Mathematical Rigor Audit and Strengthened Derivations: Chapters 27–29 of the Tau Universe Mathematical Extensions

This preprint combines Chapters 27–29 of The Tau Universe Codex: Mathematical Extensions — From Logistic Resonance to Tav Topology (Volume 4, 2026). It provides a rigorous computational audit of three central mathematical claims connecting Tav Topology to the logistic map: (1) the control parameter formula r(τ) causes τ to cancel identically; (2) the resulting value lies in the chaotic regime rather than the superstable period-6 window; (3) the orbit points are not equal to k/7. For each issue we propose concrete correction paths — including a mode-coupling derivation yielding r(τ) = 4 − C_KK/τ² and a permutation conjugacy theorem linking the orbit to 1/7 = 0.142857 — that preserve the core physical framework while restoring mathematical consistency. All numerical results are verified to 15-digit precision using IEEE 754 double-precision arithmetic. Companion papers (forthcoming on Zenodo) Corrected Mode-Coupling Derivation and Permutation Conjugacy Theorem Feigenbaum Universality and the Exact Period-6 Polynomial in τ-Space Full volume (Amazon KDP, Volume 4, 2026) The Tau Universe Codex: Mathematical Extensions — From Logistic Resonance to Tav Topology