Pipeline-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.

Exact smooth coupled counterpoised dust and observables — Examples

1 · Prerequisites

2 · Summary

These examples evaluate central fields and observables in the exact regular sphere and specialize the dilute estimates to $0 < δ \leq 1 / 10$ . The calculations describe ideal coupled model fields and do not report experiments, simulated measurements, stability, or a freely falling test-dust limit.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

Everywhere regular counterpoised sphere

Physical scope

Central values and proper integrals in one exact smooth coupled charged-dust model profile. This is an ideal conditional calculation, not an observation, particle-composition claim, or stability result.

Example

Assume ${A C}_{ω}$ as defined in The Axiom of Countable Choice ( ${A C}_{ω}$ ). Take $a = b > 0$ and $s = + 1$ in the exact A-page family. With $r^{2} = \sum_{i} (x^{i})^{2}$ , $Ψ = \sqrt{r^{2} + b^{2}}$ , and $H = 1 + b / Ψ$ , the center has $H (0) = 2$ , $\nabla H (0) = 0$ , and $Δ H (0) = - 3 / b^{2}$ . Its central rest-mass density and scalar curvature are $ρ (0) = \frac{3 c^{2}}{32 π G_{N} b^{2}}, S_{g} (0) = \frac{3}{4 b^{2}} .$ The observer electric field vanishes at the center. The complete model remains smooth with strictly positive density for every finite $r$ and has infinite spatial extent. Its proper mass and signed proper charge are $c^{2} b / G_{N}$ and $Q_{s} c^{2} b / G_{N}$ . There is no point source at the center; no stability, real-particle composition, or empirical realization is asserted.

Facts & Assumptions

Given: The indexed ${A C}_{ω}$ premise [A1], $b > 0$ , $a = b$ , $s = + 1$ , the selected SI conventions, and the exact A-page theorem.

[F1]

Under ${A C}_{ω}$ , Exact smooth coupled counterpoised dust and observables supplies the exact smooth coupled fields, $H = 1 + a / \sqrt{r^{2} + b^{2}}$ , $Δ H = - 3 a b^{2} / (r^{2} + b^{2})^{5 / 2}$ , $ρ = - c^{2} Δ H / (4 π G_{N} H^{3})$ , $E_{i} = - s c^{2} H_{i} / (Q_{s} H^{2})$ , $S_{g} = - 2 Δ H / H^{3}$ , and the proper mass and charge integrals. It asserts ideal model deductions only.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) is the indexed premise that every family $(X_{n})_{n \in N}$ of nonempty sets has a function $f$ on $N$ with $f (n) \in X_{n}$ for each $n$ . It is assumed here because the exact A-page theorem [F1] requires it; the finite substitutions below add no stronger choice principle.

Verification

1.1givenA1F1algebra

Since $b > 0$ , $r^{2} + b^{2} > 0$ everywhere. The derivative of the spherical profile in [F1] is $H_{i} = - b x^{i} / (r^{2} + b^{2})^{3 / 2}$ , which vanishes at $x = 0$ . Substituting $a = b$ into its Laplacian gives $Δ H (0) = - 3 b^{3} / b^{5} = - 3 / b^{2}$ , while $H (0) = 1 + b / b = 2$ . Thus the profile is regular at the center and is within [F1]'s positive smooth domain.

2.1givenA1F1step 1.1algebra∎

Substitution into [F1]'s density and curvature formulas yields $ρ (0) = 3 c^{2} / (4 π G_{N} \cdot 8 b^{2}) = 3 c^{2} / (32 π G_{N} b^{2})$ and $S_{g} (0) = 6 / (8 b^{2}) = 3 / (4 b^{2})$ . Since every $H_{i} (0) = 0$ , its observer electric components are zero there. The positive density for all finite $r$ and the proper mass/charge values are the exact spherical conclusions of [F1]; they make no observation or point-source claim.

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

Quantitative dilute comparison of actual coupled solutions

Physical scope

Exact coordinate metric and density inequalities in a restricted dilute subfamily of the smooth coupled model. Bounds compare ideal mathematical fields and are not experimental or numerical measurements.

Example

Assume ${A C}_{ω}$ as defined in The Axiom of Countable Choice ( ${A C}_{ω}$ ). Fix $b > 0$ and $0 < δ \leq 1 / 10$ , and set $a = δ b$ . With $r^{2} = \sum_{i} (x^{i})^{2}$ , let $h_{0} : = b / \sqrt{r^{2} + b^{2}}$ and $ρ_{l i n} : = 3 c^{2} δ b^{3} / [4 π G_{N} (r^{2} + b^{2})^{5 / 2}]$ . The exact coupled fields of the A-page theorem satisfy, entrywise in this Cartesian chart, $∣ g_{00} + (1 - 2 δ h_{0}) ∣ \leq \frac{3}{100}, ∣ g_{i j} - (1 + 2 δ h_{0}) δ_{i j} ∣ \leq \frac{1}{100},$ and $0 \leq 1 - \frac{ρ}{ρ_{l i n}} \leq \frac{3}{10} .$ These fractions bound discrepancies in an ideal exact model family. They are not measured errors and imply no stability or small-data existence theorem.

Facts & Assumptions

Given: The indexed ${A C}_{ω}$ premise [A1], $b > 0$ , $0 < δ \leq 1 / 10$ , $a = δ b$ , and the exact A-page family.

[F1]

Under ${A C}_{ω}$ , Exact smooth coupled counterpoised dust and observables establishes that for $a = δ b$ , $0 < δ \leq 1$ , the same exact smooth coupled fields satisfy $∣ g_{00} + (1 - 2 δ h_{0}) ∣ \leq 3 δ^{2}$ , $∣ g_{i j} - (1 + 2 δ h_{0}) δ_{i j} ∣ \leq δ^{2}$ , and $0 \leq 1 - ρ / ρ_{l i n} \leq 3 δ$ , with the displayed positive density profile.

[A1]

Verification

1.1givenA1F1algebra

Since $r^{2} + b^{2} \geq b^{2} > 0$ , the function $h_{0} = b / \sqrt{r^{2} + b^{2}}$ satisfies $0 < h_{0} \leq 1$ . The chosen $a = δ b$ has $a, b > 0$ and $0 < δ \leq 1$ , so the stated profile and positive $ρ_{l i n}$ meet [F1]'s exact spherical hypotheses.

2.1givenA1F1step 1.1algebra∎

Apply [F1]'s exact bounds and use $δ^{2} \leq 1 / 100$ and $3 δ \leq 3 / 10$ , which follow from $0 < δ \leq 1 / 10$ . This gives all three inequalities in the Example. Every comparison concerns exact model fields; no measured error is being reported.