Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generated

How statement and proof provenance work

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Divergence of the point-charge field energy and the limits of mass inference

Physical scope

The truncated electric field energy of the idealised point charge in vacuum: the shell energy is $q^{2} / (8 π ε_{0}) (1 / δ - 1 / R)$ , finite for $0 < δ < R$ but divergent as the inner cutoff $δ ↓ 0$ for $q \neq 0$ . The item records this divergence as a property of the point idealisation and explicitly excludes inferences about particle rest mass, self-force, empirical measurement or inconsistency of the framework; finite-energy statements remain those of the compact-source branches.

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ) as carried by the Lebesgue and polar interfaces (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $q \in R$ , $x_{0} \in R^{3}$ , and let $E_{q}$ be the ideal point-charge field of The localized Newtonian potential and the ideal point-charge Coulomb field(d) (The electric and magnetic fields as fixed-time spatial vector fields, Framework-local SI units, the vacuum constants and the exact/measured status of their values). For $0 < δ < R$ put $U_{q} (δ, R) : = \frac{ε_{0}}{2} \int_{{δ \leq ∣ x - x_{0} ∣ \leq R}} ∣ E_{q} (x) ∣^{2} d λ_{3} (x)$ (the truncated field energy of the shell between the two spheres, Open ball, closed ball and sphere in a metric space). Then:

(a) [Exact truncated energy] for all $0 < δ < R$ , $U_{q} (δ, R) = \frac{q^{2}}{8 π ε_{0}} (\frac{1}{δ} - \frac{1}{R});$

(b) [Divergence] $\lim_{R \to \infty} U_{q} (δ, R) = q^{2} / (8 π ε_{0} δ)$ for every $δ > 0$ , and for $q \neq 0$ $\lim_{δ ↓ 0} U_{q} (δ, R) = + \infty for every R > 0;$ so the field energy of the ideal point charge is locally infinite at its own location, and any finite-energy treatment of a point-like charge must modify the source/field near the singularity or introduce an explicit cutoff;

(c) [Limits of the mass inference] the identity is a divergence of the field energy of the idealised point source in the framework's SI conventions. It does not assert that a physical particle has infinite rest mass, does not identify field energy with rest mass, contains no measurement, and involves neither a self-force nor a contradiction claim for classical electrodynamics: the framework supplies no mass-energy identification for a charged particle, and the smooth compact-source field/assembly energy of Finite-source assembly energy equals the electric field energy remains finite. The compact $L^{1}$ branch of The localized Newtonian potential and the ideal point-charge Coulomb field retains its locally integrable potential and distributional Poisson equation; it does not guarantee finite field energy for arbitrary $L^{1}$ densities. The point source itself is the idealisation of Distributional Maxwell equations with point and stationary-sheet sources(b) (The static and pure electrostatic regimes and their fixed-time reading(e)); its energy divergence is a mathematical property of that idealisation, not an empirical result.

Facts & Assumptions

Given: The Axiom of Countable Choice, $q \in R$ , $x_{0} \in R^{3}$ , the point-charge field $E_{q} (z) = q z / (4 π ε_{0} ∣ z ∣^{3})$ off $z = 0$ of the Coulomb solution, and $0 < δ < R$ .

[F1]

The ideal point-charge field satisfies $∣ E_{q} (z) ∣ = ∣ q ∣ / (4 π ε_{0} ∣ z ∣^{2})$ for $z \neq 0$ ; its value at the pole is arbitrary and immaterial; the compact-source field/assembly energy of a $C^{1}$ compactly supported density is finite and equal to $\frac{1}{2} \int ρ φ$ , and the compact $L^{1}$ branch keeps a locally integrable potential with distributional Poisson equation but asserts no finite field energy (The localized Newtonian potential and the ideal point-charge Coulomb field, Finite-source assembly energy equals the electric field energy, The electric and magnetic fields as fixed-time spatial vector fields, Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F2]

The polar formula computes radial integrals, the sphere area is $σ (S^{2}) = 4 π$ from $∣ S^{n - 1} ∣ = n ∣ B_{1} ∣$ and $∣ B_{1} ∣ = 4 π / 3$ , and the fundamental theorem evaluates $\int_{δ}^{R} r^{- 2} d r$ as the increment of the primitive $- 1 / r$ on compact subintervals, the Riemann and Lebesgue values agreeing for the continuous integrand; a singleton or the pole sphere set ${x_{0}}$ is null and does not affect the integral (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, The polar surface set function on the unit sphere, The closed form for the volume of the unit $n$ -ball, Agreement with the existing polar sphere measure, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, Continuity and derivatives of positive-base real powers, Riemann–Lebesgue comparison for distribution test integrands, A nonnegative integral over a null set vanishes).

[F3]

The nonnegative integral is monotone and homogeneous, the integral of a radial density is the polar integral, and monotone convergence passes increasing limits; the point source is the idealisation of the weak point-source definition and of the static-to-distributional reading (Monotonicity and nonnegative homogeneity of the nonnegative integral, Integral over a measurable subset, Integrable real and complex functions, and their integrals, Monotone convergence for the integral, Continuous functions on Euclidean spaces are Borel measurable, Distributional Maxwell equations with point and stationary-sheet sources, The static and pure electrostatic regimes and their fixed-time reading, Open ball, closed ball and sphere in a metric space, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The finite and reverse triangle inequalities for a norm; and for $n \geq 1$ every norm $N$ on $R^{n}$ satisfies $N (x) \leq C ∥ x ∥_{1}$ and is Lipschitz, hence continuous, for $d_{2}$ ).

Proof

technique · direct

1.1givenF1F2F3

For $z \neq 0$ the field satisfies $∣ E_{q} (z) ∣^{2} = q^{2} / (16 π^{2} ε_{0}^{2} ∣ z ∣^{4})$ by [F1]; the shell ${δ \leq ∣ x - x_{0} ∣ \leq R}$ excludes $x_{0}$ because $δ > 0$ , and the integrand is continuous on the shell, hence Borel and locally bounded there; the value assigned to $E_{q}$ at $x_{0}$ does not affect the integral since ${x_{0}}$ is null ([F1], [F2], [F3]).

2.1step 1.1F2F3

Polar integration about $x_{0}$ : substituting $x = x_{0} + z$ and using the polar formula in $z$ with the radial density $q^{2} / (16 π^{2} ε_{0}^{2} r^{4})$ gives $\int_{{δ \leq ∣ x - x_{0} ∣ \leq R}} ∣ E_{q} (x) ∣^{2} d λ_{3} (x) = \frac{q^{2}}{16 π^{2} ε_{0}^{2}} \int_{δ}^{R} r^{- 4} \cdot 4 π r^{2} d r = \frac{q^{2}}{4 π ε_{0}^{2}} \int_{δ}^{R} r^{- 2} d r,$ where $σ (S^{2}) = 4 π$ and the radial integrals are taken over the compact interval $[δ, R]$ ([F2], [F3]).

3.1step 2.1F2F3

The fundamental theorem with the primitive $r \mapsto - r^{- 1}$ of $r \mapsto r^{- 2}$ gives $\int_{δ}^{R} r^{- 2} d r = 1 / δ - 1 / R$ ([F2]); substituting into step 2.1 and multiplying by $ε_{0} / 2$ yields $U_{q} (δ, R) = \frac{ε_{0}}{2} \cdot \frac{q^{2}}{4 π ε_{0}^{2}} (\frac{1}{δ} - \frac{1}{R}) = \frac{q^{2}}{8 π ε_{0}} (\frac{1}{δ} - \frac{1}{R}),$ which is clause (a); the quantity is finite for $0 < δ < R$ and of the sign of $q^{2}$ ([F3]).

4.1step 3.1F2F3

Clause (b): letting $R \to \infty$ in the exact formula gives $\lim_{R \to \infty} U_{q} (δ, R) = q^{2} / (8 π ε_{0} δ)$ for every $δ > 0$ , and letting $δ ↓ 0$ gives $\lim_{δ ↓ 0} U_{q} (δ, R) = + \infty$ whenever $q \neq 0$ , while for $q = 0$ both expressions vanish identically; the limits are read off the continuous elementary functions $R \mapsto 1 / R$ and $δ \mapsto 1 / δ$ on $(0, \infty)$ ([F2], [F3]).

5.1step 4.1F1F3∎

Clause (c): the computation used only the idealised point field of [F1] and the radial integration of [F2]; it yields a divergence of the truncated field energy of that idealisation. No identification of field energy with rest mass, no self-force, no measurement and no inconsistency claim is made; the smooth compact-source energy of Finite-source assembly energy equals the electric field energy remains finite by [F1], the compact $L^{1}$ branch keeps only its locally integrable potential and distributional Poisson equation, and the point source is the idealisation of the weak point-source definition together with the static-to-distributional reading [F3].

Divergence of the point-charge field energy and the limits of mass inference

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources