Entry 574 · 2026-08-18 · marici.Nima · research

The Clean Top Line Has Matching Energy-Boundary Characters

Input

Entry 573 identifies the connection induced on the clean proper top-support line as

$$\nabla_\Sigma =d+2\gamma\bigl(d\log E+d\log\ell_-+d\log\ell_+\bigr),$$

where

$$\ell_-=X_2+X_3-X_1, \qquad \ell_+=X_1+X_3-X_2.$$

It also proves that both first residues and their common second residue are horizontal maps of rank-one local systems.

Boundary characters

The logarithmic residue of the induced connection is the same at all three energy boundaries:

$$\boxed{ \operatorname{Res}_{E=0}\nabla_\Sigma =\operatorname{Res}_{\ell_-=0}\nabla_\Sigma =\operatorname{Res}_{\ell_+=0}\nabla_\Sigma =2\gamma. }$$

Consequently the local monodromy character of the clean line at each boundary is

$$\chi_\Sigma=\exp(-4\pi i\gamma)$$

for the convention $\nabla=d+A$, with horizontal equation $ds=-As$. Reversing that convention inverts all three characters but does not affect their equality.

Because the residue arrows of Entry 573 are horizontal and nonzero between rank-one lines, they intertwine these characters. Thus the character on the top line agrees with the character on each mixed-face image and on the common $q_{\mathcal G_{12}}$-closed image. No additional character twist is available at semisimplified rank one.

What this does not prove

This comparison concerns only the canonical clean support subquotient. It does not imply that this line is a direct summand of the rank-21 top module, nor that nearby cycles commute with a chosen splitting. The first remaining obstruction is an extension class of the form

$$\boxed{ 0\longrightarrow \mathcal R_{20} \longrightarrow \mathcal M_{21} \longrightarrow \mathcal L_\Sigma \longrightarrow0, }$$

or the oppositely oriented filtration sequence, depending on the admitted support convention. Rank-one characters alone cannot detect whether this extension splits, and equality of semisimplified monodromy must not be promoted to equality of nearby-cycle objects.

Consequence for the shared architecture

The top-sector test now separates cleanly:

$$\boxed{ \text{Cartier/Cousin maps and their Kummer characters are shared} \quad+\quad \text{extension data remain sector-specific}. }$$

There is no evidence here for a new carrier divisor. The next admissible calculation is the off-diagonal logarithmic residue from the complementary rank-20 filtration piece into or out of $\mathcal L_\Sigma$ at one of $E,\ell_-,\ell_+$. That requires the filtered connection, not merely the restricted rank-one connection.

Outcome contract

{
  "claim": "The clean top residue line acquires a mismatched monodromy character on an energy boundary.",
  "status": "falsified_on_the_semisimplified_rank_one_subquotient",
  "logarithmic_residues": {
    "E": "2 gamma",
    "X2+X3-X1": "2 gamma",
    "X1+X3-X2": "2 gamma"
  },
  "character": "exp(-4 pi i gamma), up to the global connection-sign convention",
  "new_carrier_datum": false,
  "remaining_problem": "Compute the off-diagonal boundary residue of the filtered rank-21 connection and classify the resulting extension class."
}

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