A PT -Symmetric RuellePerronFrobenius Framework for the Riemann Hypothesis: Pseudo-Hermitian Metric Rigidity, Kato Exceptional Points, and BáezDuarte Density
Abstract
The HilbertPólya paradigm and the NymanBeurlingBáezDuarte L 2 (0, 1) approximation criterion represent two of the deepest functional-analytic avenues toward the Riemann Hypothesis (RH). Yet, taken in isolation, each faces a severe structural obstruction: classical self-adjoint dilation operators suer from continuous spectrum pathologies and nontrace-class perturbations, whereas the nite BáezDuarte Gram matrices G N exhibit superpolynomial ill-conditioning (κ(G N ) ≥ e c √ N ) that obscures the asymptotic cancellation of Möbius weights. In this treatise, we unify these two perspectives into a single dynamical and pseudo-Hermitian framework whose primary objective is to establish a rigorous mathematical and structural equivalence for the Riemann Hypothesis. First, working on the scale-invariant Hilbert space H = L 2 (R + , x -1 dx), we show that the antilinear paritytime involution PT composed of multiplicative inversion (Pf )(x) = f (1/x) and complex conjugation (T f )(x) = f (x) has the critical line Re(s) = 1/2 as its exact Mellin xedpoint manifold, while the BáezDuarte fractional-part kernel k(x, y) = ρ(x/y) -ρ(y/x) emerges naturally as the antisymmetric generator of the classical Gauss continued-fraction map T (x) = {1/x}. Second, by lifting the dynamics to the HardyHilbert space H 2 (D), we replace the ill-conditioned Gram matrix and the unregularized continuum perturbation with the nuclear RuellePerronFrobenius transfer operator L s , which we meromorphically continue across the critical transfer axis Re(s) = 1/4 via Hurwitz zeta regularization with explicit singular-value control up to the boundaries of the strip 0 < Re(s) < 1/2. Third, utilizing Mostafazadeh's pseudo-Hermitian metric theory (η s = e -Qs ) alongside Mayer's Fredholm determinant identity det(I -L s ) = Ω(s)/ζ(2s), we prove that any hypothetical o-critical zero ρ 0 = 1/2 + δ + it 0 (δ ̸ = 0) forces a spontaneous PT -symmetry breaking transition through a Kato exceptional point, triggering an unbounded divergence in the metric condition number ∥η s ∥ op ∥η -1 s ∥ op → ∞. Fourth, we establish a strict tripartite structural equivalence between the Riemann Hypothesis, the uniform boundedness of the PT -metric resolvent functional in the weighted Plancherel norm, and the logarithmic vanishing d 2 N = O(1/ ln N ) of the BáezDuarte projection distance. Finally, via a WeinsteinAronszajn rank-one factorization det Fred (I -L s ) = det Fred (I -R s )W (s), we prove unconditionally that the parabolic core operator R 1/4+it/2 is absolutely trace-class on Re(s) = 1/4 with O(n -3/2 ) branch decay and satises unconditional L 2 oscillatory phase cancellation, thereby reducing the innitedimensional spectral problem to a single scalar characteristic function W (s).
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BibTeX
@misc{arbai2026symmetric,
title = {{A PT -Symmetric RuellePerronFrobenius Framework for the Riemann Hypothesis: Pseudo-Hermitian Metric Rigidity, Kato Exceptional Points, and BáezDuarte Density}},
author = {Aziz Arbai},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05852361v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A PT -Symmetric RuellePerronFrobenius Framework for the Riemann Hypothesis: Pseudo-Hermitian Metric Rigidity, Kato Exceptional Points, and BáezDuarte Density
AU - Aziz Arbai
PY - 2026
DA - 2026/09/21
PB - HAL
UR - https://hal.science/hal-05852361v1
AB - The HilbertPólya paradigm and the NymanBeurlingBáezDuarte L 2 (0, 1) approximation criterion represent two of the deepest functional-analytic avenues toward the Riemann Hypothesis (RH). Yet, taken in isolation, each faces a severe structural obstruction: classical self-adjoint dilation operators suer from continuous spectrum pathologies and nontrace-class perturbations, whereas the nite BáezDuarte Gram matrices G N exhibit superpolynomial ill-conditioning (κ(G N ) ≥ e c √ N ) that obscures the asymptotic cancellation of Möbius weights. In this treatise, we unify these two perspectives into a single dynamical and pseudo-Hermitian framework whose primary objective is to establish a rigorous mathematical and structural equivalence for the Riemann Hypothesis. First, working on the scale-invariant Hilbert space H = L 2 (R + , x -1 dx), we show that the antilinear paritytime involution PT composed of multiplicative inversion (Pf )(x) = f (1/x) and complex conjugation (T f )(x) = f (x) has the critical line Re(s) = 1/2 as its exact Mellin xedpoint manifold, while the BáezDuarte fractional-part kernel k(x, y) = ρ(x/y) -ρ(y/x) emerges naturally as the antisymmetric generator of the classical Gauss continued-fraction map T (x) = {1/x}. Second, by lifting the dynamics to the HardyHilbert space H 2 (D), we replace the ill-conditioned Gram matrix and the unregularized continuum perturbation with the nuclear RuellePerronFrobenius transfer operator L s , which we meromorphically continue across the critical transfer axis Re(s) = 1/4 via Hurwitz zeta regularization with explicit singular-value control up to the boundaries of the strip 0 < Re(s) < 1/2. Third, utilizing Mostafazadeh's pseudo-Hermitian metric theory (η s = e -Qs ) alongside Mayer's Fredholm determinant identity det(I -L s ) = Ω(s)/ζ(2s), we prove that any hypothetical o-critical zero ρ 0 = 1/2 + δ + it 0 (δ ̸ = 0) forces a spontaneous PT -symmetry breaking transition through a Kato exceptional point, triggering an unbounded divergence in the metric condition number ∥η s ∥ op ∥η -1 s ∥ op → ∞. Fourth, we establish a strict tripartite structural equivalence between the Riemann Hypothesis, the uniform boundedness of the PT -metric resolvent functional in the weighted Plancherel norm, and the logarithmic vanishing d 2 N = O(1/ ln N ) of the BáezDuarte projection distance. Finally, via a WeinsteinAronszajn rank-one factorization det Fred (I -L s ) = det Fred (I -R s )W (s), we prove unconditionally that the parabolic core operator R 1/4+it/2 is absolutely trace-class on Re(s) = 1/4 with O(n -3/2 ) branch decay and satises unconditional L 2 oscillatory phase cancellation, thereby reducing the innitedimensional spectral problem to a single scalar characteristic function W (s).
ER -