%% \documentclass[preprint,review,12pt]{elsarticle} %% Use the options 1p,twocolumn; 3p; 3p,twocolumn; 5p; or 5p,twocolumn %% for a journal layout. \documentclass[final,3p,times]{elsarticle} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{url} \usepackage{hyperref} \hypersetup{colorlinks=true,allcolors=blue} \usepackage{amssymb} \usepackage{amsthm} \usepackage{amsmath} \usepackage{mathtools} \usepackage{caption} \usepackage{microtype} \usepackage{enumitem} \usepackage{xcolor} \usepackage{float} \usepackage{graphicx} \usepackage{tikz} \usetikzlibrary{arrows.meta,positioning,calc} \theoremstyle{plain} \newtheorem{Theorem}{Theorem}[section] \newtheorem{Lemma}[Theorem]{Lemma} \newtheorem{Proposition}[Theorem]{Proposition} \newtheorem{Corollary}[Theorem]{Corollary} \newtheorem{Hypothesis}[Theorem]{Hypothesis} \theoremstyle{definition} \newtheorem{Definition}[Theorem]{Definition} \newtheorem{Remark}[Theorem]{Remark} \journal{Hexagon} \begin{document} \begin{frontmatter} \title{Nested Squares and Distinct-Prime Goldbach Partitions} \author{Frank Vega} \address{Information Physics Institute, 840 W 67th St, Hialeah, FL 33012, USA} \ead{vega.frank@gmail.com} \begin{abstract} The distinct-prime Goldbach conjecture asserts that for every $N\geq 4$ there is $M\in A_N=[1,N-3]$ with $N-M$ and $N+M$ both prime; geometrically, nested squares of side lengths $N$ and $M$ have an L-shaped difference of semiprime area $(N-M)(N+M)$. With $C_N$ and $D_N$ the sets of $M\in A_N$ for which $N-M$, respectively $N+M$, is prime, the conjecture for $2N$ is $C_N\cap D_N\neq\varnothing$. We embed this centered condition in a straddling one: $C_N\cap D_N$ lies in the set $E_N$ of half-differences $(Q-P)/2\in A_N$ of primes $20$, and with explicit constants we prove $|E_N|>N/760$ for every $N\geq 3275$. Consequently, for all large $N$ and all but $O_D(N/(\log N)^D)$ primes $P0$ (Proposition~\ref{prop:almost-all}), which we deduce from Mikawa's theorem on the Hardy--Littlewood asymptotic for prime pairs with almost all differences~\cite{mikawa1991prime}. With explicit constants, $|E_N|>N/760$ for every $N\geq 3275$ (Theorem~\ref{thm:linear}); the proof combines the Cauchy--Schwarz inequality, an exact identity for the additive energy of the primes below and above $N$ (Lemma~\ref{lem:energy}), Siebert's explicit sieve bound~\cite{siebert1976montgomery}, and the estimates of Dusart~\cite{dusart1998autour} and of Rosser and Schoenfeld~\cite{rosser1962approximate}. \item[\emph{(2)}] \emph{Goldbach partitions up to a difference of two primes.} For all large $N$, $|C_N\cap E_N|=\pi(N-1)-1-O_D(N/(\log N)^D)$; equivalently, for all but $O_D(N/(\log N)^D)$ primes $3\leq P0$, where $H(N)=\log^2(2N)-|A_N\setminus E_N|$ (Proposition~\ref{prop:pigeonhole-CE}). \item[\emph{(3)}] \emph{The centering step.} In contrast with $E_N$, $|C_N\cap D_N|<(N-3)-\log^2(2N)$ for every $N\geq 34$ (Proposition~\ref{prop:HR-negative}). The prime gap around $N$ and the prime gap below $2N$ force missing values of $E_N$ (Lemmas~\ref{lem:gap} and~\ref{lem:gap-top}); consequently $H(N)>0$ for all large $N$ would imply $p'-p<2\log^2(2p+2)+2$ for consecutive primes $p0$, $|E_N|\geq 0.9906\,N$ for $N\geq 3275$, the ratios of $R(2N)$ to the Hardy--Littlewood prediction, and the share of missing values of $E_N$ forced by prime gaps, which decreases to $0.2698$ on $(2^{19},2^{20}]$ (Section~\ref{sec:computational}). \end{enumerate} The symmetric reformulation~\cite{oeisA082467}, its geometric form (Section~\ref{sec:geometric}), and the verified range (Proposition~\ref{prop:verified}, which combines the verification of Oliveira e Silva, Herzog and Pardi~\cite{oliveira2014empirical} with our computation) are the setting of these results. By Proposition~\ref{prop:verified}, the distinct-prime Goldbach conjecture is equivalent to $C_N\cap D_N\neq\varnothing$ for every $N>2\cdot 10^{18}$, and it is not proved here. The remainder of this paper is organised as follows. Section~\ref{sec:prelim} collects the analytic number theory prerequisites used in the quantitative discussion. Section~\ref{sec:geometric} presents the geometric setting and the set $E_N$, and proves Theorems~\ref{thm:linear} and~\ref{thm:positive-proportion}. Section~\ref{sec:computational} presents computational evidence. Section~\ref{sec:main} records the rigorous consequences, the open centered problem, and the small cases. Section~\ref{sec:conclusion} summarises the results and lists open questions. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Preliminaries: Prime Distribution Results}\label{sec:prelim} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this section we collect the results from analytic number theory that are used in the quantitative discussion. We use explicit estimates of Dusart~\cite{dusart1998autour} and Rosser--Schoenfeld~\cite{rosser1962approximate}. \subsection{Primes in Short Intervals} The following result guarantees the existence of at least one prime in every sufficiently short interval. It is the principal tool for controlling the density of primes in $(N,2N)$. \begin{Proposition}[{\cite[Th\'eor\`eme~1.9, p.\,35]{dusart1998autour}}] \label{prop:dusart-short} For every real number $x \geq 3275$, there exists a prime $p$ satisfying \[ x < p \leq x\!\left(1 + \frac{1}{2\ln^2 x}\right). \] \end{Proposition} Proposition~\ref{prop:dusart-short} is derived from the following bound on consecutive primes. \begin{Proposition}[{\cite[Proposition~1.10, p.\,34]{dusart1998autour}}] \label{prop:dusart-consec} For $k \geq 463$ (equivalently, $p_k \geq 3299$), the consecutive primes $p_k$ and $p_{k+1}$ satisfy \[ p_{k+1} \leq p_k\!\left(1 + \frac{1}{2\ln^2 p_k}\right). \] \end{Proposition} The proof of Proposition~\ref{prop:dusart-short} in~\cite{dusart1998autour} proceeds by using Proposition~\ref{prop:dusart-consec} for all $x \geq p_{463} = 3299$ and then verifying the claim computationally for $3275 \leq x < 3299$. \subsection{Bounds on the Prime-Counting Function} We use the following bounds of Dusart~\cite{dusart1998autour} and of Rosser and Schoenfeld~\cite{rosser1962approximate} for $\pi(x)$, the number of primes not exceeding $x$. \begin{Proposition}[{\cite[Th\'eor\`eme~1.10, p.\,36]{dusart1998autour}}] \label{prop:dusart-upper} The following inequalities hold: \begin{enumerate} \item[\emph{(1)}] $\displaystyle \frac{x}{\ln x}\left(1 + \frac{1}{\ln x}\right) \leq \pi(x)$ \quad for all $x \geq 599$. \item[\emph{(2)}] $\displaystyle \pi(x) \leq \frac{x}{\ln x}\left(1 + \frac{1.2762}{\ln x}\right)$ \quad for all $x > 1$. \end{enumerate} \end{Proposition} \begin{Proposition}[{\cite[Corollary~1, (3.5)]{rosser1962approximate}}] \label{prop:rs-pi} For every real number $x\geq 17$, \[ \pi(x)>\frac{x}{\ln x}. \] \end{Proposition} Throughout this paper, $\log$ denotes the natural logarithm (i.e., $\log \equiv \ln$). \subsection{Primes in Dyadic Intervals} We shall also use the following classical explicit form of Bertrand's postulate due to Rosser and Schoenfeld~\cite{rosser1962approximate}. \begin{Proposition}[{\cite[Corollaries~3.8--3.9]{rosser1962approximate}}]\label{prop:rs-dyadic} For every real number $x\geq 41/2$, \[ \frac{3x}{5\log x}<\pi(2x)-\pi(x)<\frac{7x}{5\log x}. \] \end{Proposition} \subsection{An Elementary Inequality} \begin{Lemma}\label{lem:elementary} For every real number $x\geq 356$, \[ \frac{x}{\log x+2}-2>\log^2(2x). \] \end{Lemma} \begin{proof} Let $f(x)=\frac{x}{\log x+2}-2-\log^2(2x)$ for $x>1$, and write $L=\log x$. Then \[ f'(x)=\frac{L+1}{(L+2)^2}-\frac{2(L+\log 2)}{x}, \] so $f'(x)>0$ if and only if $x(L+1)>2(L+\log 2)(L+2)^2$. Since $\log 2<1$, we have $2(L+\log 2)(L+2)^2<2(L+1)(L+2)^2$, and therefore $f'(x)>0$ whenever $x>2(L+2)^2$, that is, whenever \[ h(L):=L-\log 2-2\log(L+2)>0. \] For $L>0$ we have $h'(L)=1-\frac{2}{L+2}>0$, so $h$ is increasing on $(0,\infty)$. At $L=\log 356$ we have $2(L+2)^2<124.1<356=e^{L}$, so $h(\log 356)>0$. Hence $h(L)>0$ for every $L\geq \log 356$, and $f$ is increasing on $[356,\infty)$. Finally, \[ \frac{356}{\log 356+2}-2>43.206 \quad\text{and}\quad \log^2(712)<43.140, \] so $f(356)>0$, and therefore $f(x)>0$ for every $x\geq 356$. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Geometric Construction and Reformulation}\label{sec:geometric} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We now develop the geometric framework used in the reformulation. \subsection{Nested Squares and Semiprime Areas} Consider a square $S_N$ with integer side length $N \geq 4$, having area $N^2$. Inside $S_N$, inscribe a smaller square $S_M$ with side length $M$, where $1 \leq M \leq N-3$, sharing the bottom-left corner with $S_N$. The region between $S_N$ and $S_M$ forms an L-shaped annulus with area \[ N^2 - M^2 = (N - M)(N + M). \] Define $P = N - M$ and $Q = N + M$. The bounds on $M$ translate to constraints on $P$ and $Q$: $M \geq 1$ gives $P \leq N - 1$ and $Q \geq N + 1$, while $M \leq N - 3$ gives $P \geq 3$. Thus $3 \leq P \leq N-1$ and $Q \geq N+1$, with $P < Q$ since $M \geq 1$. \subsection{Connection to Goldbach Partitions} The sum and difference of $P$ and $Q$ are: \begin{align*} P + Q &= (N - M) + (N + M) = 2N \geq 8, \\ Q - P &= (N + M) - (N - M) = 2M. \end{align*} Since both the sum and difference are even, $P$ and $Q$ have the same parity. For both to be prime with $P \geq 3$, they must both be odd primes, hence distinct. The area $N^2 - M^2 = P \cdot Q$ is a semiprime (product of exactly two primes) if and only if both $P$ and $Q$ are prime. \subsection{The Geometric Equivalence} \begin{Theorem}[Geometric Goldbach Variant]\label{thm:geometric} The following are equivalent for all $N \geq 4$: \begin{enumerate} \item[\emph{(i)}] The even integer $2N$ can be written as the sum of two distinct primes. \item[\emph{(ii)}] There exists $M \in [1, N-3]$ such that $P = N - M$ and $Q = N + M$ are both prime. \item[\emph{(iii)}] The L-shaped region between squares $S_N$ and $S_M$ (sharing a corner) has area $(N-M)(N+M)$ that is a semiprime for some $M \in [1, N-3]$. \end{enumerate} \end{Theorem} \begin{proof} \textbf{(i) $\Rightarrow$ (ii):} If $2N = p + q$ with distinct primes $p < q$, set $M = (q-p)/2$. Since $p$ and $q$ are distinct odd primes (as $2N \geq 8$), both $(q+p)/2 = N$ and $(q-p)/2 = M$ are positive integers. We have $P = N - M = p$ and $Q = N + M = q$, both prime. To verify $M \in [1, N-3]$: distinctness gives $q - p \geq 2$, so $M \geq 1$; and $M \leq N - 3$ is equivalent to $p \geq 3$, which holds since $p$ is an odd prime. \textbf{(ii) $\Rightarrow$ (iii):} Immediate, as $N^2 - M^2 = (N-M)(N+M)$ with both factors prime. \textbf{(iii) $\Rightarrow$ (i):} If $N^2 - M^2=(N-M)(N+M)$ is a semiprime and $M \in [1, N-3]$, then both factors $N-M$ and $N+M$ are greater than $1$. Hence the two displayed factors must be prime, and their sum is $2N$. Since $M\geq 1$, they are distinct. \end{proof} \begin{figure}[H] \centering \begin{tikzpicture}[ scale=1.5, every node/.style={font=\small}, >=stealth ] \def\N{5} \def\M{2} \def\P{3} \def\Q{7} % Outer square \draw[thick, black] (0,0) rectangle (\N, \N); \node[below left] at (0,0) {$O$}; % Inner square \draw[thick, black] (0,0) rectangle (\M, \M); % Dashed region \draw[dashed, thick, black] (\M, \M) rectangle (\N, \N); % Shade L-region \fill[gray!30, opacity=0.7] (\M,0) rectangle (\N,\N); \fill[gray!30, opacity=0.7] (0,\M) rectangle (\M,\N); % Labels \node[above] at (\N/2, \N) {$N$}; \node[above] at (\M/2, \M) {$M$}; \node[right] at (\N, \N/2) {$N$}; \node[right] at (\M, \M/2) {$M$}; % P and Q \node[below right, red] at (\N/2 + \M/2, -0.2) {$P = N - M = 3$}; \draw[red, thick, ->] (\M,0) -- (\N,0); \node[above left, blue] at (-0.2, \N/2 + \M/2) {$Q = N + M = 7$}; \draw[blue, thick, ->] (0,\M) -- (0,\N); % Area \node[black] at (\N/2 + \M/2, \N/2 + \M/2) {$N^2 - M^2 = 21 = 3 \times 7$}; \node[below, gray, font=\footnotesize] at (\N/2, -0.8) {Example: $N=5$, $M=2$ gives $2N = 10 = 3 + 7$}; \end{tikzpicture} \caption{The geometric construction for $N=5$, $M=2$: The L-shaped region has area $N^2 - M^2 = 25 - 4 = 21 = 3 \times 7$, a semiprime. The factors $P=3$ and $Q=7$ are both prime and sum to $2N=10$, providing the Goldbach partition $10 = 3 + 7$.} \label{fig:geometric} \end{figure} \subsection{Reformulation as a Centered Set Intersection Problem}\label{subsec:setintersection} Let \[ A_N=\{1,2,\ldots,N-3\} \] be the admissible range for $M$. For each $N \geq 4$, define two centered subsets of $A_N$: \begin{itemize} \item \textbf{Lower-prime candidate set:} $C_N = \{M\in A_N \mid N-M \text{ is prime}\}$. \item \textbf{Upper-prime candidate set:} $D_N = \{M\in A_N \mid N+M \text{ is prime}\}$. \end{itemize} Both centered sets lie in a single parity class. Indeed, if $N-M$ or $N+M$ is an odd prime, then $M$ has parity opposite to $N$. Thus \[ C_N,D_N\subseteq B_N:=\{M\in A_N\mid M\not\equiv N\pmod 2\}. \] \begin{Proposition}[Centered intersection criterion]\label{prop:intersection} The distinct-prime Goldbach conjecture holds for the even integer $2N$ ($N \geq 4$) if and only if $C_N \cap D_N \neq \emptyset$. \end{Proposition} \begin{proof} \textbf{($\Rightarrow$)} Suppose $2N = p + q$ with distinct primes $p < q$. Set $M = (q-p)/2$. As in the proof of Theorem~\ref{thm:geometric}, $M \in A_N$, $p=N-M$, and $q=N+M$. Hence $M\in C_N\cap D_N$. \textbf{($\Leftarrow$)} Suppose $M \in C_N \cap D_N$. Then $N-M$ is prime by the definition of $C_N$, and $N+M$ is prime by the definition of $D_N$. Their sum is $2N$, and they are distinct because $M\geq 1$. \end{proof} \subsection{The Half-Difference Diagnostic Set}\label{subsec:diagnostic} It is useful computationally to compare the centered sets with a set of half-differences arising from straddling prime pairs. Define \[ E_N = \left\{ \frac{Q-P}{2}\in A_N \;\middle|\; 22$, $P(q-3)/2$ lies in $A_N\setminus E_N$; if $q\leq 2N-3$, there are $(2N-3-q)/2$ such values. \end{Lemma} \begin{proof} If $M=(Q-P)/2$ with primes $2(q-3)/2$ are $M=(q-3)/2+1,\ldots,N-3$, and there are $N-3-(q-3)/2=(2N-3-q)/2$ of them. \end{proof} \subsection{Nearby Even Numbers Represented by $E_N$}\label{subsec:nearby-even} Each element of $E_N$ arises from a Goldbach partition of a nearby even number. If $M\in E_N$, then there exist odd primes $P,Q$ such that \[ 2Q_2>\cdots>Q_r\}. \] If $2N-3$ is prime, then $2N=3+(2N-3)$ is already a centered Goldbach partition with $M=N-3$. For estimates of $r$, the only possible prime counted by $\pi(2N)-\pi(N)$ but not by $\pi(2N-3)-\pi(N)$ is $2N-1$. Hence \[ \pi(2N)-\pi(N)-1\leq r\leq \pi(2N)-\pi(N). \] Proposition~\ref{prop:rs-dyadic} therefore gives, for $N\geq 41/2$, \[ \frac{3N}{5\log N}-10$ define \[ \rho_L(d)=\#\{(P,P')\in L_N^2\mid P'-P=d\}, \qquad \rho_U(d)=\#\{(i,j)\mid i0}\rho_L(d)\rho_U(d), \] which counts the quadruples $(P,P',Q_i,Q_j)$ with $P,P'\in L_N$, $i2}\left(1-\frac{1}{(p-1)^2}\right),\qquad \kappa(h)=\prod_{\substack{p\mid h\\ p>2}}\frac{p-1}{p-2}. \] For every real number $x>1$ and every even integer $h\neq 0$, \[ \#\{p\leq x \mid p \text{ and } p+h \text{ are prime}\}\leq 16\,\omega\,\kappa(h)\,\frac{x}{\log^2 x}. \] \end{Proposition} For $M\in A_N$ define \[ \nu_N(M)=\#\{(P,Q)\in L_N\times U_N\mid Q-P=2M\}. \] Every pair $(P,Q)\in L_N\times U_N$ gives $(Q-P)/2\in E_N$, as shown in Section~\ref{subsec:upper-prime-count}. Hence $\nu_N$ vanishes outside $E_N$, and \[ \sum_{M\in E_N}\nu_N(M)=a_N\,r. \] \begin{Lemma}[Energy identity]\label{lem:energy} For every $N\geq 4$, \[ \sum_{M\in A_N}\nu_N(M)^2=a_N\,r+2\,\mathcal C_N. \] \end{Lemma} \begin{proof} The left-hand side counts the quadruples $(P,Q,P',Q')\in L_N\times U_N\times L_N\times U_N$ with $Q-P=Q'-P'$, that is, with $P'-P=Q'-Q=:h$. The value $h=0$ gives $P=P'$ and $Q=Q'$, hence $a_N r$ quadruples. For $h>0$ the number of quadruples is $\rho_L(h)\rho_U(h)$, and exchanging $(P,Q)$ with $(P',Q')$ maps the quadruples with difference $h$ bijectively onto those with difference $-h$. \end{proof} \begin{Lemma}[Energy bound]\label{lem:energy-bound} Let \[ G=\prod_{p>2}\left(1+\frac{2p-3}{p(p-2)^2}\right). \] For every $N\geq 4$, \[ \mathcal C_N\leq 256\,\omega^2 G\,\frac{N^3}{\log^4 N}. \] Moreover, $\omega<0.6614$ and $G<2.658$. \end{Lemma} \begin{proof} All primes in $L_N\cup U_N$ are odd, so $\rho_L(h)=0$ unless $h$ is even with $2\leq h\leq N-4$. For such $h$, Proposition~\ref{prop:siebert} with $x=N$ and with $x=2N$ gives \[ \rho_L(h)\leq 16\,\omega\,\kappa(h)\frac{N}{\log^2 N},\qquad \rho_U(h)\leq 16\,\omega\,\kappa(h)\frac{2N}{\log^2 (2N)}\leq 32\,\omega\,\kappa(h)\frac{N}{\log^2 N}, \] since every pair counted by $\rho_U(h)$ consists of primes $Q2). \] Since $g\geq 0$, for every $K\geq 1$ \[ \sum_{k\leq K}\kappa(k)^2=\sum_{d\leq K}g(d)\left\lfloor\frac{K}{d}\right\rfloor\leq K\sum_{d\geq 1}\frac{g(d)}{d}=K\,G. \] Taking $K=N/2$ gives the bound for $\mathcal C_N$. Every factor of $\omega$ lies in $(0,1)$, so $\omega$ is at most the product of its factors with $3\leq p\leq 97$, which is less than $0.66138$. For $G$, the product of its factors with $3\leq p\leq 97$ is less than $2.63013$. For $p\geq 101$ we have $\frac{2p-3}{p(p-2)^2}<\frac{2}{(p-2)^2}$ and $p-2$ is odd, so, using $\log(1+y)\leq y$, \begin{align*} \log\prod_{p\geq 101}\left(1+\frac{2p-3}{p(p-2)^2}\right)&<2\sum_{\substack{m\geq 99\\ m \text{ odd}}}\frac{1}{m^2}\leq 2\left(\frac{1}{99^2}+\frac{1}{2}\int_{99}^{\infty}\frac{dt}{t^2}\right)\\ &=2\left(\frac{1}{9801}+\frac{1}{198}\right)<0.01031. \end{align*} Hence $G<2.63013\,e^{0.01031}<2.658$. \end{proof} \begin{Theorem}[Linear lower bound]\label{thm:linear} For every integer $N\geq 3275$, \[ |E_N|>\frac{N}{760}. \] \end{Theorem} \begin{proof} Write $L=\log N$ and $\lambda=\log 2$. By the Cauchy--Schwarz inequality and Lemma~\ref{lem:energy}, \[ (a_N r)^2=\Bigl(\sum_{M\in E_N}\nu_N(M)\Bigr)^2\leq |E_N|\sum_{M\in A_N}\nu_N(M)^2=|E_N|\,(a_N r+2\mathcal C_N). \] \emph{Lower bound for $a_N r$.} By Proposition~\ref{prop:rs-pi} with $x=N-1\geq 17$, $a_N=\pi(N-1)-1>\frac{N-1}{L}-1$, so \[ a_N>\frac{N}{L}\,\mathcal A(N),\qquad \mathcal A(N)=1-\frac{1+L}{N}. \] As shown in Section~\ref{subsec:upper-prime-count}, $r\geq \pi(2N)-\pi(N)-1$. Proposition~\ref{prop:dusart-upper}\emph{(1)} with $x=2N\geq 599$ and Proposition~\ref{prop:dusart-upper}\emph{(2)} with $x=N$ give \[ r\geq \frac{2N}{L+\lambda}\left(1+\frac{1}{L+\lambda}\right)-\frac{N}{L}\left(1+\frac{1.2762}{L}\right)-1=\frac{N}{L}\,\mathcal B(N), \] where, using $\frac{2L}{L+\lambda}=2-\frac{2\lambda}{L+\lambda}$, \[ \mathcal B(N)=1-\frac{2\lambda}{L+\lambda}+\frac{2L}{(L+\lambda)^2}-\frac{1.2762}{L}-\frac{L}{N}. \] Both $\mathcal A$ and $\mathcal B$ are increasing in $N$ for $N\geq 3275$. For $\mathcal A$ this holds because $(1+\log x)/x$ is decreasing for $x>1$. For $\mathcal B$, the term $-L/N$ is increasing for $N>e$, and the derivative with respect to $L$ of the remaining terms is \begin{align*} \frac{2\lambda(\lambda+1)-(2-2\lambda)L}{(L+\lambda)^3}+\frac{1.2762}{L^2}&>-\frac{(2-2\lambda)L}{L^3}+\frac{1.2762}{L^2}\\ &=\frac{1.2762-(2-2\lambda)}{L^2}>0, \end{align*} since $2-2\lambda<0.614$; here the first numerator is negative for $L\geq\log 3275$. At $N=3275$, $\mathcal A>0.99722$ and $\mathcal B>0.89174$. Hence, for every $N\geq 3275$, \[ a_N r>\alpha\,\frac{N^2}{L^2},\qquad \alpha=0.8892. \] \emph{Conclusion.} The function $t\mapsto t^2/(t+K)$ is increasing for $t>0$, and decreasing in $K>0$. With $X=\alpha N^2/L^2$ and, by Lemma~\ref{lem:energy-bound}, $2\mathcal C_N\leq \overline K=\beta N^3/L^4$, where $\beta=512\cdot 0.6614^2\cdot 2.658<595.33$, we obtain \[ |E_N|\geq\frac{(a_N r)^2}{a_N r+2\mathcal C_N}>\frac{X^2}{X+\overline K}=\frac{\alpha^2 N}{\beta\,(1+\delta_N)},\qquad \delta_N=\frac{X}{\overline K}=\frac{\alpha L^2}{\beta N}. \] The function $t\mapsto \log^2 t/t$ is decreasing for $t>e^2$, so $\delta_N\leq \delta_{3275}<3\cdot 10^{-5}$. Therefore \[ |E_N|>\frac{0.8892^2}{595.33\cdot 1.00003}\,N>0.001328\,N>\frac{N}{760}. \] \end{proof} By definition, $M\in E_N$ if and only if there is a prime $p$ with $\max\{2,N-2M\}0$. For all sufficiently large $x$ and every $y$ with $2x\leq y\leq x^{3-\varepsilon}$, all but $O(x(\log x)^{-A})$ integers $k$ with $1\leq k\leq x$ satisfy \[ \Psi(y,2k)=\mathfrak S(2k)(y-2k)+O\!\left(\frac{y}{(\log y)^{B}}\right), \] where the implied constants depend only on $\varepsilon$, $A$ and $B$. \end{Proposition} We use Proposition~\ref{prop:mikawa} only for $2x\leq y\leq 4x$. \begin{Proposition}[Almost all half-differences]\label{prop:almost-all} For every $D>0$, \[ |A_N\setminus E_N|\ll_D\frac{N}{(\log N)^D}. \] In particular, $|E_N|=N-o(N)$. \end{Proposition} \begin{proof} Write $L=\log N$ and $a=\lceil N/L^D\rceil$, and apply Proposition~\ref{prop:mikawa} with $\varepsilon=1/2$, $A=D$ and $B=D+1$. The terms of $\Psi(y,2k)$ with $y\leq 2N$ in which $n$ or $n-2k$ is a prime power $p^j$ with $j\geq 2$ contribute $O(\sqrt N L^3)$, since there are $O(\sqrt N L)$ such prime powers up to $2N$ and each term is at most $L^2+O(L)$. \emph{The range $aa$; moreover $P$ is odd, because $Q$ is odd and $2k$ is even, so $P>2$. Thus $k\in E_N$. \emph{The range $N/22$ as before. Thus $k\in E_N$. \emph{Conclusion.} The remaining integers $k\in A_N$ satisfy $k\leq a$ or $k>N-3-a$, and there are at most $2a$ of them. Hence $|A_N\setminus E_N|\leq 2a+O(NL^{-D})\ll_D N/L^D$. \end{proof} \begin{Corollary}[Goldbach partitions up to a difference of two primes]\label{cor:goldbach-diff} For every $D>0$, \[ |C_N\cap E_N|=\pi(N-1)-1+O_D\!\left(\frac{N}{(\log N)^D}\right). \] Equivalently, for all but $O_D(N/(\log N)^D)$ primes $P$ with $3\leq P(N-1)/\log N-1$ by Proposition~\ref{prop:rs-pi}, which exceeds the error term for large $N$. \end{proof} Proposition~\ref{prop:almost-all} is deduced from Proposition~\ref{prop:mikawa} and gives no explicit value of $N$ from which a stated bound holds. Theorem~\ref{thm:linear} gives the explicit bound $|E_N|>N/760$ for every $N\geq 3275$. \subsection{A Positive Proportion of Centered Partitions}\label{subsec:positive-proportion} The same energy method applies to the centered count \[ R(2N)=|C_N\cap D_N|=\#\{p\mid 3\leq p\frac{X}{9600}. \] \end{Theorem} \begin{proof} Write $L=\log X$ and $s=X/L$, and let $S$ be the set of odd primes not exceeding $2X$. Let \[ T=\sum_{4\leq N\leq X}R(2N),\qquad V=\sum_{4\leq N\leq X}R(2N)^2. \] By the Cauchy--Schwarz inequality, the number of $N$ with $R(2N)\geq 1$ is at least $T^2/V$. \emph{Lower bound for $T$.} Every pair of odd primes $p\frac{(s-2)^2}{2}. \] \emph{Upper bound for $V$.} The square $R(2N)^2$ counts pairs of representations $2N=p+q=p'+q'$ with $p2X$. As in the proof of Lemma~\ref{lem:energy-bound}, $\sum_{2\leq h\leq 2X,\,h \text{ even}}\kappa(h)^2\leq XG$. Hence \[ V\leq \pi(2X)^2+2\cdot 1024\,\omega^2\frac{X^2}{L^4}\,XG=\pi(2X)^2+K\frac{X^3}{L^4},\qquad K=2048\,\omega^2 G. \] By Proposition~\ref{prop:dusart-upper}\emph{(2)}, and since $y\mapsto \frac1y\bigl(1+\frac{1.2762}{y}\bigr)$ is decreasing, $\pi(2X)\leq \frac{2X}{L}\bigl(1+\frac{1.2762}{L}\bigr)$. \emph{Conclusion.} Since $X^3/L^4=s^4/X$, \[ \frac{T^2}{V}>\frac{(s-2)^4/4}{4s^2\bigl(1+\frac{1.2762}{L}\bigr)^2+K s^4/X}=\frac{X\,(1-2/s)^4}{4\bigl(K+e(X)\bigr)}, \] where $e(X)=\frac{4L^2}{X}\bigl(1+\frac{1.2762}{L}\bigr)^2$. For $X\geq 10^6$, $s$ is increasing and $e(X)$ is decreasing, so $(1-2/s)^4>0.99988$ and $e(X)<0.00092$. By Lemma~\ref{lem:energy-bound}, $K<2048\cdot 0.6614^2\cdot 2.658<2381.3$. Therefore \[ \frac{T^2}{V}>\frac{0.99988}{4\cdot 2381.30092}\,X>\frac{X}{9600}. \] \end{proof} Montgomery and Vaughan~\cite{montgomery1975exceptional} proved that, for some $\delta>0$ and all sufficiently large $X$, at most $X^{1-\delta}$ even integers $n\leq X$ are not sums of two primes. Theorem~\ref{thm:positive-proportion} is weaker in its conclusion, but it is explicit and holds for every $X\geq 10^6$. By Proposition~\ref{prop:verified}, its conclusion is trivial for $X\leq 2\cdot 10^{18}$. \subsection{A Half-Difference Coverage Statistic}\label{subsec:gap} For the diagnostic set $E_N$, define \[ H(N) = \log^2(2N) - \bigl((N-3) - |E_N|\bigr). \] Rearranging, $H(N)>0$ is equivalent to \[ |E_N| > (N-3) - \log^2(2N). \] Thus $H(N)>0$ holds exactly when fewer than $\log^2(2N)$ admissible values are missing from $E_N$. \begin{Hypothesis}[Half-Difference Density Hypothesis]\label{hyp:half-difference-density} For every integer $N>10^6$, we have $H(N) > 0$. \end{Hypothesis} \begin{Hypothesis}[Centered Prime-Correlation Hypothesis]\label{hyp:centered} For every integer $N>2\cdot 10^{18}$, the centered prime-pair count \[ R(2N):=|C_N\cap D_N| \] is positive. \end{Hypothesis} \begin{Remark}[Status of the hypotheses] By Theorem~\ref{thm:cramer} and the computation of Section~\ref{sec:computational}, Hypothesis~\ref{hyp:half-difference-density} implies the Cram\'er-type bound $p'-p<2\log^2(2p+2)+2$ for all consecutive primes $p 0$ for \emph{every} $N \in [4, 10^6]$, providing empirical support for Hypothesis~\ref{hyp:half-difference-density}. Second, the minimum of $H(N)$ on $I_m$ strictly increases for $2\leq m\leq 16$; the minimum on $I_{17}$ is smaller than the minimum on $I_{16}$, and the minima increase again from $I_{17}$ to $I_{19}$. Third, the largest exceptional set on $I_{19}$ has $140$ elements, at $N=668364$, where $\log^2(2N)>198$. \begin{table}[H] \centering \caption{Minimum $H(N)$, largest exceptional set $A_N\setminus E_N$ (with the $N$ attaining it), and minimum $|E_N|/N$ on the dyadic intervals $I_m$. Note that $H(N) > 0$ for all tested values.} \label{tab:gaps} \footnotesize \setlength{\tabcolsep}{4pt} \begin{tabular}{|c|c|c|r|c|c|} \hline \textbf{$m$} & \textbf{Range $I_m$} & \textbf{$N$ at min $H$} & \textbf{Min $H(N)$} & \textbf{Max $|A_N\setminus E_N|$ (at $N$)} & \textbf{Min $|E_N|/N$} \\ \hline 2 & $[4,8]$ & 5 & 4.301898 & 2 (7) & 0.200000 \\ 3 & $(2^{3},2^{4}]$ & 9 & 7.354249 & 2 (11) & 0.545455 \\ 4 & $(2^{4},2^{5}]$ & 19 & 10.232033 & 6 (29) & 0.684211 \\ 5 & $(2^{5},2^{6}]$ & 61 & 14.078618 & 9 (61) & 0.803279 \\ 6 & $(2^{6},2^{7}]$ & 73 & 17.836335 & 10 (113) & 0.863014 \\ 7 & $(2^{7},2^{8}]$ & 151 & 20.608977 & 13 (165) & 0.900662 \\ 8 & $(2^{8},2^{9}]$ & 269 & 23.537165 & 16 (269) & 0.929368 \\ 9 & $(2^{9},2^{10}]$ & 541 & 28.812111 & 24 (986) & 0.957486 \\ 10 & $(2^{10},2^{11}]$ & 1327 & 33.154668 & 29 (1327) & 0.975377 \\ 11 & $(2^{11},2^{12}]$ & 2161 & 35.081569 & 35 (2161) & 0.982416 \\ 12 & $(2^{12},2^{13}]$ & 7069 & 42.329014 & 49 (7069) & 0.991389 \\ 13 & $(2^{13},2^{14}]$ & 14138 & 44.057758 & 61 (14138) & 0.994665 \\ 14 & $(2^{14},2^{15}]$ & 17863 & 46.906582 & 64 (29443) & 0.996305 \\ 15 & $(2^{15},2^{16}]$ & 43331 & 55.271687 & 74 (43331) & 0.998006 \\ 16 & $(2^{16},2^{17}]$ & 66239 & 58.102490 & 90 (107525) & 0.998732 \\ 17 & $(2^{17},2^{18}]$ & 206699 & 54.240920 & 113 (206699) & 0.999315 \\ 18 & $(2^{18},2^{19}]$ & 484853 & 55.019283 & 135 (484853) & 0.999576 \\ 19 & $(2^{19},2^{20}]$ & 668364 & 58.971771 & 140 (668364) & 0.999749 \\ \hline \end{tabular} \end{table} \subsection{Centered Partitions and the Pigeonhole Count}\label{subsec:centered-data} The run of \texttt{experiment.py} gives the following results for every $N\in[4,2^{14}]$. \begin{enumerate} \item[\emph{(1)}] $R(2N)\geq 1$. Hence every even integer $2N$ with $8\leq 2N\leq 2^{15}$ is the sum of two distinct primes, and, by Lemma~\ref{lem:CD-in-E}, $C_N\cap E_N\supseteq C_N\cap D_N\neq\emptyset$. \item[\emph{(2)}] The pigeonhole count $|C_N|>|A_N\setminus E_N|$ holds for every $N$ except $N=5$ and $N=7$. For these two values, $C_5\cap E_5=\{2\}$ and $C_7\cap E_7=\{4\}$. \item[\emph{(3)}] $(C_N\cap E_N)\setminus D_N\neq\emptyset$ for $16{,}377$ of the $16{,}381$ values of $N$, namely for every $N$ except $N\in\{4,5,7,8\}$. The first elements are $M=3$ for $N=6$, $M=6$ for $N=9$, $M=5$ for $N=10$, $M=4$ for $N=11$ and $M=9$ for $N=12$; for instance, $3\in C_6\cap E_6$, while $6+3=9$ is composite. \item[\emph{(4)}] Among the integers $2\leq N\leq 10^6$, the inequality of Lemma~\ref{lem:elementary} fails at $N=355$ and holds for every $N\geq 356$; the bound $356$ of Lemma~\ref{lem:elementary} is therefore the least possible integer threshold. \end{enumerate} For comparison with $H(N)$, the minima of the centered statistic $H_R(N)$ of Section~\ref{subsec:HR-negative} on the intervals $I_m$ are given in Table~\ref{tab:centered}. The minimum is positive on $I_2$ and $I_3$, and negative on every interval $I_m$ with $4\leq m\leq 13$; by Proposition~\ref{prop:HR-negative}, $H_R(N)<0$ for every $N\geq 34$. \begin{table}[H] \centering \caption{Minimum $H_R(N)$ values on the dyadic intervals $I_m$.} \label{tab:centered} \small \begin{tabular}{|c|c|c|r|} \hline \textbf{Interval ($m$)} & \textbf{Range $I_m$} & \textbf{$N$ achieving min} & \textbf{Min $H_R(N)$} \\ \hline 2 & $[4,8]$ & 7 & 3.96 \\ 3 & $(2^{3},2^{4}]$ & 16 & 1.01 \\ 4 & $(2^{4},2^{5}]$ & 31 & $-8.97$ \\ 5 & $(2^{5},2^{6}]$ & 64 & $-34.46$ \\ 6 & $(2^{6},2^{7}]$ & 128 & $-86.25$ \\ 7 & $(2^{7},2^{8}]$ & 256 & $-203.08$ \\ 8 & $(2^{8},2^{9}]$ & 511 & $-441.98$ \\ 9 & $(2^{9},2^{10}]$ & 1024 & $-937.87$ \\ 10 & $(2^{10},2^{11}]$ & 2047 & $-1923.82$ \\ 11 & $(2^{11},2^{12}]$ & 4096 & $-3935.80$ \\ 12 & $(2^{12},2^{13}]$ & 8191 & $-7953.83$ \\ 13 & $(2^{13},2^{14}]$ & 16369 & $-16029.92$ \\ \hline \end{tabular} \end{table} \subsection{The Exceptional Set and the Linear Bound}\label{subsec:exceptional-data} The minimum of $|E_N|/N$ over $3275\leq N\leq 10^6$ is $0.990603$; Theorem~\ref{thm:linear} gives $|E_N|/N>1/760$ for every $N\geq 3275$, and Proposition~\ref{prop:almost-all} gives $|E_N|/N\to 1$. For each $N$ in Table~\ref{tab:gaps} at which the minimum of $H(N)$ is attained, the script lists the missing values $A_N\setminus E_N$. For $N\leq 2161$ the complete lists show that every missing value satisfies $M\leq 50$ or $N-3-M\leq 14$. For instance, for $N=1327$ the missing values are $M=1,\ldots,19$, $22$, $25$, $28$, $34$, $43$, $50$, $1316$, $1319$, $1323$ and $1324$. The primes around $1327$ are $p^-=1321$ and $p^+=1361$, so Lemma~\ref{lem:gap} forces exactly the missing values $M\leq 19$ below $(p^+-p^-)/2=20$. Likewise, the gaps $p^+-p^-=26$ around $N=2161$ and $p^+-p^-=24$ around $N=541$ force the missing values $M=1,\ldots,12$ and $M=1,\ldots,11$, and these are the initial runs of missing values for these two $N$. At the other end, Lemma~\ref{lem:gap-top} forces the final runs. The largest primes below $2N$ for $N=269$, $541$ and $2161$ are $q=523$, $1069$ and $4297$, which force $M=261,\ldots,266$, $M=534,\ldots,538$ and $M=2148,\ldots,2158$. For $N=269$ and $N=541$ these are all the missing values near $N-3$; for $N=2161$ the only other missing value with $N-3-M\leq 14$ is $M=2144$. \subsection{Comparison with the Hardy--Littlewood Prediction}\label{subsec:hl-data} Let \[ \mathrm{HL}(2N)=\omega\prod_{\substack{p\mid N\\ p>2}}\frac{p-1}{p-2}\int_{2}^{2N-2}\frac{dt}{\log t\,\log(2N-t)}, \] with $\omega$ as in Proposition~\ref{prop:siebert}. The script \texttt{large\_range.py} computes the integral by $64$-point Gauss--Legendre quadrature and $\omega$ from the primes up to $2\cdot 10^6$, which gives $\omega\approx 0.6601618372$. Table~\ref{tab:hl} gives, on each interval $I_m$, the minimum of $R(2N)$ and the minimum, maximum and mean of $R(2N)/\mathrm{HL}(2N)$. Since $R(2N)\geq 1$ for every $N\in[4,10^6]$, every even integer $2N$ with $8\leq 2N\leq 2\cdot 10^6$ is the sum of two distinct primes. On $I_{19}$ the ratio lies between $0.9657$ and $1.0296$, with mean $0.9982$. \begin{table}[H] \centering \caption{Minimum of $R(2N)$ (with the $N$ attaining it) and the minimum, maximum and mean of $R(2N)/\mathrm{HL}(2N)$ on the dyadic intervals $I_m$.} \label{tab:hl} \footnotesize \setlength{\tabcolsep}{4pt} \begin{tabular}{|c|c|c|c|c|c|} \hline \textbf{$m$} & \textbf{Range $I_m$} & \textbf{Min $R(2N)$ ($N$)} & \textbf{Min ratio} & \textbf{Max ratio} & \textbf{Mean ratio} \\ \hline 2 & $[4,8]$ & 1 (4) & 0.2497 & 0.8705 & 0.5112 \\ 3 & $(2^{3},2^{4}]$ & 2 (9) & 0.3711 & 0.6800 & 0.5565 \\ 4 & $(2^{4},2^{5}]$ & 1 (19) & 0.2806 & 1.1702 & 0.6706 \\ 5 & $(2^{5},2^{6}]$ & 2 (34) & 0.4264 & 1.0380 & 0.7407 \\ 6 & $(2^{6},2^{7}]$ & 4 (76) & 0.5738 & 1.0863 & 0.7970 \\ 7 & $(2^{7},2^{8}]$ & 6 (139) & 0.5280 & 1.0909 & 0.8530 \\ 8 & $(2^{8},2^{9}]$ & 9 (271) & 0.6300 & 1.1322 & 0.8890 \\ 9 & $(2^{9},2^{10}]$ & 16 (556) & 0.6960 & 1.1621 & 0.9258 \\ 10 & $(2^{10},2^{11}]$ & 26 (1039) & 0.7159 & 1.1448 & 0.9463 \\ 11 & $(2^{11},2^{12}]$ & 45 (2104) & 0.7833 & 1.1361 & 0.9624 \\ 12 & $(2^{12},2^{13}]$ & 75 (4276) & 0.7754 & 1.1131 & 0.9750 \\ 13 & $(2^{13},2^{14}]$ & 131 (8311) & 0.8673 & 1.1129 & 0.9820 \\ 14 & $(2^{14},2^{15}]$ & 222 (16519) & 0.8807 & 1.0906 & 0.9887 \\ 15 & $(2^{15},2^{16}]$ & 406 (32789) & 0.9093 & 1.0711 & 0.9918 \\ 16 & $(2^{16},2^{17}]$ & 707 (65551) & 0.9181 & 1.0570 & 0.9946 \\ 17 & $(2^{17},2^{18}]$ & 1279 (132274) & 0.9424 & 1.0529 & 0.9962 \\ 18 & $(2^{18},2^{19}]$ & 2291 (262436) & 0.9536 & 1.0385 & 0.9975 \\ 19 & $(2^{19},2^{20}]$ & 4130 (525166) & 0.9657 & 1.0296 & 0.9982 \\ \hline \end{tabular} \end{table} \subsection{Missing Values Forced by Prime Gaps}\label{subsec:forced-data} For each $N$, let $F(N)$ be the number of $M\in A_N$ that Lemma~\ref{lem:gap} or Lemma~\ref{lem:gap-top} places in $A_N\setminus E_N$, and let $U(N)=|A_N\setminus E_N|-F(N)$ be the number of the remaining missing values. The script \texttt{large\_range.py} computes $F(N)$ and $U(N)$ for every $N\in[4,10^6]$ from the primes around $N$ and the largest prime below $2N$, and confirms $F(N)\leq |A_N\setminus E_N|$ for every such $N$. Table~\ref{tab:forced} gives, on each interval $I_m$, the largest values of $|A_N\setminus E_N|$, $F(N)$ and $U(N)$, the mean of $U(N)$, and the share $\sum F(N)/\sum|A_N\setminus E_N|$ of the missing values that the two lemmas force. The share of forced values strictly decreases with $m$, from $1$ on $I_2$ to $0.2698$ on $I_{19}$, while the mean of $U(N)$ increases from $0$ to $41.545$. On $I_6$ and on every $I_m$ with $8\leq m\leq 14$, every $N$ has at least one missing value that the two lemmas do not force, and on every $I_m$ with $15\leq m\leq 19$ the share of $N$ with $U(N)=0$ is below $0.00005$. No $N\leq 10^6$ has $F(N)\geq\log^2(2N)$ or $U(N)>\log^2(2N)$, and the largest value of $U(N)/\log^2(2N)$ over $1000\leq N\leq 10^6$ is $0.6263$, at $N=484857$. \begin{table}[H] \centering \caption{Largest exceptional set, largest number $F(N)$ of forced missing values, largest number $U(N)$ of the remaining missing values (with the $N$ attaining it), mean of $U(N)$, and share of forced missing values on the dyadic intervals $I_m$.} \label{tab:forced} \footnotesize \setlength{\tabcolsep}{4pt} \begin{tabular}{|c|c|c|c|c|c|c|} \hline \textbf{$m$} & \textbf{$I_m$} & \textbf{Max $|A_N\setminus E_N|$} & \textbf{Max $F$} & \textbf{Max $U$ ($N$)} & \textbf{Mean $U$} & \textbf{Share} \\ \hline 2 & $[4,8]$ & 2 & 2 & 0 (4) & 0.000 & 1.0000 \\ 3 & $(2^{3},2^{4}]$ & 2 & 2 & 1 (12) & 0.125 & 0.9091 \\ 4 & $(2^{4},2^{5}]$ & 6 & 4 & 4 (30) & 0.812 & 0.7111 \\ 5 & $(2^{5},2^{6}]$ & 9 & 7 & 5 (60) & 1.375 & 0.6562 \\ 6 & $(2^{6},2^{7}]$ & 10 & 9 & 6 (128) & 2.125 & 0.6344 \\ 7 & $(2^{7},2^{8}]$ & 13 & 11 & 9 (166) & 3.555 & 0.5221 \\ 8 & $(2^{8},2^{9}]$ & 16 & 14 & 9 (311) & 4.648 & 0.4868 \\ 9 & $(2^{9},2^{10}]$ & 24 & 17 & 15 (702) & 7.020 & 0.4439 \\ 10 & $(2^{10},2^{11}]$ & 29 & 21 & 20 (1373) & 8.673 & 0.4420 \\ 11 & $(2^{11},2^{12}]$ & 35 & 25 & 26 (3083) & 10.956 & 0.4092 \\ 12 & $(2^{12},2^{13}]$ & 49 & 27 & 35 (7072) & 14.507 & 0.3706 \\ 13 & $(2^{13},2^{14}]$ & 61 & 50 & 42 (14249) & 16.576 & 0.3619 \\ 14 & $(2^{14},2^{15}]$ & 64 & 45 & 51 (24683) & 20.686 & 0.3350 \\ 15 & $(2^{15},2^{16}]$ & 74 & 52 & 57 (63199) & 23.915 & 0.3218 \\ 16 & $(2^{16},2^{17}]$ & 90 & 53 & 69 (107449) & 27.847 & 0.3072 \\ 17 & $(2^{17},2^{18}]$ & 113 & 69 & 90 (250433) & 32.279 & 0.2922 \\ 18 & $(2^{18},2^{19}]$ & 135 & 83 & 119 (484857) & 36.733 & 0.2803 \\ 19 & $(2^{19},2^{20}]$ & 140 & 93 & 115 (854722) & 41.545 & 0.2698 \\ \hline \end{tabular} \end{table} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Rigorous Consequences and the Open Centered Problem}\label{sec:main} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{What Follows from the Short-Interval Estimate} The results of Dusart imply useful lower bounds for primes above $N$, and hence for the diagnostic set $E_N$. \begin{Proposition}[Primes in $(N,2N)$]\label{prop:many-upper} For every integer $N\geq 3275$, the interval $(N,2N)$ contains at least $\lfloor \log^2 N\rfloor-1$ primes. \end{Proposition} \begin{proof} Starting with $x_0=N$, form consecutive intervals \[ \bigl(x_j,\; x_j(1+1/(2\log^2 x_j))\bigr],\qquad j=0,1,2,\ldots, \] until the first index $m+1$ for which $x_{m+1}\geq 2N$. By Proposition~\ref{prop:dusart-short}, each of the first $m$ intervals contains a prime lying in $(N,2N)$. Since $x_j\leq 2N$ and $\log x_j\geq \log N$ throughout the construction, each interval has length at most $N/\log^2 N$. Reaching total length $N$ therefore requires at least $\log^2 N$ such interval lengths. Thus $m\geq \log^2 N-1$, and the number of primes in $(N,2N)$ is at least $\lfloor \log^2 N\rfloor-1$. \end{proof} \begin{Corollary}[Baseline half-difference coverage]\label{cor:half-difference-baseline} For every integer $N\geq 3275$, \[ |E_N|\geq \lfloor \log^2 N\rfloor-2. \] \end{Corollary} \begin{proof} Let $Q$ be a prime in $(N,2N)$. If $Q\leq 2N-3$, then with $P=3$ the value $M=(Q-3)/2$ lies in $A_N$ and belongs to $E_N$. Distinct primes $Q$ give distinct values of $M$. The only possible prime in $(N,2N)$ excluded by the admissible range in this construction is $Q=2N-1$, which would give $M=N-2$. Proposition~\ref{prop:many-upper} therefore gives the stated lower bound. \end{proof} Theorem~\ref{thm:linear} gives the linear bound $|E_N|>N/760$ for the same range of $N$, while Hypothesis~\ref{hyp:half-difference-density} asks for $|E_N|>(N-3)-\log^2(2N)$. The centered proof target is the stronger prime-correlation condition $R(2N)=|C_N\cap D_N|>0$. \begin{Proposition}[A pigeonhole consequence of $H(N)>0$]\label{prop:pigeonhole-CE} For $N\geq 356$, if $H(N)>0$, then \[ C_N\cap E_N\neq\emptyset. \] \end{Proposition} \begin{proof} The condition $H(N)>0$ gives \[ |A_N\setminus E_N|<\log^2(2N). \] On the other hand, $|C_N|=\pi(N-1)-1\geq \pi(N)-2$. By Proposition~\ref{prop:rs-pi}, since $N\geq 356>17$, \[ \pi(N)>\frac{N}{\log N}>\frac{N}{\log N+2}, \] and by Lemma~\ref{lem:elementary}, \[ |C_N|>\frac{N}{\log N+2}-2>\log^2(2N). \] Hence $C_N$ cannot be contained in $A_N\setminus E_N$, and so $C_N\cap E_N$ is nonempty. \end{proof} Proposition~\ref{prop:pigeonhole-CE} gives $C_N\cap E_N\neq\emptyset$ with an explicit range of $N$ under the condition $H(N)>0$, which the computation of Section~\ref{sec:computational} verifies for $4\leq N\leq 10^6$. Corollary~\ref{cor:goldbach-diff} gives the same conclusion for all sufficiently large $N$ without this condition. In both cases, at least one prime $N-M$ below $N$ has its half-difference $M$ realized by a straddling prime pair; a distinct-prime Goldbach partition of $2N$ requires a prime $N-M$ for which the centered pair $(N-M,N+M)$ itself is such a pair. \subsection{Prime Gaps and the Coverage Statistic}\label{subsec:gaps-H} Lemma~\ref{lem:gap} turns every long prime gap into many missing values of $E_N$. This gives a lower bound for the exceptional set along a subsequence, and it shows what positivity of $H(N)$ would imply for prime gaps. \begin{Theorem}[Coverage implies a Cram\'er-type bound]\label{thm:cramer} Let $N_0\geq 4$, and suppose that $H(N)>0$ for every integer $N\geq N_0$. Then every pair of consecutive primes $pN$. Let $g=p'-p$, which is even. By Lemma~\ref{lem:gap}, the values $M=1,\ldots,g/2-1$ lie in $A_N\setminus E_N$; they lie in $A_N$ because Bertrand's postulate gives $p'<2p$, hence $g0$ is equivalent to $|A_N\setminus E_N|<\log^2(2N)$, we get $g/2-1<\log^2(2p+2)$. \end{proof} Cram\'er~\cite{cramer1936order} conjectured that $p'-p=O(\log^2 p)$ for consecutive primes $p0$ for every $4\leq N\leq 10^6$, Theorem~\ref{thm:cramer} shows that Hypothesis~\ref{hyp:half-difference-density} implies $p'-p<2\log^2(2p+2)+2$ for all consecutive primes $p0$ such that, for all sufficiently large $X$, \[ \max_{p_n\leq X}(p_{n+1}-p_n)\geq c_0\,F(X),\qquad F(X)=\frac{\log X\,\log\log X\,\log\log\log\log X}{\log\log\log X}, \] where $p_n$ denotes the $n$-th prime. \begin{Corollary}[Large exceptional sets]\label{cor:long-gaps} There is an absolute constant $c>0$ such that $|A_N\setminus E_N|\geq c\,F(N)$ for infinitely many integers $N$. In particular, $|A_N\setminus E_N|$ is unbounded. \end{Corollary} \begin{proof} For large $X$, choose consecutive primes $p_np_{n+1}-p_n\geq c_0F(X)$, so $N\to\infty$ as $X\to\infty$, and infinitely many $N$ occur. Since $F$ is increasing for large arguments, $N\leq X+1$ gives $F(N)\leq F(X+1)$, and $F(X+1)\sim F(X)$. Hence $|A_N\setminus E_N|\geq c\,F(N)$ for every fixed $c0, \] because $(\ell-1)/\ell^2\leq 1/4$ and $(\ell-2)/\ell^3\leq 1/27$ for $\ell>0$, and $2\log(2x)/x$ is decreasing for $x\geq 8$. Hence $f$ is increasing on $[8,\infty)$. At $x=34$, \[ \frac{34}{\log 34}\left(1+\frac{1.2762}{\log 34}\right)<13.132\quad\text{and}\quad \log^2 68<17.805, \] so $f(34)>34-3-13.132-17.805>0.06$. Therefore $f(N)>0$ and $H_R(N)<0$ for every $N\geq 34$. \end{proof} \subsection{The Gap to a Complete Proof}\label{subsec:proof-gap} The missing ingredient is a uniform lower bound for the centered binary prime correlation \[ R(2N)=|C_N\cap D_N| =\#\{p0 \qquad (N>2\cdot 10^{18}). \] Equivalently, the set of primes below $N$ and its reflection across $N$ must intersect. By Corollary~\ref{cor:goldbach-diff} and Lemma~\ref{lem:CD-in-E}, for all large $N$ the set $C_N\cap E_N$ has $\pi(N-1)-1-O_D(N/(\log N)^D)$ elements, and $C_N\cap D_N=C_N\cap E_N\cap D_N$ is the subset of those $M$ for which $N+M$ is prime, that is, for which the centered pair $(N-M,N+M)$ realizes the half-difference $M$. The conjecture for $2N$ asks for one such $M$. By Proposition~\ref{prop:HR-negative}, $|C_N\cap D_N|<(N-3)-\log^2(2N)$ for every $N\geq 34$, whereas $|E_N|=N-o(N)$ by Proposition~\ref{prop:almost-all}. A strong sufficient form would be an explicit Hardy--Littlewood-type lower bound \[ R(2N)\geq c\,\frac{N}{\log^2 N} \prod_{\substack{p\mid N\\ p>2}}\frac{p-1}{p-2} \] with a positive constant $c$ and an error term small enough to keep the right-hand side positive for every $N>2\cdot 10^{18}$. A weaker but sufficient target is simply a proven inequality $R(2N)\geq 1$ for all $N>2\cdot 10^{18}$. \subsection{The Verified Range}\label{subsec:verified} \begin{Proposition}[Verified range]\label{prop:verified} For every integer $N$ with $4\leq N\leq 2\cdot 10^{18}$, $R(2N)\geq 1$. \end{Proposition} \begin{proof} Oliveira e Silva, Herzog and Pardi verified the even Goldbach conjecture for all even integers up to $4\cdot 10^{18}$~\cite{oliveira2014empirical}, and the least prime of a Goldbach partition of every such integer is at most $9781$~\cite{oliveira2014goldbachweb}. Hence $2N=p+q$ with primes $p\leq 9781$ and $q=2N-p$. If $N>9781$, then $p2\cdot 10^{18}$. \subsection{Small Cases}\label{subsec:small} The first cases illustrate the centered sets explicitly: \begin{itemize} \item \textbf{$N=4$} ($2N=8$): $C_4=\{1\}$ and $D_4=\{1\}$, giving $8=3+5$. \item \textbf{$N=5$} ($2N=10$): $C_5=\{2\}$ and $D_5=\{2\}$, giving $10=3+7$. \item \textbf{$N=6$} ($2N=12$): $C_6=\{3,1\}$ and $D_6=\{1\}$, giving $12=5+7$. \item \textbf{$N=7$} ($2N=14$): $C_7=\{4,2\}$ and $D_7=\{4\}$, giving $14=3+11$. \item \textbf{$N=8$} ($2N=16$): $C_8=\{5,3,1\}$ and $D_8=\{3,5\}$, giving $16=3+13$. \item \textbf{$N=9$} ($2N=18$): $C_9=\{6,4,2\}$ and $D_9=\{2,4\}$, giving $18=5+13$. \item \textbf{$N=10$} ($2N=20$): $C_{10}=\{7,5,3\}$ and $D_{10}=\{1,3,7\}$, giving $20=3+17$. \item \textbf{$N=11$} ($2N=22$): $C_{11}=\{8,6,4\}$ and $D_{11}=\{2,6,8\}$, giving $22=3+19$. \item \textbf{$N=12$} ($2N=24$): $C_{12}=\{9,7,5,1\}$ and $D_{12}=\{1,5,7\}$, giving $24=5+19$. \end{itemize} For $13\leq N\leq 2^{14}$, the computation of Section~\ref{sec:computational} verifies at least one centered intersection value $M\in C_N\cap D_N$. \begin{Remark}[Computational verification] Our implementation verified the existence of centered Goldbach partitions for all even integers $2N$ with $8\leq 2N\leq 2\cdot 10^6$. The same computations produced the values reported in Tables~\ref{tab:gaps}, \ref{tab:centered} and~\ref{tab:hl}. \end{Remark} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Conclusion}\label{sec:conclusion} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We have embedded the distinct-prime Goldbach condition $C_N\cap D_N\neq\emptyset$, which asks for a prime pair centered at $N$, in the condition defining the half-difference set $E_N$, which asks only for a prime pair straddling $N$. The equivalence between a distinct-prime Goldbach partition of $2N$, a nested-square semiprime configuration, and the condition $C_N\cap D_N\neq\emptyset$ is the classical symmetric form of the problem. The straddling condition holds for almost all $M$, and as a consequence $2N$ has Goldbach partitions up to a difference of two primes; the centering step remains open. \subsection*{Summary of Main Results} Proposition~\ref{prop:verified} gives $R(2N)\geq 1$ for every $4\leq N\leq 2\cdot 10^{18}$, and Theorem~\ref{thm:positive-proportion} proves unconditionally that $R(2N)\geq 1$ for more than $X/9600$ integers $N\leq X$, for every $X\geq 10^6$. Theorem~\ref{thm:linear} proves unconditionally that $|E_N|>N/760$ for every $N\geq 3275$, Proposition~\ref{prop:almost-all}, deduced from a theorem of Mikawa, gives $|A_N\setminus E_N|\ll_D N/(\log N)^D$, and Corollary~\ref{cor:goldbach-diff} shows that, for all large $N$ and all but $O_D(N/(\log N)^D)$ primes $P0$ for all large $N$ would imply a Cram\'er-type bound for prime gaps, Corollary~\ref{cor:long-gaps} shows that $|A_N\setminus E_N|$ is unbounded, and Proposition~\ref{prop:HR-negative} shows $H_R(N)<0$ for every $N\geq 34$. The computation of Section~\ref{sec:computational} verifies $R(2N)\geq 1$ and $H(N)>0$ for $4\leq N\leq 10^6$, gives $|E_N|\geq 0.9906\,N$ for $3275\leq N\leq 10^6$, and shows $(C_N\cap E_N)\setminus D_N\neq\emptyset$ for every $N\leq 2^{14}$ except $N\in\{4,5,7,8\}$. \subsection*{Status of the Conjecture} The distinct-prime Goldbach conjecture is not proved in this paper. The paper proves that $C_N\cap E_N$ contains almost all of $C_N$ for all large $N$ (Corollary~\ref{cor:goldbach-diff}), and that it is nonempty for every $N\geq 356$ with $H(N)>0$ (Proposition~\ref{prop:pigeonhole-CE}). The step needed for a proof is the centering step: one must show that for some such $M$, the centered upper endpoint $N+M$ is prime, i.e., \[ C_N\cap D_N\neq\emptyset. \] Equivalently, by Proposition~\ref{prop:verified}, one must prove the positivity of $R(2N)=|C_N\cap D_N|$ for every $N>2\cdot 10^{18}$. \subsection*{The Centered Gap} The proof gap is the centered correlation estimate $R(2N)>0$. The statistic \[ H(N)=\log^2(2N)-((N-3)-|E_N|) \] measures half-difference coverage and is positive for every $N\leq 10^6$, while the centered statistic $H_R(N)$ is negative for every $N\geq 34$ (Proposition~\ref{prop:HR-negative}). The prime-gap obstructions of Lemmas~\ref{lem:gap} and~\ref{lem:gap-top} force $26.98\%$ of the missing values of $E_N$ on $(2^{19},2^{20}]$ (Table~\ref{tab:forced}). A proof of the distinct-prime Goldbach conjecture along these lines requires $|C_N\cap D_N|\geq 1$ for every $N>2\cdot 10^{18}$. \subsection*{Relation to the Classical Goldbach Conjecture} The framework addresses the variant requiring \emph{distinct} primes, thus excluding $4=2+2$ and $6=3+3$. Extending the geometric construction to allow $P=Q$ would require handling the case $M=0$, which lies outside the present admissible interval. \subsection*{Open Questions} Several natural questions remain. By Proposition~\ref{prop:almost-all}, $|E_N|=N-o(N)$, but with no explicit range of $N$. For which explicit $c<1$ and $N_0$ can one prove $|E_N|\geq cN$ for every $N\geq N_0$? Theorem~\ref{thm:linear} gives $c=1/760$ and $N_0=3275$, and the computation gives $|E_N|\geq 0.9906\,N$ for $3275\leq N\leq 10^6$. Which arithmetic structure accounts for the missing values of $E_N$ that Lemmas~\ref{lem:gap} and~\ref{lem:gap-top} do not force? Can one prove an explicit lower bound for $R(2N)$, either directly or through a centering theorem for prime-pair differences? What is the exact asymptotic behaviour of $H(N)$, and does $|A_N\setminus E_N|$ exceed $\log^2(2N)$ for infinitely many $N$? By Theorem~\ref{thm:cramer}, the latter holds if $p'-p\geq 2\log^2(2p+2)+2$ for infinitely many consecutive primes $p