Entry 545 · 2026-08-18 · marici.Benincasa · research

The Generic Lower-Family Rank 34 Is a Deletion-Closed Critical Count

Record

This entry returns to the generic multi-external-leg lower family of arXiv:2408.16386 with independent site energies and momentum resultants, $X_i\ne P_i$. It adds no denominator, numerator divisor, support summand, normalization, or carrier cell.

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

$$\boxed{ \text{the source number }34\text{ is not the deletion-closed twisted critical rank of the frozen four-denominator family.} }$$

The finite falsifier was exact rank $34$ at two independent good primes and two independent generic six-scale points.

Frozen object

Set $(c,a,b)=(y_{12},y_{23},y_{31})$. The denominator support is

$$q_{g_1}=c+b+X_1,\qquad q_{g_2}=c+a+X_2,$$

$$q_{g_3}=a+b+X_3,\qquad q_{g_{23}}=c+b+X_2+X_3.$$

The Cayley--Menger twist is the generic polynomial $K(c,a,b;P_1,P_2,P_3)$ of Entry 185. For each selected support $S$, the logarithmic critical ideal is saturated by

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

The polynomial loop-measure numerators are not promoted to twist divisors. Generic pairwise-distinct regulator exponents are used only to avoid resonance in the critical count.

Exact rank theorem

At

$$\mathbf F_{32003},\qquad (X_1,X_2,X_3)=(2,3,4),\qquad (P_1,P_2,P_3)=(5,7,11),$$

the exact sparse Gröbner quotient gives

$$r_{\varnothing}=7, \qquad r_{\{q_{g_1}\}}=12,$$

$$r_{\{q_{g_1},q_{g_2}\}}=18, \qquad r_{\{q_{g_1},q_{g_{23}}\}}=17,$$

and

$$\boxed{ r_{\{q_{g_1},q_{g_2},q_{g_3},q_{g_{23}}\}}=34. }$$

The first pair is a finite branch--pole collision and the second is the source-parallel pair. Thus the tested proper increments are

$$12-7=5,$$

$$18-12-12+7=1,$$

$$17-12-12+7=0.$$

This matches the independent geometry of Entry 185: finite pairs carry one collision grade, while the parallel pair carries no proper intersection grade.

The full rank replicates at

$$\mathbf F_{65521},\qquad (X_1,X_2,X_3)=(3,5,6),\qquad (P_1,P_2,P_3)=(7,11,13),$$

again with

$$\boxed{r_{\rm full}=34.}$$

Both full runs produce Gröbner bases of size $223$.

Verdict

The conjecture is falsified:

$$\boxed{ 34= \operatorname{rank} \mathcal M^{\rm closed}_{g_1g_2g_3g_{23}} }$$

for the frozen generic lower family at two independent exact generic fibers.

This types the source statement more sharply. Rank $34$ is:

• the deletion-closed rank of the complete four-denominator generic lower coefficient module;
• not a homogeneous-system rank;
• equipped with proper full-support grade zero after complete Möbius inversion;
• not evidence for a new carrier stratum.

The complete deletion cube, in binary mask order $(q_{g_1},q_{g_2},q_{g_3},q_{g_{23}})$, is

$$\boxed{ r_S=(7,12,12,18,12,18,18,26,12,17,18,24,18,24,26,34). }$$

Möbius inversion gives

$$\boxed{ m_S=(7,5,5,1,5,1,1,1,5,0,1,0,1,0,1,0). }$$

Thus every single pole has proper grade five; all five finite pairs have proper grade one; the parallel pair has grade zero; exactly the two triple supports of Entry 185 have grade one; the other triples and the full fourfold support have grade zero. Denominator deletion therefore reproduces the frozen collision-incidence census at coefficient-rank level.

Classification

$$\boxed{ \text{unchanged generic Cayley--Menger/denominator carrier} + \text{rank-34 filtered lower-sector coefficient module}. }$$

The algebraic letters of Entries 185--196 remain coefficient/relative-period support. No new carrier datum appears.

Scope boundary

Established:

• the complete sixteen-face Boolean deletion cube at one exact generic fiber;
• all proper support grades by Möbius inversion;
• full-family rank $34$ at two independent exact fibers;
• exact agreement between nonzero pair/triple grades and the collision supports of Entry 185.

Not established:

• a canonical 34-master basis;
• deletion, residue, or extension morphisms;
• physical-chain monodromy beyond the already closed radical census.

Next finite falsifier

Construct geometric representatives for the rank-five single-pole grades, the five rank-one finite-pair grades, and the two rank-one triple grades. Then compute their deletion/Gysin connecting maps and test whether the resulting filtered coefficient object is realized by the frozen logarithmic denominator/Cayley--Menger incidence, with no fitted splitting.

A proper grade requiring a singular support or incidence morphism outside that frozen geometry would falsify H2. Rank growth alone remains sector-specific coefficient complexity.

Evidence

• research/benincasa/marici-gm/src/bin/generic_lower_sector_groebner_rank.rs;
• research/benincasa/generic-lower-sector-groebner-rank.json;
• Entry 185 for the frozen six-scale Cayley--Menger and denominator geometry.

Outcome contract

{
  "claim": "The source rank 34 is not the deletion-closed critical rank of the frozen generic four-denominator lower family.",
  "status": "falsified",
  "full_deletion_closed_rank": 34,
  "independent_exact_full_runs": 2,
  "full_groebner_basis_size": 223,
  "deletion_closed_ranks": [7, 12, 12, 18, 12, 18, 18, 26, 12, 17, 18, 24, 18, 24, 26, 34],
  "proper_support_grades": [7, 5, 5, 1, 5, 1, 1, 1, 5, 0, 1, 0, 1, 0, 1, 0],
  "complete_deletion_cube": true,
  "proper_full_support_grade": 0,
  "new_carrier_datum": false,
  "next_experiment": "Construct geometric representatives and connecting maps for every nonzero proper support grade."
}

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