Integral Quadruples, Three-Square Representations, and Fixed-Point Density

We examine ten interrelated entries in the On-Line Encyclopedia of Integer Sequences concerning primitive integral quadruples with prescribed sum and sum of squares, and unrestricted counts of the same type. An integral Hadamard correspondence reduces six primitive counting problems (A278081--A278086) to representations by three squares; the classical three-square formula then gives multiplicativity in the appropriate parameters and all prime-power factors. The same method evaluates A354766 and A354778, while a parity-sensitive transformation expresses every entry of the two-variable triangle A354777 in terms of the three-square representation function. For A354766, we characterize all positive integers satisfying $a(n)=n$, exhibit an explicit infinite family, and derive #≤ x:a(n)=n\~ Cx/√log x, where C is an explicit Euler product. Consequently the fixed points have natural density zero, a conclusion that differs from the positive-density conjecture recorded in OEIS. A Burnside argument also relates the orbit count A105970 to A278083 and primitive representations by u²+2v². We also derive a Dirichlet generating function, a quadratic summatory estimate with an explicit error order, the asymptotic odd--even distribution of fixed points, and density of the normalized values $a(n)/n$ in the positive real numbers. The arithmetic identities are proved using classical representation theory and analytic number theory, while finite computations provide reproducible numerical checks.