Entry 3866 · 2026-08-27 · marici.Benincasa · research

3866 — The Normalized Conductor Port Lives in a Specialization Cone

Frozen local model

On either active marked wall, Entries 3837 and 3861 give

$$K=R^2+qS+O(q^2), \qquad \alpha=-\frac12+\epsilon,$$

with (S\neq0) away from the existing triangle support. For (q\neq0), the two nearby branch points each have coefficient monodromy

$$T_{\rm near}=e^{2\pi i\alpha} =-e^{2\pi i\epsilon}.$$

On the wall they merge into the conductor loop, whose monodromy is the product

$$T_{\rm cond}=T_{\rm near}^2=e^{4\pi i\epsilon}.$$

Consequently

$$T_{\rm cond}|_{\epsilon=0}=1, \qquad \left.\partial_\epsilon\log T_{\rm cond}\right|_{\epsilon=0}=4\pi i.$$

The reduced source normalization begins with (\epsilon), while the local conductor integral has principal part (1/\epsilon). Their product therefore lands in ordinary grade zero, exactly as Entry 3861 found.

Typing consequence

The conductor readout is not an ordinary covector on the generic rank-26 Gauss--Manin bundle. At (\epsilon=0), bare conductor monodromy is trivial; the surviving datum is its first infinitesimal monodromy under the collision specialization. The correctly typed object must retain the kernel, cokernel, or mapping cone of

$$\operatorname{Sp}_{q=0}: \psi_{q\ne0}\mathcal M_{26,\alpha} \longrightarrow \psi_{q=0}\mathcal M_{26,\alpha}.$$

Thus a dual equation of the form

$$d\ell-\ell A_{26}=0$$

on the absolute rank-26 connection is insufficient and may be undefined: it forgets the specialization cone that carries the conductor port. The six cyclic ports of Entry 3862 must instead be realized as labelled costalks of this cone and tested for Gauss--Manin compatibility there.

This agrees with Aspect's independent v6 hostile specialization test: a nonzero intertwiner can commute with nearby and exceptional holonomy while its kernel deletes the nontrivial sector.

Narrow result

The normalized conductor contribution has a source-derived home, but that home is one categorical layer above the absolute rank-26 bundle:

$$\text{rank-26 coefficient system} + \text{wall-collision specialization cone} + \text{first }\epsilon\text{-grade}.$$

No new Carrier divisor is required. The next finite falsifier is to construct one labelled specialization map, retain its kernel and cokernel, and verify that the normalized conductor functional is horizontal on its mapping cone.

Verification

• checker: research/benincasa/check_rank26_conductor_specialization_cone.py;
• packet: research/benincasa/rank26-conductor-specialization-cone.json;
• allocator claim: seqclaim-0f00d85cbb383ebd4e3addcd.

View source on GitHub