A Chebyshev Trace Criterion for the Riemann Hypothesis

We give a self-contained synthesis of an order-three descent of the Riemann xi function and isolate the exact point at which its earlier positivity criteria become a boundedness criterion. The anharmonic group generated by w(s) = (s-1)/s and sigma(s) = 1-s has an invariant q(s) whose orbit product of xi descends to an entire function K(q^2) of order 1/2 and genus zero. Nontrivial critical-line zeros are transported to (-infinity, -27/4), while K(x) > 0 for x >= 0. This yields one local moment sequence whose equivalent forms are an unshifted Hankel tower, a Hausdorff moment sequence, a compact Jacobi realization, a two-index beta lattice (Pascal form), and a completely monotone boundary.

We then make a lossless triangular change of basis. For the normalized zero nodes, the resulting Chebyshev trace u_n is an integer linear combination of the local moments and therefore a finite residue functional over a single exceptional orbit. Writing u_0 = h_0 > 0, we prove RH is equivalent to |u_n| <= h_0 for all n, equivalent to sup_n |u_n| < infinity, equivalent to limsup |u_n|^(1/n) <= 1. The reverse implication is carried by one rigidity fact: escaping poles cannot cancel, so an off-line zero becomes an uncancellable pole inside the unit disk and forces exponential coefficient growth.

Finally, a uniform Bernstein-Walsh theorem shows that Chebyshev is exponent-optimal, up to an absolute factor, among all degree-n probes retaining finite three-point locality. A scale-dependent Mobius scan improves the conformal exponent, and we prove exactly why that improvement costs finite locality. The infinite verification remains infinite; what changes is the form of failure, from a hidden sign defect to an explicit interior singularity. A direct proof of the completed prime-archimedean trace bound would imply RH; that inequality is not proved here.

1. 1. Introduction: what changes, and what does not1
2. 2. The descent is forced3
3. 3. From zero location to the positivity tower5
4. 4. The lossless Chebyshev coordinate8
5. 5. Koebe uniformization and the generating function10
6. 6. Boundedness, poles, and Green escape11
7. 7. Carathéodory, Schur, Clark measure, and a Hermite–Biehler lift16
8. 8. A Wiener variance law19
9. 9. Trace filters on the von Mangoldt measure20
10. 10. Numerical illustration22
11. 11. Context, contribution, and the remaining problem23