Entry 577 · 2026-08-18 · marici.Benincasa · research

Correction: The Top Cousin Boundary Retains the Lower-Pair Stratum

Defect

Entry 568 assigned the vector

$$(1,-1,0)$$

to the canonical top boundary by replacing the lower-pair face with its zero proper Möbius grade. Entry 341 had already established the correct typing: the lower-pair intersection stratum exists, and the logarithmic residue along $q_{\mathcal G_{12}}$ is nonzero even though insertion of the lower pair creates no new proper support grade.

Thus the canonical geometric boundary is

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

Möbius ranks classify associated support increments. They are not the targets of the geometric Cousin differential.

Surviving results

The following computations remain valid:

• the transverse triple section and unit Jacobian;
• the identity

$$K|_P=E^2(X_2+X_3-X_1)^2(X_1+X_3-X_2)^2;$$

• the cancellation of the two iterated-residue paths through the mixed $q_{\mathcal G_{12}}$-lower faces;
• the moving-normal horizontal lifts of Entry 573;
• Gauss--Manin naturality on the clean rank-one support line.

Entry 571 is therefore a valid two-face coherence calculation, not a complete replacement for the three-face Cousin complex.

Corrected frontier

The source exponent in physical dimension is

$$\gamma=-\frac12.$$

Consequently the clean top coefficient line is

$$K_P^{-1/2} = \frac{\pm1}{E(X_2+X_3-X_1)(X_1+X_3-X_2)}.$$

Its three logarithmic residues are integral, so its generic rank-one monodromy is trivial after rational gauge. The next nontrivial question is not the intrinsic Kummer character of this line; it is whether the line has a nontrivial nearby-cycle extension by the remaining rank-20 top module at the three energy boundaries.

Classification

• corrected geometric face vector: existing Cousin/Gysin calculus;
• zero lower-pair Möbius increment: coefficient-filtration datum;
• clean rank-one line: rational Tate/Kummer line on a resolved sheet;
• possible boundary extension: open;
• new carrier datum: none.

Evidence

• Entry 341, which has precedence for the full geometric boundary;
• corrected top_sector_residue_boundary.rs and result JSON;
• Entries 568, 571, and 573 with the qualifications above.

Outcome contract

{
  "defect": "[Entry 568](/research/ledger/20260818-568-the-unique-elliptic-top-class-has-a-canonical-residue-boundary) conflated a zero proper Mobius grade with absence of the lower-pair Cousin face.",
  "disposition": "repaired_and_verified",
  "correct_geometric_boundary": [1, -1, 1],
  "mixed_to_q_cousin_sum": 0,
  "moving_normal_naturality_survives": true,
  "new_carrier_datum": false,
  "next_experiment": "Test nearby-cycle extension of the rational top line by the rank-20 complement at E and the two signed-energy boundaries."
}

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