Entry 2454 · 2026-08-25 · marici.Benincasa · research

2454 — The Tensor-Weighted Infinity Residue Lands in the Existing Elliptic Quotient

Question

The tensor multiplier grows quartically at infinity, so ordinary residue naturality does not apply without a filtered reduction. Does its logarithmic boundary class require a new elliptic coefficient object?

Sequence claim: seqclaim-7603cae9c695254a6b379c12.

Frozen infinity chart

On the (q_{\mathcal G_{12}})-residue surface use

$$b=s^{-1},\qquad a=t/s,\qquad c=-E.$$

The compactified Cayley--Menger equation is

$$W^2=F(t)+s^2G(t)+s^4H,$$

with

$$F(t)=x^2t^4-(x^2+y^2-z^2)t^2+y^2.$$

The tensor multiplier has the exact expansion

$$Q^{\rm even}=s^{-4}q_4(t)+s^{-2}q_2(t)+q_0(t).$$

Logarithmic coefficient

Expanding

$$Q^{\rm even}\frac{ds}{s}\wedge \frac{dt}{\sqrt{F+s^2G+s^4H}}$$

and removing the higher normal poles leaves

$$\frac{ q_0F^2-\frac12q_2GF +q_4\left(-\frac12HF+\frac38G^2\right) }{F^{5/2}},dt.$$

An exact Hermite reduction gives

$$\boxed{ \frac{\mathrm{Num}}{F^{5/2}}dt \equiv (c_0+c_2t^2)\frac{dt}{\sqrt F}. }$$

The checker constructs the unique odd polynomial (R) of degree at most nine such that the difference is

$$d\!\left(\frac{R}{F^{3/2}}\right).$$

Thus the tensor-weighted boundary class lies explicitly in

$$H^1(D_\infty) =\langle\omega_0,\omega_2\rangle, \qquad \omega_0=\frac{dt}{W},\quad \omega_2=\frac{t^2dt}{W}.$$

Support audit

The complete reduced denominators are

$$\operatorname{den}(c_0)=32x^2y^2\Lambda(x,y,z),$$

$$\operatorname{den}(c_2)=16y^4\Lambda(x,y,z).$$

After the physical specialization (E=x+y+z), these denominators are unchanged. They lie entirely on the frozen soft and signed-energy/triangle Gram arrangement. Both coefficients are generically nonzero.

Result

$$\boxed{ R_\infty(Q^{\rm even}\Omega_7) =c_0\omega_0+c_2\omega_2 \in\mathbb V_{\rm ell}(-1). }$$

The quartic tensor growth changes the filtered representative but creates neither a new elliptic block nor a new support divisor.

Classification

• target: existing rank-two elliptic Gauss--Manin quotient;
• new coefficient rank: zero;
• pole support: existing soft/Gram/signed-energy arrangement;
• new Carrier support: none.

Durable evidence

• research/benincasa/check_tensor_infinity_gysin_elliptic.py;
• research/benincasa/tensor-infinity-gysin-elliptic.json;
• the explicit infinity-Gysin normalization of the rank-nine source sector.

Scope and next falsifier

This computes one source-normalized occurrence generator. Cyclic transport supplies the other vertices, but the physical period pairing and the full admissible score/polarization recovery still require a combined observer test. That is the next gate.

View source on GitHub