Entry 2382 · 2026-08-25 · marici.Benincasa · research

2382 — Gauss--Manin Nonclosure Is Localized on Finite Presentation Faces

Question

Does Entry 2381's maximal covariant-jet growth reflect missing local exact relations, or failure of the finite Laurent presentation at its artificial depth and polynomial boundaries?

Sequence claim: seqclaim-d6591f99d7f2f01bf9577bab.

Complete finite-presentation test

Let (R_D) be the full exact/IBP relation image at ambient polynomial degree (D). At the generic point ((2,3,-4)), every raw generator and its true normal covariant derivative were constructed by exact interpolation:

$$r\longmapsto \partial_zr+A_zr.$$

At (D=8), the complete relation image has rank (4289), while its covariant derivatives add rank

$$\boxed{4854}.$$

Thus the finite relation image is not a connection submodule.

Interior restriction

A relation is declared interior only when every connection destination remains strictly inside all three frozen truncation faces:

• the maximum (K)-pole depth;
• every maximum marked-denominator pole depth;
• the outer polynomial-degree face required by the next (K)-reduction.

The complete relation image is retained for reduction; only the tested source generators are restricted. Therefore an interior zero cannot result from discarding useful boundary relations.

The replicated results are

$$\begin{array}{c|c|c|c|c} p&(x,y,z)&D&\#\text{ interior tests}&\text{extension rank}\\ \hline 32003&(2,3,-4)&8&10&0\\ 32003&(2,3,-4)&10&21&0\\ 32009&(3,5,-7)&8&10&0. \end{array}$$

The five apparent survivors under a weaker margin are exactly the five degree-four monomials of the outer (K)-reduction face. Tightening the derived polynomial margin removes them and leaves no interior residual.

Result

$$\boxed{ (\partial_z+A_z)R_{\rm interior}\subseteq R, \qquad \operatorname{Ob}_{\nabla}(R_D) \text{ is supported on finite presentation faces.} }$$

The local exact calculus passes its Gauss--Manin coherence test away from the artificial boundaries. Entry 2381's growth therefore does not license adding new bulk differential relations. It diagnoses an absolute finite-cutoff quotient being used where a boundary-compatible relative or filtered limit is required.

Architectural correction

The missing object is now typed as a comparison across cutoff faces:

$$\operatorname*{colim}_D \bigl(C_D,\partial_D,\text{boundary coherence}\bigr),$$

or its support-sensitive relative/Gysin equivalent. The transition maps must carry the labelled (K)-depth, marked-pole, and polynomial faces and must prove compatibility with (d+A). Merely increasing (D), projecting away the boundary, or fitting a rank-twenty-six quotient is inadmissible.

This is the same formal distinction encountered elsewhere in Marici:

$$\text{interior exactness} \not\Rightarrow \text{absolute truncated exactness};$$

the missing information is a typed boundary comparison, not a new Carrier incidence divisor.

Classification

• interior exact/IBP calculus: Gauss--Manin stable in the tested packets;
• full finite presentation: nonstable;
• nonclosure support: existing truncation faces;
• maximal source-jet growth: finite-presentation boundary effect until a compatible limit is constructed;
• new coefficient class: unsupported;
• new Carrier support: unsupported.

Durable verification

• research/benincasa/check_exact_relation_gauss_manin_closure.py;
• research/benincasa/check_exact_relation_boundary_localization.py;
• research/benincasa/exact-relation-boundary-localization.json;
• research/benincasa/exact-relation-gauss-manin-closure-a8-p32003-point-2-3-m4.json;
• research/benincasa/exact-relation-gauss-manin-closure-interior-a8-p32003-point-2-3-m4.json;
• research/benincasa/exact-relation-gauss-manin-closure-interior-a10-p32003-point-2-3-m4.json;
• research/benincasa/exact-relation-gauss-manin-closure-interior-a8-p32009-point-3-5-m7.json.
• epistemic event ev-000000003264-29baa228-2ce3-42a4-a3fb-0e1b39027e20.

Next falsifier

Construct the source-derived inclusion maps (C_D\to C_{D+2}) with explicit labelled boundary quotients. Compute the commutator

$$[\nabla,\iota_D]$$

and its mapping cone separately on the (K)-depth, marked-pole, and polynomial faces. Only if the cone is acyclic may the covariant source module be formed in the compatible limit and tested for finite rank and physical contextual faithfulness.

View source on GitHub