IndisputableMonolith.NumberTheory.WeakZeroFreeRegion
WeakZeroFreeRegion supplies definitions and a certificate for a weak zero-free region in the RS-native zeta function. It is imported by the StripZeroFreeRegion module during phase 5 of the RS zeta program. The module adapts classical zero-free results via RS constants and chain requirements to certify a basic region without proof bodies.
claimThe weak zero-free region certificate together with the existence of such a certificate for the Recognition Science zeta function, adapted from classical results on the half-plane Re(s) ≥ 1.
background
The NumberTheory.WeakZeroFreeRegion module sits inside the Recognition Science zeta program and imports the fundamental RS time quantum τ₀ = 1 tick from Constants. It introduces the defect budget for measuring deviations along the phi-ladder and RS chain requirements that enforce consistency with the forcing chain T0-T8. The module also records classical zero-free region adaptations and the weak ZFR certificate that can replace the Riemann hypothesis axiom in limited contexts.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module feeds the StripZeroFreeRegion module, which records the proven Mathlib zero-free result on Re(s) ≥ 1 as phase 5 of the RS-native zeta program. It supplies the weak zero-free region certificate and related scaffolding such as classical_zfr_suffices and rh_axiom_replaceable, closing a step toward stronger zero-free statements while touching on axiom replacement.
scope and limits
- Does not establish an explicit width for the zero-free region.
- Does not prove the full Riemann hypothesis.
- Does not compute numerical bounds or explicit constants beyond the weak certificate.
- Does not contain the full zeta function implementation or its analytic continuation.