Entry 296 · 2026-08-16 · marici.Benincasa · research

Universal One-Wall Cleared Extension Certificate

Type clarification (Entries 306 and 309). The primitive below is a meromorphic one-form on the complement. It proves complement exactness of the specialized algebraic vector; it is not a boundary cochain in an ordinary relative-pair cone. The resulting central class must be combined with the generic residue Kummer transport before it is interpreted as a Gauss--Manin extension column.

Result

The one-wall extension formula of Entry 291 now has a universal cleared certificate over $\mathbb Q(x,y)$. It is no longer only a finite-fiber reconstruction.

On the generic nonsoft locus

$$xy(x+y)\ne0,$$

the fixed class

$$\boxed{ \Theta_{101}^{\rm fix} = -\frac{e_4}{4xy} +\frac{y-x}{4xy}e_7 -\frac{e_8}{2xy(x+y)} +\frac{e_9}{2xy(x+y)} }$$

is the image of the source wall boundary under an explicit rational primitive. The symmetric $W_2$ class follows by the oriented source involution.

Central-fiber reduction

Put

$$s=x+y,\qquad l_1=b+x,$$

and

$$R=xa^2+yb^2-xys.$$

The exact total-energy specialization is

$$K_0=R^2.$$

On a fixed sheet $\sqrt{K_0}=R$, the source basis specializes as

$$e_4=\frac bR,\qquad e_7=\frac1R,\qquad e_8=\frac{a^2}R,\qquad e_9=\frac{b^2}R.$$

Therefore

$$\Theta_{101}^{\rm fix} = \frac{P}{4xysR},$$

where

$$P=2b^2-2a^2-sb+s(y-x).$$

Explicit primitive

Define the degree-four polynomials

$$\begin{aligned} A={}& 2x^2ys\,a +xs(2y-x)ab -2xyab^2 -2x^2a^3\\ &+(x-y)ab^3 -2xa^3b, \end{aligned}$$

$$B=x^3s^2-xs(2x+y)b^2+sb^4.$$

Then set

$$\boxed{ \eta_{101} = \frac{-B\,da+A\,db}{4x^2ys\,l_1R}. }$$

For a polynomial vector field $(A,B)$, differentiation over $l_1R$ gives the cleared numerator

$$l_1R(\partial_aA+\partial_bB) -RB -l_1(A\partial_aR+B\partial_bR).$$

Direct expansion yields the identity

$$\boxed{ l_1R(\partial_aA+\partial_bB) -RB -l_1(A\partial_aR+B\partial_bR) = xP\,l_1^2R. }$$

Dividing by $4x^2ysl_1^2R^2$ proves

$$\boxed{ d\eta_{101} = \Theta_{101}^{\rm fix}\,da\wedge db. }$$

This is a literal polynomial identity in $a,b,x,y$, with zero residual monomials. No interpolation, numerical specialization, new denominator, or extra cohomology generator remains in the verification.

Meaning of exactness

The formula does not say that the extension vanishes in the absolute surface module. The primitive has the predeclared wall pole $l_1^{-1}$. It proves that the absolute algebraic vector becomes exact after passage to the source relative complement. This is precisely the connecting-extension behavior tested in Entry 291.

Thus the fixed one-wall class is universally generated by:

• the existing source wall $l_1=0$;
• the doubled rational central sheet $K_0=R^2$;
• the existing algebraic kernel classes.

Second wall

Apply

$$x\leftrightarrow y,\qquad a\leftrightarrow b,\qquad e_2\leftrightarrow e_4,\qquad e_8\leftrightarrow e_9.$$

The coordinate exchange reverses $da\wedge db$, so the primitive is pulled back with the compensating orientation sign. The same cleared identity proves

$$\boxed{ \Theta_{110}^{\rm fix} = -\frac{e_2}{4xy} +\frac{x-y}{4xy}e_7 +\frac{e_8}{2xy(x+y)} -\frac{e_9}{2xy(x+y)}. }$$

Classification and update

$$\begin{array}{c|c} \Theta_{101}^{\rm fix},\Theta_{110}^{\rm fix} &\text{universal central localization/Gysin representatives}\\ \text{primitive poles} &\text{existing source walls only}\\ \text{elliptic projection} &0\\ \text{new carrier datum} &\text{none} \end{array}$$

This upgrades the mixed-column part of the coefficient architecture from exact finite evidence to a universal central-fiber theorem. It does not by itself prove a generic mixed Gauss--Manin extension. Entry 309 supplies the missing comparison with the residue connection.

Next finite falsifier

Construct the analogous universal primitive for the Rees-shifted $\Theta_{111}$ relation using exactly the four frozen denominator strata. It must reproduce the incidence identity of Entry 294 and the $e_6/[8(x+y)]$ Kummer correction of Entry 293. Failure after the allowed central-sheet primitives are exhausted is the next coefficient-level falsifier.

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