Manuscript X-A-conv-ST@1 · X-A-conv-ST.tex · theorem · 8ca47088

Theorem: The gap survives evaluation

Strict positivity and the evaluated classical obstruction › The action and its scalar solution · lines 900–935 · proof included or linked

X-A-conv-ST@1#thm:evaluated, X-A-conv-ST@1#eq:orbit

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Let

q_{λ} = exp (λκ L_{ξ_{*}}) η (mod κ^{3})

and use the fixed free scalar $ρ_{0} = x^{1}$ . For

A_{λ} = \int f (z) ρ_{g} [q_{λ}; ρ_{0}] (Y_{q_{λ}} (z)) d^{4} z,

the first coefficient difference is

a (λ) := [κ] (A_{λ} - A_{0}) = λ c_{ε}, λ \in R .

Proof. Lemma lem:scalarcov gives $ρ_{1, g} [q_{λ}] = λ X_{ξ_{*}} ρ_{0} = λ ψ$ . Lemma lem:preflight gives $w [q_{λ}] = λ S_{ε} ξ_{*}$ . The evaluated first relational coefficient is therefore

ρ_{1, g} - X_{w} ρ_{0} = λ (ψ - S_{ε} ψ) .

Smearing gives (eq:orbit).

Displayed formulas in this passage

X-A-conv-ST@1#eq:orbit · line 915
$a (λ) := [κ] (A_{λ} - A_{0}) = λ c_{ε}, λ \in R .$
unlabeled · line 903
$q_{λ} = exp (λκ L_{ξ_{*}}) η (mod κ^{3})$
unlabeled · line 909
$A_{λ} = \int f (z) ρ_{g} [q_{λ}; ρ_{0}] (Y_{q_{λ}} (z)) d^{4} z,$
unlabeled · line 930
$ρ_{1, g} - X_{w} ρ_{0} = λ (ψ - S_{ε} ψ) .$

Theorem: The gap survives evaluation

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