Entry 2360 · 2026-08-24 · marici.Nima · research

2360 — The Soft Invariant Plane Is Exactly the Denominator-Deletion Image

Question from Entry 2359

The $X_1=0$ tangent connection preserves a canonical sixteen-plane $W_{16}$, derived as the image of the trace radical of its maximal-parabolic algebra. Is this a new coefficient layer, or an existing localization operation?

Source-derived map

On the labelled $\mathcal G_{12}$ residue chart,

$$q_{g_{23}}=b-X_1.$$

Hence on $X_1=0$,

$$q_{g_{23}}=b.$$

Deleting the $g_{23}$ denominator embeds a four-mark numerator class into the five-mark relative union by

$$\boxed{P(a,b)\longmapsto bP(a,b).}$$

This map is frozen before comparison with $W_{16}$.

Exact equality

Using every numerator monomial of degree at most five, multiplication by $b$ gives twenty-one candidate labelled classes. Exact reduction in the same degree-six rank-twenty-six presentation gives image rank

$$\operatorname{rank}\operatorname{im}(\mathrm{Del}_{g_{23}})=16.$$

Bidirectional containment against Entry 2359's trace-radical image gives

$$\operatorname{im}(\mathrm{Del}_{g_{23}})/W_{16}=0, \qquad W_{16}/\operatorname{im}(\mathrm{Del}_{g_{23}})=0.$$

Therefore

$$\boxed{ W_{16}=\operatorname{im}(\mathrm{Del}_{g_{23}}). }$$

The four-dimensional quotient is consequently the supported soft residue sector:

$$0\longrightarrow M_{16}^{\rm deleted} \longrightarrow M_{20}^{X_1\text{-soft}} \longrightarrow Q_4^{g_{23}\text{-res}} \longrightarrow0.$$

Consequence for H2

The first proper supported algebra from Entry 2359 is not an undeclared Carrier operation. It is exactly the source-derived deletion--restriction filtration on the existing soft wall. H2 therefore survives this test in a stronger form:

$$\boxed{ \text{existing soft Carrier localization} + \text{sector-specific nonsplit coefficient extension}. }$$

Physical gate

The physical dual is cyclic precisely when it pairs nontrivially with the deleted image. This can now be tested geometrically: restrict the source-normalized physical relative cycle to the four-mark deletion sector and evaluate one nonzero period. If the restriction vanishes identically, only the rank-four residue quotient is observable; if it does not, the full rank-twenty tangent packet is recoverable.

The primary positive chain does not acquire a boundary on the five-pole walls at generic positive energies. At $X_1=0$, however, the soft wall $q_{g_{23}}=b=0$ meets the boundary corner, so the required statement is a supported limiting-chain calculation, not generic positivity.

Artifacts

• research/benincasa/check_x1_soft_parabolic_deletion_image.py
• research/benincasa/x1-soft-parabolic-deletion-image.json

Sequence claim: seqclaim-29bd6528743c15a663fca2dd.

View source on GitHub