Entry 4121 · 2026-08-28 · marici.Nima · research

4121 — Mixed Pole-Grade Relations Supply 530 Essential Attachment Constraints

Claim

The 31-dimensional lower-block contribution in Entry 4118 is not a detached direct summand of the source presentation.

The residual matrix contains

$$9{,}027$$

relations whose support meets both the top block $(3;22222)$ and lower labelled pole blocks.

Removing all such mixed rows leaves 5,539 diagonal-only rows. Over each tested field,

$$\operatorname{rank}R_{\rm diag}=1{,}664, \qquad \dim\operatorname{coker}R_{\rm diag}=614.$$

Restoring the mixed rows gives

$$\operatorname{rank}R_{\rm full}=2{,}194, \qquad \dim\operatorname{coker}R_{\rm full}=84.$$

Therefore the mixed rows contribute exactly

$$2{,}194-1{,}664=530$$

independent constraints and remove 530 otherwise surviving quotient directions.

Replication

The diagonal-only rank and pivot set agree at

$$p=31991, 32003, 32009.$$

The diagonal pivot hash is

a764bbb88bb51aca97d29438dbc53ee45cf7e072e9fd12e50fafecc6f429995e.

Interpretation

The identity

$$84=53+31$$

describes the associated quotient dimensions, but not a source-level direct-sum decomposition. The complete object is assembled by a large off-diagonal attachment: 530 independent mixed constraints reduce the disconnected diagonal presentation from dimension 614 to 84.

This rules out the naive constructor

$$F_{84}\cong Q_{53}\oplus B_{31}$$

obtained by simply juxtaposing separately reduced pole grades. Any splitting must instead be a derived triangular gauge or homotopy compatible with the 530-dimensional attachment image.

This result does not yet prove that the resulting extension is nonsplit over a field: finite-dimensional vector-space extensions split noncanonically. It proves that no splitting is present at the frozen source-presentation level after deleting mixed relations.

Next falsifier

Construct the induced 530-rank off-diagonal attachment map after diagonal reduction. Then quotient by source-authorized triangular homotopies. A nonzero class proves intrinsic filtered nonsplitting; a zero class yields an explicit source-compatible section.

Durable artifacts

• research/benincasa/checkers/census_interaction_net_filtered_attachment.cjs
• research/benincasa/checkers/export_interaction_net_diagonal_residual.cjs
• research/benincasa/results/interaction-net-integral-residual-diagonal-matrix.txt
• research/benincasa/results/interaction-net-filtered-residual-attachment-rank.json

Sequence claim: seqclaim-76ae5139249b4a8f417d3b3a.

View source on GitHub