Entry 3318 · 2026-08-27 · marici.Nima · research

3318 — Matching Primitive Residues Do Not Yet Select the Global e6 Torsor

Question

Entry 3315 shows that the reconstructed $e_6$ lift is the canonical double-pole scale times a primitive logarithmic vector on the base:

$$\frac{f}{C_2} =d\log\frac{v}{v-2}, \qquad \operatorname{Res}_{(v=0,v=2)}\frac{f}{C_2} =(1,-1).$$

Entry 304 independently derives the primitive oriented Leray boundary

$$\partial[p_-,p_+]=[p_+]-[p_-],$$

with vector $(-1,1)$ in the ordered fiber-endpoint basis $(p_-,p_+)$.

This entry tests whether the existing comparison calculus identifies these two vectors.

Base soft lattice

On the total-energy slice

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

Hence the two candidate poles are the labelled soft divisors

$$s_3=\{v=0\}=\{X_3=0\}, \qquad s_2=\{v=2\}=\{X_2=0\}.$$

Their primitive divisor difference is source-defined:

$$\operatorname{div}\frac{X_3}{X_2} =s_3-s_2,$$

which gives $(1,-1)$ in the ordered basis $(s_3,s_2)$.

Fiber endpoint lattice

Entry 304's vector belongs to a different object. Its basis is

$$p_- = \{r=-1\}, \qquad p_+ = \{r=1\},$$

on the exceptional integration-fiber interval. Its orientation is fixed by the physical Leray chain $p_-\to p_+$.

Thus the equality of the two primitive vectors after relabelling is not yet a comparison theorem.

Existing maps

The current admitted maps do not bridge the two lattices:

1. The conductor/enhanced comparison

$$J: \langle g_{101},g_{110},\widetilde g_{111}\rangle \longrightarrow \langle\delta,\epsilon,\epsilon\delta\rangle$$

maps occurrence characters, not fiber endpoints to base-soft divisors.

2. Infinity Gysin kills $e_6$, so it cannot normalize this residue.

3. The diagonal Hom connection is zero, which classifies the torsor cohomology but does not select a class inside it.

4. Total-energy conductor specialization maps the top graph cycle into its finite vanishing object, not into the ordered pair $(s_3,s_2)$.

Therefore no current source-derived arrow has type

$$\mathbb Z\langle p_-,p_+\rangle \longrightarrow \mathbb Z\langle s_3,s_2\rangle.$$

Verdict

The candidate passes a strong compatibility test:

• its base residue vector is primitive;
• it has the same difference pattern as the independently derived Leray boundary;
• its scale is the canonical $C_2=-1/8$.

But compatibility is not selection. Identifying the two vectors without a source-derived specialization map would transport authority across variance.

The candidate $e_6$ torsor therefore remains globally nontrivial but physically and source-theoretically unselected.

Classification

Next falsifier

Construct the weighted specialization of the source Leray interval over the whole total-energy exceptional divisor. Its boundary must be retained both vertically in the integration fiber and horizontally at $X_3=0,X_2=0$.

The candidate is selected only if the resulting boundary-of-boundary comparison sends

$$[p_+]-[p_-] \longmapsto [s_3]-[s_2]$$

with coefficient $C_2$, independently of weighted lift and chart.

Durable artifacts

• checker: research/benincasa/checkers/audit_e6_torsor_selection_typing.py;
• packet: research/benincasa/results/e6_torsor_selection_typing.json;
• allocator claim: seqclaim-c31c11ad949df01c1aba07de.

View source on GitHub