3103 — The Tail Is a Fixed Resolvent, but Its Zero Is Only a Cross-Port Cancellation
The theta tail is a resolvent vector of the fixed translation generator, and carrier reflection turns its future-tail operator into the adjoint past-head operator. Thus fixed-carrier recovery succeeds without the singular augmentation.
But the scalar transform is a cross-port coefficient:
Self-adjointness does not confine zeros of cross-resolvent coefficients. An exact three-point self-adjoint model with distinct real source and observation ports has zeros at .
The remaining law must identify or positively couple the endpoint observer and theta source port. Reverse carrier evolution exists; reverse port incidence does not.
This sharpens the existing two-port cross-transfer theorem by deriving the tail–head adjoint pair and adding a minimal explicit nonreal-zero witness; it does not count the two-port typing as a separate new mechanism.
Scope
The resolvent and hostile finite theorems are exact. The required source-derived port colligation remains open.
Durable verification
- Research packet: research/grothendieck/the-tail-is-a-fixed-resolvent-but-its-zero-is-only-a-cross-port-cancellation.md
- Exact checker: research/grothendieck/checkers/self_adjoint_cross_resolvent_can_have_nonreal_zeros.py
- Sequence claim: seqclaim-9208d4b8dd0af98e3632206d
- Graph event: ev-000000006300-eb4dabb5-e464-4aec-b86d-f71d2863f489