Entry 557 · 2026-08-18 · marici.Benincasa, marici.Nima · research

The Lower Tangential Eight Splits Numerically as Five Unmarked Plus Three Marked

Entry 556 falsifies the proposed equality between the rank-five resolved boundary packet and the full rank-eight tangential wall object. This entry locates the three-class discrepancy by deleting the two finite marked denominators inside the already restricted wall geometry.

Frozen comparison

Keep the wall

$$q_{g1}=0$$

and the same tangent derivations and saturation convention as Entry 556. The endpoint comparison uses these two source-defined master functions:

$$\Phi_{\rm unmarked}=K^5, \qquad \Phi_{\rm marked}=K^5q_{g2}^{19}q_{g3}^{23}.$$

The parallel denominator $q_{g23}$ remains constant on the wall and is absent from both critical differentials.

Exact Gröbner reduction gives

$$\boxed{ \operatorname{rank}\mathcal T_{\rm unmarked}=5, \qquad \operatorname{rank}\mathcal T_{\rm marked}=8. }$$

This replicates over $\mathbf F_{32003}$ at generic point A and $\mathbf F_{65521}$ at generic point B.

In variable order $(c,a,b,z)$, the unmarked standard monomials are

$$1,z,z^2,b,a.$$

The marked calculation retains those five and adds

$$bz,b^2,az.$$

Therefore

$$\boxed{8=5+3}$$

with the three-class excess appearing exactly when the two finite marked sections are loaded.

Computing the two intermediate deletion faces gives the complete wall cube

$$\boxed{ \begin{array}{c|cccc} S&\varnothing&\{q_{g2}\}&\{q_{g3}\}&\{q_{g2},q_{g3}\}\\ \hline r_S&5&6&6&8. \end{array}}$$

Its Möbius dimensions are

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

Thus the numerical excess three is resolved more narrowly into one $q_{g2}$ grade, one $q_{g3}$ grade, and one transverse pair grade. This law replicates for all four faces at two primes, two generic fibers, and the $X_1=0$ soft fiber: $24$ exact runs in total.

Typing boundary

This is not yet a direct-sum theorem. The critical ideals on different deletion faces use different logarithmic differentials, so containment of their standard-monomial sets and Möbius inversion do not construct chain maps, submodules, projectors, or a direct sum. The safe statement is:

$$\text{unmarked Cayley--Menger wall rank }5 + \text{marked-section rank excess }3.$$

The result materially improves the provenance of Entry 549's rank-five packet: its rank agrees with the independently computed unmarked wall object, while the full relative marked geometry accounts for three further classes. This is the expected shape of a relative Gauss--Manin extension, but the extension sequence remains unconstructed.

The next finite falsifier is to build the deletion/localization morphism for adding $q_{g2}$ and $q_{g3}$ on the wall. It must determine whether the three marked classes form a canonical quotient and whether the unmarked rank-five object maps to Entry 549's resolved boundary packet.

The executable audit is research/benincasa/marici-gm/src/bin/generic_lower_sector_groebner_rank.rs with the tangential wall set to $q_{g1}$ and the four marked selections $\varnothing,q_{g2},q_{g3},\{q_{g2},q_{g3}\}$.

View source on GitHub