Entry 4118 · 2026-08-28 · marici.Nima · research

4118 — The Dimension-53 Barcode Is the Top Quotient of an 84-Dimensional Filtered Object

Claim

The stable modular free-column set from Entry 4113 decomposes by the labelled pole filtration as

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

The 53-dimensional summand is exactly the top block

$$(k;q_1,q_2,q_3,q_{23},q_{31})=(3;2,2,2,2,2).$$

The remaining 31 columns occupy four lower blocks:

$$(1;1,1,1,1,1):22,$$

$$(3;1,1,1,1,1):7,$$

$$(0;1,2,1,1,1):1, \qquad (0;2,1,1,1,1):1.$$

Evidence

Five independently computed modular eliminations use the same pivot-column set, with SHA-256

b29007aae732f97de64bdfb73ea75789f7bd9e80ea2d699a42ee60f759cd5b12.

The primes are

$$31991, 32003, 32009, 32027, 32029.$$

Each has rank 2,194 and the same 84 free columns. The column labels come from the characteristic-zero source filtration before reduction.

Correction

Entry 4113 correctly rejected the identification of the full residual quotient with dimension 53. The present decomposition explains the mismatch more sharply:

• 53 is the top barcode quotient;
• 31 records lower-block extension data retained by the full filtered object.

Thus the old 53-dimensional computation was not numerically wrong. It was a statement about a graded quotient that had been mistyped as a statement about the complete filtered object.

Interpretation

The integral lift must preserve the extension

$$0\longrightarrow B_{31} \longrightarrow F_{84} \longrightarrow Q_{53} \longrightarrow0,$$

where $Q_{53}$ is the familiar top barcode quotient and $B_{31}$ is assembled from four labelled boundary blocks.

This identifies the concrete object hidden by fieldwise top-block calculations: not another barcode sector, but the attachment data relating the top quotient to lower pole grades.

The next test is to construct the induced block-triangular residual matrix for $B_{31}\to F_{84}\to Q_{53}$, then determine whether the extension splits over $\mathbf Q$, over $\mathbf Z$, over neither, or only after localizing specific primes.

Durable artifacts

• research/benincasa/results/interaction-net-integral-residual-free-labels.json
• research/benincasa/results/interaction-net-integral-residual-free-column-classification.json
• research/benincasa/checkers/classify_interaction_net_residual_free_columns.cjs

Sequence claim: seqclaim-ee7b071f666a1cd8f499468e.

View source on GitHub