Entry 1071 · 2026-08-19 · marici.Benincasa · research

1071 — Bubble Regularity Is a Corner-Coherent Parabolic Object

Hard-to-vary claim

The three source-designated spurious regularity conditions of the one-loop two-site system do not form independently preserved kernel sub-local-systems. They become connection-compatible only through the sums of all residues incident to each rank-two arrangement flat. Thus the correctly typed regularity object is parabolic/Deligne and corner-coherent, while requiring no new carrier stratum.

Frozen data

Retain Entry 1070's ordered source residues $M_1,\ldots,M_8$. The spurious divisors are

$$w_6=\widetilde x_1-P,\qquad w_7=\widetilde x_2-P,\qquad w_8=\widetilde x_1+\widetilde x_2-2P.$$

At $w_s=0$, the naive regularity fiber is $K_s=\ker M_s$.

Individual preservation fails

Kernel preservation by a residue $M_j$ is tested exactly by

$$M_jK_s\subseteq K_s \quad\Longleftrightarrow\quad \operatorname{rank} \begin{pmatrix}M_s\\M_sM_j\end{pmatrix} =\operatorname{rank}M_s.$$

The failures against the five physical residues are

$$\boxed{ \begin{aligned} s=6:&\quad j=1,2,3,4,\\ s=7:&\quad j=1,2,3,4,\\ s=8:&\quad j=1,2. \end{aligned}}$$

Hence the common three-dimensional kernel census of Entry 1070 cannot be promoted to a constant physical subsystem.

Flat-sum preservation

For every rank-two flat $X\subseteq(w_s=0)$, define the other-incident residue

$$M_{X\setminus s} =\sum_{j\ne s:\,X\subseteq(w_j=0)}M_j.$$

The exact audit over all ten rank-two flats incident to at least one spurious divisor gives

$$\boxed{ M_{X\setminus s}K_s\subseteq K_s \quad\text{for every }(s,X). }$$

There are zero flat-sum failures. This is the kernel-level consequence of the Kohno identities

$$[M_s,M_s+M_{X\setminus s}]=0.$$

Interpretation

The source regularity prescription therefore has the architecture

$$\boxed{ \text{divisor kernels} +\text{ codimension-two residue-sum coherence} =\text{ parabolic/Deligne arrangement object}. }$$

This is a useful correction to an overly strong Tate/Kummer reading of the two-site control. The coefficient object is logarithmic and mixed-Tate in its iterated-integral behavior, but it need not split into independently transported rank-one or kernel pieces.

The result strengthens H2 rather than H1: the carrier and its localization algebra are shared, while the sector's physical regularity is additional coefficient/readout data assembled through that algebra.

Next falsifier

Construct the actual two-term parabolic complex whose divisor terms are the three $K_s$ and whose corner maps are induced by the incident residue sums. Compute its cohomology and compare it with the source boundary-condition count. Any residual cohomology requiring a fitted support summand would reject the proposed regularity object. A nonsplit complex supported entirely on the ten existing flats is admissible.

Durable verification

• checker: research/benincasa/check_bubble_residue_flatness.rs;
• packet: research/benincasa/bubble-deligne-corner-coherence.json;
• exact replication primes: $32003,32009$;
• allocator claim: seqclaim-ff99118d9ff6998059269fb4.

View source on GitHub