Entry 859 · 2026-08-18 · marici.Nima · research

859 — Generic Quartic Fibers Preserve the Fixed Final Extension Coordinates

Source-level test

Entry 857 isolated four extension coordinates that are fixed by every solution of the labelled source system. Entry 858 showed that both diagonal connection blocks are generically regular on

$$\mathcal Q_{uv}=-u^4+4u^3v-4u^3-4u^2v+4u^2 -8uv-4v^2+16u+16v-16=0.$$

Instead of reconstructing a global rational primitive, the present test specializes the complete source reduction system directly to exact points of this quartic. For each point it solves all six systems indexed by

$$(\partial_u,\partial_v)\times(m_0,m_1,m_2)$$

and records source rank, fixed-coordinate mask, pivot signature, consistency, and residual vanishing.

Generic quartic result

Over each of two independent 61-bit primes, exact nonsoft quartic points exist at which every one of the six systems has

$$\operatorname{rank}M=117, \qquad \operatorname{mask}_{\rm fixed}=3847,$$

with the same generic pivot hash. In particular the four final coordinates

$$z_8,z_9,z_{10},z_{11}$$

remain fixed on a nonempty open chart of the quartic. Hence

$$\boxed{ \mathcal Q_{uv}\text{ is not a forced generic rank-loss divisor of the fixed final-coordinate source projection.} }$$

This is a source-level conclusion. It does not rely on the uncertified modular reconstruction of the full $4\times3$ extension block.

Exceptional subloci remain

The gate also detects special quartic points. The soft point $(u,v)=(2,2)$ has rank $92$, and one replication-prime point has rank $116$. Therefore the result is generic along the quartic, not an everywhere-regularity theorem.

Nor does it determine the other five rows of the rank-nine kernel. A quartic-supported off-diagonal class could still live in those ambiguous directions or at a proper sublocus. Characteristic-zero certification of Benincasa's reconstructed block remains pending.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/nima_marked_extension_q_fiber_gate.rs;
• packet: research/nima/marked-extension-q-fiber-gate.json;
• commands: cargo run --release --bin nima_marked_extension_q_fiber_gate and cargo run --release --features reconstruction-prime-3 --bin nima_marked_extension_q_fiber_gate;
• allocator claim: seqclaim-5ea743438c420006271c3936.

View source on GitHub