A Boundary-Layer Calculus for Chebyshev Prime Filters
Blaize Rouyea · Corey Bourgeois
abstract
We isolate a local scaling phenomenon associated with a Chebyshev trace construction for the Riemann xi function. Near the exceptional point s = 2, the orbital weight W(s) = 27/(4 q(s)^2) has a double pole. This forces the ordinary Chebyshev generating radius to collapse. After the diagonal scaling v = lambda t^2, x = mu/t, t = s - 2, the collapse resolves into two distinct microscopic objects depending on the order of operations.
Formal summation before residue extraction gives a geometric resolvent with critical value lambda = 3/16, whereas legal residue extraction first produces factorially regularized prime-filter diagonals. The leading diagonal sums to a hyperbolic-sine master function. Successive diagonals are obtained from it by polynomial differential operators in the dilation generator D = (z/2) d/dz. We give the first five scaling functions and an exact coefficient mechanism generating all further diagonals. The result is local and asymptotic; no claim toward a proof of the Riemann hypothesis is made.
contents
- 1. Introduction1
- 2. Setup and statement of the local result2
- 3. The collapsing Chebyshev radius3
- 4. Residue-first scaling and the master function3
- 5. Diagonal differential calculus4
- 6. An exact mechanism for all diagonals6
- 7. Interpretation and scope7
topics
cite
@article{rouyea2026boundarylayer,
title={A Boundary-Layer Calculus for Chebyshev Prime Filters},
author={Rouyea, Blaize and Bourgeois, Corey},
year={2026},
note={Draft},
eprint={2608.boundary-layer-calculus-chebyshev-prime-filters},
archivePrefix={alphaXiv}
}