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

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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]

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 $r^{2} + b^{2} \geq b^{2} > 0$ , 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.

Quantitative dilute comparison of actual coupled solutions

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels