Using the microscopic model Hamiltonian, the equations for order parameter and chemical potential have been obtained for a system of Fermi-particles interacting with phonons which permit an analysis of its superconducting properties to be made for an arbitrary fermion concentration. The solution of the equations is given that makes it possible to trace the crossover from a local pair regime to Cooper pairing. The type of superconducting behavior is shown to depend on both the position of the Fermi level with respect to a maximum phonon energy and strength of indirect interactions between fermions.
This Stage–1 publication introduces and formalizes Photonic Reverberant Information (PRI), a theoretical and experimentally testable extension to classical optical scattering models. The work provides explicit mathematical definitions of the Anti–Manifest Field and PRI fields, demonstrates compatibility with Maxwellian and Green’s function–based scattering theory, and proposes measurable optical signatures that distinguish PRI from conventional transient scattering effects. The manuscript presents three primary contributions: A formal PRI field construction, including a persistence kernel that characterizes reverberant photonic information following photon–matter interaction. Derived measurable quantities—intensity persistence, spatial entropy behavior, and structural correlation—that enable experimental discrimination of PRI effects. A complete ultrafast-optics experimental design for PRI validation using existing laboratory instrumentation. This Stage–1 release establishes PRI as a falsifiable hypothesis suitable for peer review and experimental investigation. It provides a theoretical foundation for subsequent Stage–2 experimental studies and Stage–3 extensions to broader photonic information systems.
Pseudorandom functions (PRFs) are one of the most fundamental primitives in classical cryptography. On the other hand, in quantum cryptography, it is possible that PRFs do not exist but their quantum analogues could exist, and still enabling many applications including SKE, MACs, commitments, multiparty computations, and more. Pseudorandom unitaries (PRUs) [Ji, Liu, Song, Crypto 2018], pseudorandom isometries (PRIs) [Ananth, Gulati, Kaleoglu, Lin, Eurocrypt 2024], and pseudorandom function-like state generators (PRFSGs) [Ananth, Qian, Yuen, Crypto 2022] are major quantum analogs of PRFs. PRUs imply PRIs, and PRIs imply PRFSGs, but the converse implications remain unknown. An important open question is whether these natural quantum analogues of PRFs are equivalent. In this paper, we partially resolve this question by ruling out black-box constructions of them: 1. There are no black-box constructions of
O(logλ)-ancilla PRUs from PRFSGs. 2. There are no black-box constructions of
O(logλ)-ancilla PRIs with
O(logλ) stretch from PRFSGs. 3. There are no black-box constructions of
O(logλ)-ancilla PRIs with
O(logλ) stretch from PRIs with
Ω(λ) stretch. Here,
O(logλ)-ancilla means that the generation algorithm uses at most
O(logλ) ancilla qubits. PRIs with
s(λ) stretch is PRIs mapping
λ qubits to
λ+s(λ) qubits. To rule out the above black-box constructions, we construct a unitary oracle that separates them. For the separations, we construct an adversary based on the quantum singular value transformation, which would be independent of interest and should be useful for other oracle separations in quantum cryptography.