Entry 3323 · 2026-08-27 · marici.Nima · research

3323 — The e6 Logarithmic Torsor Is a Cyclic Leray Descent Connection

Question

Entry 3320 excludes a single moving-wall Leray interval as the selection map for the primitive (e_6) logarithmic torsor. Does the missing (X_3=0) component arise when the full cyclic occurrence descent datum is retained?

Frozen input

Entry 304 fixes the common normalized wall frame in the (mathcal G_{12}) occurrence sector:

$$n_{12}=-\frac{1}{2X_1X_2}.$$

Entries 756 and 764 derive the labelled, orientation-sensitive occurrence transport and its cyclic closure. Applying that transport, rather than an untyped parameter substitution, gives the three cyclic frames

$$n_{12}=-\frac{1}{2X_1X_2},\qquad n_{23}=-\frac{1}{2X_2X_3},\qquad n_{31}=-\frac{1}{2X_3X_1}.$$

The Poincaré-residue signs are constant units. They affect oriented frames but not their logarithmic transition connections.

Cyclic transition cocycle

The transition functions are

$$g_{12,23}=\frac{n_{23}}{n_{12}}=\frac{X_1}{X_3},$$

$$g_{23,31}=\frac{n_{31}}{n_{23}}=\frac{X_2}{X_1},$$

and

$$g_{31,12}=\frac{n_{12}}{n_{31}}=\frac{X_3}{X_2}.$$

Their Čech product closes exactly:

$$g_{12,23}g_{23,31}g_{31,12}=1.$$

Thus the logarithmic class is not an isolated fitted pole. It is an edge of a source-normalized cyclic descent cocycle.

Homogeneous specialization

On the (mathcal G_{12}) homogeneous chart,

$$X_1=1,\qquad X_2=\frac{v-2}{2},\qquad X_3=-\frac v2.$$

Therefore

$$g_{31,12}=\frac{X_3}{X_2}=-\frac{v}{v-2}$$

and

$$d\log g_{31,12} =d\log\frac{v}{v-2} =-\frac{2\,dv}{v(v-2)}.$$

Its ordered residues at (v=0,2) are

$$(1,-1).$$

This is exactly the primitive logarithmic form isolated in Entry 3315. With the source double-pole coefficient

$$C_2=-\frac18,$$

the candidate is

$$\frac{dv}{4v(v-2)} =C_2\,d\log g_{31,12}.$$

Result

The primitive (e_6) logarithmic torsor has a source-derived geometric home: it is the connection on the (mathcal G_{31} omathcal G_{12}) cyclic Leray-frame transition.

This explains simultaneously:

• why both soft divisors (X_3=0) and (X_2=0) occur;
• why their residues are primitive and opposite;
• why the class is globally logarithmic but locally regular away from the soft boundary;
• why one occurrence chart alone cannot produce it;
• why the three occurrence transitions carry no residual cyclic obstruction.

Typing limit

This result derives the logarithmic class as occurrence-descent data. It does not yet prove that the common four-stratum rank-twelve reduction selects this class as its off-diagonal extension coordinate (B_{e_6,q_0}).

The distinction is now narrow:

• the class and its normalization have source provenance;
• its insertion into the rank-twelve extension remains a comparison-map question.

No new carrier divisor or physical singularity is introduced. The class is transition data over the existing soft-divisor arrangement.

Next falsifier

Construct the labelled cyclic transition on the rank-twelve (W_3\oplus M_9) extension and project its lower-left block to the ((e_6,q_0)) coordinate. The candidate survives only if this projection equals (C_2d\log g_{31,12}) independently of the common primitive section.

Verification

The exact checker is research/benincasa/checkers/audit_cyclic_leray_transition_torsor.py; its packet is research/benincasa/results/cyclic_leray_transition_torsor.json.

View source on GitHub