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Local potentials and smooth circle-bundle connection data — Examples

1 · Prerequisites

2 · Summary

Scope

This leaf page works through the normalized north/south patch data at the smallest nonzero integer flux sector. The calculation checks the transition winding and the nonexistence of a global potential under the exact hypotheses of Sphere winding, nonexact curvature and flat holonomy; it reports no empirical magnetic-monopole observation or quantum inference.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-generatedVerification: AI-generatedOpen item page →

Nonexact sphere flux and circle winding

Example

Use the conditional, still-escalated draft result Sphere winding, nonexact curvature and flat holonomy with $b = κ_{0} / 2$ . Its authored supplier status and remaining external closure obligation are recorded below; no independent proof acceptance is asserted. The local potentials are

A_{N} = \frac{κ_{0}}{2} (1 - \cos θ) d ϕ, A_{S} = - \frac{κ_{0}}{2} (1 + \cos θ) d ϕ .

They have common curvature $F = \frac{κ_{0}}{2} \sin θ d θ \land d ϕ$ , flux $2 π κ_{0}$ , and transition $g_{N S} = e^{- i ϕ}$ of winding number $- 1$ under the north-to-south convention. The example is a mathematical circle-connection patch model. It is not an observation of a magnetic monopole and makes no quantum or empirical claim.

Facts & Assumptions

Given: The exact punctured-domain two-form, smooth patch potentials, circle normalization and transition convention of Sphere winding, nonexact curvature and flat holonomy.

[A1]

This example retains the reserved EMG-MG01 dependency interfaces: finite-real supremum and infimum domain; finite-valued metric and pseudometric conventions with metric nonnegativity proved; rational-root and real-power domains; the sequence criterion under exactly ACω; boundaryless metric and geodesic completeness domains; and set-sized objects/definable-class schemas under power-set and separation. Their exact hypotheses and conventions are at anchors #supremum-domain, #metric-definition, #metric-nonnegativity, #power-domain, #heine-choice, #boundaryless-domain, and #category-size in research/extended-frameworks-2026-10-03/einstein-maxwell-models/foundational-replacements.md. Those typed carriers are now authored earlier on emg-mg02, preserving the exact interfaces. This verification asserts no independent acceptance.

[L1]

Sphere winding, nonexact curvature and flat holonomy: conditionally, the displayed sphere flux is $4 π b$ , the local potentials differ by $- 2 b d ϕ$ , and the normalized transition is $e^{- 2 i b ϕ / κ_{0}}$ with integer winding when single-valued. The source is an authored draft with complete local calculations; no independent acceptance is asserted.

Verification

1.1givenA1L1algebra

Under the exact planned conditional interfaces in [A1], substitute $b = κ_{0} / 2$ into the flux $4 π b$ from [L1] to obtain $\int_{S^{2}} F = 2 π κ_{0}$ and hence $\int_{S^{2}} F / (2 π κ_{0}) = 1$ .

2.1givenL1step 1.1algebra

The same substitution gives $A_{S} - A_{N} = - κ_{0} d ϕ$ and $g_{N S} = e^{- i ϕ}$ . As $ϕ$ runs from $0$ to $2 π$ , this transition makes one negative turn, so its winding number is $- 1$ .

3.1givenL1step 2.1algebra∎

The transition is single-valued because $e^{- i (ϕ + 2 π)} = e^{- i ϕ}$ ; the local forms are smooth at their respective poles and both differentiate to $F$ . By Sphere winding, nonexact curvature and flat holonomy, the nonzero flux rules out a global potential. The calculation is conditional mathematical verification, not evidence about nature.