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

Theorem: Strict convolution gap

Strict positivity and the evaluated classical obstruction › The exact positive gap · lines 717–757 · proof included or linked

X-A-conv-ST@1#thm:positive, X-A-conv-ST@1#eq:ceps, X-A-conv-ST@1#eq:energy

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For the frozen data (eq:witness),

c_{ε} := \int f (x) (ψ (x) - S_{ε} ψ (x)) d^{4} x = \frac{1}{2 N} \int_{R^{4}} ϱ_{ε} (a) \int_{R^{4}} (ψ (x) - ψ (x - a))^{2} d^{4} x d^{4} a > 0.

Therefore the frozen coordinate and relational prescriptions fail first-order covariance on $g_{C}$ .

Proof. Expanding the squared difference and using translation invariance of Lebesgue measure gives

\int (ψ (x) - ψ (x - a))^{2} d^{4} x = 2 \int ψ^{2} - 2 \int ψ (x) ψ (x - a) d^{4} x .

Integration against the normalized $ϱ_{ε}$ , followed by division by $2 N$ , gives (eq:ceps).

For every $a \neq = 0$ , the inner integral is positive. If it vanished, continuity would imply $ψ (x) = ψ (x - a)$ everywhere. Iteration would make a nonzero compactly supported function invariant under arbitrarily large translations, which is impossible. The mollifier is strictly positive on $0 < ∣ a ∣_{E} < ε$ , a set of positive measure. This proves strict positivity.

For $ξ = ξ_{*}$ and $ϕ = x^{1}$ , (eq:firstdefect) is exactly $c_{ε}$ .

Displayed formulas in this passage

X-A-conv-ST@1#eq:ceps · line 720
$c_{ε} := \int f (x) (ψ (x) - S_{ε} ψ (x)) d^{4} x = \frac{1}{2 N} \int_{R^{4}} ϱ_{ε} (a) \int_{R^{4}} (ψ (x) - ψ (x - a))^{2} d^{4} x d^{4} a > 0.$
X-A-conv-ST@1#eq:energy · line 720
$c_{ε} := \int f (x) (ψ (x) - S_{ε} ψ (x)) d^{4} x = \frac{1}{2 N} \int_{R^{4}} ϱ_{ε} (a) \int_{R^{4}} (ψ (x) - ψ (x - a))^{2} d^{4} x d^{4} a > 0.$
unlabeled · line 737
$\int (ψ (x) - ψ (x - a))^{2} d^{4} x = 2 \int ψ^{2} - 2 \int ψ (x) ψ (x - a) d^{4} x .$

Theorem: Strict convolution gap

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