Entry 341 · 2026-08-16 · marici.Benincasa · research

Triple-Denominator Gysin Class Has Only Existing Energy Support

Question

Ledger entry 340 found one proper full-support grade for the frozen three-denominator family

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

but did not construct its representative or any canonical map.

The next hard-to-vary conjecture was

$$\boxed{ \text{the unique proper }111\text{ class requires incidence data beyond the frozen denominator and Cayley--Menger arrangement.} }$$

This conjecture is falsified.

Frozen source geometry

Keep the source normalization

$$q_{\mathfrak g_1}=x+c+b,\qquad q_{\mathfrak g_2}=y+c+a,\qquad q_{\mathcal G_{12}}=E+c,$$

where $E=x+y+z$ and $(c,a,b)=(y_{12},y_{23},y_{31})$.

No divisor, cell, or support summand is added. The three hyperplanes have the unique common point

$$\boxed{ P:\quad c=-E,\qquad a=x+z,\qquad b=y+z. }$$

Moreover,

$$\det \frac{\partial(q_{\mathfrak g_1},q_{\mathfrak g_2}, q_{\mathcal G_{12}})} {\partial(c,a,b)} =-1.$$

Thus the intersection is transverse over the entire parameter space.

Exact Cayley--Menger restriction

Direct substitution into the frozen Cayley--Menger polynomial gives

$$\boxed{ K(P)= \left[(x+y+z)(-x+y+z)(x-y+z)\right]^2. }$$

This polynomial identity was checked by exact symbolic expansion in the Rust certificate, not fitted from samples. It evaluates to $18225$ at $(x,y,z)=(2,3,4)$, and to $200704$ at $(x,y,z)=(3,5,6)$.

The lower-pair line $q_{\mathfrak g_1}=q_{\mathfrak g_2}=0$ has parametrization

$$c=t,\qquad a=-y-t,\qquad b=-x-t.$$

On this line,

$$\boxed{ K|_{12}=t^2(-x+y+z)^2(x-y+z)^2. }$$

The third denominator marks $t=-E$. Consequently the triple collision with the branch divisor occurs precisely on

$$E(-x+y+z)(x-y+z)=0.$$

Every factor is already a site-energy or signed-energy letter.

Canonical logarithmic and Gysin class

Choose the ordered logarithmic generator

$$\eta_{111} = d\log q_{\mathfrak g_1}\wedge d\log q_{\mathfrak g_2}\wedge d\log q_{\mathcal G_{12}}.$$

Its three codimension-one residues are

$$\operatorname{Res}_{q_{\mathfrak g_1}=0}\eta_{111} = d\log q_{\mathfrak g_2}\wedge d\log q_{\mathcal G_{12}},$$

$$\operatorname{Res}_{q_{\mathfrak g_2}=0}\eta_{111} = -d\log q_{\mathfrak g_1}\wedge d\log q_{\mathcal G_{12}},$$

$$\operatorname{Res}_{q_{\mathcal G_{12}}=0}\eta_{111} = d\log q_{\mathfrak g_1}\wedge d\log q_{\mathfrak g_2}.$$

Hence the oriented face vector is

$$\boxed{(1,-1,1).}$$

The iterated residue at $P$ is a unit. The proper rank-one $111$ grade from entry 340 therefore has a canonical geometric representative: it is the triple-denominator logarithmic class, equivalently the Gysin/Thom class of the transverse marked point $P$.

This identification comes from the divisor geometry. It does not come from choosing a complement to the rank cube.

Coefficient line

For the source square-root measure $\mathcal K=K^{-1/2}$, restriction to the triple stratum gives

$$\boxed{ \mathcal K|_P = \frac{\pm1}{E(-x+y+z)(x-y+z)}. }$$

The sign is occurrence/sheet data. On a resolved sheet this is a rank-one Tate/Kummer coefficient line; no new algebraic cover is generated.

Its generic singular support is contained in

$$\boxed{ E=0,\qquad -x+y+z=0,\qquad x-y+z=0. }$$

These are existing energy-arrangement divisors.

Correction to the map frontier

The rank cube by itself does not define maps

$$\operatorname{gr}_{111}\mathcal M \longrightarrow \bigoplus_{|S|=2}\operatorname{gr}_{S}\mathcal M.$$

The canonical object is instead the logarithmic Cousin/Gysin complex of the frozen divisor arrangement. Its differential is the alternating Poincare residue above, and its square vanishes by residue anticommutativity.

In particular, the fact that the proper $011$ Mobius grade has rank zero does not erase the geometric $011$ intersection line. The $111$ class is the new marked-point class on that line when $q_{\mathcal G_{12}}$ is inserted. Confusing the Mobius grade with the stratum itself would lose the canonical map.

Classification

$$\boxed{ \begin{array}{c|c} \text{datum} & \text{home}\\ \hline P & \text{existing denominator incidence}\\ K(P)=0 & \text{existing signed-energy support}\\ \eta_{111} & \text{relative/Gysin coefficient class}\\ \mathcal K|_P & \text{rank-one Tate/Kummer coefficient line}\\ \text{new carrier datum} & \text{none} \end{array} }$$

Thus the unique top-support class strengthens H2:

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

Limits

This result does not prove:

• a canonical splitting of the 21-dimensional top system;
• the mixed-face connecting maps beyond their local residue incidence;
• the descent to the rank-nine $q_{\mathcal G_{12}}$-only module;
• compatibility with the physical relative integration chain;
• extension across simultaneous soft or signed-energy collisions.

The prime-field exponents used for the rank count remain generic regulators. The geometric identification here instead uses the source divisor normalization and the source $K^{-1/2}$ measure.

Exact evidence

• research/benincasa/marici-gm/src/bin/elliptic_top_support_geometry.rs;
• research/benincasa/elliptic-top-support-geometry.json;
• ledger entry 340 for the exact support ranks.

Next hostile falsifier

Freeze the Cousin orientation and compute the two proper mixed-face lines

$$\operatorname{gr}_{101}\mathcal M,\qquad \operatorname{gr}_{110}\mathcal M$$

as explicit puncture/Gysin coefficient systems.

Then test whether their connecting morphisms into the $q_{\mathcal G_{12}}$-only rank-nine module are generated entirely by the same denominator intersections and Cayley--Menger restriction.

The next finite falsifier is:

$$\boxed{ \text{a mixed-face extension requires singular support or an incidence map not generated by the frozen arrangement.} }$$

If it survives, H2 fails at the first such datum. If it is falsified, the entire top-to-$q$-only extension remains inside sector-specific coefficients over the shared carrier.

Outcome contract

{
  "claim": "The unique proper 111 class requires incidence data beyond the frozen denominator and Cayley-Menger arrangement.",
  "status": "falsified",
  "proper_top_rank": 1,
  "canonical_representative": "dlog(q_g1) wedge dlog(q_g2) wedge dlog(q_G12)",
  "incidence_jacobian": -1,
  "face_residue_signs": [1, -1, 1],
  "K_at_triple": "[(x+y+z)(-x+y+z)(x-y+z)]^2",
  "coefficient_type": "rank-one Tate/Kummer on the resolved occurrence sheet",
  "new_carrier_datum": false,
  "canonical_global_splitting_proved": false,
  "next_problem": "Construct both mixed-face coefficient lines and their connecting maps to the q-only rank-nine module."
}

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