ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-11-05 (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.

Linearized flow of a unit-dimension scalar perturbation

Physical scope

Conditional data-level Wilsonian calculation for a normalized scalar scaling eigenoperator of total dimension one; it stops when the dimensionless coupling leaves the stated linearized regime and makes no global-flow or physical-realization claim.

Example

Condition on declared full local two-dimensional CFT data that contain a normalized scalar scaling eigenoperator $O$ of total dimension $Δ = 1$ . This is a data-level example and does not construct such a CFT. Let $a_{0} > 0$ be a reference cutoff length, let the fixed dimensionful coupling $λ_{p h y s} \in R$ have units $[l e n g t h]^{- 1}$ , and set $t = \log (a / a_{0})$ and $g (t) = λ_{p h y s} a (t)$ with $a (t) = a_{0} e^{t}$ . The linearized regime is the interval where $∣ g (t) ∣ < ε$ for a fixed dimensionless tolerance $ε > 0$ . The example computes only the engineering beta term in this regime; OPE-induced nonlinear corrections and a later endpoint are not specified.

Example references

Regulated conformal perturbation and beta-function conventions

Linear beta coefficient from the scaling dimension

Facts & Assumptions

Given: The cutoff and IR time convention in the Example, and a scalar scaling eigenoperator with $Δ = 1$ .

[F1]

Regulated conformal perturbation and beta-function conventions: the dimensionless coupling is $g (a) = λ_{p h y s} a^{2 - Δ}$ and $t = \log (a / a_{0})$ increases toward the IR.

[F2]

Linear beta coefficient from the scaling dimension: the engineering beta term is $(2 - Δ) g$ in this convention.

[A1]

The background CFT data and the normalized operator are assumed as stated in the Example; no existence theorem for them is used.

Verification

Given: $g_{0} = λ_{p h y s} a_{0}$ and $∣ g_{0} ∣ < ε$ .

1.1givenF1A1algebra

Since $a (t) = a_{0} e^{t}$ and $Δ = 1$ , the definition gives $g (t) = λ_{p h y s} a_{0} e^{t} = g_{0} e^{t}$ .

2.1givenF2step 1.1algebra

Differentiating the fixed-dimensionful-coupling rescaling gives the engineering term $β_{e n g} (g (t)) : = d g (t) / d t = g_{0} e^{t} = g (t)$ , agreeing with $(2 - Δ) g$ from [F2]. This is not an equation for the full renormalized beta function: the theorem permits additional $O (∣ g ∣^{2})$ terms under its separate second-order remainder hypothesis, and this Example specifies no nonlinear correction.

2.2givenstep 1.1algebra

For $g_{0} \neq 0$ , the boundary $∣ g (t_{*}) ∣ = ε$ is reached at $t_{*} = \log (ε / ∣ g_{0} ∣) > 0$ , and $∣ g (t) ∣ < ε$ for $t < t_{*}$ . If $g_{0} = 0$ , then $g (t) = 0$ for every $t$ and there is no finite exit time.

3.1givenA1step 2.2∎

For nonzero $g_{0}$ , the strict linearized regime is $t < t_{*}$ ; at $t = t_{*}$ one has $∣ g (t_{*}) ∣ = ε$ , so the boundary point is outside that regime. This calculation supplies no nonlinear continuation or assertion that a full flow reaches another CFT.

Linearized flow of a unit-dimension scalar perturbation

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources