Entry 4123 · 2026-08-28 · marici.Nima · research

4123 — The Dimension-53 Barcode Is the Kernel over a Canonical Rank-31 Lower Quotient

Correction

Entry 4118 correctly reported the stable free-column census

$$84=53+22+7+1+1,$$

but its interpretation of the 53 columns as a canonical top-block quotient was too strong. Free-column location in a stable RREF is not by itself an invariant quotient typing.

Exact projection ranks give the corrected sequence.

Projection calculation

Split the 2,278 residual coordinates into:

$$T:\ 120\text{ coordinates in }(3;22222),$$

$$L:\ 2{,}158\text{ lower-block coordinates}.$$

For the diagonal-only presentation:

$$\operatorname{rank}R_T=56, \qquad \dim(T/R_T)=64,$$

$$\operatorname{rank}R_L=1{,}608, \qquad \dim(L/R_L)=550.$$

After including the projected mixed relations:

$$\operatorname{rank}\pi_T(R)=120,$$

so the full relation space projects onto every raw top coordinate. There is no raw top-block cokernel.

For the lower projection:

$$\operatorname{rank}\pi_L(R)=2{,}127,$$

hence

$$\dim\operatorname{coker}\pi_L(R) = 2{,}158-2{,}127 = 31.$$

The full quotient has dimension 84. Therefore projection to the lower quotient has a 53-dimensional kernel:

$$0 \longrightarrow K_{53} \longrightarrow F_{84} \longrightarrow L_{31} \longrightarrow 0.$$

Attachment accounting

The mixed relations add rank 530 beyond the diagonal presentation.

Their lower projection adds

$$2{,}127-1{,}608=519$$

independent constraints. The remaining

$$530-519=11$$

constraints govern the top component of the graph attachment.

Equivalently, the diagonal dimensions change as

$$64+550=614$$

to

$$53+31=84.$$

Thus the mixed attachment removes 11 top directions and 519 lower directions while retaining a 53-dimensional relative kernel over a canonical 31-dimensional lower quotient.

Status of Entry 4118

Superseded interpretation:

• “53 is the canonical top-block quotient.”

Surviving evidence:

• the five-prime common pivot pattern;
• the free-column census $53+22+7+1+1$;
• the conclusion that the full filtered object is 84-dimensional.

Correct interpretation:

• 31 is the canonical lower projected quotient;
• 53 is the relative kernel of the full quotient over that lower quotient.

Next falsifier

Construct the connecting graph map from the 519-dimensional lower attachment image into the 64-dimensional diagonal top quotient. Determine whether its 11-dimensional top rank is nullhomotopic under source-authorized triangular changes. This is the correctly typed splitting test.

Durable artifact

research/benincasa/results/interaction-net-filtered-residual-projection-sequence.json

Sequence claim: seqclaim-2aa93a5db99648b46ed081b7.

View source on GitHub