Encoding-Relative Variance (ERV) Barrier (With Implications for P vs NP)

Additional technical notes and experimental reports related to the IVT, corridor width, and variance-mediated phase transitions are available on Zenodo.To locate the full body of related work, please use the Zenodo search box and enter the author name with quotation marks: "De Jesus, Elias" Using quotation marks ensures accurate retrieval of all associated records and avoids fragmentation due to name variations. Polynomial-time computability appears to be a representational notion rather than an intrinsic invariant. The distinction between P and NP may only acquire well-defined meaning at the level of encoding-quantified relations, where diagonalization becomes available. Whether any nontrivial invariant survives at that level remains an open question. We introduce Encoding-Relative Variance (ERV) as a structural barrier to a broad class of approaches to the \mathsf{P} vs.\ \mathsf{NP} problem. ERV formalizes the observation that many proposed diagnostics of computational hardness—based on variance propagation, geometric structure, or spectral properties—depend on the chosen polynomial-time encoding of problem instances rather than on the underlying language itself. Through a sequence of falsification probes and analytic constructions, we show that neither variance-based persistence measures nor spectral observables derived from constraint–variable incidence operators are invariant under polynomial-time re-encodings. Across problem families in \mathsf{P}, \mathsf{NP}-complete, and linear classes, encoding-induced variation systematically dominates asymptotic scaling effects. We formalize ERV as a barrier theorem, identify the minimal axioms it forbids, and situate it alongside relativization, natural proofs, and algebrization as an independent obstruction. The ERV barrier implies that any proof strategy for \mathsf{P} vs.\ \mathsf{NP} relying on instance-level geometric, spectral, or variance-collapse diagnostics is structurally misaligned with the encoding-independent nature of complexity classes. This work reframes the search for viable approaches by sharply restricting the space of admissible observables and proof techniques.