Entry 1074 · 2026-08-20 · marici.Benincasa · research

1074 — The Source Bubble Boundary Factors Exactly through Degree Zero

Hard-to-vary claim

The primary two-site construction does not assign independent boundary values to the three spurious divisors. It imposes regularity on one master vector at their common point

$$(\widetilde x_1,\widetilde x_2,P)=(1,1,1).$$

Consequently its boundary map factors exactly through the degree-zero cohomology of Entry 1073's corner complex:

$$\boxed{ B_{\rm source} =K_6\cap K_7\cap K_8 \xrightarrow{\sim} H^0(C^\bullet_{\rm sp}), \qquad B_{\rm source}\longrightarrow H^1(C^\bullet_{\rm sp})=0. }$$

The intrinsic rank-one corner $H^1$ is therefore not an additional physical bubble boundary condition. It measures the obstruction to gluing arbitrary divisor-local regular data that the source never introduces.

Frozen source prescription

Appendix B of arXiv:2408.16386v2 gives the canonical six-master system and states that:

• $\mathcal J_1$ and $\mathcal J_5$ are fixed by direct integration;
• $\mathcal J_2,\mathcal J_4,\mathcal J_6$ are fixed order by order in $\epsilon$ by imposing regularity at $(\widetilde x_1,\widetilde x_2,P)=(1,1,1)$;
• $\mathcal J_4$ also admits a direct-integration match at $(0,0,1)$.

At the regularity point, the three source-labelled spurious letters vanish simultaneously:

$$\widetilde x_1-P=0, \qquad \widetilde x_2-P=0, \qquad \widetilde x_1+\widetilde x_2-2P=0.$$

Thus source regularity requires a single boundary value $v\in V$ to obey

$$M_6v=M_7v=M_8v=0.$$

Exact boundary space

In the ordered source basis

$$(\mathcal J_1,\ldots,\mathcal J_6),$$

a characteristic-zero basis of the common kernel is

$$\boxed{ \begin{aligned} b_1&=(0,0,2,1,0,0),\\ b_2&=(-4,-2,-2,0,1,0),\\ b_3&=(8,4,4,0,0,1). \end{aligned}}$$

Direct substitution into all three published residue matrices gives

$$M_sb_i=0 \qquad (s=6,7,8;\ i=1,2,3).$$

The basis has rank three over both replication primes and therefore spans the three-dimensional common kernel computed in Entry 1073.

Factorization through the corner complex

Recall

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

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

The source supplies the diagonal map

$$\Delta: B_{\rm source}\longrightarrow C^0, \qquad v\longmapsto(v,v,v).$$

It is a chain map because

$$d\Delta(v)=(0,0).$$

Moreover,

$$\operatorname{im}\Delta = K_6\cap K_7\cap K_8 = H^0(C^\bullet_{\rm sp}).$$

Since the source map lands in cycles in degree zero and introduces no independent divisor-local cochain, its induced component in $H^1=\operatorname{coker}d$ is identically zero.

Interpretation

Entry 1073's rank-one $H^1$ remains a real coefficient-theoretic corner class. This entry narrows its role:

$$\boxed{ H^1 = \text{obstruction to independent local gluing}, \quad \text{not a source-selected physical boundary value}. }$$

This is not a fitted projection. The factorization is forced by the source's use of one common kinematic point and one common master vector. No relative homotopy is needed to kill the line; the physical boundary map never enters its degree.

Classification:

• existing carrier: the common spurious flat $[6,7,8]$;
• source-selected coefficient data: the rank-three $H^0$;
• unselected derived coefficient data: the rank-one $H^1$;
• new carrier datum: none.

Next falsifier

Test whether this exact pattern persists for another source-defined polylogarithmic arrangement sector: one common physical boundary vector should factor through degree zero of the labelled corner complex, while higher cohomology records only failures of independently assigned local data. A source prescription that canonically pairs with higher corner cohomology would falsify that extrapolation without falsifying H2.

Durable verification

• primary source: arXiv:2408.16386v2, Appendix B;
• checker: research/benincasa/check_bubble_parabolic_complex.rs;
• packet: research/benincasa/bubble-physical-boundary-factorization.json;
• replication primes: $32003,32009$;
• allocator claim: seqclaim-d75f530607bf9d401e537b4c;
• epistemic event: ev-000000000754-c12ffc39-6fa6-4cb6-8ce4-26afcb010bb7.

View source on GitHub