TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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

Statement

Let $ρ > 0$ be a dimensionless scale and put $t = \log ρ \in R$ . Take $I \subseteq R$ to be an interval containing $0$ . Let $g : I \to R$ be a differentiable real coupling and let $b, g_{0} \in R$ be dimensionless constants; require $g$ to satisfy the truncated scalar initial-value problem $\frac{d g}{d t} = b g^{2}, g (0) = g_{0} .$ A solution through $(0, g_{0})$ is maximal if there is no strictly larger interval $J ⊋ I$ and solution $G : J \to R$ of the same initial-value problem with $G ∣_{I} = g$ . If $c : = b g_{0} = 0$ , the unique maximal real solution is $g (t) = g_{0}$ on $R$ . If $c \neq 0$ , the unique maximal solution through $t = 0$ is $g (t) = \frac{g_{0}}{1 - b g_{0} t}$ on $(- \infty, c^{- 1})$ when $c > 0$ and on $(c^{- 1}, \infty)$ when $c < 0$ .

For the formal beta-series calculation, let $u$ be an indeterminate and work in $R [[u]]$ . Write $β (u) = b u^{2} + O (u^{3})$ and let $Φ (u) = u + a u^{2} + O (u^{3})$ be an invertible formal change of dimensionless coupling, with dimensionless $a \in R$ . The notation $O (u^{k})$ means an element of $u^{k} R [[u]]$ , so its terms have degree at least $k$ ; the real trajectory $g (t)$ is distinct from the formal indeterminate $u$ . Put $\tilde{u} = Φ (u)$ , let $Φ^{'}$ be the termwise formal derivative, and define the transformed formal vector field in $R [[\tilde{u}]]$ by $\tilde{β} (\tilde{u}) = (Φ^{'} (u) β (u)) ∣_{u = Φ^{- 1} (\tilde{u})} .$ The quadratic coefficient of $\tilde{β}$ is $b$ . The real ODE is exact only for the displayed truncation; neither it nor the formal scheme-change statement determines $b$ from a QFT graph or bounds omitted higher-order terms. Its pole is a property of this truncated equation, not a theorem about an ultraviolet completion.

Statement references

First-order systems, initial value problems, and solutions on intervals

Facts & Assumptions

Given: $b, g_{0} \in R$ , $c = b g_{0}$ , dimensionless $ρ > 0$ with $t = \log ρ$ , and the formal change $Φ (u) = u + a u^{2} + O (u^{3})$ in $R [[u]]$ .

[A1]

The vector field is the prescribed truncation $F (t, g) = b g^{2}$ on $D = R^{2}$ ; no higher-order beta coefficients or measured coupling are specified.

[A2]

The formal beta transformation is defined in $R [[u]]$ by push-forward under the invertible formal series $Φ$ ; its termwise derivatives and substitutions are coefficientwise, and no formal series is assumed to converge.

[F1]

The Statement takes $ρ > 0$ dimensionless and $t = \log ρ$ , and separates the real trajectory $g (t)$ from the formal indeterminate $u \in R [[u]]$ . The scalar coefficient $b$ is prescribed independently; no renormalization cocycle or QFT coefficient calculation is assumed.

[F2]

First-order systems, initial value problems, and solutions on intervals: a solution on an interval is a differentiable real function satisfying the stated equation and initial condition.

[F3]

Local Lipschitz continuity in the state variable, locally uniform in time and parameters: a vector field is locally Lipschitz in the state variable when one finite Lipschitz constant works on each compact time-state cylinder.

[F4]

Picard-Lindelöf local existence and uniqueness for first-order systems: a continuous vector field locally Lipschitz in the state variable has a unique local solution, and two solutions with the same initial data agree on every common interval containing the initial time.

Proof

technique · direct verification of the rational solution, ODE uniqueness, and formal re-expansion

Given: [A1]–[A2] and [F1]–[F4].

1.1A1F1F2algebra

Under [A1], [F1] fixes $t = \log ρ$ as the dimensionless scale coordinate and the vector field is the prescribed $F (t, g) = b g^{2}$ . If $c = 0$ , the constant function $g (t) = g_{0}$ has derivative zero and $b g_{0}^{2} = 0$ , so it solves the IVP on all of $R$ . If $c \neq 0$ , set $γ (t) = g_{0} / (1 - c t)$ on the interval specified in the Statement. For every interior $t$ , $\frac{γ (t + h) - γ (t)}{h} = \frac{g_{0} c}{(1 - c t) (1 - c (t + h))} ⟶ \frac{g_{0} c}{(1 - c t)^{2}} = b γ (t)^{2},$ and $γ (0) = g_{0}$ , so it solves the IVP.

2.1A1F2F3F4step 1.1

The vector field $F (t, x) = b x^{2}$ is continuous on the open domain $D = R^{2}$ . On any compact time-state cylinder with $∣ x ∣, ∣ y ∣ \leq R$ , $∣ F (t, x) - F (t, y) ∣ = ∣ b ∣ ∣ x - y ∣ ∣ x + y ∣ \leq 2 ∣ b ∣ R ∣ x - y ∣,$ so it is locally Lipschitz in the state variable by [F3]. Uniqueness from [F4] shows that every solution through $(0, g_{0})$ agrees with step 1.1 on its common interval. If $c \neq 0$ , any strictly larger interval containing the stated interval must contain $c^{- 1}$ . A solution $G$ on such an interval would be continuous at $c^{- 1}$ , while uniqueness forces $G = γ$ on the original interval and $∣ γ (t) ∣ \to \infty$ as $t$ approaches $c^{- 1}$ from within it, a contradiction. When $c = 0$ , the constant solution already exists on all of $R$ . These facts establish the stated maximal intervals.

3.1A2algebra∎

Termwise differentiation in $R [[u]]$ gives $Φ^{'} (u) = 1 + 2 a u + O (u^{2})$ . Since $Φ$ has linear term $u$ , recursive coefficient matching gives $Φ^{- 1} (\tilde{u}) = \tilde{u} + O ({\tilde{u}}^{2})$ , hence $u^{2} = {\tilde{u}}^{2} + O ({\tilde{u}}^{3})$ . Therefore $Φ^{'} (u) β (u) = b u^{2} + O (u^{3})$ and re-expression in $\tilde{u}$ yields $\tilde{β} (\tilde{u}) = b {\tilde{u}}^{2} + O ({\tilde{u}}^{3})$ . The coefficient $b$ is unchanged; no coefficient at higher order is determined here.

Remarks

The adjacent Low-momentum matching has a controlled remainder is a separate spectral remainder estimate under its low-energy band hypothesis. It supplies no estimate for omitted terms in this beta truncation. This theorem derives no numerical QED beta coefficient and identifies no scheme parameter with a measured observable. The context of Formal scaling and renormalization cocycle motivates the logarithmic scale convention; the present theorem uses that convention directly and assumes no existence of a renormalization map or time-ordered system.

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

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

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