ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generated

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Nonbounding sphere carries source-free charge flux

Example

On a sphere-containing dyonic RN domain with $q_{m} \neq 0$ , the global closed field has nonzero flux through each sphere $S_{r}$ . Each such sphere is a nonbounding finite smooth singular cycle in the field domain; local north/south potentials exist but no global real one-form potential does. On a finite full-sphere KN exterior annulus, the same conclusion holds conditionally for a constant dual rotation with $q \sin β \neq 0$ , under the exact current batch-21 draft interface. Neither statement gives a magnetic source, charge quantization, or an empirical observation.

Facts & Assumptions

Given: The RN sphere product and its nonzero magnetic parameter; for the KN comparison, the current batch-21 dyonic-flux interface on a specified finite exterior annulus.

[F1]

Dyonic RN gauges and global exactness obstruction: the current A-page item proves $\int_{S_{r}} F = 4 π q_{m}$ and gives smooth local north/south potentials. Its proof is directly supplied using RN data, Hodge calculus and finite-chain Stokes.

[F2]

Stokes theorem for smooth singular chains: for finite smooth singular chains $c$ and smooth $η$ , $\int_{c} d η = \int_{\partial c} η$ .

[L1]

Dyonic Kerr-Newman parameters and conserved electric/magnetic flux is a batch-21 draft, not an accepted proof. Its exact current Statement gives, on a full-sphere exterior annulus with $R_{0} > 4 ∣ M ∣$ and $Δ > 0$ , the flux parameters $q_{e} = q \cos β$ , $q_{m} = q \sin β$ for the constant dual rotation $F_{β} = \cos β F + \sin β ⋆_{g} F$ , and conservation between finite homologous exterior spheres. Its Proof constructs finite sphere and shell cycles. Closedness is supplied separately by [L2], not asserted to be part of this Statement. The flux use remains conditional on the exact draft interface and its proof status.

[L2]

Fully verified electric Kerr-Newman electrovacuum, Statement, asserts $d F = d (⋆_{g} F) = 0$ on the full-sphere exterior $r > R_{0} > 4 ∣ M ∣$ . Classical electrovac dual rotation, Statement, gives $d F_{β} = 0$ for a constant dual rotation of a smooth source-free geometric electrovac datum. Restriction to the specified annulus preserves these equations. Both are current mathematical drafts; this use retains their unaccepted proof status.

[L3]

Poincare lemma for differential forms on star shaped domains: every smooth closed $k$ -form on a star-shaped open Euclidean domain is exact for $k \geq 1$ .

[A1]

The RN domain avoids $r = 0$ . Its sphere cycle is constructed in the proof of [F1]. In the KN case use the stated finite exterior annulus and the finite sphere and shell cycles constructed in the exact current proof interface [L1]; choose a constant $β$ with $q \sin β \neq 0$ .

Verification

1.1F1F2A1algebra

In RN, [F1] gives $\int_{S_{r}} F = 4 π q_{m} \neq 0$ . If $S_{r} = \partial c$ for a finite smooth singular 3-chain in the field domain, then $d F = 0$ by the local potential identities in [F1] and $d^{2} = 0$ . Applying [F2] would give $4 π q_{m} = \int_{S_{r}} F = \int_{c} d F = 0,$ a contradiction. Thus $S_{r}$ is nonbounding.

1.2L1L2L3F2A1algebra

The conditional flux interface [L1] gives $\int_{S_{r}} F_{β} = 4 π q \sin β \neq 0$ . By [L2], the unrotated field is a smooth source-free geometric electrovac datum on the full-sphere exterior, and its constant dual rotation is closed on the annulus. If $S_{r} = \partial c$ for a finite smooth singular 3-chain there, [F2] gives $\int_{S_{r}} F_{β} = \int_{c} d F_{β} = 0$ , a contradiction. A global smooth real one-form potential would likewise give zero flux by Stokes and $\partial S_{r} = 0$ . The north and south stereographic sphere charts, each omitting one pole, give product coordinate domains $R \times (R_{0}, R_{1}) \times R^{2}$ , which are convex and hence star-shaped about any of their points. Pull back $F_{β}$ to either domain, apply [L3] with $k = 2$ , and transport the resulting primitive back. This supplies smooth real patch potentials covering the annulus. Thus the KN sphere is nonbounding and its field is locally but not globally exact, conditionally on [L1-L2].

2.1F1F2step 1.1A1algebra

If a global smooth potential $A$ existed, then [F2] and $\partial S_{r} = 0$ would give $4 π q_{m} = \int_{S_{r}} d A = \int_{\partial S_{r}} A = 0,$ contradicting step 1.1's nonzero flux. The smooth patch potentials in [F1] show that the obstruction is global exactness, not local existence.

3.1F1F2L1L2step 1.1step 1.2step 2.1∎

The RN conclusion follows from the current RN local-gauge proof and finite-chain Stokes. The KN conclusion remains conditional on the exact batch-21 flux, full-sphere verification and duality interfaces and their draft proofs. Neither yields a regular magnetic source or a quantization condition.

Remarks

The field is a globally defined closed two-form even when it is not globally exact. The excluded center is not filled in by this argument.

Nonbounding sphere carries source-free charge flux

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources