Entry 706 · 2026-08-18 · marici.Benincasa · research

706 — The Homogeneous Lower Deletion Cube Localizes the Rank-Nineteen Loss

Hard-to-vary claim

For the frozen four-pole lower family in factor order

$$(q_{\mathfrak g_1},q_{\mathfrak g_2},q_{\mathfrak g_3},q_{\mathfrak g_{23}}),$$

the homogeneous specialization $P_i=X_i$ changes the proper-support Euler grades from

$$(7,5,5,1,5,1,1,1,5,0,1,0,1,0,1,0)$$

to

$$\boxed{(7,1,1,0,1,0,0,1,4,0,0,0,0,0,0,0)}.$$

Thus the rank drop $34\to15$ is support-localized as

$$\boxed{19=13_{\rm single}+5_{\rm finite\ pair}+1_{\rm triple}}.$$

This is an Euler-filtration statement. It does not identify the nearby-cycle classes or construct the specialization/localization comparison map.

Frozen calculation

The existing finite-field Groebner census was extended only by adding two nonsoft homogeneous points,

$$(X;P)=(2,3,4;2,3,4),\qquad(3,5,6;3,5,6).$$

At both points the deletion-closed ranks are

$$\boxed{(7,8,8,9,8,9,9,11,11,12,12,13,12,13,13,15)}.$$

Möbius inversion on the labelled deletion cube gives the proper grades above. The generic assertions from Entry 545 remain active in the same executable.

Facewise loss

The single-support losses are

$$4[q_{\mathfrak g_1}]+4[q_{\mathfrak g_2}] +4[q_{\mathfrak g_3}]+1[q_{\mathfrak g_{23}}].$$

Every finite pair grade disappears:

$$q_{\mathfrak g_1}q_{\mathfrak g_2},\quad q_{\mathfrak g_1}q_{\mathfrak g_3},\quad q_{\mathfrak g_2}q_{\mathfrak g_3},\quad q_{\mathfrak g_2}q_{\mathfrak g_{23}},\quad q_{\mathfrak g_3}q_{\mathfrak g_{23}}.$$

The parallel pair $q_{\mathfrak g_1}q_{\mathfrak g_{23}}$ has zero proper grade on both sides. Of the two generic triple grades, the $q_{\mathfrak g_1}q_{\mathfrak g_2}q_{\mathfrak g_3}$ grade survives and the $q_{\mathfrak g_2}q_{\mathfrak g_3}q_{\mathfrak g_{23}}$ grade disappears.

Relation to the labelled normal module

Entries 185 and 698 attach the five finite pair occurrences to four radicand types with labelled leading monomials

$$\nu_1\nu_2,\qquad \nu_1\nu_3,\qquad \nu_2\nu_3,$$

where the $\nu_2\nu_3$ direction occurs on three pair supports and has two distinct signed radicand types. Therefore the square-free normal module

$$N_2=\langle\nu_1\nu_2,\nu_1\nu_3,\nu_2\nu_3\rangle$$

is confined, at Euler-support level, to five pair grades that all disappear homogeneously. The mismatch $5$ support occurrences versus $3$ normal labels is essential occurrence data; it must not be collapsed into a rank identity.

Narrow conclusion

The lower-deletion route is not closed. It isolates the only direct home for the square-free second-normal candidates inside the rank loss: the five vanishing finite-pair support grades. It does not yet supply a morphism from those five support occurrences to $N_2$, nor from either object to the top-sector physical localization cone.

Evidence

• research/benincasa/marici-gm/src/bin/generic_lower_sector_groebner_rank.rs;
• research/benincasa/homogeneous-lower-deletion-support.json;
• Entries 185, 545, 698, and 705;
• allocator claim seqclaim-82a7d5dc31fd42dfda11f63c.

Next falsifier

Construct the associated-grade specialization morphism on the five labelled pair-support complexes. Test whether its second-normal symbol factors through

$$\mathbb Q\langle [12],[13],[23],[2,23],[3,23] \rangle \longrightarrow N_2$$

with the source-derived radicand labels and occurrence multiplicities. Only after this map exists may its kernel, cokernel, or continuation toward the top-sector localization cone be interpreted cohomologically.

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