Claim Summary · Conditionality Footer

Claim Summary

Section-by-section structural enumeration with epistemic-status labels.

1 (The Phase). DERIVATION. U(1) is the gauge group from the Born rule’s phase freedom. Not imposed; derived from Axiom S via AP09.

2 (The Pre-state is One). DERIVATION. The pre-state is one. The phase is global. The path sum is globally U(1)-covariant (Proposition).

3 (The Connection). DERIVATION USING STANDARD MATHEMATICS. Aμ is the U(1) principal bundle connection over M. The connection is the dictionary — it expresses the pre-state’s global phase as read locally.

4 (Electromagnetism). DERIVATION. Theorem 1: ε is charged. Theorem 2: ε forces non-trivial connection and curvature. Theorem 3: action principle from path sum. Corollary: Fμν ≠ 0 in regions containing ε. Maxwell’s equations as the unique gauge-invariant local action at leading derivative order.

5 (The Non-actualised). DERIVATION. Theorem 4 (Phase Coherence Partition): the pre-state’s phase coherence partitions exhaustively into connection curvature on M and entanglement in ℋ, with ε as sole transfer mechanism. Gap D: structurally addressed by Theorem 4; formal conserved-balance theorem held open (KS-31 LIVE — STRUCTURAL).

6 (Charge). DERIVATION USING STANDARD MATHEMATICS. Theorem 5 (Charge Quantisation): all U(1)_EM charges are integer multiples of ε’s charge. Compactness of U(1) + Peter–Weyl + ε is minimum (Axiom B). Gap E: CLOSED.

7 (The Photon). DERIVATION. Boson (AP11), massless (U(1) exact, gauge-invariant mass term forbidden), spin 1 (Aμ is a 1-form), propagates at c (Axiom C + Maxwell wave equation).

8 (Summary of Derivation). SYNTHESIS. Derivation chain from axioms to electromagnetism enumerated step by step.

Conditionality Footer

Dependencies

Framework: the four axioms {S, B, R, C} from Notebook I; AP05 (the structural derivation of ε); AP06 (the Landauer bound, AP06 section 4, and the leakage framework for ε ≡ η); AP08 (Einstein’s field equations and the Lovelock-type uniqueness argument); AP09 (complex Hilbert space, Born rule, Schrödinger equation, path sum, σ = complex conjugation); AP10 (D = 4); AP11 (fermion/boson distinction, spin); AP12 (uncertainty principle, ℏ); AP13 (decoherence, environment); AP14 (time-slicing of the path sum); The Lock (Edition 04; ε = electron at chemical-biological scale); AP20 (Embedding Hypothesis, Quantum–Records Algebra).

Conditional on: None. EH and QRA proved in AP20. All other dependencies are derived corpus results.

Dependents

AP27 (Harmonics — internal state space and gauge-group structure). AP16 (Electroweak Break — U(1)×SU(2) breaking). AP19 (Direction — SU(3) colour). AP24 (Residual — multi-projection structure of ε; KS-4 treatment). AP14 (Correction — Planck-scale corrections, one-loop amplitude finiteness). AP28 (Constant — G via channel-counting using ε = α_em).

Kill switches closed by this paper

KS-28 (Phase-group uniqueness): CLOSED — MATHEMATICAL. The symmetry of a circle is U(1).

KS-29 (Maxwell uniqueness): CLOSED — MATHEMATICAL. Standard classification of gauge-invariant operators by mass dimension.

Gaps closed: Gap B (minimal coupling, Theorem 2a); Gap E (charge quantisation formal derivation, Theorem 5); Gap F (bundle construction and action principle, Theorems 1–3 and Corollary). Gap D (entanglement–connection) is structurally addressed by Theorem 4; the formal conserved-balance theorem is held open (KS-31 LIVE — STRUCTURAL).

Kill switches that remain live

KS-30 (Phase globality): LIVE — STRUCTURAL. Targets the assumption that the pre-state’s phase is globally coherent.

KS-31 (Entanglement–connection identification): LIVE — STRUCTURAL. Theorem 4 establishes the partition by exhaustion and identifies ε as sole transfer mechanism, but the conserved-balance theorem the ledger framing asserts is held open.

KS-4 (α ≈ 1/137): LIVE — HARD. Existing corpus kill switch. AP15 identifies α_em as the coupling strength but does not derive its value. Carried forward to AP24 and AP28.

Structural debts owed

Gap A (the value of α_em). AP15 derives that ε couples to the connection with some strength. It does not derive the numerical value ≈ 1/137. The path to closure runs through AP24 and AP28. [AP15 scope: not closed; carried by KS-4]

Items held open without claiming debt status

Uniqueness of ε as transfer agent across all gauge sectors. Bundle topology extension of section 3 logic to SU(2) and SU(3) (developed in AP16, AP19, AP27).