π as a Non-Geometric Closure Readout: From FDC Residues to Metric Circle Ratios (PCR v1)

This paper introduces Pi as a non-geometric closure readout within the Finite Distinguishability Closure (FDC) framework. The aim is not to provide a new numerical proof of the Leibniz formula, but to reverse the explanatory order usually assigned to Pi: rather than treating the Euclidean circle as the source of Pi, the paper treats the circle ratio as a downstream metric readout of a prior closure constant. Starting from the FDC axioms of finite distinguishability, non-privileged representation, single serial ledger structure, distinction preservation, distinction-generation exclusion, and mapping stability, the paper reconstructs a parameterless Leibniz-type closure residue. Under the External Completion Admissibility Principle (ECAP), this rational Cauchy-determinable residue yields an ECAP-readout value SFDC and the associated closure constant piFDC = 4 SFDC. The paper then separates three layers of interpretation: piFDC as the FDC-forced closure readout constant, piₐn as the one-variable analytic readout given by 4∫₀¹ dt/ (1+t²), and pigeom as the downstream Euclidean metric-circle ratio. The analytic identification piFDC = piₐn is established through standard one-variable analysis, while the metric identification pigeom = piₐn = piFDC is treated as a bridge theorem under an explicitly supplied metric readout layer. The result is a structural placement of Pi within the MOF–FDC–PFC–ARG program: Pi is not introduced from a primitive spatial circle, but appears as the analytic and metric readout of a finite closure residue. The paper also states the remaining limitations, including the external status of ECAP, the Archimedean readout condition, and the externally supplied arc-length functional.