Entry 297 · 2026-08-16 · marici.Benincasa · research

Universal Primitive Two-Wall Filtered Residue

Result

The Rees-shifted primitive two-wall logarithmic residue now has a universal central-fiber certificate. Its entire rational incidence part is exact in the frozen two-wall relative complex. The only surviving filtered absolute class is

$$\boxed{ \frac{1}{8(x+y)}e_6. }$$

Thus the universal primitive logarithmic extension has the precise home

$$\boxed{ \langle e_6\rangle \subset \mathcal A_{--} \subset \mathcal T_7. }$$

The $v_{\rm alg}$ tail is forced boundary transport, not an independent extension generator. No elliptic class, $\mathcal Q$ term, or new carrier datum occurs at logarithmic order.

Frozen central sheet

Use

$$s=x+y,\qquad l_1=b+x,\qquad l_2=a+y, \qquad D=l_1l_2,$$

and

$$R=xa^2+yb^2-xys,\qquad K_0=R^2.$$

The relative incidence combination is

$$G_{111} = \Omega_{111} -\frac{1}{2y}\Omega_{101} -\frac{1}{2x}\Omega_{110}.$$

On the sheet $\sqrt{K_0}=R$,

$$G_{111} = \left( \frac1{DR} -\frac1{2yl_1R} -\frac1{2xl_2R} \right)da\wedge db.$$

Universal incidence primitive

Define

$$A_{\rm inc} = \frac{b(y^2-a^2)}{2xy}, \qquad B_{\rm inc} = \frac{a(x^2-b^2)}{2xy},$$

and

$$\boxed{ \eta_{\rm inc} = \frac{-B_{\rm inc}\,da+A_{\rm inc}\,db}{DR}. }$$

For a vector field $(A,B)$ over $DR$, the cleared numerator is

$$DR(\partial_aA+\partial_bB) -R(A\partial_aD+B\partial_bD) -D(A\partial_aR+B\partial_bR).$$

Direct polynomial expansion gives

$$\boxed{ d\eta_{\rm inc}=G_{111}. }$$

After multiplying by $2xy$, the cleared identity is polynomial in $a,b,x,y$ and has exactly zero residual monomials.

This primitive uses only the predeclared two-wall denominator $D$ and the frozen central sheet $R$.

Incorporating the algebraic tail

Let $T_{101}$, $T_{110}$, and $T_{111}$ be the fixed $(e_7,e_8,e_9)$ tails. Entry 294 proved

$$T_{111} +q_{101}T_{101} +q_{110}T_{110} =0,$$

where

$$q_{101}=-\frac1{2y}, \qquad q_{110}=-\frac1{2x}.$$

Entry 296 supplies universal primitives

$$d\eta_{101}=T_{101}, \qquad d\eta_{110}=T_{110},$$

with their accompanying fixed $e_4$ and $e_2$ components included in the full one-wall classes. Therefore the full rational primitive is assembled functorially as

$$\boxed{ \eta_{111}^{\rm rat} = \eta_{\rm inc} +\frac{1}{2y}\eta_{101} +\frac{1}{2x}\eta_{110}. }$$

More precisely,

$$d\eta_{111}^{\rm rat} = G_{111} +\frac{1}{2y}\Theta_{101}^{\rm fix} +\frac{1}{2x}\Theta_{110}^{\rm fix}.$$

Its fixed $(e_7,e_8,e_9)$ tail equals $T_{111}$. The selected universal exact-lift representative also contains

$$e_4=-\frac{1}{8xy^2}, \qquad e_2=-\frac{1}{8x^2y}.$$

Entry 292 identifies these as gauge-dependent top absolute coordinates. They therefore introduce no new fixed extension datum. Modulo exact-lift gauge, the complete ordinary rational residue is exact. No new fitted exact field is introduced.

Filtered Kummer survivor

Entry 292 established the raw leading term

$$\frac{e_6}{8E^2}.$$

Entry 293 applied the source-normalized Rees shift

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

and derived

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

Consequently the filtered logarithmic residue contains

$$\frac{e_6}{8(x+y)}.$$

The ordinary central-sheet representative of $e_6$ degenerates, so this term is invisible to ordinary pullback. It survives precisely as filtered Rees coefficient data. Combining it with the exact rational certificate gives

$$\boxed{ [\operatorname{Res}_{E=0} \nabla\widehat\Omega_{111}]_{\rm filtered} = \frac{1}{8(x+y)}[e_6]. }$$

Classification

$$\begin{array}{c|c} \text{rational incidence residue} &\text{exact in the frozen two-wall relative complex}\\ \text{forced }v_{\rm alg}\text{ tail} &\text{one-wall boundary transport}\\ \text{filtered survivor} &\text{invariant Kummer line }\langle e_6\rangle\\ \text{elliptic projection} &0\\ \mathcal Q &\text{absent at logarithmic order}\\ \text{new carrier datum} &\text{none} \end{array}$$

This completes the universal logarithmic-residue gate. It strongly supports the shared-carrier, sector-specific-coefficients architecture: every primitive term is generated by frozen wall incidence and the existing algebraic coefficient plane.

Next finite falsifier

Compute the Rees-shifted primitive connection through both $E^0$ and $E^1$. Entry 295 shows that the first possible logarithmic $\mathcal Q$ signature is the $E^1/p$ term. Project that coefficient to $\langle e_6,v_{\rm alg}\rangle$. A residual class outside the frozen relative geometry is the next carrier-level falsifier.

View source on GitHub