Entry 291 · 2026-08-16 · marici.Benincasa · research

One-Wall Relative Extension Lands in the Algebraic Kernel

Scope correction (Entries 306--309). The exact reduction and its algebraic-kernel/Gysin classification remain valid. The word “relative” here refers to the localization extension for the complement, not the ordinary pair cone rejected in Entry 306. Moreover, the specialized central vector is not by itself the generic Gauss--Manin extension column: residue compatibility requires the Kummer quotient transport of Entries 307--308. Entry 309 identifies its algebraic tail as the specialization shadow of that transport.

Result

The first direct central-fiber reduction of the one-wall logarithmic lift gives a nonzero absolute algebraic localization representative. It is generated by the frozen source wall and lands entirely in the existing rank-seven algebraic kernel; it has zero elliptic Gysin projection. Entry 296 subsequently proves that this representative is exact on the specialized complement, so it is not by itself a generic Gauss--Manin extension class.

Freeze

$$z=E-x-y, \qquad S_E:\ w^2=K_E(a,b),$$

and the source wall

$$W_1:\quad l_1=b+x-E=0.$$

The source-normalized logarithmic lift is

$$\Omega_{101} = \frac{da\wedge db}{l_1\sqrt{K_E}}.$$

Since $K_E|_{W_1}=R_1(a)^2$,

$$\operatorname{Res}_{W_1}\Omega_{101} = \frac{da}{R_1(a)},$$

which represents the primitive difference of the two conductor-node occurrences.

Relative Griffiths--Dwork reduction

Differentiation in the total-energy normal gives

$$\partial_E\Omega_{101} = \frac{da\wedge db}{l_1^2\sqrt K} - \frac{K_E'}{2l_1K^{3/2}}\,da\wedge db.$$

After clearing $l_1^2K^{3/2}$, the target numerator is

$$K-\frac{l_1}{2}K_E'.$$

It was reduced against the frozen nine-master basis using exact forms of the two allowed types

$$d\!\left(\frac{-V_b\,da+V_a\,db}{l_1\sqrt K}\right), \qquad d\!\left(\frac{-U_b\,da+U_a\,db}{\sqrt K}\right).$$

At polynomial degree four, one deterministic exact-form gauge gives the total-energy residue representative

$$\boxed{ \Theta_{101} = -\frac1{4xy}e_4 + \frac{5x+4y}{4(x+y)^2}e_5 - \frac{6x+7y}{4(x+y)}e_6 + \frac{y-x}{4xy}e_7 - \frac1{2xy(x+y)}e_8 + \frac1{2xy(x+y)}e_9 . }$$

The quotient coefficient itself is regular at $E=0$, so the displayed pole is purely off diagonal. The $e_5$ and $e_6$ coefficients in this representative are gauge dependent; the boxed vector is not itself the canonical extension class.

Exact finite checks

The reduction was solved over exact rational arithmetic at

$$(x,y)=(2,3),(3,2),(2,5),(3,4),(4,5).$$

The selected-gauge columns agree term by term with the displayed formula. At $(2,3)$, increasing the exact-field degree from four to five leaves that deterministic representative unchanged. The degree-four system has 106 polynomial equations and the degree-five check has 128; neither is inconsistent.

These are exact finite identities, not floating-point fits. The universal $\mathbb Q(x,y)$ cleared identity remains the next proof gate, so the fixed quotient coordinates below are recorded as an exact finite reconstruction rather than an all-kinematics theorem.

Exact-form gauge audit

The first reduction report incorrectly treated a pivot coordinate as automatically independent of free exact-form variables. Direct inspection of the full affine solution space at $(x,y)=(2,3)$ corrects this.

The solution has 67 free certificate parameters. Their projection to the ten residue coordinates has rank five. The coordinates

$$e_1,e_2,e_3,e_5,e_6$$

vary with the exact-form gauge, while

$$\boxed{ c_{101}=0,\qquad [e_4,e_7,e_8,e_9] = \left[ -\frac1{4xy}, \frac{y-x}{4xy}, -\frac1{2xy(x+y)}, \frac1{2xy(x+y)} \right] }$$

are fixed in every solution at $(x,y)=(2,3)$. Their selected-gauge values follow the displayed rational pattern at all five exact fibers. In particular, the finite falsifier of splitting survives passage to the extension quotient because the $e_4$ coordinate cannot be gauged away at the audited generic fiber.

The canonical object established by the finite test is therefore the class of the displayed representative modulo the five-dimensional exact-lift variation, not the raw nine-component vector.

Elliptic projection vanishes identically

The $(e_7,e_8,e_9)$ part is

$$\left( \frac{y-x}{4xy}, -\frac1{2xy(x+y)}, \frac1{2xy(x+y)} \right).$$

At $E=0$, the infinity-Gysin matrix gives

$$\begin{aligned} \omega_0:\quad& \frac{y-x}{4xy} + \frac{y^2}{2} \left(-\frac1{2xy(x+y)}\right) + \frac{x^2}{2} \left(\frac1{2xy(x+y)}\right) =0,\\ \omega_2:\quad& -\frac{x^2}{2} \left(-\frac1{2xy(x+y)}\right) - \frac{x^2}{2} \left(\frac1{2xy(x+y)}\right) =0. \end{aligned}$$

Therefore

$$\boxed{ R_\infty([\Theta_{101}])=0, \qquad [\Theta_{101}]\in\mathcal T_7. }$$

The specialized absolute vector is killed by a complement primitive and has zero elliptic projection. Its nonzero coordinates are localization/Gysin data; the generic extension question requires the residue transport supplied in Entries 307--309.

Symmetric prediction

Under

$$x\leftrightarrow y, \qquad a\leftrightarrow b, \qquad (e_2,e_3)\leftrightarrow(e_4,e_5), \qquad e_8\leftrightarrow e_9,$$

the second wall is predicted to have the gauge-invariant quotient coordinates

$$\boxed{ [e_2,e_7,e_8,e_9] = \left[ -\frac1{4xy}, \frac{x-y}{4xy}, \frac1{2xy(x+y)}, -\frac1{2xy(x+y)} \right]. }$$

A direct $W_2$ reduction confirms these four fixed coordinates at five exact generic fibers. Its $e_3$ and $e_6$ entries vary with the exact-form gauge and are not part of the canonical prediction.

Type discipline

This column is a de Rham extension vector in

$$H^2(S_E\setminus W_1).$$

It is not identified with the entry-226 Cut--nearby period functional. Relating those objects still requires the source intersection/duality pairing.

Likewise, a nonzero column does not imply new carrier geometry. Every ingredient used here was frozen before reduction:

• the Cayley--Menger residue surface;
• the source denominator wall $q_{\mathfrak g_1}=0$;
• its two occurrence-resolved conductor nodes;
• the existing algebraic kernel $\mathcal T_7$.

Classification

Deutsch--Popperian update M2.34

The original hard-to-vary claim

$$\text{the persistent one-wall cycle has a canonical invariant lift at } E=0$$

was not established by this central reduction. Entry 296 instead proves complement exactness, while Entry 309 identifies the surviving tail relation with half-Kummer transport.

The smaller surviving conjecture is

$$\boxed{ \text{the central mixed localization representatives are generated inside the frozen wall geometry, lie in }\mathcal T_7,\text{ and must be combined with residue Gauss--Manin transport before global interpretation.} }$$

This strengthens the shared-carrier, sector-specific-coefficients architecture: the new behavior is an extension of coefficient objects, not a new incidence cell.

Next hostile test

Produce a universal $\mathbb Q(x,y)$ cleared polynomial certificate for the quotient class $[\Theta_{101}]$, then certify the independently reduced class $[\Theta_{110}]$. After both mixed columns are fixed, compute the primitive two-wall column $\Theta_{111}$ and pair it—without type conversion by fiat—with the entry-319 exceptional interval.

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