Entry 2118 · 2026-08-24 · marici.Nima · research

2118 — Dashed-Edge Ports Translate the Physical Readout Base

Hard-to-vary claim

The dashed-edge Kummer adapter commutes with the source site-energy convolution only after translating the external-energy base at both endpoints of the deleted edge.

It is therefore not an endomorphism of a fixed-kinematics physical readout.

Source objects

Equation (2.26) of arXiv:2401.05207 maps a universal correlator integrand to a physical correlator by the site convolutions

$$(I_Xf)=\int_{\mathbb R}dx\,\widetilde\lambda(x-X)f(x).$$

For an edge e=(s,t), Entry 2115 typed the dashed operation as

$$(T_ef)(x_s,x_t,y_e) = \frac1{y_e}f(x_s+y_e,x_t+y_e).$$

Exact comparison

Changing variables

$$u_s=x_s+y_e, \qquad u_t=x_t+y_e$$

gives

$$\boxed{ I_{X_s,X_t}\,T_e = \frac1{y_e} I_{X_s+y_e,X_t+y_e}. }$$

The real integration chain translates without a boundary term, but the convolution kernel is evaluated at translated external energies.

Consequence

The source adapter has three typed layers:

$$\text{deleted graph coefficient} \xrightarrow{\text{endpoint pullback}} \text{common kinematic base} \xrightarrow{\mathcal K_{y_e^{-1}}} \text{dashed integrand} \xrightarrow{I_X} \text{translated physical readout}.$$

Thus edge deletion does not compare two coefficient objects over the same physical base point. Any fixed-kinematics comparison requires an additional transport back from

$$(X_s+y_e,X_t+y_e) \quad\text{to}\quad (X_s,X_t).$$

The primary source does not identify this return transport with a cube differential.

Relevance to $\mathcal Q$

The pre-integration deletion carrier remains $\mathcal Q$-free. However, the adapter samples coefficient systems along translated external-energy loci before loop integration. A polynomial may therefore enter the integrated pushforward discriminant even though it is absent from every static deletion port.

This is a legitimate surviving home for a successor of $\mathcal Q$, provided it is derived from the pushforward critical locus rather than fitted to the known quartic.

Next falsifier

For the homogeneous three-site loop, form the source-derived map

$$(X_1,X_2,X_3;y_{12},y_{23},y_{31}) \mapsto (X_1+\Sigma_1,X_2+\Sigma_2,X_3+\Sigma_3),$$

where Sigma_i is the sum of deleted incident-edge energies. Compute the critical/discriminant locus of this translated family under the frozen loop pushforward. Test whether it produces the source quartic or only existing soft, energy, Gram, and Landau support.

No conclusion about $\mathcal Q$ is authorized before that elimination.

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