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

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Scaling yields the twisted composition law

Statement

Assume Countable Choice.

Let $Z_{ρ}$ be the mass-indexed formal renormalization map defined for the scale action $σ_{ρ}$ , $ρ > 0$ . For all $ρ, τ > 0$ , $Z_{ρ τ} = Z_{ρ} \circ (σ_{ρ} Z_{τ} σ_{ρ}^{- 1}) .$ If $σ_{ρ} Z_{τ} σ_{ρ}^{- 1} = Z_{τ}$ for every $ρ, τ > 0$ , this reduces to $Z_{ρ τ} = Z_{ρ} \circ Z_{τ}$ . No one-parameter subgroup law is asserted without this condition.

Statement references

Formal scaling and renormalization cocycle

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Facts & Assumptions

Given: Countable Choice, and the scale-indexed systems and formal maps in Formal scaling and renormalization cocycle, with $ρ, τ > 0$ .

[A1]

Assume Countable Choice The Axiom of Countable Choice ( ${A C}_{ω}$ ) as required by the scalar contraction contract used to define the comparison maps.

[F1]

Formal scaling and renormalization cocycle: on a map family $A = (A_{m})_{m \geq 0}$ , the mass- $m$ component of its induced scale conjugate is $(σ_{ρ} A σ_{ρ}^{- 1})_{m} = C_{ρ}^{*} \circ A_{ρ^{- 1} m} \circ (C_{ρ}^{*})^{- 1}$ , and $σ_{ρ τ} = σ_{ρ} σ_{τ}$ . In particular, at each $m \geq 0$ , $C_{ρ}^{*} \circ S_{ρ^{- 1} m} \circ (C_{ρ}^{*})^{- 1} = S_{m} \circ Z_{ρ, m}$ , where $Z_{ρ} = (Z_{ρ, m})_{m \geq 0}$ and each $Z_{ρ, m}$ is the unique local formal map with identity linear term supplied by the finite-renormalization comparison proposition.

Proof

technique · direct substitution followed by uniqueness

Given: [A1] and [F1].

1.1A1F1algebra

Under [A1], conjugating the defining equation for $Z_{τ}$ by $σ_{ρ}$ and using the group law for $σ$ gives $σ_{ρ τ} S σ_{ρ τ}^{- 1} = σ_{ρ} (S \circ Z_{τ}) σ_{ρ}^{- 1} = (σ_{ρ} S σ_{ρ}^{- 1}) \circ (σ_{ρ} Z_{τ} σ_{ρ}^{- 1}) = S \circ Z_{ρ} \circ (σ_{ρ} Z_{τ} σ_{ρ}^{- 1}) .$ The conjugated map is local, support-preserving, Poincare-covariant, and has identity linear term because $σ_{ρ}$ is an automorphism normalizing the Poincare action and $Z_{τ}$ has those properties.

2.1A1F1step 1.1algebra∎

The left side of step 1.1 also equals $S \circ Z_{ρ τ}$ by [F1]. Both maps composed with $S$ satisfy the same finite-renormalization conditions, so uniqueness in [F1] gives $Z_{ρ τ} = Z_{ρ} \circ (σ_{ρ} Z_{τ} σ_{ρ}^{- 1}) .$ When the conjugation action is trivial the factor in parentheses is $Z_{τ}$ , giving the stated special case.

Remarks

The mass label is part of the scale action. Suppressing it in the notation does not turn the cocycle into a group homomorphism; the required action on the renormalization maps remains explicit in the formula.

Scaling yields the twisted composition law

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources