Bernoulli numbers and the integer group determinants of $C_p\times C_p$

For a prime $p\ge 7$, let $S(C_p\times C_p)$ be the set of integer group determinants and $e(p)$ the least exponent for which $p^{e(p)}\mathbf{Z}\subseteq S(C_p\times C_p)$. Let $r_p$ be the least integer $r\ge 1$ satisfying $(2r+1)i>r(p-1)$ for every irregular pair $(p,2i)$ with $r\mid i$. We prove the unconditional inclusion $p^{p+3+2r_p}\mathbf{Z}\subseteq S(C_p\times C_p)$. Equality $e(p)=p+3+2r_p$ holds when Vandiver’s conjecture holds for $p$ and $r_p\le (p-5)\mathrm{/}2$, hence for every prime $7\le p<2^{31}$. The first exception to $e(p)=p+5$ is $p=103$, where $e(103)=116$. A polynomial obstruction from Bernoulli numbers supplies the lower bound, and circular units give the construction. For regular primes, we characterize the entire determinant set through explicit affine systems over ${\mathbf{F}}_p$ and principality conditions in $\mathbf{Z}[{\zeta }_p]$. For regular $p\ge 11$, prime cofactors cannot occur at valuation $p+3$; at valuation $p+4$ they have natural density $(p+1)\mathrm{/}(h{p}^{(p-3)\mathrm{/}2})$, where $h$ is the class number of $\mathbf{Q}({\zeta }_p)$. We also show that $p^{p+5}$ is the least absolute value of a nonzero $p$-divisible determinant for these regular primes.