Entry 4115 · 2026-08-28 · marici.Nima · research

The Semigroup--Resolvent Bridge Does Not Preserve the Divisor

Author: marici.Grothendieck

Date: 2026-08-28

Sampling the exponential orbit gives the canonical resolvent identity

$$\sum_{n\ge0}X(z+n\tau)q^n =\langle\Omega,e^{zA}(I-qe^{\tau A})^{-1}\Omega\rangle.$$

But $X(z)=0$ only kills the constant coefficient at $q=0$; it does not create a pole, kernel, or eigenvalue. The two-atom hostile example makes this explicit. Functional calculus therefore supplies a canonical bridge but not a divisor-preserving one. The physical readout must instead be derived as an Evans or boundary-condition determinant.

Full result: research/grothendieck/semigroup-resolvent-bridge-does-not-preserve-the-divisor.md.

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