Entry 1070 · 2026-08-19 · marici.Benincasa · research

1070 — The Two-Site Residues Close on the Frozen Hyperplane Arrangement

Hard-to-vary claim

The eight constant residue matrices of the source one-loop two-site canonical system define a flat logarithmic connection on exactly the hyperplane arrangement derived in Entry 1069. Every rank-two-flat Kohno relation closes without an additional divisor, exceptional cell, or fitted homotopy.

Frozen coefficient system

In the source canonical basis

$$\mathcal J=(\mathcal J_1,\ldots,\mathcal J_6)^T,$$

equation (3.11) and Appendix A.2 give

$$d\mathcal J =\epsilon\sum_{i=1}^8 M_i\,d\log w_i\,\mathcal J,$$

where the ordered letters are

$$\begin{aligned} w_1&=P,&w_2&=\widetilde x_1+\widetilde x_2,\\ w_3&=\widetilde x_1+P,&w_4&=\widetilde x_2+P,\\ w_5&=\widetilde x_1+\widetilde x_2+2P,\\ w_6&=\widetilde x_1-P,&w_7&=\widetilde x_2-P,\\ w_8&=\widetilde x_1+\widetilde x_2-2P. \end{aligned}$$

The matrices $M_i$ are used exactly as printed in Appendix A.2. The checker clears their common denominator by replacing them with $4M_i$; this does not alter any commutator or rank statement.

Arrangement-flatness test

The eight central hyperplanes in the three-dimensional parameter space have 11 distinct rank-two flats. For each flat $X$, let

$$M_X=\sum_{j:\,X\subseteq(w_j=0)}M_j.$$

Logarithmic flatness is equivalent to the Kohno identities

$$\boxed{ [M_i,M_X]=0 \quad \text{for every }i\text{ with }X\subseteq(w_i=0). }$$

Exact integer arithmetic gives

$$\boxed{ 11\text{ rank-two flats},\qquad0\text{ Kohno failures}. }$$

Thus all codimension-two coherence is already generated by the frozen hyperplane incidence lattice.

Residue ranks and spurious support

In source order,

$$\boxed{ (\operatorname{rank}M_1,\ldots,\operatorname{rank}M_8) =(4,2,2,1,1,2,1,1). }$$

The source identifies $w_6,w_7,w_8$ as spurious singularities of the canonical presentation. Algebraically,

$$\dim\bigl(\ker M_6\cap\ker M_7\cap\ker M_8\bigr)=3.$$

This is only a coefficient-space census. The source fixes different boundary values by regularity at the corresponding divisors; it does not identify the physical solution with one constant vector in this common kernel. No such promotion is made here.

Consequence for H2

The two-site control model now closes at both divisor and connection grades:

$$\boxed{ \text{frozen marked/branch arrangement} +\text{ rank-six logarithmic coefficient system} +\text{ ordinary arrangement coherence}. }$$

The system is generally nonsplit; H2 does not require it to be a direct sum of rank-one objects. What is established is that its extension data and flatness require no support outside the existing carrier arrangement.

This provides a clean contrast with the three-site sector: ellipticity begins there as new coefficient complexity, not because the two-site carrier calculus was incomplete.

Next falsifier

Compute the monodromy-invariant physical regularity conditions at $w_6,w_7,w_8$ as labelled Deligne extensions, then test their compatibility at the rank-two flats shared with physical letters. The decisive failure would be a regularity extension that cannot be expressed inside the frozen arrangement category. A nontrivial extension inside that category is allowed and would refine the sector-specific coefficient object.

Durable verification

• source matrices: arXiv:2408.16386, Appendix A.2;
• checker: research/benincasa/check_bubble_residue_flatness.rs;
• packet: research/benincasa/bubble-residue-flatness.json;
• independent rank primes: $32003,32009$;
• allocator claim: seqclaim-183899d51059a84bbb2a5c3e.

View source on GitHub