This paper develops an exact polygon geometry for the Mobius function by studying the oriented area of the polygonal walk whose steps are the complex numbers mu (n) times the unit complex number e^ (2 pi i n theta) as n ranges from 1 to N. The central result is an exact identity expressing the oriented area of this Mobius walk polygon as one-half the finite sine transform of the truncated binary pair correlations: the area at angle theta equals one-half the sum over lags h from 1 to N minus 1 of the lag-h pair correlation of the Mobius function times the sine of 2 pi h theta. The pair correlation at lag h is the sum of mu (n) mu (n plus h) over n from 1 to N minus h. Four exact consequences follow immediately. A Fourier inversion formula recovers each individual pair correlation from the area function by integrating the area against the corresponding sine. A Parseval identity equates eight times the integral of the squared area to the sum of squared pair correlations, establishing a precise equivalence between the quadratic size of the area function and the quadratic size of the pair-correlation sequence. A cosine expansion of the squared modulus of the walk endpoint, combined with the Parseval identity, yields an exact fourth-moment identity: the integral of the fourth power of the modulus of the walk equals the square of the squarefree count plus sixteen times the integral of the squared area. Special-angle evaluations show the area vanishes at theta equal to 0 and 1/2 for all N, and give an alternating combination of odd-lag correlations at theta equal to 1/4. The exact rational-angle character decomposition of the walk at theta equal to a/q is proved, expressing the walk as a linear combination of twisted Mertens sums weighted by Gauss-type sums over characters modulo q, plus an explicit error term supported on the non-coprime residues. The error term is bounded by the floor of N/p for prime moduli. The necessity of this nonunit error term is established: any expansion using only multiplicative characters vanishes on nonunits and is therefore incomplete. Geometric reformulations of classical analytic number theory results follow. The prime number theorem is equivalent to the statement that the walk endpoint at theta equal to 0 is o (N). The Riemann Hypothesis is equivalent to the endpoint being O (N^ (1/2 plus epsilon) ) for every positive epsilon. A fixed-window Chowla criterion is proved: for fixed window size H, the H individual pair correlations each being o (N) is equivalent to the supremum of the truncated area polynomial being o (N), which is in turn equivalent to the L² norm of the truncated area polynomial being o (N²). An explicit remark identifies the scaling obstruction that prevents a global o (N²) pointwise bound from recovering individual correlations, since a single correlation of size cN contributes only an O (N) sine mode that is still o (N²). The paper records the exact companion results from the digit-polygon theory: the Dirichlet-energy identity expressing the signed area of the digit polygon of 1/p in base b as negative one-quarter the sum of squared lag-2 differences of the digit sequence, and the spectral formula expressing the same area as a negative weighted sum of squared discrete Fourier transform coefficients of the residue orbit with explicit spectral weights. A structural comparison table identifies the parallel: both theories diagonalize exact quadratic area energies after Fourier analysis, the digit polygon using the cyclic discrete Fourier transform on the finite orbit group and the Mobius walk using the continuous sine basis on the circle. The relationship to the geometric sieve for Artin's conjecture is noted, with an explicit statement that no theorem currently converts the digit-side deficiency into Mobius pair correlations.
Kevin Fathi (Sun,) studied this question.