Recovery Witnesses for Suzuki's Screw Function: Service-Clock Geometry and Episode-Minimum Certificates — Part 4
Abstract
Suzuki proved that the Riemann hypothesis (RH) is equivalent to the nonnegativity of an explicit continuous function Ψ, the negative of the screw function of the Riemann zeta function. Between consecutive prime powers Ψ is strictly convex, a directed-rounding certificate proves Ψ(t) > 0 for 0 < t ≤ log 10¹⁰, and the infinite tail reduces to one reserve inequality V_q ≥ 0 at every active prime-power event q, one immediately after which Ψ decreases. We show that within each episode of decrease only the last of these inequalities matters. In the service clock τ = A′(t), where A is the smooth part of Ψ, the workload Y = −Ψ′ decreases at unit speed and jumps up by Λ(q)/√q at each prime power. The last active event q before Y returns to zero, the recovery witness, is characterized by the window A′(log q) < ∑_(n≤q) Λ(n)/√n ≤ A′(log Q), with Q the next prime power, and its reserve is the minimum of Ψ over the episode. With the unconditional exclusion of a never-ending episode (Part 5), RH holds if and only if V_q ≥ 0 at every recovery witness that recovers beyond a verified positive base. An exact occupation formula bounds the difference between recovery counts and the normalized time that Y spends just above zero by explicit endpoint counts; strong convexity gives two-sided bounds for the final descent, a certificate for single-event episodes and a prime-gap test for multi-event witnesses. Directed 128-bit arithmetic shows that only 102,896 of the 25,614,164 active events up to 10⁹ are recovery witnesses and, in the complete census of the finite certificate, finds exactly 364,764 strict recoveries through 10¹⁰; ball arithmetic resolves the critical 106-event episode. Proving at every recovery witness that a nonnegative smoothed von Mangoldt sum does not exceed an explicit capacity remains open and is equivalent to RH.
Related
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- Classical plane geometry by coordinate certificates: Desargues, Monge, Simson, Napoleon and Lami in Lean 4
- Compact Weil Event Cells from Suzuki's Screw Function: Parity Polarization, Exact Enclosures, and Uniform Gram Certificates
- Recovery Set Structures and Service Rates of Codes Obtained by the Plotkin-type Construction
- Self-Reflection Resilience-Recovery Activity Promotion Training for Emergency Medical Service Clinicians: A Randomized Controlled Pilot Trial of a Novel Digital Intervention
BibTeX
@misc{mittermeier2026recovery,
title = {{Recovery Witnesses for Suzuki's Screw Function: Service-Clock Geometry and Episode-Minimum Certificates — Part 4}},
author = {Rainer Andreas Mittermeier},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23401342},
url = {https://zenodo.org/records/23401342},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Recovery Witnesses for Suzuki's Screw Function: Service-Clock Geometry and Episode-Minimum Certificates — Part 4
AU - Rainer Andreas Mittermeier
PY - 2026
DA - 2026/10/18
PB - Zenodo
DO - 10.5281/zenodo.23401342
UR - https://zenodo.org/records/23401342
AB - Suzuki proved that the Riemann hypothesis (RH) is equivalent to the nonnegativity of an explicit continuous function Ψ, the negative of the screw function of the Riemann zeta function. Between consecutive prime powers Ψ is strictly convex, a directed-rounding certificate proves Ψ(t) > 0 for 0 < t ≤ log 10¹⁰, and the infinite tail reduces to one reserve inequality V_q ≥ 0 at every active prime-power event q, one immediately after which Ψ decreases. We show that within each episode of decrease only the last of these inequalities matters. In the service clock τ = A′(t), where A is the smooth part of Ψ, the workload Y = −Ψ′ decreases at unit speed and jumps up by Λ(q)/√q at each prime power. The last active event q before Y returns to zero, the recovery witness, is characterized by the window A′(log q) < ∑_(n≤q) Λ(n)/√n ≤ A′(log Q), with Q the next prime power, and its reserve is the minimum of Ψ over the episode. With the unconditional exclusion of a never-ending episode (Part 5), RH holds if and only if V_q ≥ 0 at every recovery witness that recovers beyond a verified positive base. An exact occupation formula bounds the difference between recovery counts and the normalized time that Y spends just above zero by explicit endpoint counts; strong convexity gives two-sided bounds for the final descent, a certificate for single-event episodes and a prime-gap test for multi-event witnesses. Directed 128-bit arithmetic shows that only 102,896 of the 25,614,164 active events up to 10⁹ are recovery witnesses and, in the complete census of the finite certificate, finds exactly 364,764 strict recoveries through 10¹⁰; ball arithmetic resolves the critical 106-event episode. Proving at every recovery witness that a nonnegative smoothed von Mangoldt sum does not exceed an explicit capacity remains open and is equivalent to RH.
ER -