A Constructible Tower Approaching π: Explicit Compass-and-Straightedge Approximants under Transparent Algebraic Acceleration

This revised mathematical note presents an explicit sequence of compass-and-straightedge constructible polygonal half-perimeters approaching the transcendental constant π, together with rational Richardson–Romberg extrapolation. Every finite entry of the resulting triangular tableau remains constructible. The analysis derives an explicit leading diagonal-error law, establishes the corrected quadratic digit-growth coefficient log₁₀(2), and develops refined asymptotic terms for the convergence rate. A reproducible 650-decimal-digit numerical verification accompanies the manuscript, including computed diagonal approximants, error tables, and comparisons with the leading asymptotic model. The work distinguishes finite algebraic constructibility from convergence to a nonconstructible limit. It does not claim an exact compass-and-straightedge construction of π, a new value for π, or a record-speed numerical algorithm. The original manuscript dates from May 24, 2026; the authoritative v1.3 revision is dated October 29, 2026. The accompanying archive includes the five-page manuscript, LaTeX source, verification code, numerical output, reproducibility instructions, and SHA-256 checksums.