# AP27 — DIVERGED, no source of record

**Checked 7 September 2026** by the paragraph-level test of record. The closest
candidate is `AP27_The_Harmonics_v2.docx` (found in Formatted), and it is **not** the published
text: 25 paragraph(s) in the source are absent from the
published PDF, and 2 published sentence(s) are absent from
the source.

**Until a matching source is produced, the published PDF's text is the source for
AP27, 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 | `/AP27_The_Harmonics.pdf` |
| sha256 | `b6b78c49ff9ea3cbf4b9fd1fd71055595c904a18e9bb97f88b95386200cb5013` |
| Closest candidate | `AP27_The_Harmonics_v2.docx` (Formatted) — NOT of record |

## In the source, absent from the published paper

- The argument is a single move. Axiom S provides two sectors L and P, connected by the involution σ. The break ε is the boundary between them. When ε propagates on the manifold as a quantum field, it necessarily carries an internal state encoding its relationship to each sector. That state lives in a two-dimensional complex Hilbert space HEW ≅ C2 (Lemma 1). The choice of basis on HEW is unphysical 
- L and P are the two sectors of Axiom S, connected by the involution σ: L ↔ P. ε is the break - the minimal asymmetry, the boundary between sectors, the object that writes records on the manifold (Axiom R). HEW is the internal electroweak state space of ε: HEW ≅ C2. It is distinct from spin space (AP11), colour space (AP19), and the overall phase (AP15). |L⟩ and |P⟩ form a basis for HEW - abstract 
- U(2) is the group of 2×2 unitary matrices, the full gauge group on HEW before decomposition. SU(2) is the traceless (relative-orientation) part of U(2), candidate for weak isospin with generators T1, T2, T3 (the Pauli matrices). U(1)Y is the determinant part of U(2), candidate for hypercharge with generator Y. U(1)EM is the post-breaking electromagnetic remnant (AP15): Q = T3 + Y/2. Chirality is a
- Axiom S (two sectors). The structural ground of AP27. L and P are the two sectors; σ is the involution; ε is the boundary between them. AP27 reads the two-sector structure as the source of ε's internal electroweak state space.
- Axiom S states that reality has two sectors, L and P, connected by the involution σ: L ↔ P. The break ε is the boundary between them - the object that connects the two sectors and, through Axiom R, writes records on the manifold.
- The break is not purely in L. It is not purely in P. It is the boundary: the thing that sits between and connects them. When ε propagates on the manifold as a quantum field, it necessarily carries information about its relationship to both sectors.
- Proof. (i) Axiom S provides exactly two sectors: L and P. No more, no fewer. The break connects them. On the manifold, a propagating field arising from ε carries a state that encodes how ε relates to each sector. Since there are two sectors, the state has two components: one for ε's relationship to L, and one for ε's relationship to P.
- (iii) The state is a vector (zL, zP) ∈ C2. The inner product on HEW is physical: it determines transition amplitudes via the Born rule (AP25). The internal state space is HEW ≅ C2.
- Note: states in HEW are not merely classical labels "L or P". The break can exist in coherent superpositions of |L⟩ and |P⟩. A general state α|L⟩ + β|P⟩ is physically meaningful. Such superpositions exhibit interference. This is what gives rise to the full continuous U(2) gauge freedom rather than a discrete Z2 swap.
- Note: HEW is distinct from all previously derived spaces. In fibre bundle language: the spinor bundle, the colour bundle, and the electroweak bundle are three distinct fibre bundles over the manifold, with structure groups SU(2)spin, SU(3), and U(2) respectively. Their fibres are different vector spaces. Their connections are independent.
- The two basis vectors |L⟩ and |P⟩ label the break's relationship to each sector. The involution σ maps L ↔ P, so the labelling is ambiguous: σ swaps the basis. This is a discrete (Z2) ambiguity. But the full gauge redundancy is larger than Z2.
- All physical observables on HEW depend on inner products: transition amplitudes are |⟨φ|ψ⟩|2 (Born rule, AP25), and expectation values are ⟨ψ|A|ψ⟩. Any transformation on HEW that preserves all inner products is physically undetectable. The group of inner-product-preserving transformations on C2 is U(2). The involution σ is one element of U(2). But U(2) contains all inner-product-preserving transfo
- Lemma 2 (Electroweak Gauge Freedom). The choice of basis on HEW at each point of the manifold is unphysical. Demanding consistency under local basis changes forces a gauge connection with structure group U(2).
- Proof. (i) A local basis change on HEW at point x preserves the inner product on C2 (the inner product is physical: Born rule, AP25). The group of such transformations is U(2). The discrete involution σ ∈ U(2) motivates the basis ambiguity; the Born rule promotes it from discrete to continuous by requiring invariance under all inner-product-preserving transformations.
- Proposition 1 (Electroweak Decomposition). U(2) on HEW decomposes as U(2) ≅ SU(2) × U(1)Y. SU(2) acts on the relative orientation of the two sector-components (weak isospin). U(1)Y acts on the overall phase of the doublet (hypercharge).
- Proposition 2 (Chiral Coupling Selection). The gauge connection on HEW couples to left-chiral fermion fields as doublets and to right-chiral fermion fields as singlets.
- The involution σ maps L ↔ P. By AP22, σ acts as CPT on the manifold. CPT conjugation interchanges PL and PR: it maps left-chiral fields to right-chiral fields. Axiom R breaks CPT as a dynamical symmetry - the break goes forward in time, not backward. The two chiral projections are NOT dynamically equivalent.
- Sector-relationship freedom (how ε's relationship to L/P is described) → SU(2) × U(1)Y.
- Claim. Lemma 1 derives the internal electroweak state space HEW ≅ C2 from Axiom S's two-sector structure.
- Claim. Lemma 1 explicitly constructs HEW as an internal space distinct from spin space. The SU(2) derived in AP27 acts on HEW and is the weak isospin group, separate from the Lorentz spin SU(2) of AP11.
- Test. Demonstrate that the derived SU(2) on HEW is identical to the Lorentz spin SU(2) of AP11. Failure: the paper has derived the wrong group.
- 3 (Gauge Freedom). DERIVED. Lemma 2: the local basis on HEW is unphysical; demanding consistency under local basis changes forces a gauge connection with structure group U(2).
- HEW - Internal electroweak state space of ε. HEW ≅ C2. Distinct from spin space (AP11), colour space (AP19), and overall phase (AP15).
- |L⟩, |P⟩ - Basis for HEW. Abstract internal labels, not spacetime directions.
- U(2) - Group of 2×2 unitary matrices. The full gauge group on HEW before decomposition.

## In the published paper, absent from the source

- colour space (AP19), and the overall phase (AP15). | ⟩ and |
- sectors, the state has two components: one for ε's relationship to , and one for

*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.*