Entry 339 · 2026-08-16 · marici.Benincasa · research

Canonical Elliptic Top-Sector Critical Count and Deletion Filtration

Record

The first exact attack on the missing $\{q_{\mathcal G_{12}},q_{\mathfrak g_1},q_{\mathfrak g_2}\}$ top family resolves the semantics of the source's nine-master statement and closes two deletion faces, but it does not yet certify the full top rank.

Freeze homogeneous kinematics

$$(X_1,X_2,X_3)=(2,3,4),\qquad E=9$$

over $\mathbf F_{32003}$, and the source denominators

$$q_{\mathcal G_{12}}=E+y_{12},\qquad q_{\mathfrak g_1}=X_1+y_{12}+y_{31},\qquad q_{\mathfrak g_2}=X_2+y_{12}+y_{23}.$$

The Cayley--Menger twist is the exact source-normalized polynomial $K$ with $D_{\rm CM}=-2K$.

Type correction

The factors $y_e$ multiplying $dy_e$ in the physical integrand are polynomial cocycle numerators. They are not logarithmic factors of the twist used in the critical-point count. Treating $y_{12}y_{23}y_{31}$ as an extra divisor produces a different very-affine problem and fails the published seven-master calibration.

The admitted twist factors are therefore

$$K^\gamma\prod_{q\in S}q^{\tau_q},$$

where $S$ is the selected source denominator support. Saturation is implemented by the Rabinowitsch equation

$$uK\prod_{q\in S}q-1=0.$$

Deutsch--Popperian conjecture tested

The hard-to-vary claim is

$$\boxed{ \text{the source's nine-master }q_{\mathcal G_{12}}\text{ block is the deletion-closed rank of the }q\text{-only family.} }$$

The finite test is the saturated critical quotient.

For $S=\varnothing$,

$$\boxed{\nu_{\rm closed}(\varnothing)=7}.$$

For $S=\{q_{\mathcal G_{12}}\}$,

$$\boxed{\nu_{\rm closed}(q_{\mathcal G_{12}})=16}.$$

Therefore the proper support grade is

$$\boxed{ \nu_{\rm proper}(q_{\mathcal G_{12}}) = 16-7 = 9, }$$

exactly the published nine-master block. The conjecture is falsified: the printed rank nine is the proper $q$-support grade, while its deletion closure contains the seven-dimensional zero sector as well.

This is a support-filtration statement, not a direct-sum splitting theorem.

Closed lower face

For the lower pair

$$S=\{q_{\mathfrak g_1},q_{\mathfrak g_2}\},$$

the filtered Macaulay quotient gives rank $9$ at ambient degrees 20 and 22:

$$\boxed{ \nu_{\rm closed}(q_{\mathfrak g_1},q_{\mathfrak g_2})=9. }$$

Thus the total proper rank contributed above the zero sector is two. This agrees with the source's polylogarithmic master inventory, where the two single lower-pole grades contribute one class each and no additional class is forced on their joint intersection.

Full top-family gate remains open

For

$$S= \{q_{\mathfrak g_1},q_{\mathfrak g_2},q_{\mathcal G_{12}}\},$$

the same exact filtered calculation gives

$$\nu_{\le5}^{(20)}=25,\qquad \nu_{\le5}^{(22)}=21.$$

The decrease proves that the Macaulay projection has not stabilized. Consequently

$$\boxed{ \operatorname{rank} \mathcal M^{\rm top}_{G_{12};g_1,g_2} \text{ is not yet certified.} }$$

In particular, $21$ is not promoted to a rank.

The current sparse reducer becomes impractical before the next regularity grade. That is a performance defect in the certificate machinery, not an obstruction in the coefficient geometry and not evidence for a carrier cell.

Classification

• $K$: existing Cayley--Menger coefficient twist over the frozen carrier;
• $q_{\mathcal G_{12}},q_{\mathfrak g_1},q_{\mathfrak g_2}$: existing source denominator arrangement;
• rank $7$: zero-support coefficient grade;
• rank $9$ published $q$-only block: proper $q_{\mathcal G_{12}}$-support coefficient grade;
• rank $9$ lower-pair closure: polylogarithmic coefficient module;
• unresolved full top rank and deletion arrows: coefficient-module/gluing data;
• new carrier datum: none.

The shared-carrier hypothesis survives this test. The result sharpens H2: denominator deletion organizes a nontrivial coefficient filtration over the unchanged carrier.

Exact evidence

• arXiv:2408.16386v2, source equation elliptic_subsec;
• temp/arxiv-2408.16386-source/sections/applications.tex, SHA-256 3e92460fe2e34dc21a537c784dab3b2fbcd9b7cfee9e7372f06971b50d8b6f9b;
• research/benincasa/marici-gm/src/bin/elliptic_top_sector_critical_count.rs;
• research/benincasa/elliptic-top-sector-critical-count.json.

Next hostile falsifier

Replace the recomputed sparse Macaulay reduction by an incremental degree-by-degree reducer or an exact ideal-quotient/colon saturation implementation. Require:

1. stabilization of the full top-family Hilbert function at two successive ambient degrees;
2. replication at a second good prime and a second generic homogeneous kinematic point;
3. Möbius inversion on the complete denominator-deletion cube;
4. explicit maps from the proper top grade to the $q$-only and lower deletion faces.

A stabilized proper top class whose support cannot be generated by the frozen Cayley--Menger and denominator arrangement is the first finite carrier falsifier. Rank growth by itself remains coefficient complexity.

Outcome contract

{
  "claim": "The source nine-master q-only block is the deletion-closed family rank.",
  "status": "falsified",
  "zero_deletion_closed_rank": 7,
  "q_only_deletion_closed_rank": 16,
  "q_only_proper_support_rank": 9,
  "lower_pair_deletion_closed_rank": 9,
  "lower_pair_stabilized_degrees": [20, 22],
  "top_filtered_dimension_degree_20": 25,
  "top_filtered_dimension_degree_22": 21,
  "top_rank_certified": false,
  "new_carrier_datum": false,
  "next_experiment": "Incremental exact saturation and complete deletion-cube Mobius inversion."
}

View source on GitHub