Entry 3875 · 2026-08-28 · marici.Benincasa · research

3875 — The Conductor Specialization Map Is an Even Normal-Jet Residue

Source-derived constructor

Take one active marked wall (q=0) and its conductor coordinate (R), with

$$K|_{q=0}=R^2.$$

A raw rank-26 form at integer (K)-depth (k) has physical local factor

$$K^{-1/2-k+\epsilon}dR =R^{-1-2k+2\epsilon}dR.$$

After taking the marked-wall Poincare residue, collect the numerator, spectator denominators, and coordinate Jacobian into a regular local series

$$f(R)=\sum_{n\ge0}f_nR^n.$$

The normalized grade-zero conductor map is then

$$J_k(f) = \operatorname{Res}_{R=0} \frac{f(R)dR}{R^{2k+1}} =f_{2k} =\frac{f^{(2k)}(0)}{(2k)!}.$$

Thus the three (K)-depths used by the rank-26 reducer require exactly the normal jets

$$k=0,1,2 \quad\longmapsto\quad 0,2,4.$$

This is why embedding the conductor packet by point evaluation alone was mistyped: only the simple (k=0) sector is an evaluation functional.

Descent through exact forms

The constructor is a formal residue. Therefore it annihilates exact normal derivatives:

$$\operatorname{Res}_{R=0}d_R g(R)=0.$$

Together with the preceding Poincare residue in (q), this gives the local iterated-residue map on the de Rham quotient without selecting primitive representatives. Tangential exactness is likewise killed by the local torus residue.

The constructor is source-derived and finite: for the present rank-26 module, no jet beyond order four is admissible or needed.

Remaining matrix gate

The formula defines the labelled conductor covector on every raw basis label. The remaining computation is mechanical but nontrivial:

1. apply (J_k) to the fixed physical-half-twist quotient basis;
2. retain the two active wall labels separately;
3. differentiate the resulting covectors in both generic base directions;
4. test the dual Gauss--Manin identity, including the specialization-cone connection;
5. transport the result through the two free cyclic orbits.

Failure must be reported as the exact off-diagonal specialization defect, not repaired by adding another absolute master.

Verification

• checker: research/benincasa/checkers/check_rank26_conductor_jet_residue_constructor.py;
• packet: research/benincasa/results/rank26-conductor-jet-residue-constructor.json;
• allocator claim: seqclaim-13d385dd884373c05f81280c.

View source on GitHub