# AP22 — DIVERGED, no source of record

**Checked 7 September 2026** by the paragraph-level test of record. The closest
candidate is `AP22_The_Ledger_FINAL.docx` (found in Formatted), and it is **not** the published
text: 28 paragraph(s) in the source are absent from the
published PDF, and 5 published sentence(s) are absent from
the source.

**Until a matching source is produced, the published PDF's text is the source for
AP22, 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 | `/AP22_The_Ledger.pdf` |
| sha256 | `be14c17fa9e1c8bb79bdcd2b09548a74b003d8341ff09d7488be815c58675a6b` |
| Closest candidate | `AP22_The_Ledger_FINAL.docx` (Formatted) — NOT of record |

## In the source, absent from the published paper

- Axiom S supplies the involution σ (two sectors, L ↔ P) and, with it, the σ-boundary. Axiom B supplies the break ε, which has no σ-image - the uncancellable entry that keeps the ledger from closing. Axiom R kills the white-hole region (records cannot be unwritten), collapsing the four-region Kruskal structure. Axiom C forces the fold and bounds propagation at c, fixing the causal structure in which
- σ - the involution mapping L ↔ P (Axiom S). On the manifold, maps matter ↔ antimatter.
- L, P - the two sectors of the pre-state. L → exterior manifold (propagation, 1-poles, matter). P → interior of horizons (fold, 0-poles, antimatter).
- Axiom S defines the pre-state as two sectors, L and P, perfectly mapped by the involution σ. Perfect symmetry: 1:1.
- It requires only that for every element in L, there exists a σ-image in P. It does not require that the image is reachable by an observer in L.
- It operates entirely within the L-sector and does not require the σ-boundary to be crossed.
- (A1) On the manifold (AP20, AS = manifold), the two sectors L and P must be expressed as two regions. The boundary between them is the σ-boundary: the surface where the sector character changes.
- (A2) AP17 identifies the sector characters: L is the 1-condition (propagation - signals can escape to arbitrary distance). P is the 0-condition (fold - Axiom C forces compactification; signals cannot escape).
- (A5) The σ-boundary separates the L-sector (1-condition) from the P-sector (0-condition). The event horizon separates the region where outward propagation is possible from the region where it is not. These are the same surface.
- Region IV would be a second copy of L - a second exterior. But σ connects one L to one P.
- Therefore J maps aL → aP. The modular conjugation now targets the correct region.
- For a von Neumann algebra a with a cyclic and separating vector |Ω⟩, there exists a unique pair (J, Δ) where J is an anti-linear isometric involution (the modular conjugation) and Δ is a positive self-adjoint operator (the modular operator), determined by the polar decomposition of the Tomita operator S: a|Ω⟩ ↦ a|Ω⟩.
- The key properties: J|Ω⟩ = |Ω⟩, JaJ = a′ (J maps the algebra to its commutant), and Δ^{it} generates the modular automorphism group.
- By §4.2, the topology is two-sector, so the commutant of aL is aP.
- Therefore the Sewell / Kay-Wald results apply: J maps aL → aP and J = CPT (up to spatial rotation about the radial axis).
- (P2) σ̂ is anti-linear. By Part A, σ maps L (exterior) to P (interior) at the horizon. The Killing vector field ∂/∂t is timelike in the exterior and spacelike in the interior.
- Formal proof of anti-linearity. The Tomita operator S for the pair (aL, |ΩHH⟩) is defined by S(a|ΩHH⟩) = a|ΩHH⟩ for all a ∈ aL.
- (P3) σ̂ maps aL to aP. By Part A, σ maps L (exterior) to P (interior).
- Therefore σ̂ aL σ̂ = a′L, which is the defining property of a modular conjugation. ✓
- Given the algebra aL and the vector |ΩHH⟩, there is exactly one anti-linear isometric involution satisfying JaLJ = a′L and J|ΩHH⟩ = |ΩHH⟩.
- σ̂ satisfies (P1)-(P4): it is an anti-linear involution that maps aL to a′L and preserves the vacuum. J satisfies the same properties and is uniquely determined.
- Proof. The involution σ maps L ↔ P. The notation "1:1" denotes a bijection: every element in L has exactly one image in P, and every element in P has exactly one pre-image in L.
- Because ε has no reflection in the P-sector, the 1:1 cannot perfectly close. The splinter holds the door open. Matter remains outside the horizon and antimatter remains inside, held apart by the structural geometry.
- Let |m⟩ be a state in the visible manifold (L-sector). Let σ̂ represent the σ-involution as an operator on quantum states. By Theorem 1, σ̂ = CPT at the event horizon.
- Proof. By Theorem 1, the exterior manifold is the L-sector and the interior of every horizon is the P-sector.
- By Axiom S, σ is a bijection between L and P (σ(L) = P, σ(P) = L): every element in L has a unique image in P and conversely.
- The interior of every event horizon is the P-sector (the 0-pole, the black of the Eye). Theorem 1 derives this identification: the event horizon is the σ-boundary, and σ̂ = CPT there.
- Claim. The black-hole interior is the P-sector; evaporation should reflect the conjugation - either net antibaryon number in the radiation, or a remnant.

## In the published paper, absent from the source

- at the actualisation event: matter propagates outward (the -sector, 1-poles),
- LP - the two sectors of the pre-state. → exterior manifold (propagation, 1-
- (A1) On the manifold (AP20, AS = manifold), the two sectors and
- event horizon (boundary of causal past of +), not Schwarzschild coordinates.
- or identification of Region IV with (making it the same region, not a new one).

*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.*