- C has exactly two field automorphisms fixing R pointwise - the identity and conjugation (any such automorphism sends i to ±i). σ is non-trivial - it exchanges the two sectors - so σ ≠ identity. Therefore σ = conjugation. Mathematical fact.
- Test: Demonstrate a construction from R + C + EH that does not produce a vector space, equally motivated from the same axiom-derived ingredients.
- Superposition: BRIDGE DERIVATION. Linearity from R + C + EH (KS-Q.7). Complex amplitudes from Lorentzian signature (KS-Q.4).

## In the published paper, absent from the source

- from monoid + + faithful embedding is motivated but not proven unique.
- AP11 (The Spin): The 2 of Axiom S read on the rotation group.
- Complex numbers form a field, , with addition, multiplication, and inverses for both.
- The set of such functions - all maps from {r1, ..., r } to - is n, a complex vector space by
- a -module, and every finitely generated -module is a vector space.
- The identification σ ↔ complex conjugation is proven unique. has exactly

*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.*