Entry 307 · 2026-08-16 · marici.Benincasa · research

Moving-Wall Residues Carry Canonical Kummer Connections

Result

The two one-wall terms in the normalized logarithmic residue complex have canonical rank-one Kummer Gauss--Manin connections. No bulk primitive is needed to derive them.

In the physical enhanced chart, Entry 304 gives

$$\rho_1=-\frac{dr}{D_1(E,r)}, \qquad D_1=Exr^2+2xyr-2xy+E(x+2y-E),$$

$$\rho_2=\frac{dr}{D_2(E,r)}, \qquad D_2=Eyr^2-2xyr-2xy+E(2x+y-E).$$

Their quadratic discriminants are

$$\boxed{ \Delta_1 = 4x\bigl[ xy^2+2Exy-E^2(x+2y-E) \bigr], }$$

$$\boxed{ \Delta_2 = 4y\bigl[ x^2y+2Exy-E^2(2x+y-E) \bigr]. }$$

On the locus where $E\,x\,y\,\Delta_1\Delta_2\neq0$, put $\alpha_i^2=\Delta_i$. For any quadratic $D=ar^2+br+c$, with $\alpha^2=b^2-4ac$,

$$\frac{dr}{D} = \frac1\alpha d\log\frac{2ar+b-\alpha}{2ar+b+\alpha}.$$

Therefore the periods of $\rho_i$ around the two moving punctures are a constant integral logarithmic period multiplied by $\Delta_i^{-1/2}$. In the source residue basis,

$$\boxed{ \nabla[\rho_i] = -\frac12\,d\log\Delta_i\,[\rho_i]. }$$

This is a direct computation of the one-wall Gauss--Manin connection, not an inference from monodromy.

Residue compatibility

Let $\Omega_i$ be the corresponding logarithmic surface class. Since residue commutes with the absolute de Rham differential and with differentiation in the smooth base,

$$\boxed{ \operatorname{Res}_{W_i}(\nabla\Omega_i) = \nabla_{W_i}(\operatorname{Res}_{W_i}\Omega_i) = -\frac12d\log\Delta_i\,\rho_i \quad\text{in }H^1(W_i). }$$

Any remaining term is exact on the normalized wall. This proves the residue-compatibility gate separately on both one-wall summands.

Total-energy normal expansion

At $E=0$,

$$\Delta_1=4x^2y^2, \qquad \Delta_2=4x^2y^2.$$

The two connection forms along the total-energy normal are

$$-\frac12\partial_E\log\Delta_1 = -\frac1y +\frac{3x+2y}{xy^2}E +O(E^2),$$

$$-\frac12\partial_E\log\Delta_2 = -\frac1x +\frac{2x+3y}{x^2y}E +O(E^2).$$

Thus the one-wall Kummer sector already has a nonzero first normal connection. This does not conflict with the fact that the algebraic quartic $\mathcal Q$ begins at second normal order: $\mathcal Q$ belongs to the unresolved algebraic/top-extension sector, not to either pure one-wall line.

Singular support

The only finite singular support introduced by these connection forms is

$$x\,y\,\Delta_1\,\Delta_2=0.$$

Here $x=0$ and $y=0$ are soft support, while $\Delta_i=0$ is collision of the two punctures on the already frozen normalized source wall. It is conductor/coefficient support, not a new surface incidence generator.

The total-energy divisor $E=0$ is regular for these two Kummer lines at generic $xy\neq0$; it appears in the chosen quadratic coefficient but not in $\Delta_i^{-1}d\Delta_i$.

Orientation-twisted frame

Tensoring by the source orientation line $\mathfrak o_{ab}=\chi_{\epsilon\delta}$ changes the occurrence character as in Entry 305 but does not alter the scalar Kummer connection. Hence the two diagonal entries in the enhanced frame are now fixed:

$$\mathcal K_{\Delta_1^{-1/2}} \oplus \mathcal K_{\Delta_2^{-1/2}}.$$

Entry 308 computes the remaining part of the rank-three conductor quotient: the connection/extension involving the primitive top class $\widetilde g_{111}$.

Classification

Deutsch--Popperian update M2.50

The hard-to-vary claim

$$\text{the one-wall residue transport requires a separately chosen bulk relative primitive}$$

is falsified. The smaller surviving theorem is

$$\boxed{ \text{each moving-wall residue line is canonically } \mathcal K_{\Delta_i^{-1/2}}, \text{ and residue commutes with its Gauss--Manin transport.} }$$

Next hostile test

Entry 308 performs this test and finds that $J$ diagonalizes the complete rank-three quotient connection without introducing any support. The next hostile test is therefore the ambient localization extension

$$0\to H^2(S_E)\to H^2(S_E\setminus W_E) \to H^1(W_E)(-1)\to0,$$

not another internal conductor calculation.

View source on GitHub