Relative Value Phases on Semantic Phase Spaces

This preprint develops a device- and architecture–independent theory of relative value phases on semantic phase spaces built from observation geometries. Starting from a previous framework where semantic phase transitions arise as sharp thresholds in Poisson–Boolean models on Ahlfors-regular metric–measure spaces, this work asks how “value” can be assigned not to individual tokens but to macroscopic semantic phases in a representation-invariant way. Part (A) introduces a semantic phase category and value evaluation bases inspired by the Theory of Relativity of Theories (TRoT). Value assignments are treated as monotone, order-enriched maps from phases into a quantale-like value lattice. Under TRoT-style base transformations and Blum-type device-independent complexity, a short categorical argument yields a representation-invariant value order theorem: the induced partial order on semantic phases is preserved under all admissible changes of representation and evaluation basis. Part (B) combines phase-sensitive order parameters from continuum percolation with a locally constant (or at least locally analytic) complexity penalty. This yields value assignments whose raw values exhibit genuine phase transitions at the same critical parameters as the underlying semantic system. The non-analyticity is proven to arise entirely from semantic structure, not from the complexity term. Part (C) is the main technical contribution. Within an idealised but mathematically transparent class of stabilising, local, symmetric, and approximately additive valuation functionals on Poisson point processes, a token-level collapse theorem shows that the expected valuation is real-analytic in the intensity parameter and therefore cannot encode percolation-type non-analyticities. In contrast, cluster-level valuations built from continuum percolation order parameters naturally exhibit non-analytic behaviour. Combining these results yields a phase-level gap theorem: any representation-invariant, phase-sensitive value topology that is constant on each phase (in expectation) cannot be realised by token-level valuation in this class and must, at minimum, be attached to clusters and phases.