The Vitruvian Approximation: Rational Circle-to-Square via the G-Cycle and the Principia Orthogona Selection Functional

Abstract We formalize a minimal instance of the C→K→F→U operator chain from the Generative Contact Mechanics (GCM) framework acting on rational circle data. The paper pursues three honest goals. First, we show that one G-cycle applied to a rational circle of diameter d produces a rational square of area (8d/9)² — recovering the Rhind Papyrus approximation as an explicit computation rather than a new derivation. Second, we examine what the dm³ stability conditions (μ_max < 0, ε₀ = 1/3, τ = 2) actually constrain in the rational approximation setting, finding that they confirm admissibility of the Rhind compression but do not select it — a genuine open problem. Third, we propose an explicit selection functional _rat acting on rational approximants to π, motivated by the contact- 𝒞 geometric structure, and state the key conjecture: that the minimizers of _rat 𝒞 with bounded denominator are exactly the continued-fraction convergents of π. All Lean 4 statements are included; the main conjecture is marked sorry in the tradition of Axiom 9 (Honest Incompleteness). This work does not claim to solve the classical squaring problem or to circumvent transcendence results; it operates entirely in the rational approximation regime. The paper is part of a series whose domain applications include plasma reconnection [B3CH3], supernova remnant dynamics [B3TEX], and galactic merger attractors including the Milkomeda fixed point [B3CH7]; the present paper provides a minimal algebraic foundation for such applications in the rational approximation setting. Keywords: rational approximation, squaring the circle, Rhind Papyrus, continued fractions, contact geometry, dm³, TO/TOGT, Lean 4, honest incompleteness