One Euler Product, Many Coordinates: Bernoulli, Sine, Zeta Continuation, and Radial Layer-Counting as Shared Analytic Infrastructure

Several small constants recur across elementary number theory, analytic continuation, lattice visibility, and geometric normalization. This note separates two cases. In the first case, repeated constants are structurally meaningful because the same analytic identity can be written down. Examples include the Euler product for \(\zeta(s)\), the prime-zeta decomposition of \(\log\zeta(s)\), the Bernoulli generating function \(u/(e^u-1)\), Euler's sine product, and the zeta functional equation linking sine, Gamma, and Bernoulli values. In the second case, constants are merely collisions in a small denominator pool. The first-shell Beal value \(\chi(3,3,4)=-1/12\) and the zeta value \(\zeta(-1)=-1/12\) are numerically equal but arise from different mechanisms. Likewise, the denominator \(84\) in a 45-degree cone void and the intermediate denominator \(84\) in \(\zeta(-13)=-7/84=-1/12\) are not structurally linked. The rule used throughout is: no shared identity, no shared mechanism. The result is a cooled atlas of genuine shared analytic infrastructure, not a claim that unrelated systems prove or explain one another.