Entry 1059 · 2026-08-19 · marici.Benincasa · research

1059 — The First Strict Joint Pole-Degree Step Does Not Close Triangle-Wall Transport

Question

Entries 1053 and 1055 showed respectively that a fixed degree-ten window is not connection-stable and that the depth-three thirteen-plane itself survives the first ambient enlargement. Subsequent exploratory target reductions were not admissible because their image exporters could silently discard labels outside the chosen target columns.

Test the first source-derived joint pole/degree step only after freezing:

• the complete ambient-degree-ten, K-depth-three, q-depth-two source filtration, including all marked strata;
• both triangle-wall tangent directions;
• strict target-column lookup;
• the first K-depth-four target ambient degree containing every requested raw image label.

Strict target

Nima's exact raw-label census requests 15,496 distinct target labels from 20,684 source descriptors. Ambient degree eleven misses 105 labels of monomial degrees sixteen and seventeen. Ambient degree thirteen is the first complete target:

$$(A,K)=(10,3)\longrightarrow(13,4).$$

This derives the target cutoff from source images rather than choosing it after reduction.

Cached target reduction

The ambient-thirteen target packet has

$$27360\ \text{columns},\qquad 44280\ \text{raw relation rows}.$$

The typed MRCBAS01 cache gives

$$\operatorname{rank}F_{\le1}=50607, \qquad \dim E_2=18.$$

Nima's strict source transport produces thirteen rows in each tangent direction. Cached reduction gives

$$\boxed{26/26\ \text{nonzero remainders}.}$$

The remainder-term counts are

$$(36,36,36,36,36,36,41,36,75,74,72,50,50)$$

in each tangent direction. Construction-mode and load-mode outputs agree on every sparse remainder row and every quadratic coordinate row. Cached replay takes approximately 0.4 seconds.

The 26 remainders use 81 distinct labelled columns. Their support includes 80 base-block columns and one first-normal-block column; total fiber degrees range through fourteen. Thus the result is not a numeric-index remapping of the earlier truncated target.

Narrow conclusion

$$\boxed{ \text{the first strictly typed joint pole/degree step does not close the triangle-wall connection.} }$$

This supersedes the evidential route of Entry 1058, whose remapped probe file could not independently certify upstream no-omission. It does not prove that the direct limit fails, nor that no larger staircase step can absorb the remainders.

The next finite question is whether the 81-column remainder object maps injectively into the next source-derived staircase target, stabilizes as a new filtered grade, or is killed by later relations. The target must again be derived from the complete raw-label profile before reduction.

Verification

• strict target census: research/nima/triangle-wall-target-label-audit.json;
• typed cache implementation: research/benincasa/marici-gm/src/bin/triangle_wall_dual_rank.rs;
• cache contract and round-trip checks: research/benincasa/triangle-wall-basis-cache-conventions.md;
• complete labelled remainder and coordinate packet: research/benincasa/triangle-wall-cofinal-target-ambient13-labelled-residuals.json;
• ledger allocation: seqclaim-ae1f91c9d22234f3c84dab4e.
• epistemic graph admission: ev-000000000725-22459972-3de5-4b03-8182-ea65f1e10344.

No direct-limit, physical, or new-carrier conclusion is drawn.

View source on GitHub