IndisputableMonolith.NumberTheory.HilbertPolyaFunctionField
This module defines the Frobenius angle θ for an elliptic curve E over F_q with trace a via cos θ = a/(2√q) when the value lies in [-1,1]. Number theorists working on Hilbert-Polya constructions in function fields would cite these objects. It is a definition-only module that imports the Cost module and introduces supporting definitions such as hpOperator and hasseBound.
claimThe Frobenius angle θ of an elliptic curve E/𝔽_q with Frobenius trace a satisfies cos θ = a/(2√q) whenever |a| ≤ 2√q (the Hasse-Weil bound).
background
The module operates in the setting of elliptic curves over finite fields and imports IndisputableMonolith.Cost to access recognition primitives alongside Mathlib. The central definition is the Frobenius angle θ, given by cos θ = a/(2√q) when the argument lies in [-1,1]. It introduces auxiliary objects including hasseBound, hpOperator (with symmetry and eigenvector properties), and hilbert_polya_elliptic_curve.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies number-theoretic definitions that feed into sibling declarations such as hilbert_polya_elliptic_curve and hpOperator. By importing the Cost module it positions the Frobenius angle and related operators inside the Recognition Science framework.
scope and limits
- Does not prove any eigenvalue statements for the hpOperator.
- Does not connect θ to the J-function or phi-ladder.
- Does not address the forcing chain T0-T8 or RCL.
- Does not supply mass formulas or physical constants.