phase5_current_state
The theorem records that the Riemann zeta function has no zeros for real part at least 1 and that the existence of the named strip bridge implies a logarithmic zero-free region for large imaginary part. Researchers tracing the Recognition Science zeta program would cite it as the Phase 5 state that separates proven boundary results from the remaining analytic bridge. The proof is a direct term-mode pairing of the half-plane lemma with the bridge implication.
claimThe Riemann zeta function satisfies $r(s) = 0$ for no $s$ with real part at least 1. If the strip zero-free bridge holds, then there exist $c > 0$ and $T > 1$ such that $r(s) = 0$ is impossible whenever $|$imaginary part of $s| > T$ and real part of $s$ exceeds $1 - c / $log of that imaginary part.
background
Phase 5 of the RS-native zeta program records the proven zero-free region on Re(s) >=1 and names the logarithmic strip as the next bridge object. The StripZeroFreeBridge is the proposition that a LogZeroFreeStrip exists, which supplies the classical de la Vallee-Poussin shape. The module does not assert the full critical-strip result but packages the bridge so that later steps can see the irreducible analytic content needed for RH-quality closure.
proof idea
The proof is a one-line term that constructs the conjunction by feeding the half-plane result riemannZeta_ne_zero_re_ge_one into the first slot and the bridge-derived implication riemannZeta_ne_zero_in_log_strip into the second slot.
why it matters in Recognition Science
This declaration marks the current unconditional results available to the Recognition Science zeta program and feeds the critical-strip zero-free bridge required for closure of the recovered RH chain. It isolates the gap between the boundary region Re >=1 and the open right half of the critical strip, noting that the full strip theorem follows from Mathlib's formal Riemann hypothesis but is not inhabited here.
scope and limits
- Does not prove zero-freeness inside the critical strip.
- Does not supply explicit numerical constants for the logarithmic region.
- Does not close the full Riemann hypothesis.
- Does not link the zero-free region to RS physical constants or forcing chain steps.
formal statement (Lean)
96theorem phase5_current_state :
97 (∀ s : ℂ, 1 ≤ s.re → riemannZeta s ≠ 0) ∧
98 (StripZeroFreeBridge → ∃ c T : ℝ, 0 < c ∧ 1 < T ∧
99 ∀ s : ℂ, T ≤ |s.im| →
100 1 - c / Real.log |s.im| < s.re →
101 riemannZeta s ≠ 0) := by
proof body
Term-mode proof.
102 exact ⟨fun s hs => riemannZeta_ne_zero_re_ge_one hs,
103 riemannZeta_ne_zero_in_log_strip⟩
104
105/-! ## 4. Phase 7: critical-strip zero-free bridge
106
107The `LogZeroFreeStrip` above is the classical de la Vallée-Poussin shape and
108is not enough to close the recovered RH chain. The chain is built around
109witnessed defect sensors with `1/2 < Re(ρ) < 1`, so vacuous closure requires
110zero-freeness on the open right half of the critical strip. That bridge is
111the actual analytic input sitting between the recovered chain and a
112million-dollar theorem.
113
114We name it explicitly here. We do not inhabit it. We do prove that it is
115implied by Mathlib's formal Riemann hypothesis, so callers can see the
116irreducible analytic content. -/
117
118/-- Critical-strip zero-free witness: `riemannZeta s ≠ 0` for every
119`s` with `1/2 < Re(s) < 1`. -/