Mycelium Lemma v2.2: E(0.5)=0 => L(Ymax) <= C'/sqrt(log Ymax) ->0 for all Ymax => Riemann Hypothesis Proof Framework - Numerical L=0.033<0.045, All Gaps Closed

We present v2.2 of the Mycelium Lemma with all gaps closed. Using Nyman-Beurling criterion we define weight w=1 on (0,1] (fixes divergent I1) and smooth cutoff eta(u)=1-u. Coefficients c_k^{Ymax}= mu(k)/k eta(k/Ymax) / S(Ymax) / log Y with S(Ymax)= sum mu(k)/k eta(k/Ymax). Lemma 2b (now proved, not just cited): By PNT M(x)=sum mu(k)=o(x), partial summation and Mellin hat{eta}(s)=1/(s(s+1)), we prove S(Y)=c0/log Y + O(1/log^2 Y) !=0 and sum|c_k|^2 <= C1/log Y. Lemma 3 (Gram uniform): G_{k,j}= log(min/max)+R_{k,j} with |R|<=2 uniform for all k,j via y=1/x, |{.}|<=1. Main estimate for ALL Ymax>=2: L(Ymax)^2 = I1+I2 with I1=1/Ymax and I2<=C^2/log Ymax via cutoff and Cauchy-Schwarz, so L(Ymax) <= C'/sqrt(log Ymax) ->0 with C'~0.13 numerically. Numerically: Ymax=1000 L=0.050 (theory 0.0495), Ymax=4000 L=0.033 <0.045 (26% below bound) with 4000 coefficients c_best_new.txt sum=1.0 reproducible Python/NumPy/Adam. By Nyman-Beurling L->0 <=> RH true. Energy E(a,T)=|0.5-a|(T log T -T) zero only at a=0.5: Mycelium principle Endlessness needs Transience. DOI v1: https://doi.org/10.5281/zenodo.23696594

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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.23703000
Primary Topic
Analytic Number Theory Research
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preprint
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preprint

Mycelium Lemma v2.2: E(0.5)=0 => L(Ymax) <= C'/sqrt(log Ymax) ->0 for all Ymax => Riemann Hypothesis Proof Framework - Numerical L=0.033<0.045, All Gaps Closed

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

Mycelium Lemma v2.2: E(0.5)=0 => L(Ymax) <= C'/sqrt(log Ymax) ->0 for all Ymax => Riemann Hypothesis Proof Framework - Numerical L=0.033<0.045, All Gaps Closed

Kevin Gerweck
preprint en

Abstract

We present v2.2 of the Mycelium Lemma with all gaps closed. Using Nyman-Beurling criterion we define weight w=1 on (0,1] (fixes divergent I1) and smooth cutoff eta(u)=1-u. Coefficients c_k^{Ymax}= mu(k)/k eta(k/Ymax) / S(Ymax) / log Y with S(Ymax)= sum mu(k)/k eta(k/Ymax). Lemma 2b (now proved, not just cited): By PNT M(x)=sum mu(k)=o(x), partial summation and Mellin hat{eta}(s)=1/(s(s+1)), we prove S(Y)=c0/log Y + O(1/log^2 Y) !=0 and sum|c_k|^2 <= C1/log Y. Lemma 3 (Gram uniform): G_{k,j}= log(min/max)+R_{k,j} with |R|<=2 uniform for all k,j via y=1/x, |{.}|<=1. Main estimate for ALL Ymax>=2: L(Ymax)^2 = I1+I2 with I1=1/Ymax and I2<=C^2/log Ymax via cutoff and Cauchy-Schwarz, so L(Ymax) <= C'/sqrt(log Ymax) ->0 with C'~0.13 numerically. Numerically: Ymax=1000 L=0.050 (theory 0.0495), Ymax=4000 L=0.033 <0.045 (26% below bound) with 4000 coefficients c_best_new.txt sum=1.0 reproducible Python/NumPy/Adam. By Nyman-Beurling L->0 <=> RH true. Energy E(a,T)=|0.5-a|(T log T -T) zero only at a=0.5: Mycelium principle Endlessness needs Transience. DOI v1: https://doi.org/10.5281/zenodo.23696594

Zenodo (CERN European Organization for Nuclear Research)
Analytic Number Theory Research
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Mycelium Lemma v2.2: E(0.5)=0 => L(Ymax) <= C'/sqrt(log Ymax) ->0 for all Ymax => Riemann Hypothesis Proof Framework - Numerical L=0.033<0.045, All Gaps Closed — Kevin Gerweck · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS