Computing the Riemann Hypothesis: Thorin Measures, Pólya Densities and Hitting Times, version 1.1.0

Version 1.1.0

This version adds two chapters after Chapter 13 and before the appendices: Chapter 14, "The Pólya density as a maximum order statistic and the Thorin measure of the clock", after the first author's manuscript of October 2026, and Chapter 15, "The Thorin phase diagram", after a working report of the authors of October 2026. The chapters, sections, theorems, equations, exercises and problems of version 1.0.0 keep their numbers; only the last section of Appendix A, on the reruns, moves up one number, and the page numbers change.

What Chapter 14 adds:

• The Pólya kernel of the Riemann Xi function as the density of the square root of a sum of two independent copies of the maximum of independent exponential variables, and the Khinchin factorisation of the tilted maximum through the zeta distribution.
• Wald couples, with the logistic law as a normal scale mixture with Kolmogorov mixing read through the maximum, and the clock in a Wald couple with the Pólya variable.
• A criterion for the Thorin condition in the variable -x-iη: the clock is a generalised gamma convolution if and only if a smoothed Thorin density is nonnegative, and a zero off the critical line shows as an atom of negative mass at an explicit point.
• An explicit generalised gamma convolution within a rigorously bounded Laplace distance of the clock, built from the zeros verified up to height 3 x 10^12 and with no assumption on the zeros above it; the bound is computed in ball arithmetic.
• The term of one zero in the Thorin density of the basepoint family in the variable λ, and a finite form of the basepoint density below α = 1 with explicit pole terms and a Fejér taper.
• A proof that the clock is not a maximum of independent exponential variables, and the Gumbel norming constant of the clock from Gnedenko's extreme value theory.
• The finite stages of the maximum and their symmetrisations, with the heights up to which their zeros lie on the critical line.
• Hyperbolic complete monotonicity of the clock to a finite order, and the equivalence of the Riemann Hypothesis with membership of the clock in the Bondesson class.
• Open problems and a table of the numerical checks behind the chapter.

The chapter comes with twelve new scripts (code/maxwald_.py), the data they produce (data/maxwald_.json) and nine figures, and with twelve new bibliography entries. The catalog of results has a unit for the chapter and the repository a summary of it.

What Chapter 15 adds:

• A classification of the computations of the book by what they can see above the height to which the zeros are verified, and the three routes that reach every height.
• The threshold theorem along the de Bruijn Newman flow: at every basepoint alpha >= 1/2 the clock of xi_t is a generalised gamma convolution exactly when no zero of xi_t lies to the right of alpha.
• The Thorin phase diagram: de Bruijn's strip theorem in Thorin form, which carries the certified points (t, alpha) along parabolas and gives the de Bruijn Newman constant as the minimum of t + 2(alpha - 1/2)^2 over them, with the known results placed on it (the critical strip, Polymath15's Lambda <= 0.22, Platt and Trudgian's Lambda <= 0.2, and a conditional half plane).
• The Thorin density at positive time, computed from Pólya's integral with no primes and no zeros, and how little it differs from that at time zero on the windows computed.
• A certificate for one point of the diagram with explicit constants, and its finite part checked in ball arithmetic below height 160.
• Two experiments on where the missing uniform step could come from: the Weil Gram matrix on compactly supported test functions and the motion of the zeros along the flow, with a collision of two zeros under the backward flow located and bounded.
• The cost of the flow route, read from the table of conditional bounds of Polymath15: the verification height that each bound on Lambda needs, and what that implies for bounds below 0.15.
• A programme, five open problems and fourteen exercises.

The chapter comes with seven new scripts (code/phase_.py), the data they produce (data/phase_.json) and five figures, described in a new section of Appendix A, and with two new bibliography entries, the authors' working report and Chapter 21 of their draft on the Langlands conjectures. The catalog of results has a unit for the chapter and the repository a summary of it.

Outside the two chapters the text changed in six places: two sentences of Chapter 1 that describe the new chapters, one sentence in the introduction to Chapter 8 that points to Chapter 14, one clause of the preface that credits the manuscript and the working report behind them, one row of Appendix A that names the scripts of Chapter 14, the new section of Appendix A for the scripts of Chapter 15, and one sentence of Appendix A on the runs of the new scripts.