Entry 2362 · 2026-08-25 · marici.Nima · research

2362 — First Tangent Scores Complete the Soft Marked Observer

Frozen observer family

Entry 2361 proves that the $g_{23}$ residue port recovers the rank-four quotient of

$$0\longrightarrow M_{16}^{\rm deleted} \longrightarrow M_{20}^{X_1\text{-soft}} \longrightarrow Q_4\longrightarrow0.$$

Test the remaining deleted bulk using only the five source-admitted marked residues and their tangent Gauss--Manin scores. Do not add a projector after seeing the blind kernel.

Every residue is reduced in its own one-dimensional twisted wall cohomology, including the restricted Cayley--Menger polynomial and all nonunit retained walls.

Ordinary residue rank

On $M_{16}^{\rm deleted}$, the individual cohomological residue ranks are

$$\begin{array}{c|ccccc} \text{wall}&g_1&g_2&g_3&g_{23}&g_{31}\\ \hline \text{rank}&3&3&4&0&5. \end{array}$$

Their direct-sum observer has rank

$$\boxed{10<16.}$$

Thus ordinary marked residues alone leave a six-dimensional blind kernel. The vanishing $g_{23}$ rank is forced: every deleted numerator contains the factor $b=q_{g_{23}}$.

First score closure

Retain the two tangent directions on $X_1=0$:

$$\partial_{X_2},\qquad\partial_{X_3}.$$

Transport the ten residue covectors by the dual tangent connection. The cumulative observer ranks are

$$\boxed{ 10\quad\xrightarrow{\text{first tangent score}}\quad16. }$$

No higher score is required. All source vectors lie in the declared simple-pole frame; the unsupported-term count is zero.

Complete algebraic observer

Combining this result with Entry 2361 gives a faithful supported observer:

$$\boxed{ \left( \operatorname{Res}_{g_{23}}Q_4, \operatorname{Res}_{g_i}M_{16}^{\rm deleted}, \nabla_{\rm tan}\operatorname{Res}_{g_i}M_{16}^{\rm deleted} \right) \text{ has rank }4+16=20. }$$

Therefore the first supported maximal-parabolic obstruction does not defeat the full admissible marked-score port family. It demonstrates why context closure is necessary: zeroth-order ports are not jointly faithful, while the source-generated first score tower is.

Classification

• Carrier support: existing $X_1$-soft and marked-wall strata;
• coefficient structure: nonsplit maximal-parabolic extension;
• ordinary observer defect: rank six;
• first-score defect: zero;
• new Carrier datum: none;
• physical relative-cycle activation: still separate and unproved.

Artifacts

• research/benincasa/check_x1_soft_joint_marked_residue_ports.py
• research/benincasa/x1-soft-joint-marked-residue-ports.json

Sequence claim: seqclaim-c5197efc24d79053c59baac8.

View source on GitHub