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A Critical Hardy-Space Contraction for the Cubic Chebyshev Trace

Blaize Rouyea · Corey Bourgeois

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abstract

We continue the cubic descent and Chebyshev trace program of the authors. The earlier work produces a finite-local trace (u_n) whose boundedness is equivalent to the Riemann Hypothesis. The present paper does not prove RH. Instead, it proves an unconditional contraction theorem for the exact prime-filter hierarchy and isolates the arithmetic obstruction that remains.

The critical Cayley coordinate chi = (s-2)/(s+1) maps the critical line to |chi| = 1. In this coordinate the singular Chebyshev branch factors through an explicit Schur function phi with phi(0) = 16/27. The trace filters admit an exact Laguerre-Hardy representation, from which we prove that the supremum over n of the weighted critical integral is finite. The first two weighted critical moments are then O(n) and O(n^2), and the prime-square quadratic variation is polynomially bounded.

A complementary S_3 Fourier calculation gives the exact identity B_0(1 - beta_1 beta_2) = 6 and identifies the normalized standard-channel Green defect with the Chebyshev-Krein U-port. On the real invariant axis the completed source has a rank-one negative future component. The completed endpoint residue is then represented by a coefficient of H G phi^(n-1), where all filter growth is controlled and every possible interior exponential instability is carried by a pole of xi'/xi. An explicit symmetric, positive, real-log-convex countermodel with off-symmetry zeros shows why the real-axis contraction cannot be promoted by a generic maximum principle. Thus the remaining pole exclusion is left explicit rather than hidden inside a Hardy-space assumption.

contents

  1. 1. Introduction1
  2. 2. Minimal setup from the cubic descent and trace3
  3. 3. The Krein block and the no-gain formulation3
  4. 4. An S_3 current compression4
  5. 5. An unconditional source contraction on the real invariant axis6
  6. 6. The critical Cayley coordinate7
  7. 7. The cubic Schur multiplier8
  8. 8. Critical Laguerre-Hardy representation of the trace filters10

topics

number theoryriemann hypothesishardy spaceschur functionchebyshev polynomialskrein spaceprime filters

cite

@article{rouyea2026hardycontraction, title={A Critical Hardy-Space Contraction for the Cubic Chebyshev Trace}, author={Rouyea, Blaize and Bourgeois, Corey}, year={2026}, note={Draft}, eprint={2608.hardy-space-contraction-chebyshev-trace}, archivePrefix={alphaXiv} }