Encoding-Invariant Variance Amplification and the Failure of Encoding-Invariant Collapse: A Diagnostic Probe

We report a pair of controlled computational probes investigating whether variance propagation in constraint satisfaction problems is invariant under polynomial-time re-encoding. Variance behavior is quantified using a persistence gradient, defined as the logarithmic derivative of relational variance with respect to instance size. In the first probe, we study XOR-SAT under three distinct encodings: canonical XOR clauses, expanded CNF gadget encodings, and linear-algebraic representations over GF(2). Across all encodings, relational variance exhibits consistent amplification with a strictly positive persistence gradient, indicating encoding-invariant variance amplification at fixed constraint density. In the second probe, we test for encoding-invariant variance collapse in graph reachability using local-transition, matrix-powering, and CSP-style encodings. In contrast to XOR-SAT, the sign of the persistence gradient varies across encodings, with amplification observed under local-transition encodings and weak collapse or stability under global encodings. This falsifies the hypothesis that variance collapse is encoding-invariant. Taken together, these results demonstrate a structural asymmetry: variance amplification can be an intrinsic property of a constraint operator, while variance collapse is observer-dependent and algorithm-specific. The findings clarify why variance collapse cannot define tractability, and why encoding-invariant amplification is a more robust diagnostic for structural hardness. No claims are made regarding complexity class separations.