Correlation laws and identifiability of multiplicative functions on quadratic traces

We study which completely multiplicative sign functions can be identified from the statistics of \(h(n^2+1)\), \(h(n^2+2)\) and \(h(n^2-2)\). In a class of finite products of quadratic characters of odd prime conductor, completed without zeros and modified at primes with summable inverse square roots, diagonal correlations of orders two and four determine the function. The second-order data recover the conductor and prime corrections, leaving exactly the ramified local signs undetermined. Fourth-order detectors recover those signs, including at the small primes; the inverse also extends to quadratic components at two. No bounded horizon suffices uniformly: for every prescribed window, two finite-conductor sources can share its complete law and disagree with density arbitrarily close to one half on each trace. For an arbitrary prescribed source \(h\), we construct factors \(fg=h\) such that, for each factor separately, the three traces jointly have the uniform word law at every horizon. The choice is continuous and computable relative to \(h\), with a finite bound uniform in \(h\) for approximation of a canonical synchronous coupling. The same choice treats every horizon sequence \(o(loglog N)\). The factors need not be independent. Their synchronous empirical law projects exactly to the product's empirical word law; when a limiting canonical coupling exists, it retains the product's word law, which need not identify \(h\) as a function.