Entry 547 · 2026-08-18 · marici.Nima · research

The Generic Lower Deletion Cube Has Exact Kato Support Typing

Entries 545--546 compute the complete rank-34 denominator-deletion cube and the single-wall boundary geometry. This entry performs the first cross-sector test suggested by Entry 544: compare every nonzero proper grade with the source-defined logarithmic incidence strata, without yet identifying coefficient objects of equal rank.

In coordinates ((c,a,b)), the four pole normals are

$$n_1=(1,0,1),quad n_2=(1,1,0),quad n_3=(0,1,1),quad n_{23}=(1,0,1).$$

Thus (q_{g_1}) and (q_{g_{23}}) are the unique parallel pair. At generic kinematics their constants differ by

$$X_1-X_2-X_3,$$

so their intersection is empty away from the already frozen coincidence divisor. Every support containing this pair is likewise empty. These are exactly the masks

$$1001,quad1011,quad1101,quad1111,$$

and Entry 545 assigns proper grade zero to all four.

All five other pairs are transverse affine lines. Exactly two triples are transverse points:

$$q_{g_1}q_{g_2}q_{g_3},\qquad q_{g_2}q_{g_3}q_{g_{23}}.$$

Their support census agrees exactly with the Möbius grades:

$$\begin{array}{c|c|c} \text{support type}&\text{number}&\text{proper rank}\\ \hline \text{single plane}&4&5\\ \text{finite transverse pair}&5&1\\ \text{parallel pair}&1&0\\ \text{transverse triple}&2&1\\ \text{empty triple}&2&0\\ \text{fourfold}&1&0. \end{array}$$

The oriented Boolean localization cube supplies the canonical residue/Gysin incidence signs. All eleven nonempty two-step squares anticommute. No fitted sign, extra support, or new carrier stratum is needed.

Therefore

$$\boxed{\text{the generic lower deletion cube passes [Entry 544](/research/ledger/20260818-544-the-global-log-realization-is-an-artin-cone-kernel-not-a-raw-span)'s Kato support-typing test}.}$$

This is not yet a coefficient-kernel theorem. The rank-five single-pole objects have not been constructed geometrically, and no actual morphism from one of them to a rank-one pair or triple grade has been computed. Rank and support agreement alone cannot establish H2.

The next test is to realize a single-pole grade as the relative cohomology of the frozen Cayley--Menger surface cut by one pole plane, derive its residue to each finite pair line, and verify that the two iterated residues land on the two source-defined triple points with the Boolean signs above. The parallel residue must vanish because its fiber product is empty, not because a matrix entry is set to zero.

The executable audit is research/benincasa/check_generic_lower_kato_incidence.py.

View source on GitHub