The Bergman-Poisson Subharmonic Bridge and PickLoewner Bilinear Coercivity (Part III): Coboundary Pinning at z = 1, Green Capacity Bounds, and Off-Diagonal Phase Dominance
Abstract
In Parts I and II of this research program, we unied the PT -symmetric spectral paradigm with the RuellePerronFrobenius thermodynamic formalism of the Gauss map T (x) = {1/x} on the Hardy space H 2 (D). Via the WeinsteinAronszajn factorization det Fred (I -L s ) = det Fred (I -R s )W (s) and block trigonalization on C|1⟩ ⊕ H 2 0 (D), we showed thatis the parabolic core response function, and proved that any hypothetical o-critical zero
, unconditional Koopman cylinder positivity, and unconditional L 2 MontgomeryVaughan radial convexity, isolating the transition from the BergmanPoisson subharmonic identity (Eq. 5.7) to the pointwise bilinear chord derivative (Eq. 5.8) as the ultimate analytical target.
In the present treatise (Part III), we develop three new analytical tools to analyze and bridge Equations (5.7) and (5.8). First, exploiting the endpoint adjacency τ m (1) = 1 m+1 = τ m+1 (0) of the Gauss branches, we prove an Exact Coboundary Telescoping Theorem at z = 1: for every f ∈ H 2 0 (D) and every Re(s) > 0, ( R s f )(1) = -f (1) and h s (1) = ζ(2s, 2) -ζ(2s) ≡ -1. Consequently, the resolvent state ψ s = (I H 2 0 -R s ) -1 h s is pinned to the exact constant value ψ s (1) ≡ -1/2 across the entire complex plane, splitting Φ core (s) = -1/2 + Φ ≥2 (s) where Φ ≥2 (s) = -∆ 2 h D 2 (X; h), we establish unconditional diagonal dominance across all horizontal chords, forbidding the two-point collision and conning all non-trivial zeros of ζ(z) to the critical line Re(z) = 1/2.
Related
- The BergmanPoisson Subharmonic Bridge and PickLoewner Bilinear Coercivity (Part III): Coboundary Pinning at z = 1, Green Capacity Bounds, and O-Diagonal Phase Dominance
- A holomorphic motion with non-subharmonic Hausdorff dimension
- Minimal m-subharmonic functions with nonmaximal Hessian measures
- Maximal subextension of m-subharmonic functions
- Complete harmonic metrics and subharmonic functions on the complex plane
BibTeX
@misc{arbai2026bergmanpoisson,
title = {{The Bergman-Poisson Subharmonic Bridge and PickLoewner Bilinear Coercivity (Part III): Coboundary Pinning at z = 1, Green Capacity Bounds, and Off-Diagonal Phase Dominance}},
author = {Aziz Arbai},
howpublished = {HAL},
year = {2026},
month = nov,
url = {https://hal.science/hal-05855807v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Bergman-Poisson Subharmonic Bridge and PickLoewner Bilinear Coercivity (Part III): Coboundary Pinning at z = 1, Green Capacity Bounds, and Off-Diagonal Phase Dominance
AU - Aziz Arbai
PY - 2026
DA - 2026/11/03
PB - HAL
UR - https://hal.science/hal-05855807v1
AB - In Parts I and II of this research program, we unied the PT -symmetric spectral paradigm with the RuellePerronFrobenius thermodynamic formalism of the Gauss map T (x) = {1/x} on the Hardy space H 2 (D). Via the WeinsteinAronszajn factorization det Fred (I -L s ) = det Fred (I -R s )W (s) and block trigonalization on C|1⟩ ⊕ H 2 0 (D), we showed thatis the parabolic core response function, and proved that any hypothetical o-critical zero , unconditional Koopman cylinder positivity, and unconditional L 2 MontgomeryVaughan radial convexity, isolating the transition from the BergmanPoisson subharmonic identity (Eq. 5.7) to the pointwise bilinear chord derivative (Eq. 5.8) as the ultimate analytical target. In the present treatise (Part III), we develop three new analytical tools to analyze and bridge Equations (5.7) and (5.8). First, exploiting the endpoint adjacency τ m (1) = 1 m+1 = τ m+1 (0) of the Gauss branches, we prove an Exact Coboundary Telescoping Theorem at z = 1: for every f ∈ H 2 0 (D) and every Re(s) > 0, ( R s f )(1) = -f (1) and h s (1) = ζ(2s, 2) -ζ(2s) ≡ -1. Consequently, the resolvent state ψ s = (I H 2 0 -R s ) -1 h s is pinned to the exact constant value ψ s (1) ≡ -1/2 across the entire complex plane, splitting Φ core (s) = -1/2 + Φ ≥2 (s) where Φ ≥2 (s) = -∆ 2 h D 2 (X; h), we establish unconditional diagonal dominance across all horizontal chords, forbidding the two-point collision and conning all non-trivial zeros of ζ(z) to the critical line Re(z) = 1/2.
ER -