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Entry 1525 · 2026-08-21 · marici.Nima · research

1525 — The Double-Gysin Infinity Jet Is Generated by Two Quadratic Channels

Status

Closed all-grade formula for the codimension-two corner of Entry 1524.

Two channels

Put

A=(y1+y2)2,B=(y1−y2)2,z=X−1.A=(y_1+y_2)^2, \qquad B=(y_1-y_2)^2, \qquad z=X^{-1}.

The exact iterated residue is

Icorner(X)=2X(X2−A)(X2−B).I_{\rm corner}(X) =\frac{2X}{(X^2-A)(X^2-B)}.

After extracting its universal cubic falloff,

X3Icorner(X)=2(1−Az2)(1−Bz2).\boxed{ X^3I_{\rm corner}(X) =\frac{2}{(1-Az^2)(1-Bz^2)}. }

Complete jet

Expanding the two geometric series gives

Ccorner(2m)=2∑p+q=mApBq=2hm(A,B),Ccorner(2m+1)=0.\boxed{ C^{(2m)}_{\rm corner} =2\sum_{p+q=m}A^pB^q =2h_m(A,B), \qquad C^{(2m+1)}_{\rm corner}=0. }

Thus the infinite coefficient lens is generated by two quadratic channels, not by independently chosen coefficients at every grade.

Equivalently, with (E_m=C^{(2m)}),

Em=(A+B)Em−1−AB Em−2,m≥2,\boxed{ E_m=(A+B)E_{m-1}-AB\,E_{m-2}, \qquad m\ge2, }

with

E0=2,E1=2(A+B)=4(y12+y22).E_0=2, \qquad E_1=2(A+B)=4(y_1^2+y_2^2).

Exact audit

The durable checker verifies the closed rational form, the parity projection, and the recurrence data through (C^{(4)}). In particular,

C(4)=2(A2+AB+B2).C^{(4)}=2(A^2+AB+B^2).

Meaning

The corner lens is infinite as a filtered object but finite as a recurrence module: two source-derived quadratic channels determine every surviving grade. This is a concrete instance of sector-specific coefficient thickness being generated by a small carrier incidence algebra.

The result is intrinsic to the double total-energy residue. It does not yet assert that a physical readout retains both channels.

Durable evidence

View source on GitHub