IndisputableMonolith.NumberTheory.HilbertPolyaFunctionField
This module defines the Frobenius angle θ for elliptic curves E over finite fields F_q together with related operators and bounds. It supplies the number-theoretic objects needed for the Hilbert-Polya approach inside Recognition Science. The definitions rest on the imported Cost module and enforce the Hasse-Weil constraint |a| ≤ 2√q.
claimFor an elliptic curve E over F_q with Frobenius trace a, the angle θ satisfies cos θ = a / (2 √q) whenever |a| ≤ 2 √q (the Hasse-Weil bound).
background
The module sits in the NumberTheory domain and imports Mathlib for elliptic-curve infrastructure together with IndisputableMonolith.Cost for the underlying J-cost and defect measures. Its central object is the Frobenius angle θ defined by the cosine relation to the trace a when the Hasse-Weil bound holds. The setting therefore links finite-field arithmetic directly to the Recognition Composition Law and the phi-ladder.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the elliptic-curve primitives that feed the Hilbert-Polya function-field constructions, including hpOperator and hilbert_polya_elliptic_curve. It thereby connects the Hasse-Weil bound to the J-uniqueness property (T5) and the eight-tick octave (T7) of the forcing chain.
scope and limits
- Does not prove the Riemann hypothesis or any eigenvalue statement.
- Does not compute explicit spectra or characteristic polynomials.
- Does not extend beyond the Hasse-Weil interval [-1,1].
- Does not address curves of genus greater than one.