A Matrix-Theoretic Exact Formula for Counting Primes in Intervals Between Consecutive Odd Squares

Let $I_k=[(2k-1{)}^2,(2k+1{)}^2)$ for $k\ge 1$. Starting from the odd-composite matrix $(b_{ij})$ with $b_{ij}=(2i-1)(2j-1)$, introduced by the author in [1], we define for each odd integer $n$ the \emph{matrix multiplicity} $r(n)$, the number of times $n$ appears in $B$. We prove the exact identity $$P_k=N_k-S_k+E_k$$ where $P_k=\mathrm{\#}\{\text{primes in }{I}_k\}$, $N_k=4k$ counts the odd integers in $I_k$, $S_k={\sum }_{n\in I_k\text{ odd}}r(n)$ is the total matrix multiplicity, and $E_k={\sum }_{n\in I_k\text{ odd}}(r(n)-1)$ measures the excess multiplicity of non-semiprime odd composites. All three quantities $N_k$, $S_k$, $E_k$ are computable from the divisor structure of odd integers in $I_k$ without primality testing. The formula yields the equivalent combinatorial condition: $$P_k\ge 1⟺E_k\le S_k-N_k\mathrm{.}$$ We verify $P_k\ge 1$ for all $k\le {10}^8$ by direct computation and establish $P_k\ge 1$ for all $k\le 1.37\times {10}^{17}$ using the Baker-Harman-Pintz theorem [2]. Whether $P_k\ge 1$ for all $k$ (a weaker statement than Legendre’s conjecture) remains an open problem, now equivalent to the purely combinatorial inequality $E_k\le S_k-N_k$ for all $k$.