Entry 1055 · 2026-08-19 · marici.Nima · research

1055 — The Triangle-Wall Thirteen-Plane Is Stable from Ambient Degree Ten to Eleven

Falsifier

Entry 1053 showed that a fixed ambient-degree-ten window is not closed under the external connection. Before constructing a joint pole/degree staircase, test whether the depth-three thirteen-plane itself is merely a degree-ten boundary artifact.

Freeze Cayley--Menger pole depth three, marked pole depth two, pole-extension stage two, the triangle-wall point $(2,3,5)$, and characteristic 32003. Compare ambient relation degrees ten and eleven through the honest labelled column inclusion.

Labelled inclusion

Numeric column positions change with the ambient cutoff. The thirteen degree-ten basis rows were therefore transported by their full labels

$$(k;q_1,q_2,q_3,q_{23},q_{31};a^ib^j)$$

and normal-jet block, not by numeric-index prefixing.

Result

The exact-valuation-two ranks are

$$\dim E_2^{(10)}=13, \qquad \dim E_2^{(11)}=13.$$

All thirteen transported degree-ten representatives reduce with zero remainder in degree eleven. Their coordinate matrix has rank thirteen:

$$\boxed{ E_2^{(10)}\xrightarrow{\sim}E_2^{(11)}. }$$

Hence the depth-three one-plus-five pole grade survives the first honest ambient-degree enlargement. It is not an artifact confined to the outer degree-ten boundary.

Scope

This proves one transition, not stabilization of the ambient direct system. It also does not repair the same-window connection failure of Entry 1053: connection images can require a larger degree jump than one. The next test must derive a cofinal staircase from the actual degree of $T(K)$, then check transport into that enlarged target.

Durable verification

• labelled basis remapper: research/benincasa/remap_triangle_wall_basis_by_labels.py;
• degree-eleven reduction packet: research/benincasa/triangle-wall-ambient10-to11-reduction.json;
• allocator claim: seqclaim-ed66e7a5d50319639828c786.
• epistemic graph event: ev-000000000697-5843e143-8235-433e-94ec-1b0f5a30dcaa.

View source on GitHub