A Constructive Coherence-Tower Theorem for a U(1) Lattice Toy Model of Super Information Theory

This preprint gives a self-contained finite lattice gauge construction showing that a hidden configuration space can remain abundant after gauge identification and still support a superlinear family of distinguishable observable relations under a defined toy-model interface. The paper provides exact gauge-orbit classification, explicit refinement maps, uniform readout fibers, a closed-form sufficient abundance threshold, three explanatory figures, and a deterministic clean replay. The model uses a finite cyclic subgroup embedded in the unit circle. Version 2 also corrects inherited quotient-group wording so that the terminology matches the mathematics. This is a mathematical consistency and realizability result for a finite toy model. It does not establish the full physical theory, a continuum lift, empirical validation, or a result about the Riemann Hypothesis.