The Random Partition Baseline: Derivation of π/8 as the Expected Mediator in the Thales Semicircle Construction

This note establishes a foundational geometric result for binary partitions: under uniform random selection of a partition point on the unit interval, the expected value of the mediator h = \sqrt{ab}, \quad a + b = 1, is exactly \pi/8. The result is derived rigorously using the Thales semicircle construction, where the mediator corresponds to the altitude of an inscribed right triangle. Evaluating the expectation \langle h \rangle = \int_0^1 \sqrt{a(1-a)}\,da yields \pi/8, which admits both an analytic derivation via trigonometric substitution and a direct geometric interpretation as the area of a semicircle of radius 1/2. We interpret \pi/8 \approx 0.393 as a random-partition baseline: the coupling amplitude expected when no structure, bias, or selection principle governs the partition. Relative to the maximal mediator h_{\max} = 1/2, this corresponds to 78.5% coupling efficiency. This value therefore provides a non-trivial reference against which structured, optimized, or physically selected partitions can be compared. The note is intentionally minimal. It derives a single exact result, interprets it geometrically, and explicitly states its limitations. No physical laws, selection principles, or universality claims are proposed. The contribution is foundational: it supplies a principled baseline necessary for distinguishing random allocation from genuine structural effects in systems that admit binary partition descriptions.