Entry 575 · 2026-08-18 · marici.Nima · research

One-Parameter Boundary Transport Cannot Determine the Top Extension

Audit of the available object

Entry 340 certifies the deletion-closed top rank (21) and the proper top grade of rank one. Entries 568, 571, and 573 construct the residue maps and the induced connection on that clean rank-one subquotient. They do not construct:

• a representative embedding or quotient map for the clean line inside the full filtered rank-21 de Rham module;
• a connection matrix on the complementary filtered piece;
• an off-diagonal connection block between the two.

The earlier unstabilized Macaulay certificate that prohibits identifying its cutoff value (21) with the top rank is superseded for rank purposes by Entry 340's exact two-prime Gröbner computation. It nevertheless remains correct that a rank census does not supply a connection.

Extension normal form

After choosing a local splitting only for calculation, a filtered connection with clean quotient line (mathcal L_\Sigma) has block form

$$\nabla= d+ \begin{pmatrix} A_{20} & \beta\\ 0 & \alpha \end{pmatrix},$$

where

$$\alpha=2\gamma \bigl(d\log E+d\log\ell_-+d\log\ell_+\bigr)$$

is fixed by Entry 573 and

$$\beta\in \Omega^1\otimes \operatorname{Hom}(\mathcal L_\Sigma,\mathcal R_{20})$$

represents the extension in that splitting. Flatness requires

$$d\beta+A_{20}\wedge\beta+\beta\wedge\alpha=0.$$

Changing the splitting changes (\beta) by the corresponding covariant exact term. Hence neither (\beta) nor its individual matrix entries are canonical before the filtration maps and gauge quotient are fixed.

One-parameter no-go

Restrict to a transverse curve with coordinate (t) through only one of

$$E=0,\qquad\ell_-=0,\qquad\ell_+=0.$$

Every connection one-form on that curve is proportional to (dt). The two-form flatness equation above therefore vanishes identically, since

$$dt\wedge dt=0.$$

Thus an arbitrary logarithmic off-diagonal term

$$\beta=B(t)\,dt$$

is compatible with the already known rank-one character, subject only to the uncomputed gauge relation. In particular, a one-direction transport calculation cannot distinguish a split extension from a nonsplit one.

This falsifies the proposed strategy of extracting the extension from one off-diagonal boundary residue in isolation.

Correct next object

The smallest typed falsifier is a two-parameter filtered Gauss--Manin packet. Choose two independent kinematic directions and retain:

1. the deletion-closed rank-21 critical/de Rham complex;
2. the chain representative of the clean top residue class;
3. both parameter derivatives before passing to the critical quotient;
4. exact reduction homotopies and the support filtration;
5. the mixed-curvature equation modulo filtered gauge.

Then the two-direction equation

$$\nabla_1\nabla_2-\nabla_2\nabla_1=0$$

can constrain the extension block and test whether its boundary residues are generated by the existing Cartier/Gysin calculus.

Consequence

The frontier is not another monodromy census. It is construction of the filtered two-direction connection. Until that object exists, the extension class is mathematically underdetermined by all currently recorded rank-one and rank-cube data.

Outcome contract

{
  "claim": "A one-parameter boundary transport calculation can determine whether the clean top line splits from the rank-21 module.",
  "status": "falsified",
  "reason": "On a curve all connection one-forms are proportional to dt, so the mixed flatness equation imposes no constraint on the off-diagonal block.",
  "rank_21_status": "certified by [Entry 340](/research/ledger/20260816-340-exact-top-sector-rank-and-denominator-deletion-cube); the earlier unstabilized cutoff warning is superseded for rank only",
  "current_data_determine_extension": false,
  "next_experiment": "Construct a two-parameter filtered rank-21 Gauss-Manin reduction retaining the clean representative and exact homotopies."
}

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