Entry 304 · 2026-08-16 · marici.Benincasa · research

Moving-Wall Normalization Fixes the Exceptional Logarithms

Type correction (Entries 306--307). The wall-residue formulas and endpoint logarithms in this entry are unchanged. References below to a bulk relative primitive or relative Griffiths--Dwork lift are superseded: the required continuation is the Gauss--Manin connection on the logarithmic residue system, followed by Leray-tube duality.

Result

Although Entry 303 proves that the central-fiber primitives cannot be transplanted to the moving exceptional interval, the frozen source wall normalizations already determine the two endpoint logarithms canonically.

For the two mixed walls, Entry 340 gives

$$K_E(t,E-x)=R_1(t)^2, \qquad K_E(E-y,t)=R_2(t)^2.$$

On the physical $(--)$ enhanced chart their normalized wall residues have the limits

$$\boxed{ \rho_{101} =\frac{da}{R_1(a)} \longrightarrow -\frac{1}{2xy}\frac{dr}{r-1}, }$$

$$\boxed{ \rho_{110} =\frac{db}{R_2(b)} \longrightarrow -\frac{1}{2xy}\frac{dr}{r+1}. }$$

Thus the logarithmic coefficients and their occurrence-resolved locations $r=1$ and $r=-1$ are source-derived. No choice of a bulk Griffiths--Dwork primitive is involved.

Exact first wall calculation

Write $s=x+y$, $z=E-s$. The source square root on

$$W_1:\quad b=y+z=E-x$$

is

$$\begin{aligned} R_1(t) ={}&xt^2-z^3-yz^2+y^2z+y^3\\ &-2xz^2-2xyz-xy^2-x^2z-x^2y. \end{aligned}$$

Substitution of $z=E-s$ gives the exact normal form

$$\boxed{ R_1(t) = xt^2-xy^2-2xyE+(x+2y)E^2-E^3. }$$

On the $n=0$ slice of the physical weighted chart,

$$a=-y-Er,\qquad da=-E\,dr.$$

Therefore

$$\boxed{ R_1(-y-Er) = E\left[ 2xy(r-1) +E(xr^2+x+2y-E) \right]. }$$

Consequently

$$\frac{da}{R_1(a)} = -\frac{dr} {2xy(r-1)+E(xr^2+x+2y-E)}$$

and the stated $E\to0$ logarithm follows.

Exact second wall calculation

The source square root on

$$W_2:\quad a=x+z=E-y$$

reduces exactly to

$$\boxed{ R_2(t) = yt^2-x^2y-2xyE+(2x+y)E^2-E^3. }$$

On the same chart,

$$b=-x+Er,\qquad db=E\,dr,$$

so

$$\boxed{ R_2(-x+Er) = E\left[ -2xy(r+1) +E(yr^2+2x+y-E) \right]. }$$

Hence

$$\frac{db}{R_2(b)} = \frac{dr} {-2xy(r+1)+E(yr^2+2x+y-E)} \longrightarrow -\frac{1}{2xy}\frac{dr}{r+1}.$$

Integral occurrence vector

Stripping the common source normalization $-1/(2xy)$, the two wall residues are the logarithmic occurrence covectors at

$$p_+=\{r=1\}, \qquad p_-=\{r=-1\}.$$

The physical Leray interval is oriented $p_-\to p_+$. Its boundary is

$$\partial[p_-,p_+]=[p_+]-[p_-],$$

so the endpoint pairing is governed by the already established integral occurrence vector

$$\boxed{(-1,1)}$$

in the ordered basis $(p_-,p_+)$, with the common rational normalization kept in the coefficient object.

This identifies the parity/occurrence part of the eventual Stokes boundary pairing before any bulk lift is selected.

What this does and does not solve

The endpoint logarithms are now fixed. The remaining problem is to transport these wall classes by the residue-compatible Gauss--Manin connection. Different surface-complement representatives may differ by exact gauge, but they cannot move the residue poles away from $p_\pm$ or change their source normalization.

In particular, the central logarithm at $r=0$ from Entry 303 is not a third physical occurrence. It is the specialization artifact of collapsing the moving wall before performing the weighted blowup.

Classification

Deutsch--Popperian update M2.47

Entry 303 falsified direct central pullback. The smaller surviving claim is now stronger than mere existence:

$$\boxed{ \text{the frozen moving-wall normalization uniquely fixes the exceptional logarithmic boundary classes, whose continuation is their canonical residue Gauss--Manin transport.} }$$

Next hostile test

Assemble the two Kummer residue lines of Entry 307 with the primitive conductor top extension of Entries 280 and 305. The resulting connection must have no poles except soft support and the predeclared Cayley--Menger/conductor strata. Any unavoidable additional pole is the next finite carrier-level falsifier.

Publication notation repair: malformed TeX delimiters or tags are corrected for display only. Historical source bytes and research claims are unchanged.

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