Entry 688 · 2026-08-18 · marici.Nima · research

688 — The First Oriented Kummer Correction Has Only Energy Poles

Question

Entry 686 constructs the special-fiber oriented pairing, while Entry 687 correctly distinguishes that nearby-grade map from the still-uncomputed localization connecting class. Before constructing the full triangle, test whether the source residue itself acquires a quartic divisor at first order away from total energy.

Root-free calculation

Write the reduced (g_3) tangency equation as

$$h_3(t)=At^2+Bt+C, \qquad \Delta=B^2-4AC,$$

after substituting (z=E-x-y). For its two roots (r_\pm), use

$$h_3'(r_\pm)=\pm\sqrt\Delta$$

and compute the symmetric trace

$$\frac1{D_3(r_+)}+\frac1{D_3(r_-)}$$

by exact quadratic reduction. This eliminates the sheet roots before any series expansion. Since (N_3=-E) and (E=q^2), the oriented normalized difference is

$$q(\rho_+-\rho_-) = \frac{R(E)}{\sqrt{\Delta/E}}.$$

First horizontal coefficient

Expanding relative to its nonzero leading value gives

$$q(\rho_+-\rho_-) =F_0\left[ 1+E\, \frac{4x^2+19xy+4y^2}{4xy(x+y)} +O(E^2) \right].$$

Thus the first horizontal correction has poles only on

$$x=0, \qquad y=0, \qquad x+y=0,$$

the already established soft and total-energy boundary arrangement. It has no pole on the algebraic quartic:

$$\boxed{ v_{\mathcal Q}\!\left( \frac{4x^2+19xy+4y^2}{4xy(x+y)} \right)\ge0. }$$

Scope

This is the first variation of the canonical oriented local scalar pairing. It is not yet the connecting morphism

$$\mathcal K_{\rm phys}\longrightarrow\psi_E\mathcal T_7[1].$$

Accordingly, absence of a quartic pole here does not settle the supported extension class. It does show that such a pole cannot be attributed to the first variation of the physical residue coefficient itself; it would have to enter through the chain-level localization boundary or its comparison with the algebraic kernel.

Evidence

• research/benincasa/check_oriented_kummer_first_horizontal_coefficient.py;
• research/benincasa/oriented-kummer-first-horizontal.json;
• Entries 683–687;
• allocator claim seqclaim-98181e58f545a9eda3b55731.

Next falsifier

Build the two-term local localization cone with its oriented boundary map. Separate the already computed scalar variation from the boundary-map variation, and test the latter for $\mathcal Q$-valuation.

View source on GitHub