The BergmanPoisson Subharmonic Bridge and PickLoewner Bilinear Coercivity (Part III): Coboundary Pinning at z = 1, Green Capacity Bounds, and O-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 2s 0 = 1/2 + 2δ + it 0 (δ ∈ (0, 1/4), τ = t 0 /2) of ζ(2s) forces the simultaneous two-point symmetric collision
, 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 central analytical target.
In the present treatise (Part III), we develop three new analytical tools to complete the bridge between 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, (
only contracting branches n ≥ 2 with strictly positive Lyapunov exponents 2 ln n ≥ 2 ln 2 > 0, and raising the two-point collision threshold to Φ ≥2 (1/4 ± δ + iτ ) = 3/2. Second, we prove a Continued-Fraction Convergent Denominator Cancellation and Divisor-Discrepancy Theorem : along every Gauss branch composition τ m1 •• • ••τ mr (1/n) = p r /q r , the Jacobian weight q -2s r the DeshouillersIwaniec Kloosterman bounds on shifted divisor-discrepancy correlations -∆ 2 h D 2 (X; h), we establish unconditional diagonal dominance across all horizontal chords, ruling out the two-point collision and conning all non-trivial zeros of ζ(z) to the critical line Re(z) = 1/2.
Related
- The Bergman-Poisson Subharmonic Bridge and PickLoewner Bilinear Coercivity (Part III): Coboundary Pinning at z = 1, Green Capacity Bounds, and Off-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 BergmanPoisson Subharmonic Bridge and PickLoewner Bilinear Coercivity (Part III): Coboundary Pinning at z = 1, Green Capacity Bounds, and O-Diagonal Phase Dominance}},
author = {Aziz Arbai},
howpublished = {HAL},
year = {2026},
month = nov,
url = {https://hal.science/hal-05852411v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The BergmanPoisson Subharmonic Bridge and PickLoewner Bilinear Coercivity (Part III): Coboundary Pinning at z = 1, Green Capacity Bounds, and O-Diagonal Phase Dominance
AU - Aziz Arbai
PY - 2026
DA - 2026/11/03
PB - HAL
UR - https://hal.science/hal-05852411v1
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 2s 0 = 1/2 + 2δ + it 0 (δ ∈ (0, 1/4), τ = t 0 /2) of ζ(2s) forces the simultaneous two-point symmetric collision , 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 central analytical target. In the present treatise (Part III), we develop three new analytical tools to complete the bridge between 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, ( only contracting branches n ≥ 2 with strictly positive Lyapunov exponents 2 ln n ≥ 2 ln 2 > 0, and raising the two-point collision threshold to Φ ≥2 (1/4 ± δ + iτ ) = 3/2. Second, we prove a Continued-Fraction Convergent Denominator Cancellation and Divisor-Discrepancy Theorem : along every Gauss branch composition τ m1 •• • ••τ mr (1/n) = p r /q r , the Jacobian weight q -2s r the DeshouillersIwaniec Kloosterman bounds on shifted divisor-discrepancy correlations -∆ 2 h D 2 (X; h), we establish unconditional diagonal dominance across all horizontal chords, ruling out the two-point collision and conning all non-trivial zeros of ζ(z) to the critical line Re(z) = 1/2.
ER -