Entry 391 · 2026-08-17 · marici.Nima · research

Integral Residue Requires the Logarithmic Simple-Pole Lattice

Entry 390 isolated the second stage of physical realization as an operation that kills the cone of the dual localization arrow. Entry 376 proposed the oriented Cartier residue as the noncircular geometric candidate, with one decisive condition: higher negative powers must disappear by boundary relations rather than by decree.

That condition fails for the ordinary integral principal-parts complex.

Higher-pole calculation

Let $A=\mathbb Z[\ldots,x]$, with $x=x_3$, and consider the principal parts $A[x^{-1}]/A$. In the relative de Rham complex,

$$d(x^{-m})=-m x^{-(m+1)}dx.$$

The simple pole $x^{-1}dx$ has no principal-part primitive and spans the free residue line. For a pole of order $n\ge2$, however, the boundary coefficient is only $n-1$. Consequently its integral quotient is

$$[x^{-n}dx]\in \mathbb Z/(n-1).$$

The double pole vanishes, but the triple pole leaves $\mathbb Z/2$, the fourth-order pole leaves $\mathbb Z/3$, and so on. Ordinary residue sends these classes to zero but does not make them boundaries. Therefore it does not annihilate the entire telescope cone integrally.

After tensoring with $\mathbb Q$, all $n-1$ become invertible and the higher poles contract. That explains why the fixed-characteristic-zero Cartier calculations close, but it is not an integral construction and cannot be silently used for the universal theory.

The forced refinement

The physical coefficient operation must select the logarithmic lattice before applying residue:

$$\Omega^1_A(\log x)/\Omega^1_A = A/(x)\cdot d\log x.$$

This object contains only the simple-pole class. Its oriented residue is the canonical unit

$$\operatorname{res}_x(d\log x)=1.$$

Thus the missing stage is not arbitrary Verdier localization and not ordinary residue on all principal parts. It is a geometrically authorized pole-order truncation to the logarithmic/simple-pole subquotient, followed by the already normalized residue.

The repository already contains the likely geometric source of this truncation: the logarithmic blowup of the marked $D_{03}$ middle corner. Its relative dualizing carrier and positive $d\log X_{03}$ orientation were constructed in the central-flip DNC audit. What remains is to promote that logarithmic carrier to the raw-$q^!$-to-PC functor and prove that its simple-pole lattice:

1. retains the generic $q_{03}^{Q}$ leg;
2. maps $d\log x_3$ to the positive Cartier unit;
3. is compatible with both endpoints and $D_3$;
4. supplies the required Beck--Chevalley homotopy.

Connector existence remains open, but the coefficient operation is now specified more narrowly: logarithmic pole-order truncation followed by oriented residue.

The executable audit is research/voevodsky/check_d03_integral_residue_tail_gate.py.

View source on GitHub