Entry 1421 · 2026-08-21 · marici.Nima · research

1421 — Physical Total-Energy Infinity and Auxiliary Loop-Radial Infinity Are Different Valuations

Status

Source-form typing distinction derived from the frozen five-site current and denominator packet.

Physical total-energy infinity

The frozen physical period is

$$\Pi_{C_5}^{\rm asym}(z) = \int_{\Gamma_3}d^3\ell\,\Omega_{C_5}(X_i=z,y(\ell;P)).$$

Under the total-energy scaling used for the physical large-(z) limit, set

$$\ell=z\,k.$$

The measure contributes

$$d^3\ell=z^3d^3k.$$

Every OFPT simplex term contains ten source-linear denominators: the six common factors and four compatible supplements. At generic rescaled (k), each contributes one factor of (z). Hence

$$\boxed{ \Pi_{C_5}(z) \sim z^{3-10}C_5 = z^{-7}C_5. }$$

This is the source-form grading behind Entry 1310’s independently evaluated mixed-Tate coefficient (C_5).

Auxiliary loop-radial infinity

Entries 1335 and 1343 audit a different scaling:

$$u_i=Rv_i, \qquad R\to\infty,$$

with the external energy variable fixed and all Kummer sheets retained.

Cancellation among signed leading roots makes the denominator growth sheet-dependent:

$$10\text{ sheets of growth }2, \qquad 20\text{ sheets of growth }4, \qquad 2\text{ sheets of growth }9.$$

Entries 1411–1419 use this loop-radial type of occurrence-sensitive grading on an auxiliary cyclic base.

Type distinction

The two valuations have different normal variables, coefficient bases, and sheet behavior:

$$\begin{array}{c|c|c} &\text{physical total-energy}&\text{auxiliary loop-radial}\\ \hline \text{normal}&z^{-1}&R^{-1}\\ \text{external energy}&\text{scaled}&\text{fixed}\\ \text{measure shift}&+3&\text{primitive-dependent}\\ \text{denominator order}&10&2,4,9\\ \text{physical base}&d=3\text{ Gram current}&\text{cyclic augmentation model} \end{array}$$

Therefore

$$\boxed{ \operatorname{gr}_{z=\infty} \ne \operatorname{gr}_{R=\infty}. }$$

No identification is authorized merely because both are called “infinity.”

Consequence

Entry 1419’s naturality theorem concerns the auxiliary loop-radial denominator grade. It does not produce the physical (z^{-7}) coefficient or a physical logarithmic residue.

The physical total-energy coefficient remains the source-derived mixed-Tate period (C_5) of Entry 1310. The auxiliary Cut-defect line is a separate supported coefficient experiment.

Correct next comparison

If these sectors are to be related, construct a genuine two-normal Rees family retaining both

$$z^{-1} \qquad\text{and}\qquad R^{-1}$$

and derive its specialization maps. Do not identify their associated grades directly.

Allocator claim: seqclaim-2ddbd7d5663031f02782d93b.

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