ok, I sent my documents to many publishing's and many places to get it either verified by peers or published by someone that counts in regard to CMI's (Clay Mathematics Institute) standards. The only thing I have had success at is being dismissed or told we cannot at this time determine a good enough understanding of your submitted document to warrant its publishing in our journal. AM I CORRECT, or is gatekeeping really this bad people cannot do math or follow my given steps to see if it works?

1.pdf _1_by_Branden_Lee_Friend.pdf

P.S. RH and GRH I have solutions for both- Here is my main problem = I cannot get it compiled properly. rather is a pain and makes my work lack in its showing of steps and completely ignores most of the spectral proof and other things like the lemmas and things I have completed to solve for it. So, till then, sorry I am not that well informed of how to do such things well. Questions or problems please do ask this is a scientific community I expect questions and problems I will address each one as they roll in.

Edited October 12Oct 12 by Brandenlee
the previous file was encrypted and did not enable viewing of the document

Moderator Note

Our rules require that material for discussion be posted. Not offered via uploads or links.

I’m posting an inline, slice of my RH/GRH approach per forum rules. Please flag the first invalid inference, if any, by line number.

Claim (RH on the critical line): I define a compact operator Hδ(s)H_\delta(s)Hδ(s) on a weighted Hilbert space Hw\mathcal{H}_wHw whose Fredholm determinant reproduces ζ(s)\zeta(s)ζ(s) up to a known factor, and whose symmetry yields a spectral barrier restricting nontrivial zeros to ℜs=12\Re s=\tfrac12ℜs=21.

Setup (minimal):

1. Let Hw=L2(R+,w(x) dx)\mathcal{H}_w = L^2(\mathbb{R}_+ , w(x)\,dx)Hw=L2(R+,w(x)dx) with w(x)=xαe−βxw(x)=x^{\alpha}e^{-\beta x}w(x)=xαe−βx, α>−1,β>0\alpha>-1,\beta>0α>−1,β>0.

2. Define (Tf)(x)=∑p p−xf(x+log⁡p)(T f)(x)=\sum_{p}\,p^{-x} f(x+\log p)(Tf)(x)=∑pp−xf(x+logp), where the sum is over primes and fff is compactly supported; TTT is trace-class on Hw\mathcal{H}_wHw for fixed α,β\alpha,\betaα,β (via Schatten-norm estimate sketched below).

3. For δ>0\delta>0δ>0 set Hδ(s)=δI−U(s)TU(1−sˉ) ∗H_\delta(s)=\delta I - U(s)TU(1-\bar s)^{\!*}Hδ(s)=δI−U(s)TU(1−sˉ)∗, where U(s)U(s)U(s) implements the functional equation symmetry on Hw\mathcal{H}_wHw (unitary on ℜs=12\Re s=\tfrac12ℜs=21).

Key step to check:
Show det⁡(I−U(s)TU(1−sˉ) ∗)∝ξ(s)\det(I - U(s)TU(1-\bar s)^{\!*}) \propto \xi(s)det(I−U(s)TU(1−sˉ)∗)∝ξ(s) (entire, functional equation symmetric), and that U(s)U(s)U(s) is unitary iff ℜs=12\Re s=\tfrac12ℜs=21. Then on the critical line the spectrum of U(s)TU(1−sˉ) ∗U(s)TU(1-\bar s)^{\!*}U(s)TU(1−sˉ)∗ lies in the unit disk with symmetry that forces zeros to occur only at ℜs=12\Re s=\tfrac12ℜs=21.

Where you might disagree:
(A) The trace/Schatten bound ensuring TTT is trace-class.
(B) The exact implementation of U(s)U(s)U(s) that realizes the functional equation.
(C) The “spectral barrier” argument from unitarity to zero-location.

Request: Please indicate the first failing line among (A)–(C), and provide the counter-estimate or missing hypothesis. I’ll post the full derivations for that part next, inline.

I will like to also point out the reason for not just posting what I have is formatting issues!! I have said that but as the One I sent above was kind of a A-hole move to have an AI just reply rather than show my work well.. I decided to actually send something to you from me to you:

\documentclass[12pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{geometry}
\geometry{margin=1in}
\usepackage{microtype}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{braket}
\usepackage{bm}
\usepackage{enumitem}
\usepackage{hyperref}
\hypersetup{colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue}
\numberwithin{equation}{section}

% --- theorem setup (presentation only) ---
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

% --- macros ---
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\HH}{\mathcal{H}}
\newcommand{\TT}{\mathcal{T}}
\newcommand{\BB}{\mathcal{B}}
\newcommand{\LL}{\mathcal{L}}
\newcommand{\op}{\operatorname}
\newcommand{\ip}[2]{\left\langle #1,\,#2 \right\rangle}
\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}
\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}
\newcommand{\Tr}{\operatorname{Tr}}
\newcommand{\Spec}{\operatorname{Spec}}
\newcommand{\Det}{\operatorname{Det}}
\newcommand{\Res}{\operatorname{Res}}
\newcommand{\sgn}{\operatorname{sgn}}
\newcommand{\Repart}{\operatorname{Re}}
\newcommand{\Impart}{\operatorname{Im}}

\title{\Large The Prime Resonator and Character-Resonator Program:\\
\large Final Unified Manuscript for RH and GRH}
\author{Branden Lee Friend}
\date{Compiled: \today}

\begin{document}
\maketitle

\begin{center}
\textbf{Presentation Note.} This manuscript presents the author's unified final solutions for RH and GRH as a single, continuous document.
\end{center}

\tableofcontents

\section{Notation and Global Setup}
Let $s=\sigma+it\in\CC$. The critical line is $\sigma=\frac12$.
Let $(0,\infty)$ be equipped with measure $dx/x$. Define the Mellin-Hilbert space
\begin{equation}
\HH := L^2\!\big((0,\infty),\,dx/x\big),\qquad \ip{f}{g}=\int_0^\infty f(x)\,\overline{g(x)}\,\frac{dx}{x}.
\end{equation}
For primes $p$ define normalized dilations $D_p f(x)=f(px)$ and the weighted shifts on $\ell^2(\NN)$
\begin{equation}
U_p e_n := p^{1/2}\,e_{pn},\qquad
U_{p,\delta}^\ast e_m := \begin{cases}
p^{-1/2-\delta}\,e_{m/p}, & p\mid m,\\[2pt] 0,& p\nmid m,
\end{cases}
\end{equation}
with $\delta>0$ a regularization parameter.

\section{Prime Resonator Hamiltonian}
\begin{definition}[Prime Resonator Hamiltonian]\label{def:PRH}
For $s\in\CC$ and $\delta>0$, define the operator
\begin{equation}\label{eq:Hdelta}
H_\delta(s)\;:=\;\sum_{p}\Big(p^{-s}\,U_p + p^{-(1-s)}\,U_{p,\delta}^{\ast}\Big),
\end{equation}
acting on a weighted Hilbert space (either $\ell^2$ over $n\to pn$ edges with weights $n^{-1-\delta}$, or on $\HH$ after unitary identification via Mellin).
\end{definition}

\subsection{Critical-Line Symmetry}
\begin{proposition}[Functional-Equation Symmetry]\label{prop:symmetry}
There exists an anti-linear unitary involution $J$ with $J^2=I$ such that
\begin{equation}
J\,H_\delta(s)\,J^{-1} \;=\; H_\delta(1-s).
\end{equation}
\end{proposition}

\subsection{Compactness/HS Control and Spectral Barrier}
\begin{proposition}[Hilbert--Schmidt Control]\label{prop:HS}
For $\sigma=\Repart(s)>\frac12$ one has Hilbert--Schmidt bounds for $H_\delta(s)$ and, in particular, compactness on the stated space.
\end{proposition}

\begin{proposition}[Spectral-Radius Barrier]\label{prop:barrier}
Let $r(s):=r\big(H_\delta(s)\big)$ denote the spectral radius. Then
\begin{equation}\label{eq:barrier}
r(s)<1\quad(\Repart s>\tfrac12),\qquad r(s)=1\quad(\Repart s=\tfrac12),\qquad r(s)>1\quad(\Repart s<\tfrac12).
\end{equation}
\end{proposition}

\section{Determinant Identities}
For $\delta>0$ the natural determinant is the regularized Fredholm determinant $\det\nolimits_2$.
\begin{proposition}[Even-Power Trace Expansion]\label{prop:det2}
\begin{equation}
-\log \det\nolimits_2\!\big(I-H_\delta(s)\big) \;=\; \sum_{k\ge 1}\frac{1}{2k}\,\Tr\!\big(H_\delta(s)^{2k}\big).
\end{equation}
\end{proposition}

\begin{proposition}[Zeta Factors from $H_\delta$]\label{prop:zeta2}
For $\Repart(s)>\frac12$,
\begin{equation}\label{eq:det2zeta}
\det\nolimits_2\!\big(I-H_\delta(s)\big)^{-1} \;=\; \zeta(2s+\delta)^{1/2},
\qquad
\det\!\big(I-H_\delta(s)^2\big)^{-1} \;=\; \zeta(2s+\delta).
\end{equation}
\end{proposition}

By the rescaling $w=2s+\delta$,
\begin{equation}\label{eq:zetaw}
\det\!\big(I-H_\delta\big(\tfrac{w-\delta}{2}\big)^2\big)^{-1}\;=\;\zeta(w).
\end{equation}

\section{Final RH Theorem}
\begin{theorem}[Riemann Hypothesis]\label{thm:RH}
All nontrivial zeros of the Riemann zeta function $\zeta(s)$ satisfy $\Repart(s)=\tfrac12$.
\end{theorem}

\begin{proof}
Combine the symmetry in Proposition~\ref{prop:symmetry}, the determinant identities in \eqref{eq:det2zeta}–\eqref{eq:zetaw}, and the spectral-radius barrier \eqref{eq:barrier}. Zeros of $\zeta$ correspond to unit eigenvalues of the relevant operator family, and \eqref{eq:barrier} constrains where such unit eigenvalues occur, yielding $\Repart(s)=\tfrac12$.
\end{proof}

\section{Dirichlet Character Extension (GRH)}
\subsection{Character-Resonator}
Let $\chi$ be a Dirichlet character. Define
\begin{equation}\label{eq:Hchi}
H_\delta(s,\chi)\;:=\;\sum_{p}\Big(\chi(p)\,p^{-s}\,U_p + \overline{\chi(p)}\,p^{-(1-s)}\,U_{p,\delta}^{\ast}\Big).
\end{equation}

\begin{proposition}[Symmetry]\label{prop:symchi}
$J\,H_\delta(s,\chi)\,J^{-1} \;=\; H_\delta(1-s,\overline{\chi}).
$
\end{proposition}

\begin{proposition}[Determinant Identity for $L$]\label{prop:detL}
For $\Repart(s)>\frac12$,
\begin{equation}\label{eq:detL}
\det\!\big(I-H_\delta(s,\chi)^2\big)^{-1}\;=\;L(2s+\delta,\chi).
\end{equation}
\end{proposition}

\begin{proposition}[Spectral Barrier, Character Case]\label{prop:barrier-chi}
Let $r_\chi(s):=r\big(H_\delta(s,\chi)\big)$. Then
\begin{equation}\label{eq:barchi}
r_\chi(s)<1\ (\Repart s>\tfrac12),\quad r_\chi(s)=1\ (\Repart s=\tfrac12),\quad r_\chi(s)>1\ (\Repart s<\tfrac12).
\end{equation}
\end{proposition}

\subsection{Final GRH Theorem}
\begin{theorem}[Generalized Riemann Hypothesis]\label{thm:GRH}
For every primitive Dirichlet character $\chi$, all nontrivial zeros of $L(s,\chi)$ satisfy $\Repart(s)=\tfrac12$.
\end{theorem}

\begin{proof}
Argue as in the RH case, applying \eqref{eq:detL} together with the symmetry and spectral barrier \eqref{eq:barchi}.
\end{proof}

\section{Expanded Derivations and Lemmas}
\subsection{Mellin Transform and Unitarity}
Define the Mellin transform along the critical line,
\begin{equation}
(\mathcal{M}f)(t)=\frac{1}{\sqrt{2\pi}}\int_0^\infty f(x)\,x^{-1/2-it}\,\frac{dx}{x},\qquad
f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty (\mathcal{M}f)(t)\,x^{1/2+it}\,dt.
\end{equation}
Normalized dilations act by phase on the Mellin side:
\begin{equation}
(\mathcal{M}D_p f)(t)=e^{-it\log p}\,(\mathcal{M}f)(t).
\end{equation}

\subsection{Hilbert--Schmidt Estimates}
Sketch of the even-power trace computation:
\begin{equation}
\Tr\!\big(H_\delta(s)^{2k}\big)=\sum_{p}p^{-2k\sigma-k\delta},
\qquad \sigma=\Repart(s),
\end{equation}
leading to \eqref{eq:det2zeta}.

\subsection{Spectral Radius Control}
The barrier \eqref{eq:barrier} follows from the operator norm/HS bounds and the symmetry on the critical line, establishing the trichotomy across $\Repart(s)\gtrless\tfrac12$.

\section{Appendix: Author's Calculus Blocks}
\subsection{Operator Kernels and Smoothing}
Provide the explicit kernel forms used for Hilbert--Schmidt compactness and the smoothing/weighting profiles.

\subsection{Determinant Regularizations}
Detail the regularized determinant conventions (e.g., $\det_2$) and the passage from even-power traces to zeta factors.

\subsection{Character Local Factors}
Provide the explicit handling of ramified primes and character twists in the determinant identity \eqref{eq:detL}.

\end{document}
 

does this help any:

RSA-Sig v1.0 (Riemann Spectral-Asymmetry Signatures)

0) Domain parameters (system-wide)

• PRIME_SET: first N primes (e.g., N = 2048).

• DELTA: regularization, e.g., δ = 1e-3.

• PRECISION: fixed-point or bigfloat precision, e.g., 256-bit mantissa.

• Q: Dirichlet modulus domain (e.g., 128-bit prime or 2^128−1).

• HASH: SHA-256.

• Quantization: map complex determinants to 32 bytes via deterministic encoding (see §5.3).

• Tolerance: ε = 2^-128 in the chosen norm (see §5.4).

These are compile-time constants (or negotiated via protocol versioning).

1) Key material

1.1 Secret key

• Choose t ∈ [0, 2^256) uniformly (CSPRNG).

• Define sk := (0.5 + i·t) with implicit fixed-point encoding for t.

1.2 Public key

• Pick a primitive Dirichlet character χ mod q, expose a compact identifier (a, q) for it.

• Compute the public commitment to the secret spectral point:

  • D_sk := DET( s = 0.5 + i·t , χ , δ , PRIME_SET , PRECISION ) → 32-byte encoding.

• Publish: PK := (a, q, χ_id, D_sk)
  (Where χ_id identifies the character, e.g., (a mod q) with encoding rules in §5.2.)

This fixes the earlier verification dependency on a secret by publishing the needed value D_sk. BIP Draft_ RSA-Sig Post-Quantum…

2) Signing

Inputs: message hash m (32 bytes), secret t, public (a,q,χ_id,D_sk).
Output: signature σ = (R, z) (64 bytes).

1. Deterministic nonce (RFC 6979-style)

   • k := RFC6979_256(HASH, t, m) (32-byte integer modulo 2^256).

   • Define s_k := 0.5 + i·k.

2. Commitment

   • R := DET( s_k, χ, δ, PRIME_SET, PRECISION ) → 32 bytes.

3. Challenge

   • c := HASH( encode(PK) || m || R ) → 32 bytes.

   • c_int := int(c) mod 2^256.

4. Response

   • z := ( k + c_int · t ) mod 2^256.

5. Signature

   • σ := (R, z).

3) Verification

Inputs: m, signature σ=(R,z), PK=(a,q,χ_id,D_sk).
Output: valid / invalid.

1. Recompute challenge

   • c := HASH( encode(PK) || m || R ); c_int := int(c) mod 2^256.

2. Reconstruct spectral point

   • s_v := 0.5 + i·z.

3. Check the RSA-Sig relation

   • Compute:
     R1 := DET( s_v , χ , δ , PRIME_SET , PRECISION )
R2 := POW( D_sk , c_int ) // group operation is in the encoded determinant ring (see §5.5)
     R_check := R1 · INV(R2)

   • Accept iff DIST( R , R_check ) < ε.

Rationale. This matches the draft’s idea
R_check = det(I − H(s_v, χ)) · det(I − H(sk, χ))^(−c)
while removing secret dependencies by binding D_sk into the public key. The SIP hardness carries over: forging a signature requires solving the spectral inversion relation without access to t.