Dimensional Projection, Shannon Efficiency, and the Thales Geometry of Physical Constants: A Structural Hypothesis Connecting Entropy Phases, Coupling Cascades, and the Involutive Role of e

This paper proposes a structural hypothesis linking three independently developed frameworks: (i) a geometric cascade underlying the fine-structure constant within the RTLI framework, (ii) an entropy-phase description of black-hole evaporation, and (iii) the Thales semicircle construction for constrained binary partitions. Although originating in distinct physical and mathematical contexts, all three frameworks exhibit a shared pattern of progressive dimensional reduction accompanied by near-optimal preservation of a geometric mediator, defined as the geometric mean h = \sqrt{ab}. We show that mediator preservation in the range 90\%–95\% arises systematically in these systems and significantly exceeds the random-partition baseline of 78.5\%. This preservation level coincides with the unique 90\% efficiency threshold of Shannon entropy, occurring at asymmetry \chi = 1/e, a correspondence not shared by Rényi or Tsallis entropies. The Thales construction provides a natural geometric coordinate system in which this efficiency threshold acquires a simple interpretation, and normalization by e yields a set of hyperbolic functions evaluated at 1/2 as native geometric quantities. We interpret the constant e as an organizing unit of asymmetric self-reference and suggest that scalar constants such as \alpha^{-1} may be understood as numerical residues encoding the history of dimensional projection rather than as invertible geometric objects. The hypothesis is explicitly framed as falsifiable: we identify random baselines, counter-examples, failure modes, and observational regimes in which the proposed structure should not apply. No claim of universality or derivation from first principles is made. The paper establishes a constrained, testable framework intended as a foundation for further investigation into efficiency-limited geometric projections in physical systems.