Entry 861 · 2026-08-18 · marici.Nima · research

861 — The Primitive-Independent Extension Quotient Has Rank Seven

Why the coordinate mask was insufficient

Entry 857 records four individually fixed final rows of the nine-row absolute extension target. Individual coordinate invariance does not exclude invariant linear combinations among the other five rows.

The correct object is the projection of the complete source nullspace to the nine absolute coordinates. Its annihilator is the full primitive-independent quotient.

Exact projected ambiguity

Write the nine absolute coordinates as $x_0,\ldots,x_8$, and set

$$c=\frac{3u+v-2}{2}.$$

At every tested generic and generic-quartic point, for both derivatives and all three quotient generators, the projected ambiguity has the same rank-two normal form

$$\boxed{ \begin{pmatrix} 1&0&0&u&uc&0&0&0&0\\ 0&1&u+1&-1&-c&0&0&0&0 \end{pmatrix}. }$$

The formula was reconstructed from generic samples and then verified directly over two independent 61-bit primes, including exact points of $\mathcal Q_{uv}=0$.

Therefore

$$\dim\operatorname{Amb}=2, \qquad \boxed{\dim(M_9/\operatorname{Amb})=7}.$$

Three missed mixed invariants

Besides the four fixed coordinates $x_5,x_6,x_7,x_8$, the annihilator contains

$$\boxed{ \begin{aligned} I_1&=x_2-(u+1)x_1,\\ I_2&=x_3-ux_0+x_1,\\ I_3&=x_4-ucx_0+cx_1. \end{aligned} }$$

These three directions are source-derived and primitive-independent. The earlier four-row reconstruction therefore did not exhaust the invariant extension data.

Consequence for the quartic test

Entries 859--860 exclude a generic quartic pole from the four coordinate invariants, but they do not test $I_1,I_2,I_3$. Those three mixed directions are now the unique remaining source-level location for an intrinsic generic $\mathcal Q$-residue before the triangular connection gauge quotient is applied.

The next calculation should reconstruct only these three invariant functionals. This is smaller and better typed than reconstructing five ambiguous rows or a complete primitive witness.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/nima_marked_extension_invariant_quotient.rs;
• packet: research/nima/marked-extension-invariant-quotient.json;
• allocator claim: seqclaim-002100d2b2286a0357a41edf.

View source on GitHub