Entry 293 · 2026-08-16 · marici.Benincasa · research

Kummer-Normalized Rees Gauge Removes the Primitive Double Pole

Narrow result

Entry 292 found, in the raw primitive two-wall frame,

$$\nabla_E\Omega_{111}=\frac{e_6}{8E^2}+\frac{R_{\rm raw}}E+O(1).$$

The leading class is on the established invariant algebraic line $\langle e_6\rangle\subset\mathcal T_7$, whose source normalization is

$$\nabla e_6=-\frac12d\log H\,e_6,\qquad H=z^2((E^2-y^2)(E^2-x^2)+E^2z^2).$$

For fixed $x,y$, with $z=E-x-y$, the Rees lift

$$\boxed{\widehat\Omega_{111}=\Omega_{111}+\frac{e_6}{8E}}$$

has no double pole. The higher-normal term is therefore an existing-coefficient Rees extension, not a new carrier stratum.

Exact normal calculation

Put $s=x+y$ and

$$J(E)=(E^2-y^2)(E^2-x^2)+E^2(E-s)^2,\qquad H(E)=(E-s)^2J(E).$$

Then

$$J(0)=x^2y^2,\quad J'(0)=0,\quad H(0)=s^2x^2y^2,\quad H'(0)=-2sx^2y^2,$$

so

$$-\frac12\partial_E\log H\big|_{E=0}=\frac1{x+y}.$$

Consequently

$$\nabla_E\left(\frac{e_6}{8E}\right) =-\frac{e_6}{8E^2}+\frac{e_6}{8E(x+y)}+O(1).$$

This fixes both the sign of the Rees correction and the induced logarithmic coefficient

$$\boxed{\operatorname{Res}^{e_6}_{E=0} (\nabla\widehat\Omega_{111})=\frac{e_6}{8(x+y)}.}$$

Reconstructed logarithmic residue

Combining this correction with the fixed coordinates of the exact degree-four and degree-five reductions gives

$$\begin{aligned} \operatorname{Res}_{E=0}(\nabla\widehat\Omega_{111})\equiv{}& \Omega_{111}-\frac{\Omega_{101}}{2y}-\frac{\Omega_{110}}{2x} +\frac{e_6}{8(x+y)}\\ &-\frac{(x-y)^2}{8x^2y^2}e_7 +\frac{y-x}{4x^2y^2(x+y)}e_8 +\frac{x-y}{4x^2y^2(x+y)}e_9 \end{aligned}$$

modulo holomorphic absolute and exact-form gauge.

The quotient coefficients and final-block tail are exact rational reconstructions from $(x,y)=(2,3),(3,2),(2,5)$, with degree-four to degree-five stabilization at $(2,3)$. They remain finite evidence until a cleared identity over $\mathbb Q(x,y)$ is derived.

Infinity-Gysin test

At $E=0$,

$$R_\infty^{(0)}=e_7+\frac{y^2}{2}e_8+\frac{x^2}{2}e_9,\qquad R_\infty^{(2)}=-\frac{x^2}{2}(e_8+e_9).$$

The displayed tail gives

$$R_\infty^{(0)}=R_\infty^{(2)}=0.$$

Since $R_\infty(e_6)=0$, every fixed absolute residue term lies in $\mathcal T_7$. The regularization does not modify the elliptic quotient.

Classification

$$\begin{array}{c|c} E^{-2}e_6/8\text{ in raw frame}&\text{Rees gauge on existing }\mathcal T_7\\ e_6/[8(x+y)]&\text{Tate/Kummer coefficient data}\\ e_7,e_8,e_9\text{ tail}&\text{algebraic kernel }\mathcal T_7\\ \mathcal Q&\text{absent from the total-energy residue}\\ \text{new carrier incidence}&\text{none} \end{array}$$

Thus the first higher-normal correction supports the shared-carrier, sector-specific-coefficients hypothesis.

Qualifications and next falsifier

Established, conditional on the already source-normalized Kummer identification of $e_6$: cancellation of the raw double pole, the induced $e_6/[8(x+y)]$ residue, zero elliptic Gysin image, and no new carrier datum at this order.

Still open: a universal cleared reduction, uniqueness outside the normalized $e_6$ line, the regular $E^0$ extension, the home of $\mathcal Q$ there, and compatibility with the physical chain.

The next finite falsifier is to perform the Rees-shifted two-wall reduction over $\mathbb Q(x,y)$, demand the displayed universal residue, and then compute the regular term. Failure requiring a new incidence divisor rather than an existing relative/algebraic coefficient class is the next carrier-level falsifier.

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