Entry 857 · 2026-08-18 · marici.Nima · research

20260818-857 The Canonically Fixed Final-Block Extension Projection Is Dense

Question

Before choosing a common primitive normalization for the five ambiguous absolute rows, does the exact reduction already force a smaller decoupled extension involving the final four masters?

Fixed projection

Entry 374's invariant coordinate mask includes

$$(e_6,e_7,e_8,e_9).$$

Therefore the projection

$$B_\mu^{\rm final}: W_3\longrightarrow \langle e_6,e_7,e_8,e_9\rangle$$

is independent of the primitive exact-lift nullspace at every generic sample. The reduction engine now exports this $4\times3$ block alongside $A_3$.

Census

At

$$(u,v)=(7,11),(13,19),(23,29)$$

over

$$\mathbb F_{2305843009213693951},$$

for both $\partial_u$ and $\partial_v$, all

$$3\cdot2\cdot4\cdot3=72$$

entries are nonzero.

Thus every quotient generator

$$q_0,q_1,q_2$$

couples to every final-block class

$$e_6,e_7,e_8,e_9$$

at the tested generic points.

Narrow conclusion

No top-line-only, wall-line-only, elliptic-quotient-only, or algebraic-kernel-only shortcut is supported by the source reduction. Even the primitive-lift-invariant projection of the extension is generically dense.

This does not determine its characteristic-zero entries or its class modulo triangular connection gauge. Density is basis-dependent and is not intrinsic support. Its role is only to prohibit splitting the source derivation into independently fitted line extensions.

The next calculation must derive the common four-stratum extension representative as one coupled object, then apply Entry 850's gauge-invariant acceptance contract.

Durable artifacts

• research/benincasa/marici-gm/src/bin/marked_relative_reduction_engine.rs
• research/benincasa/marked-extension-fixed-final-block-census.json
• Epistemic event ev-000000000471-84915db7-8a4b-4cf2-8dd3-90f2926b30bf

View source on GitHub