Entry 306 · 2026-08-16 · marici.Benincasa · research

Logarithmic Residue Duality Replaces the Ordinary Pair Cone

Result

Entry 302 correctly falsified naive contraction of equation-(58) master coordinates, but proposed the wrong replacement complex. The frozen three-site marked system is defined by forms on

$$U_E=S_E\setminus W_E,$$

not by the ordinary relative pair $(S_E,W_E)$. Consequently its boundary map is logarithmic residue, not pullback.

For a smooth wall component the canonical sequence is

$$\boxed{ 0\longrightarrow\Omega^\bullet_{S_E} \longrightarrow\Omega^\bullet_{S_E}(\log W_E) \xrightarrow{\operatorname{Res}_{W_E}} \Omega^{\bullet-1}_{W_E} \longrightarrow0. }$$

For the reducible source wall, this is replaced by the normalization/conductor totalization already frozen in Entries 266, 268--280, 342--343.

Finite type falsifier

The forms $\eta_{101},\eta_{110},\eta_{111}^{\rm rat}$ of Entries 296--297 are meromorphic one-forms on the surface complement. They have poles along the wall. Therefore $i^*\eta$ is not a regular form on $W_E$, and these objects cannot be the boundary component of

$$\operatorname{Cone}\bigl( \Omega^\bullet(S_E)\to\Omega^\bullet(W_E) \bigr)[-1].$$

Thus the hard-to-vary claim

$$\text{the central meromorphic primitives are boundary cochains in the ordinary pair cone}$$

is falsified by type alone.

Their valid role survives: they certify complement exactness and exact-lift gauge. The actual wall objects are the residues

$$\rho_i=\operatorname{Res}_{W_i}\Omega.$$

Entry 304 computes their source-normalized exceptional limits:

$$\rho_1\longrightarrow-\frac{dr}{2xy(r-1)}, \qquad \rho_2\longrightarrow-\frac{dr}{2xy(r+1)}.$$

Correct duality

If $\gamma\subset W_i$ is a wall cycle and $T(\gamma)\subset U_E$ its source-oriented Leray tube, then locally

$$\boxed{ \int_{T(\gamma)}\Omega = 2\pi i\int_\gamma\operatorname{Res}_{W_i}\Omega, }$$

up to the fixed orientation convention. Exact-gauge cancellation therefore survives, but through residue/tube duality rather than through evaluation of a surface primitive on the wall.

The orientation character is not optional. Entry 305 proves that twisting the conductor quotient by the source line $\mathfrak o_{ab}=\chi_{\epsilon\delta}$ gives the enhanced Rees character and integral lattice exactly:

$$J= \begin{pmatrix} 2&0&1\\ 0&2&1\\ 0&0&1 \end{pmatrix}, \qquad \operatorname{im}J = \{(u,v,w):u\equiv v\equiv w\pmod2\}.$$

Hence the corrected duality uses no fitted pairing kernel and no new cell.

Classification

Deutsch--Popperian update M2.49

The stronger ordinary-pair interpretation fails. The smaller surviving claim is

$$\boxed{ \text{the enhanced/logarithmic comparison is residue-compatible Gauss--Manin transport followed by source-oriented Leray duality.} }$$

This correction changes the derived calculus used at the marked boundary; it does not change the frozen carrier.

Next hostile test

Compute the Gauss--Manin connection on the normalized residue triangle in the orientation-twisted frame and test

$$\operatorname{Res}\circ\nabla = \nabla_W\circ\operatorname{Res}.$$

Then pair the resulting wall classes with the enhanced exceptional classes through Leray tubes. The calculation must:

1. reproduce the Kummer characters and half-sum extension of Entries 280 and 305;
2. reproduce the endpoint logarithms of Entry 304;
3. introduce no singular support outside the frozen energy, conductor, Cayley--Menger, soft, and $\mathcal Q$ coefficient divisors.

A required new incidence divisor is the next finite falsifier of the shared-carrier hypothesis.

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