A Variational Principle for the Parabolic Response Function (Part II): Lyapunov Scale-Dissipation, Energy Curvature, and the Geometric Prohibition of O-Critical Zeros
Abstract
In Part I of this program, we unied the PT -symmetric spectral paradigm with the RuellePerronFrobenius ergodic theory of the Gauss continued-fraction map on the Hardy space H 2 (D). Through the WeinsteinAronszajn rank-one factorization det Fred (I -L s ) = det Fred (I -R s )W (s) and the block trigonalization of the parabolic core R s on C|1⟩ ⊕ H 2 0 (D), we showed that
is the parabolic core response function enjoying O(n -5/2 ) branch super-convergence on Re(s) = 1/4. Crucially, we proved that any hypothetical o-critical zero
In the present treatise (Part II), we develop a variational and dierential-geometric framework to rule out this two-point collision. First, utilizing the O(n -2σ-2 ) super-convergence of R s on H 2 0 (D) alongside the Uniform Non-Integrability (UNI) of the continued-fraction roof function, we establish the spectral invertibility of I H 2 0 -R s and derive the convergent NeumannHurwitz hyperbolic correlation series for Φ core (s). Second, we introduce the radial LyapunovDirichlet energy functional E τ (σ) := |Φ core (σ + iτ )| 2 across horizontal segments σ ∈ [1/4 -δ, 1/4 + δ] and combine the subharmonic modulus Laplacian ∆E τ = 4|Φ ′ core (s)| 2 ≥ 0 with parametric operator calculus. Third, we explicitly compute the Lyapunov scale-dissipation operator D s := -∂ Rs ∂σ , proving the exact remainder-free right factorization
. By intertwining D s with Mostafazadeh's pseudo-Hermitian metric η s = e -Qs , proving unconditional Koopman cylinder positivity and unconditional L 2 MontgomeryVaughan radial convexity, and formulating the BergmanPoisson subharmonic bridge, we establish that under pointwise bilinear chord coercivity, Rolle's Theorem forbids Φ core (σ +iτ ) from intersecting the Kato symmetrybreaking threshold 1 at two distinct symmetric points 1/4 ± δ + iτ , conning all non-trivial zeros to Re(z) = 1/2.
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BibTeX
@misc{arbai2026variational,
title = {{A Variational Principle for the Parabolic Response Function (Part II): Lyapunov Scale-Dissipation, Energy Curvature, and the Geometric Prohibition of O-Critical Zeros}},
author = {Aziz Arbai},
howpublished = {HAL},
year = {2026},
month = nov,
url = {https://hal.science/hal-05852389v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Variational Principle for the Parabolic Response Function (Part II): Lyapunov Scale-Dissipation, Energy Curvature, and the Geometric Prohibition of O-Critical Zeros
AU - Aziz Arbai
PY - 2026
DA - 2026/11/02
PB - HAL
UR - https://hal.science/hal-05852389v1
AB - In Part I of this program, we unied the PT -symmetric spectral paradigm with the RuellePerronFrobenius ergodic theory of the Gauss continued-fraction map on the Hardy space H 2 (D). Through the WeinsteinAronszajn rank-one factorization det Fred (I -L s ) = det Fred (I -R s )W (s) and the block trigonalization of the parabolic core R s on C|1⟩ ⊕ H 2 0 (D), we showed that is the parabolic core response function enjoying O(n -5/2 ) branch super-convergence on Re(s) = 1/4. Crucially, we proved that any hypothetical o-critical zero In the present treatise (Part II), we develop a variational and dierential-geometric framework to rule out this two-point collision. First, utilizing the O(n -2σ-2 ) super-convergence of R s on H 2 0 (D) alongside the Uniform Non-Integrability (UNI) of the continued-fraction roof function, we establish the spectral invertibility of I H 2 0 -R s and derive the convergent NeumannHurwitz hyperbolic correlation series for Φ core (s). Second, we introduce the radial LyapunovDirichlet energy functional E τ (σ) := |Φ core (σ + iτ )| 2 across horizontal segments σ ∈ [1/4 -δ, 1/4 + δ] and combine the subharmonic modulus Laplacian ∆E τ = 4|Φ ′ core (s)| 2 ≥ 0 with parametric operator calculus. Third, we explicitly compute the Lyapunov scale-dissipation operator D s := -∂ Rs ∂σ , proving the exact remainder-free right factorization . By intertwining D s with Mostafazadeh's pseudo-Hermitian metric η s = e -Qs , proving unconditional Koopman cylinder positivity and unconditional L 2 MontgomeryVaughan radial convexity, and formulating the BergmanPoisson subharmonic bridge, we establish that under pointwise bilinear chord coercivity, Rolle's Theorem forbids Φ core (σ +iτ ) from intersecting the Kato symmetrybreaking threshold 1 at two distinct symmetric points 1/4 ± δ + iτ , conning all non-trivial zeros to Re(z) = 1/2.
ER -