Entry 1465 · 2026-08-21 · marici.Benincasa · research

1465 — A Sharp Single-Time Feature Is an Evaluation Coefficient

Status

Primary-source audit of the short-sharp-step limit in Adshead--Dvorkin--Hu--Lim, arXiv:1110.3050v3. This is a one-feature/one-time-kernel statement. It does not authorize a multivertex equal-time contact prescription absent from that source.

Frozen sharp limit

The source takes

$$V(\phi)=V_0(\phi) \left[1+cF\!\left(\frac{\phi_f-\phi}{d}\right)\right], \qquad F(x)=-\tanh x,$$

and derives the sharp limit $d\to0$. Its feature derivative becomes a delta function in logarithmic conformal time at the fixed feature time $\eta_f$:

$$F'\longrightarrow 2\,\delta(\ln(\eta/\eta_f)).$$

Accordingly, every displayed source integral containing this leading feature localizes by evaluation at $\eta_f$.

Fourier typing

Suppressing the source-fixed normalization, write the localized time kernel as

$$\lambda_f(\eta) = \delta(\ln(\eta/\eta_f)) = |\eta_f|\,\delta(\eta-\eta_f).$$

With the Fourier convention of Entry 1460, its density is an oscillatory character:

$$\boxed{ \widetilde\lambda_f(\epsilon) \propto |\eta_f|e^{-i\epsilon\eta_f}. }$$

It has noncompact Fourier support, but no finite Fourier singularity or support boundary. The convolution is Fourier inversion:

$$\int_{\mathbb R}d\epsilon\, \widetilde\lambda_f(\epsilon)\psi(\epsilon) \simeq \operatorname{ev}_{\eta_f}\psi.$$

Thus this noncompact pushforward is canonically defined as a distributional evaluation functor, not as an improper ordinary integral chosen afterward.

Energy-space consequence

For a one-site oscillatory factor $e^{iE\eta}$, localization gives

$$\boxed{ e^{iE\eta_f}. }$$

This is entire in $E$. The sharp feature removes the corresponding time integration pole rather than introducing a new energy divisor. The feature scale $\eta_f$ is coefficient data controlling an entire oscillatory character.

Classification

$$\boxed{ \text{Noncompact Fourier support alone does not imply a supported Cut/pushforward defect.} }$$

The sharp single-time feature is a source-normalized evaluation coefficient over the existing labelled time/energy occurrence. It introduces neither a new carrier incidence nor a Fourier-support wall.

This is a negative falsifier for the overly broad conjecture that every noncompact Fourier density requires an excess-support correction.

Boundary of the result

At one localized interaction, evaluation commutes with restrictions and with any Cut operation not involving a separately chosen equal-time contact prescription. The source does not define a graph of several adjacent interaction vertices all pinned to the same $\eta_f$. Such a graph would meet time-ordering diagonals and require a frozen rule for $\vartheta(0)$ or its resolved replacement.

Therefore this entry does not infer multivertex Cut closure. It identifies the next legitimate hostile source:

$$\boxed{ \text{a primary multivertex sudden-transition construction with an explicit equal-time ordering/contact prescription.} }$$

Only there can a genuine supported comparison cone appear.

Provenance

• Adshead--Dvorkin--Hu--Lim, arXiv:1110.3050v3, Eqs. (1)--(2), (21)--(23);
• Entries 1460 and 1463;
• allocator claim seqclaim-dc54a6859ab9a8f6f3626e57.
• epistemic event ev-000000001567-e0971507-abf2-451d-bc75-3f265b0ea74c.

View source on GitHub