Entry 303 · 2026-08-16 · marici.Benincasa · research

Central Primitives Do Not Determine the Enhanced Stokes Pairing

Result

The proposed direct pullback of the central-fiber primitives of Entries 296--297 to the enhanced exceptional intervals is not valid. The specialized walls and the moving source walls meet the exceptional divisor at different points.

For the physical $(--)$ chart set

$$E=\tau^2,\qquad a=-y-\tau^2r,\qquad b=-x+\tau^2r-\tau^3n.$$

The central walls used in Entries 296--297 obey

$$\frac{b+x}{\tau^2}=r-\tau n, \qquad \frac{a+y}{\tau^2}=-r,$$

and therefore meet the exceptional divisor at $r=0$. The actual moving source walls obey

$$\frac{b+x-E}{\tau^2}=r-1-\tau n, \qquad \frac{a+y-E}{\tau^2}=-(r+1),$$

and meet it at $r=1$ and $r=-1$, respectively.

Thus specialization at $E=0$ and restriction to the weighted exceptional divisor do not commute for these logarithmic primitives.

Explicit finite witness from $\eta_{101}$

Entry 296 writes

$$\eta_{101} = \frac{-B\,da+A\,db} {4x^2y(x+y)(b+x)R}, \qquad R=xa^2+yb^2-xy(x+y).$$

On the $n=0$ slice of the physical chart, put $q=\tau^2$. Then

$$a=-y-qr,\qquad b=-x+qr,$$

and direct substitution gives the exact identities

$$R=(x+y)q^2r^2$$

and

$$A+B = 2x(x+y)^2q^2r^2 +(y^2-x^2)q^3r^3 +2(x+y)q^4r^4.$$

Since

$$da=-q\,dr,\qquad db=q\,dr,$$

the tangential numerator is

$$-B\,da+A\,db=q(A+B)\,dr.$$

Therefore the leading exceptional restriction of the central primitive is

$$\boxed{ \eta_{101}^{(0)} \sim \frac{1}{2xy}\frac{dr}{r}. }$$

Its logarithmic point is $r=0$. The moving wall $b+x-E=0$, however, selects $r=1$. A form with the required moving boundary behavior must be obtained from an $E$-dependent primitive; replacing $r$ by $r-1$ after the calculation would be a post hoc change of the frozen source object.

The symmetric calculation puts the central $\eta_{110}$ pole at $r=0$ while its moving wall selects $r=-1$.

Falsified claim

The hard-to-vary claim

$$\text{the central primitives can be pulled back directly to the enhanced moving-wall intervals}$$

is falsified by the three distinct exceptional points

$$r=-1,\qquad r=0,\qquad r=1.$$

Accordingly, Entry 302's stronger statement that only evaluation remained has been corrected.

Smaller surviving statement

The central/moving-wall mismatch remains a valid finite result, but Entry 306 corrects the cohomological type of the surviving comparison:

$$\boxed{ \text{moving-wall logarithmic residue} \longrightarrow \text{Leray-tube exceptional pairing}. }$$

The missing object is the residue-compatible Gauss--Manin connection on the already frozen marked complement. It is not evidence for a new carrier stratum.

Classification

Deutsch--Popperian update M2.46

The failure is narrow: central-fiber exactness does not furnish the residue class of the moving logarithmic family. The next admissible object is fixed by the residue-compatible Gauss--Manin connection, not by fitting a boundary answer.

Next hostile test

Compute the Gauss--Manin connection on the normalized logarithmic residue complex for the frozen moving denominators

$$b+x-E,\qquad a+y-E$$

and require:

1. the source normalization unchanged;
2. compatibility $\operatorname{Res}\nabla=\nabla_W\operatorname{Res}$;
3. agreement with the endpoint logarithms of Entry 304 and the orientation-twisted lattice of Entry 305;
4. no poles outside the frozen marked, Cayley--Menger, soft, and conductor divisors.

Then evaluate the Leray-tube pairing at $r=\pm1$. A forced pole on any other divisor is the next finite carrier-level falsifier.

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