POLYGON GEOMETRY OF THE M ¨OBIUS FUNCTION: SPECTRAL AREA, AD`ELIC STRUCTURE, AUTOMORPHIC LIFT, AND THE PRIME NUMBER THEOREM

This paper develops a comprehensive polygon-geometric framework for the Möbius function, unifying four companion theories. The central object is the Möbius walk polygon — the planar curve whose n-th step has signed length μ(n) and direction e^{2πinθ} — whose signed area equals the sine transform of the Möbius pair correlation R_μ(h). The paper proves that Chowla's conjecture is equivalent to the polygon having negligible area, that the Mertens function M(N) is the closure deficit at θ = 0, and that the Riemann Hypothesis is equivalent to the near-closure condition |M(N)| = O(N^{1/2+ε}). The adèlic bridge section proves that the digit polygon of 1/p and the Möbius walk polygon are local and global projections of a single adèlic object, unified via Dirichlet characters. A Character Bridge Theorem identifies a shared spectral basis in Gauss sums; a Bilinear Area Identity expresses the Möbius polygon area at rational angles as a bilinear form in Gauss sums and twisted Mertens functions; and a Local–Global Area Decomposition via the Euler product shows that the digit polygon's p-local factor assembles the global Möbius area exactly as 1/ζ(s) = ∏_p(1 − p^{−s}). A Non-Independence Theorem proves that unconditional bounds on coset energy (Artin data) constrain twisted pair correlations (Chowla data), and conversely. The automorphic section lifts the entire framework to Hecke characters on the idèle class group via Tate's thesis. A Universal Spectral Area Theorem encompasses as special cases the GL(2) spectral area theorem for modular form Fourier coefficients, the Rankin–Selberg Parseval Identity, and the Sato–Tate Area Equivalence. The final section connects polygon geometry to the Prime Number Theorem and the zero-free region. A Displacement–Area Identity relates the fourth moment of the walk's endpoint to total polygon area. A Polygon Prime Number Theorem reformulates PNT via the trigonometric positivity kernel. The Area–Zero-Free Chain Theorem (new in v3) proves that the pair correlation savings hypothesis PCS(δ) — bounding ΣR_μ(h)² ≤ CN^{3−2δ} — combined with the Nikolsky inequality for trigonometric polynomials, implies a zero-free strip for ζ(s) of width δ/2, representing a qualitative advance over the classical Vinogradov–Korobov region. The exponent hierarchy (new in v3) shows that the pair correlation exponent α interpolates continuously between the trivial bound (α = 3) and RH (α = 2). The paper also establishes a connection to the Matomäki–Radziwiłł theorem, yielding unconditional Cesàro pair correlation bounds. All identities are verified computationally for N ≤ 50,000 and p ≤ 1,000, with large-N FFT stress tests up to N = 10^7 yielding a fitted exponent α̂ ≈ 2.00, consistent with GRH and far from the trivial bound.