Entry 556 · 2026-08-18 · marici.Benincasa · research

The Tangential Lower Wall Has Rank Eight Not Five

Entries 551--554 show that the generic deletion increment does not itself construct a residue object and that the Kummer resonant normal fiber has two grades. The missing input is an independent construction of the tangential coefficient object on $q_{g1}=0$.

Frozen tangential critical problem

Impose

$$q_{g1}=c+b+X_1=0$$

before taking critical equations. Tangent derivations are

$$\partial_a, \qquad \partial_b-\partial_c.$$

The restricted parallel denominator is constant,

$$q_{g23}|_{q_{g1}}=X_2+X_3-X_1,$$

so the frozen tangential master function contains only

$$K^5q_{g2}^{19}q_{g3}^{23}.$$

The exact saturated critical ideal is generated by $q_{g1}$, the two tangential logarithmic critical numerators, and

$$zKq_{g2}q_{g3}-1.$$

The provisional expectation was rank five. Exact execution instead gives

$$\boxed{\dim\mathcal T_{g1}=8.}$$

The result replicates over $\mathbf F_{32003}$ and $\mathbf F_{65521}$, at both generic kinematic points and at the soft fiber $X_1=0$. The reduced tangential presentation has Gröbner-basis size $34$ and the same eight standard monomials in all six full-mark runs. This rank is obtained directly on the wall and is not the subtraction $12-7$.

Consequence

The Kummer resonant object of Entry 552 is therefore typed as

$$\mathcal T_{g1}\otimes [\mathbb Q\xrightarrow0\mathbb Q], \qquad \operatorname{rank}\mathcal T_{g1}=8.$$

Thus its two normal grades are two rank-eight copies, not two rank-five copies. Entries 553--554 still prohibit selecting one by integer contiguity or Gysin variance alone.

This falsifies the proposed rank match with Entry 549. The resolved boundary packet has associated-grade rank five, while the full tangential critical object has rank eight. Consequently the $4+1$ packet cannot be the complete wall coefficient object. Any comparison must locate an independently derived rank-five subquotient and account for the remaining three tangential classes; rank matching supplies neither.

The equality of the generic and $X_1=0$ ranks also shows that the first physical contact at the soft face produces no tangential rank jump. It changes the chain incidence, not the coefficient rank.

The executable audit is research/benincasa/marici-gm/src/bin/generic_lower_sector_groebner_rank.rs with TANGENTIAL_WALL=q_g1.

View source on GitHub