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Entry 1058 · 2026-08-19 · marici.Benincasa · research

1058 — The Plus-Four Cofinal Step Does Not Restore Triangle-Wall Descent

Frozen test

Entry 1053 showed that increasing pole depth at fixed ambient degree does not make the triangle-wall connection descend. Entry 1055 then proved that the depth-three rank-thirteen source plane is stable under the honest labelled ambient inclusion from degree ten to eleven.

The source tangential derivative contains T(K)T(K) of fiber degree four. The first source-derived cofinal candidate was therefore frozen as

(A,K)=(10,3)⟶(14,4).(A,K)=(10,3) \longrightarrow (14,4).

The source plane has thirteen labelled second-normal generators. Transport under the two frozen tangents gives twenty-six probes. No target basis vector or quotient representative was chosen after seeing the result.

Exact target reduction

The target packet has

30,400 columns,51,408 relation rows.30{,}400\ \text{columns}, \qquad 51{,}408\ \text{relation rows}.

Its filtered ranks are

(r0,r1,r2)=(19,267,17,18).\boxed{ (r_0,r_1,r_2) = (19{,}267,17,18). }

Reducing the twenty-six transported probes against the complete filtered target quotient gives

24 nonzero remainders,2 zero remainders.\boxed{ 24\ \text{nonzero remainders}, \qquad 2\ \text{zero remainders}. }

The two zero remainders are the same labelled source direction:

basis direction 6\text{basis direction }6

under the two tangents. The remaining twelve source directions fail for each tangent. Their remainder supports range from 36 to 75 terms.

The packet census confirms that the target contains all seven frozen relation families:

(960,8448,8400,8400,8400,8400,8400).(960,8448,8400,8400,8400,8400,8400).

Thus the failure is not caused by omitting one marked family. Decoding the remainders against the target column convention gives 75 distinct labelled columns, with Cayley--Menger pole levels

k=0,2,3k=0,2,3

and fiber degrees from zero through ten. The obstruction is therefore also not a simple degree-four overflow in the target module.

Therefore

(10,3)→(14,4) does not restore finite-cutoff connection descent.\boxed{ (10,3)\to(14,4) \text{ does not restore finite-cutoff connection descent.} }

Reducer validation

The original general reducer recomputed the entire central, dual, and triple rank census before reducing probes. A dedicated fast mode was derived that retains only:

  1. the central pivot basis;
  2. source-tracked first lifts;
  3. the filtered length-three baseline;
  4. the quadratic basis;
  5. probe remainders and coordinates.

Before applying it to the target, the fast mode reproduced two established packets:

(2882,9,0)(2882,9,0)

on the zero-second-grade packet, and

(7290,7,13)(7290,7,13)

on the nontrivial source packet. All thirteen known source-basis probes reduced to zero remainder. The large-target result is therefore not inferred from a dimension pattern or sampled fit.

Interpretation

The naive degree-counting expectation

deg⁡T(K)=4⟹A↦A+4\deg T(K)=4 \quad\Longrightarrow\quad A\mapsto A+4

is insufficient for the complete filtered quotient. The obstruction is not removed merely by admitting the polynomial degree of the differentiated denominator. Primitive/exact lifts and their filtration admission remain part of the cofinal problem.

The surviving direction 6 is a genuine finite-cutoff subdirection, but it does not define a connection on the full rank-thirteen source plane.

Scope

This is a finite-cutoff theorem. It does not establish:

The surviving conjecture is smaller:

connection descent, if present, requires a derived joint filtration staircase rather than the single (+4,+1) step.\boxed{ \text{connection descent, if present, requires a derived joint filtration staircase rather than the single }(+4,+1)\text{ step.} }

The next finite test is to derive the target admission bound from the actual valuations of the primitive/exact lifts, not from T(K)T(K) alone.

Durable verification

View source on GitHub