CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-11-04 (gpt-6-luna)

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Vanishing dimensionless controls do not force a vanishing mass ratio

Statement refuted

For every four sequences $g, a, m, M : N_{+} \to R_{> 0}$ of dimensionless real numbers, the implication $g_{n} \to 0$ and $a_{n} \to 0$ implies $m_{n} / M_{n} \to 0$ holds, without any further relation between the four sequences.

Facts & Assumptions

Given: Let $N_{+} = {1, 2, \dots}$ and work in the standard ordered real field. All sequence entries are dimensionless real numbers; no physical interpretation is imposed.

[A1]

The universal implication displayed in Statement refuted is the claim being refuted; its only hypotheses are strict positivity and the two stated control-parameter limits.

[F1]

For a real sequence $x_{n}$ , $x_{n} \to L$ means that for every $ε > 0$ there is $N \in N_{+}$ such that $n \geq N$ implies $∣ x_{n} - L ∣ < ε$ .

[F2]

For every $ε > 0$ , there is $N \in N_{+}$ with $N > 1 / ε$ (the Archimedean property of $R$ ).

[L1]

For each $n \in N_{+}$ , set $g_{n} = a_{n} = 1 / n$ and $m_{n} = M_{n} = 1 / n^{2}$ .

Counterexample

1.1givenL1algebra

Since $n \geq 1$ , both $1 / n$ and $1 / n^{2}$ are positive. Thus all four sequences in [L1] are strictly positive, and $M_{n} = 1 / n^{2} \neq 0$ makes $m_{n} / M_{n}$ well-defined for every $n$ .

1.2givenF1F2L1algebra

Given $ε > 0$ , choose $N$ as in [F2]. For every $n \geq N$ , $0 < g_{n} = a_{n} = 1 / n \leq 1 / N < ε$ . By [F1], $g_{n} \to 0$ and $a_{n} \to 0$ .

2.1givenA1F1L1step 1.1step 1.2algebra∎

For every $n \in N_{+}$ , $m_{n} / M_{n} = (1 / n^{2}) / (1 / n^{2}) = 1$ . The ratio sequence is therefore identically $1$ , so it converges to $1$ and cannot converge to $0$ (take $ε = 1 / 2$ in [F1]). The witness satisfies the antecedent of [A1] by 1.2 and violates its conclusion, refuting [A1].

Remarks

This is an abstract dimensionless control-ratio calculation motivated only by effective-field-theory bookkeeping. It supplies no compactification, stationary point, metastability result, flux or tadpole solution, or string realization.

The calculation is contextual to Controlled localized anti-D3 uplift to a full type-IIB de Sitter solution ‡ and Ten-dimensional origin of nonperturbative large-volume-potential scaling ‡ on the A page; neither open question is a premise or logical dependency.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.