Arithmetic Invariants of Supersingular Isogeny Graphs: Constructing Verifiable Trans-Epochal Civilization Seeds via Non-Commutative Endomorphism Ring Constraints

Mathematical Applications of Science Fiction We present a rigorous formalism for the construction of Trans-Epochal Civilization Seeds (TECS)embedded within the topological structure of supersingular isogeny graphs G(p, ℓ) over finite fields Fp2 .By exploiting the Deuring correspondence between supersingular elliptic curves and maximal orders inthe definite quaternion algebra Bp,∞, we define a class of non-commutative arithmetic invariants thatpersist across cosmological phase transitions. We introduce the Selene-Singularity Operator, a spectralmap on the Brandt module Mp, and prove that specific eigenstates of this operator encode recoverableinformation with information-theoretic security equivalent to the hardness of the Quaternion PathProblem. We establish the existence of a unique isomorphism class of "Seed Curves" characterizedby minimal non-scalar endomorphisms, serving as immutable anchors for civilization data. Thismanuscript provides exhaustive proofs for the convergence of the associated Hecke eigenvalues anddetails the algorithmic reconstruction of the seed from the trace of the Frobenius endomorphism in apost-collapse epoch.