This paper develops a comprehensive polygon-geometric framework for the Möbius function, unifying four companion theories into a single work. 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 (pair correlations are o (N) ) is equivalent to the polygon having negligible area, that the Mertens function M (N) is the closure deficit of the walk at θ = 0, and that the Riemann Hypothesis is equivalent to the near-closure condition |M (N) | = O (N^1/2+ε). The second part establishes that the digit polygon of 1/p and the Möbius walk polygon are local and global projections of a single adèlic object. 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 third part lifts the entire framework to Hecke characters on the idèle class group via Tate's thesis. A Universal Spectral Area Theorem subsumes 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 geometric statement that Sato–Tate equidistribution is equivalent to negligible polygon area. The final part connects polygon geometry to the distribution of primes. A Displacement–Area Identity relates the fourth moment of the walk's endpoint to total area. A Polygon Prime Number Theorem reformulates PNT via the classical trigonometric positivity kernel. A Large Sieve Displacement Bound and an Area Large Sieve convert the Vinogradov–Korobov zero-free region into exponential savings on the Möbius polygon area, far below the trivial bound. The paper identifies the area gap — the precise additional information the area invariant provides beyond any finite set of displacements — and formulates what further polygon constraints would extend the classical zero-free region. All identities are verified computationally for N ≤ 50, 000 and p ≤ 1, 000.
Kevin Fathi (Sun,) studied this question.