ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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.

A truncated Landau pole is not continuum nonexistence

Example

For dimensionless $b > 0$ and $g_{0} > 0$ , the truncated equation $d g / d t = b g^{2}$ with $g (0) = g_{0}$ has maximal solution $g (t) = g_{0} / (1 - b g_{0} t)$ for $t < t_{*} : = (b g_{0})^{- 1}$ , and $g (t) \to + \infty$ as $t ↑ t_{*}$ . Here “truncated Landau pole” means this finite endpoint $t_{*}$ of the exact solution to the truncated equation, where the real coupling diverges. The calculation supplies no estimate for omitted higher-order beta terms near $t_{*}$ and implies no nonexistence result for a continuum theory.

Example references

One-coupling truncated RG flow and leading beta-coefficient invariance

Facts & Assumptions

Given: $b > 0$ , $g_{0} > 0$ , $t_{*} = (b g_{0})^{- 1}$ , and the truncated ODE in One-coupling truncated RG flow and leading beta-coefficient invariance.

[A1]

The example concerns only the real truncated initial-value problem; no omitted beta coefficients or ultraviolet completion are specified.

[F1]

One-coupling truncated RG flow and leading beta-coefficient invariance: for the stated prescribed scalar model, the unique maximal solution is $g (t) = g_{0} / (1 - b g_{0} t)$ on $(- \infty, t_{*})$ for $b g_{0} > 0$ ; maximal means that the IVP solution has no extension to a strictly larger interval as an IVP solution.

Verification

technique · direct endpoint limit

Given: [A1] and [F1].

1.1A1F1

Under [A1], [F1] gives the denominator positive on the interval containing zero with $t < t_{*}$ , and $g$ is the exact solution of the truncated equation there.

2.1A1F1step 1.1algebra∎

As $t ↑ t_{*}$ , $1 - b g_{0} t ↓ 0$ while $g_{0} > 0$ stays fixed, so $g (t) \to + \infty$ . Since [F1] states only the truncated equation, neither this limit nor the calculation gives a bound on omitted terms or a conclusion about continuum nonexistence.

Remarks

The pole is an endpoint of this model's maximal solution. It is not a measurement and does not assert that the truncation is accurate near that endpoint.

Depends on

One-coupling truncated RG flow and leading beta-coefficient invariance