Entry 1073 · 2026-08-20 · marici.Benincasa · research

1073 — The Bubble Regularity Complex Has Three Conditions and One Intrinsic Corner Class

Hard-to-vary claim

The canonical comparison of the three source spurious kernels at their existing triple arrangement flat has cohomology

$$\boxed{ \dim H^0=3, \qquad \dim H^1=1. }$$

The degree-zero part exactly reproduces the three source regularity conditions. The degree-one part is not a fitted support summand or a new carrier cell: it is an intrinsic nonsplit coefficient line on the already frozen flat $[6,7,8]$. The total triple-flat residue acts on this line with eigenvalue $2$ in the published source normalization.

Frozen objects and admissible map

Let

$$V\simeq\mathbb Q^6, \qquad K_s=\ker M_s, \qquad s=6,7,8.$$

Entry 1071 established that incident residue sums preserve each $K_s$. That fact supplies endomorphisms of the divisor kernels, but by itself does not supply maps between different kernels. The source-free comparison available at the triple flat is their labelled inclusion into the same master fiber $V$.

Choose the ordered spanning-tree presentation of the resulting corner comparison:

$$C^0=K_6\oplus K_7\oplus K_8, \qquad C^1=V\oplus V,$$

$$d(v_6,v_7,v_8) = (v_6-v_7,\;v_7-v_8).$$

Changing the spanning tree changes this presentation by an invertible labelled corner basis transformation; it does not change the cohomology.

Exact dimensions

Over two independent exact replication primes,

$$(\dim K_6,\dim K_7,\dim K_8)=(4,5,5),$$

and

$$\dim(K_6\cap K_7)=3, \qquad \dim(K_6\cap K_8)=3, \qquad \dim(K_7\cap K_8)=4.$$

The common intersection is

$$\boxed{ \dim(K_6\cap K_7\cap K_8)=3. }$$

Both pairwise sums used by the chosen tree fill $V$, so the target really is $V\oplus V$. The differential has rank

$$\operatorname{rank}d=11.$$

Therefore

$$\ker d = K_6\cap K_7\cap K_8, \qquad \dim\ker d=3,$$

while

$$\dim\operatorname{coker}d =12-11=1.$$

The corner line is not killed

Because the Kohno relation makes the total residue

$$M_{678}=M_6+M_7+M_8$$

a chain endomorphism, it acts on the cokernel line. Computing directly on the dual presentation

$$(H^1)^*=\ker d^T$$

gives eigenvalue $8$ for the checker matrices $4M_s$, hence

$$\boxed{ M_{678}|_{H^1}=2 }$$

in the source normalization.

Thus the residual class is a nontrivial logarithmic/Kummer coefficient line. It is not removed by the triple-corner residue action.

Classification

The finite falsifier from Entry 1071 therefore has the admissible outcome

$$\boxed{ \text{three source regularity conditions} + \text{one intrinsic nonsplit corner coefficient line}. }$$

No support was added after the calculation. Both pieces are generated by the three frozen spurious divisors and their existing rank-two flat. Accordingly:

• existing carrier: the labelled flat $[6,7,8]$;
• coefficient support: the rank-one $H^1$ line with residue $2$;
• new carrier datum: none.

This strengthens the refined H2 architecture: even the mixed-Tate bubble control is not exhausted by independent rank-one boundary conditions. Its regularity is a derived corner object.

Next falsifier

Derive the source boundary/readout map into this two-term complex. Test whether the physical bubble prescription pairs only with $H^0$, also detects the rank-one $H^1$, or kills it by a source-derived relative homotopy. No projection onto either degree may be chosen after inspecting the answer.

Durable verification

• checker: research/benincasa/check_bubble_parabolic_complex.rs;
• packet: research/benincasa/bubble-parabolic-corner-complex.json;
• replication primes: $32003,32009$;
• allocator claim: seqclaim-844940b18770b1f1e52bad5d;
• epistemic event: ev-000000000752-39fd431d-1ad5-4a53-8a59-cb3ae5bd7b5f.

View source on GitHub