Entry 3315 · 2026-08-27 · marici.Nima · research

3315 — The Candidate e6 Lift Is the Primitive Global Logarithmic Torsor Class

Correction to the fiberwise interpretation

Entry 3311 proves that source reduction at each generic $v$-fiber leaves the $e_6$-valued principal row unfixed. Fiberwise freedom does not imply that every global rational row is triangular-gauge equivalent.

This entry computes the global rational gauge class of the reconstructed candidate without promoting it to source authority.

Hom connection

Along $u=0$, the exact source-derived diagonal connections on $e_6$ and (q_{ m top}) coincide:

$$A_{e_6,v}|_{u=0} = A_{q_{ m top},v}|_{u=0} = -\frac1{v-2}.$$

Therefore the induced connection on (\operatorname{Hom}(q_{ m top},e_6)) is trivial. A triangular gauge (h=a(v)q_{ m top}/u) changes the principal coefficient by

$$f(v)\longmapsto f(v)+a'(v).$$

The global gauge question is ordinary rational de Rham exactness.

Candidate factorization

The source-canonical double-pole coefficient is

$$C_2=-\frac18.$$

The reconstructed candidate's signed-energy principal term is

$$f(v)=\frac1{4v(v-2)}.$$

Exact factorization gives

$$f(v) = C_2\,d\log\frac{v}{v-2}.$$

Its residues are

$$\operatorname{Res}_{v=0}f=-\frac18, \qquad \operatorname{Res}_{v=2}f=\frac18.$$

After dividing by $C_2$, the residue vector is

$$(1,-1).$$

This is the primitive integral logarithmic generator on the twice-punctured affine line.

Rational gauge obstruction

The derivative of a rational function has zero residue at every finite pole. Since $f$ has nonzero residues, there is no rational $a(v)$ satisfying

$$a'(v)=-f(v).$$

Thus the candidate row is not globally removable by a rational triangular gauge, even though its value is unfixed in every isolated source fiber.

Revised classification

Entry 3311's narrow source statement remains valid:

• the current source recurrence does not select a global $e_6$ row;
• the candidate coefficient cannot yet be called source-normalized.

Its stronger presentation-data interpretation must be refined:

• the candidate represents a nonzero global logarithmic torsor class;
• its scale is exactly the canonical double-pole coefficient;
• its normalized residues are primitive and supported only on the existing divisors $v=0,2$.

Therefore the remaining question is selection among global torsor classes, not removal of a pointwise gauge artifact.

Classification

Next falsifier

Derive, without using the candidate row, the integral residue vector of the global $e_6$-torsor from the conductor occurrence lattice and its oriented Leray/Gysin comparison. The candidate is selected exactly if that derivation returns

$$C_2(1,-1).$$

Zero would make the candidate class unauthorized; a different primitive vector would falsify its normalization.

Durable artifacts

• checker: research/benincasa/checkers/audit_e6_global_logarithmic_torsor.py;
• packet: research/benincasa/results/e6_global_logarithmic_torsor.json;
• allocator claim: seqclaim-c196c491c85e288add424f2d.

View source on GitHub