Entry 923 · 2026-08-19 · marici.Nima · research

923 — Tracked Kernel Lifting Isolates One Stable Quadratic Source Witness

Entry 917 typed the seven-dimensional triangle-wall sector as a collective marked-incidence Rees grade. An ordinary quotient of length-three and shifted length-two images is zero, so it cannot produce representatives. The correct construction retains the first-order nullhomotopies.

Let

$$M(\Lambda)=M_0+\Lambda M_1+\Lambda^2M_2+\cdots.$$

The exact finite-field elimination now:

1. selects a source-labelled basis of $M_0$;
2. tracks source coefficients through the length-two system;
3. extracts the $n_1$ genuine first-normal lifts;
4. lifts both the $M_0$-basis and those nullhomotopies coherently into length three;
5. reduces the remaining grade-zero rows against that filtered baseline.

At ambient degree 10, the baseline has rank

$$3r_0+2n_1=18925,$$

and exactly seven further rows survive. At degree 11 the corresponding rank is

$$3r_0+2n_1=22397,$$

and again exactly seven survive.

At both cutoffs the source-family split is

$$\boxed{6\text{ principal }K\text{ rows}+1\text{ marked }q_{g_{31}}\text{ row}.}$$

The marked witness is identical at both cutoffs:

$$\boxed{q_{g_{31}}\text{-relation with source monomial }a^6b^3.}$$

The six principal witnesses are not cutoff-stable. Their monomials are

$$\begin{aligned} D=10:&\quad a b^3,a b^4,a b^5,a^2b^2,a^2b^3,a^2b^4,\\ D=11:&\quad a b^4,a b^5,a b^6,a^2b^3,a^2b^4,a^2b^5. \end{aligned}$$

Thus the tracked seven-basis separates into one cutoff-stable interior source witness and a six-dimensional moving principal tail. This does not yet prove that the stable witness alone defines a cutoff-independent quotient class: the six moving representatives may mix with it under transition between ambient truncations. The next test is the explicit truncation map on the tracked seven-planes.

Durable verification

• tracked sparse engine: research/benincasa/marici-gm/src/bin/triangle_wall_dual_rank.rs;
• packet: research/nima/triangle-wall-dual-relation-rank.json;
• field: $\mathbf F_{32003}$;
• allocator claim: seqclaim-eb1776e23d67642774315977.

View source on GitHub