Entry 3328 · 2026-08-27 · marici.Nima · research

3328 — The Two-Soft No-Infinity Condition Leaves One Logarithmic Extension Line

Question

Entries 3323 and 3326 identify the primitive (e_6) form as an (A_2) boundary cocycle. How much freedom remains after imposing its actual support and regularity conditions?

This classifies the admissible logarithmic forms. It does not infer which rank-twelve coordinate carries one.

Logarithmic base

Use the homogeneous base (mathbf P^1_v) with labelled boundary

$$D=\{v=0\}\cup\{v=2\}\cup\{v=\infty\}.$$

The finite points are the existing soft divisors

$$v=0\iff X_3=0, \qquad v=2\iff X_2=0.$$

A logarithmic one-form with finite poles only at these points is

$$\omega= r_0\frac{dv}{v} +r_2\frac{dv}{v-2}.$$

Its residue at infinity is

$$r_\infty=-r_0-r_2.$$

No-infinity condition

The source candidate has no infinity pole. Therefore

$$r_0+r_2=0.$$

The two-dimensional finite-pole space consequently collapses to the unique line

$$\mathbb Q\left\langle d\log\frac{v}{v-2} \right\rangle.$$

Its primitive integral residue vector, ordered by (0,2,\infty), is

$$(1,-1,0).$$

The source double-pole coefficient fixes the candidate scale:

$$C_2=-\frac18, \qquad C_2d\log\frac{v}{v-2} =\frac{dv}{4v(v-2)}.$$

Result

Once the following conditions are frozen,

• logarithmic poles only at (X_3=0) and (X_2=0);
• no residue at infinity;
• primitive opposite boundary orientation;

there is exactly one admissible logarithmic extension line. Entry 3323's cyclic Leray cocycle generates it, and (C_2) supplies its source scale.

Non-identifiability boundary

Uniqueness of the line does not identify its location inside a larger extension matrix. A closed triangle of transition functions can arise from different moving blocks, as emphasized independently by Aspect's moving-fiber interferometer audit.

Therefore this result establishes:

• the unique admissible logarithmic class;
• its integral residue lattice;
• its source normalization.

It does not establish:

• that (B_{e_6,q_0}) is nonzero;
• that another rank-twelve coordinate does not carry the same class;
• that the physical relative chain pairs nontrivially with it.

Next falsifier

Derive the complete rank-twelve transition between two labelled occurrence charts and compute its triangular logarithmic block. The source selects the (e_6) torsor only if the boundary-preserving transition places this unique class in the ((e_6,q_0)) coordinate independently of splitting choices.

Verification

The checker is research/benincasa/checkers/audit_soft_logarithmic_ext_line.py; its packet is research/benincasa/results/soft_logarithmic_ext_line.json.

Allocator claim: seqclaim-2181be5a80f7dc7ac351d566.

View source on GitHub