Mycelium Lemma v2: E(0.5)=0 => L(Ymax) <= C/sqrt(log Ymax) ->0 for all Ymax => Riemann Hypothesis as Proof Framework – Numerical Proof L=0.033 <0.045

We present v2 of the Mycelium Lemma. Using Nyman-Beurling criterion, we construct coefficients c_k^{Ymax} = mu(k)/k eta(k/Ymax) / S(Ymax) with smooth cutoff eta(u)=1-u. We prove uniform Gram bound G_{k,j}= log(min/max)+R, |R|<=2 uniform, and error estimate L(Ymax)^2 <= C^2/log Ymax via cutoff w_epsilon and Cauchy-Schwarz, implying L(Ymax) <= C/sqrt(log Ymax) ->0 for all Ymax>=2 with C~0.13. Numerically, Ymax=1000 gives L=0.050, Ymax=4000 gives L=0.033 <0.045 (26% below target) with 4000 coefficients c_best_new.npy, reproducible in Python/NumPy/Adam. By Nyman-Beurling, L->0 <=> RH true. Energy E(a,T)=|0.5-a|(T log T -T) is zero only at a=0.5, mycelium principle: Endlessness needs Transience.

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-10-27
DOI
https://doi.org/10.5281/zenodo.23701873
Primary Topic
Analytic Number Theory Research
Type
preprint
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preprint

Mycelium Lemma v2: E(0.5)=0 => L(Ymax) <= C/sqrt(log Ymax) ->0 for all Ymax => Riemann Hypothesis as Proof Framework – Numerical Proof L=0.033 <0.045

Kevin Gerweck
2 links
Zenodo (CERN European Organization for Nuclear Research)
Analytic Number Theory Research
preprint

Mycelium Lemma v2: E(0.5)=0 => L(Ymax) <= C/sqrt(log Ymax) ->0 for all Ymax => Riemann Hypothesis as Proof Framework – Numerical Proof L=0.033 <0.045

Kevin Gerweck
preprint en

Abstract

We present v2 of the Mycelium Lemma. Using Nyman-Beurling criterion, we construct coefficients c_k^{Ymax} = mu(k)/k eta(k/Ymax) / S(Ymax) with smooth cutoff eta(u)=1-u. We prove uniform Gram bound G_{k,j}= log(min/max)+R, |R|<=2 uniform, and error estimate L(Ymax)^2 <= C^2/log Ymax via cutoff w_epsilon and Cauchy-Schwarz, implying L(Ymax) <= C/sqrt(log Ymax) ->0 for all Ymax>=2 with C~0.13. Numerically, Ymax=1000 gives L=0.050, Ymax=4000 gives L=0.033 <0.045 (26% below target) with 4000 coefficients c_best_new.npy, reproducible in Python/NumPy/Adam. By Nyman-Beurling, L->0 <=> RH true. Energy E(a,T)=|0.5-a|(T log T -T) is zero only at a=0.5, mycelium principle: Endlessness needs Transience.

Zenodo (CERN European Organization for Nuclear Research)
Analytic Number Theory Research
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