IndisputableMonolith.NumberTheory.WeakZeroFreeRegion
The WeakZeroFreeRegion module supplies definitions and certificates for a preliminary zero-free region in the RS-native zeta analysis. Number theorists extending the Recognition Science zeta program cite it when moving from basic constants to stronger strip results. The module structures sibling declarations around WeakZFRCert and classical_zfr_suffices to support the downstream strip module.
claimThe module defines the weak zero-free region certificate $WeakZFRCert$ and the existence statement $weak_zfr_cert_exists$ that certify a preliminary region free of zeros for the zeta function under RS chain requirements.
background
The module imports Constants, where the fundamental RS time quantum satisfies τ₀ = 1 tick, and Mathlib. It operates inside the NumberTheory domain of the Recognition Science framework, which derives physics from a single functional equation and advances a zeta program in numbered phases. Sibling declarations introduce DefectBudget, RSChainRequirements, and rh_axiom_replaceable to handle zero locations weaker than the classical line Re(s) ≥ 1.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module feeds StripZeroFreeRegion, described as Phase 5 of the RS-native zeta program that records the proven Mathlib zero-free result on Re(s) ≥ 1. It supplies the preparatory weak certificate and classical_zfr_suffices needed before the strip result can replace the Riemann hypothesis axiom with RS chain requirements.
scope and limits
- Does not prove any zero-free region on its own.
- Does not contain analytic estimates beyond Mathlib imports.
- Does not address the full critical strip.
- Does not discharge the Riemann hypothesis replacement.