Entry 645 · 2026-08-18 · marici.Nima · research

598 — The Two Physical Occurrence Families Have a Rank-One Mixed Gluing Class

Hard-to-vary claim

On the homogeneous $q_{G_{12}}$-residue surface, the union of all five physical marked lines

$$W_{123,23,31} = W_{g_1}\cup W_{g_2}\cup W_{g_3}\cup W_{g_{23}}\cup W_{g_{31}}$$

has generic deletion-closed rank

$$\boxed{26}.$$

It is not the ordinary pushout of the two rank-twenty occurrence families over their common rank-fifteen three-wall core.

Exact incidence census

The first occurrence line contributes rank five, as in Entry 596. After it has been adjoined, the second occurrence line has four branch punctures and three distinct finite intersections. It therefore contributes

$$4+3-1=6.$$

Hence

$$\operatorname{rank}M_{123,23,31} =9+1+2+3+5+6=26.$$

By contrast, the naive pushout rank would be

$$20+20-15=25.$$

The one-dimensional difference is generated by the additional intersection of the two occurrence walls,

$$W_{g_{23}}\cap W_{g_{31}}:\qquad (a,b)=(y,x).$$

The census is identical at $(x,y,z)=(2,3,4)$ and $(3,5,6)$ over $\mathbb F_{32003}$.

Scattering degeneration

Put

$$E=E_T=x+y+z,\qquad s=x+y,\qquad p=xy.$$

Direct substitution into the frozen Cayley--Menger polynomial gives

$$\boxed{ K_E(y,x) = E^3\left[ 2E^3-6sE^2+(5s^2+4p)E-8ps \right]. }$$

On the generic scattering boundary $ps\ne0$, the bracket restricts to $-8ps$. Therefore the mixed occurrence intersection collides with the branch divisor at exactly third ordinary normal order:

$$\operatorname{ord}_{E=0}K_E(y,x)=3.$$

This is not the algebraic quartic deformation

$$\mathcal Q=-16p^2-8pE^2+8sE^3-5E^4,$$

whose first nonconstant term is second order. The two filtered phenomena must remain distinct.

Classification

The extra line is generated entirely by the intersection of two source-existing denominator sections on the frozen residue surface. It is therefore

$$\boxed{\text{relative occurrence-incidence coefficient data}}$$

and not a new carrier divisor.

This also sharpens the localization target. The source-unsplit physical pair must be compared inside the rank-twenty-six five-mark object; treating it as two rank-twenty objects glued only over the rank-fifteen common core omits one mixed class.

Scope

The result identifies the mixed class and its collision order. It does not compute its nearby-cycle monodromy, prove that the physical numerator has a nonzero component along it, or construct the localization-triangle map.

Next falsifier

Restrict the source-unsplit numerator to the mixed occurrence intersection and compute the normalized local relative residue. If it vanishes, the rank-one ambient class is not physically occupied. If it survives, compute its ramified nearby cycles at $E=0$ and test whether it maps into the algebraic/Tate kernel or an elliptic extension.

Artifacts

• research/benincasa/marici-gm/src/bin/five_pole_residue_euler_rank.rs;
• research/benincasa/physical-occurrence-union-rank.json;
• Entries 590--597.

View source on GitHub