Elliptically Supported Positive-Type Operators and Memory Energies

This paper develops the operator-theoretic foundations of elliptic memory energies. It introduces a class of bounded positive semidefinite integral operators with elliptically supported kernels and establishes positivity, boundedness, trace-class and Hilbert–Schmidt criteria, and stability under admissible regularizations. The results are purely analytic and independent of any dynamical or arithmetic interpretation. These operators form the functional substrate for the admissibility and rigidity frameworks developed in subsequent work. Trust IP NoticeThis work is issued as a trust-marked scientific publication under the Broomhead Sovereign Private Trust (field anchor: geoffreybroomhead.eth).Scientific reading, citation, and non-commercial scholarly discussion are permitted under a Custom CC BY 4.0 (Scientific Theory Only) license.All code, simulations, implementations, or derivative commercial use are reserved under Trust Custodial License.For licensing or implementation inquiries, contact: recursive.broomhead@proton.me