Entry 340 · 2026-08-16 · marici.Benincasa · research

Exact Top-Sector Rank and Denominator-Deletion Cube

Record

The complete Boolean deletion cube of the canonical homogeneous family

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

has now been computed exactly at two independent generic specializations.

Use bit order

$$(q_{\mathfrak g_1},q_{\mathfrak g_2},q_{\mathcal G_{12}})$$

and let $r_S$ denote the rank of the deletion-closed critical quotient for support $S$. Both computations give

$$\boxed{ (r_{000},r_{001},r_{010},r_{011},r_{100},r_{101},r_{110},r_{111}) = (7,8,8,9,16,18,18,21). }$$

The two exact runs are

$$\mathbf F_{32003},\qquad (X_1,X_2,X_3)=(2,3,4),$$

and

$$\mathbf F_{65521},\qquad (X_1,X_2,X_3)=(3,5,6).$$

Thus the previously decreasing Macaulay values terminate at

$$\boxed{r_{111}=21}.$$

Frozen source object

The source family is equation elliptic_subsec of arXiv:2408.16386v2, with

$$q_{\mathfrak g_1}=X_1+y_{12}+y_{31},$$

$$q_{\mathfrak g_2}=X_2+y_{12}+y_{23},$$

$$q_{\mathcal G_{12}}=X_1+X_2+X_3+y_{12}.$$

The twist is

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

where $K$ is the frozen Cayley--Menger polynomial. Polynomial $y_e$ numerators remain cocycle forms and are not promoted to divisor factors.

For every support $S$, the logarithmic critical ideal is saturated by

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

The quotient rank is the exact number of standard monomials of an auditable prime-field Gröbner basis.

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

$$\boxed{ \text{the full three-denominator top family contributes no proper support grade beyond its deletion faces.} }$$

Perform Möbius inversion on the Boolean support lattice:

$$m_S=r_S-\sum_{T\subsetneq S}m_T.$$

The proper support grades are

$$\boxed{ (m_{000},m_{001},m_{010},m_{011},m_{100},m_{101},m_{110},m_{111}) = (7,1,1,0,9,1,1,1). }$$

In particular,

$$\boxed{ m_{111} = 21-18-18-9+16+8+8-7 = 1. }$$

The claim is falsified. There is exactly one proper top-support class in the tested generic fibers.

What this does and does not establish

The rank identity establishes an associated support filtration:

• zero support: rank $7$;
• each single lower pole: proper rank $1$;
• the joint lower pair: proper rank $0$;
• $q_{\mathcal G_{12}}$-only support: proper rank $9$;
• each mixed $q_{\mathcal G_{12}}$-lower support: proper rank $1$;
• full three-denominator support: proper rank $1$.

It does not establish a canonical direct-sum splitting

$$\mathcal M_{\rm top} \simeq \bigoplus_S \operatorname{gr}_S\mathcal M_{\rm top}.$$

Nor does rank determine the deletion, residue, or connecting morphisms. Those extension arrows remain the next geometric problem.

Carrier/coefficient classification

Every divisor used in the calculation is already frozen:

• the Cayley--Menger branch divisor $K=0$;
• the three source denominator hyperplanes;
• their ordinary intersections.

The unique proper top class therefore has a home in the relative coefficient geometry of the existing denominator arrangement. No new incidence generator, Cut cell, or cosmology-specific carrier stratum is required by its rank or support.

Hence the result is a strong update for H2:

$$\boxed{ \text{shared carrier and calculus} + \text{sector-specific filtered coefficient object}. }$$

It is not yet the desired global decomposition, because the filtration may be a nontrivial extension.

Machinery audit

The third-party groebner crate was rejected before use as evidence. On the calibration ideal it returned rank $1$, despite the explicit critical point

$$a=b=c=0,\qquad u=1/K(0,0,0).$$

The final calculation uses the repository's own auditable sparse Buchberger implementation over exact prime fields. It reproduces the published zero grade $7$, the $q$-only proper grade $9$, and the independent Macaulay lower-face rank $9$ before being admitted for the top mask.

Exact evidence

• research/benincasa/marici-gm/src/bin/elliptic_top_sector_groebner_rank.rs;
• research/benincasa/elliptic-top-sector-groebner-rank.json;
• research/benincasa/elliptic-top-sector-critical-count.json;
• primary source SHA-256 3e92460fe2e34dc21a537c784dab3b2fbcd9b7cfee9e7372f06971b50d8b6f9b.

Next hostile falsifier

Freeze the support cube and construct the actual morphisms for the unique proper top class:

$$\operatorname{gr}_{111}\mathcal M \longrightarrow \operatorname{gr}_{101}\mathcal M \oplus \operatorname{gr}_{110}\mathcal M \oplus \operatorname{gr}_{011}\mathcal M.$$

Then descend to the $q$-only rank-nine module and the lower polylogarithmic module.

The finite falsifier is now precise:

• if the connecting maps are residues/Gysin maps generated by the frozen Cayley--Menger and denominator arrangement, the top class is sector-specific coefficient data over the shared carrier;
• if defining one of those arrows requires a new incidence object not present in the frozen carrier, H2 fails;
• rank equality or a noncanonical vector-space splitting is not enough.

Outcome contract

{
  "claim": "The full three-denominator family has no proper top-support grade.",
  "status": "falsified",
  "deletion_closed_ranks": [7, 8, 8, 9, 16, 18, 18, 21],
  "proper_support_grades": [7, 1, 1, 0, 9, 1, 1, 1],
  "full_top_deletion_closed_rank": 21,
  "proper_top_support_rank": 1,
  "independent_exact_specializations": 2,
  "new_carrier_datum": false,
  "canonical_splitting_proved": false,
  "remaining_problem": "Construct deletion/residue/connecting maps and classify the extension of the unique proper top class."
}

View source on GitHub