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Scale cocycles, finite scheme changes and controlled RG equations — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

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.