Operator-Spectral Hyperbolic Reconstruction: Spectral Curvature, su(1,1) Representations, and Poincaré Geometry
Abstract
This work develops a rigorous operator–spectral framework in which hyperbolic geometry, non‑compact su(1,1) symmetry, and the exact Poincaré metric emerge intrinsically from the interaction between a self‑adjoint operator D and a χ‑boundary projector P_chi. The χ‑boundary derivation B = [D, P_chi] acts as a geometric generator, and its compressed double commutator P_k B^2 P_k defines the spectral curvature K(k) on each spectral layer L(k). Negative curvature K(k) < 0 forces the appearance of the su(1,1) algebra and yields a discrete Bargmann representation with index q(k) > 1/2. The Central Reconstruction Theorem establishes a unique unitary equivalence U_k : L(k) → H^2(D, dμ_q(k)), mapping each hyperbolic layer to a weighted Hardy space on the Poincaré disk. Under this equivalence, the operator‑induced quadratic form Q_k(u) = reproduces the exact hyperbolic line element ds^2 = 4 dz d\bar{z} / (1 − |z|2)2. Thus, hyperbolic geometry is not assumed but reconstructed directly from operator–spectral data. A complete classification of spectral layers is obtained through the invariant triple I(k) = (K(k), G(k), q(k)), where G(k) is the spectral gradient. The Spectral Equivalence Theorem shows that two hyperbolic layers are unitarily equivalent if and only if their invariant triples coincide. The invariants are stable under compact and χ‑stable perturbations satisfying [V, P_chi] ∈ S_1, ensuring robustness of the hyperbolic regime in the Calkin algebra B(H) / S_infinity. An extended analysis demonstrates how this operator–spectral architecture forms the mathematical core of the ZEBTS‑SUPER 9.0 paradigm. The pair (D, P_chi) acts as the universal generator of χ‑reality, encoding the triune structure Absolute–Boundary–Matter, determining χ‑curvature, χ‑collapse, χ‑vacuum modes, χ‑flows, and the emergence of χ‑geometry. The invariant triple I(k) becomes the χ‑signature of vacuum modes, while the su(1,1) representation provides the algebraic engine of χ‑dynamics. The reconstruction of the Poincaré metric corresponds to the emergence of χ‑spacetime, and χ‑stable perturbations describe physical interactions preserving χ‑geometry. The results establish a unified operator–spectral foundation for hyperbolic geometry, non‑compact symmetry, χ‑vacuum structuring, and the ontological architecture of ZEBTS‑SUPER, demonstrating that spacetime, curvature, matter, and physical laws arise as inevitable consequences of operator–boundary interactions in Hilbert space.
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BibTeX
@misc{mukha2026operatorspectral,
title = {{Operator-Spectral Hyperbolic Reconstruction: Spectral Curvature, su(1,1) Representations, and Poincaré Geometry}},
author = {Anatolii Mukha},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = aug,
doi = {10.5281/zenodo.21820948},
url = {https://doi.org/10.5281/zenodo.21820948},
note = {CC BY},
}
RIS
TY - UNPB
TI - Operator-Spectral Hyperbolic Reconstruction: Spectral Curvature, su(1,1) Representations, and Poincaré Geometry
AU - Anatolii Mukha
PY - 2026
DA - 2026/08/06
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.21820948
UR - https://doi.org/10.5281/zenodo.21820948
AB - This work develops a rigorous operator–spectral framework in which hyperbolic geometry, non‑compact su(1,1) symmetry, and the exact Poincaré metric emerge intrinsically from the interaction between a self‑adjoint operator D and a χ‑boundary projector P_chi. The χ‑boundary derivation B = [D, P_chi] acts as a geometric generator, and its compressed double commutator P_k B^2 P_k defines the spectral curvature K(k) on each spectral layer L(k). Negative curvature K(k) < 0 forces the appearance of the su(1,1) algebra and yields a discrete Bargmann representation with index q(k) > 1/2. The Central Reconstruction Theorem establishes a unique unitary equivalence U_k : L(k) → H^2(D, dμ_q(k)), mapping each hyperbolic layer to a weighted Hardy space on the Poincaré disk. Under this equivalence, the operator‑induced quadratic form Q_k(u) = reproduces the exact hyperbolic line element ds^2 = 4 dz d\bar{z} / (1 − |z|2)2. Thus, hyperbolic geometry is not assumed but reconstructed directly from operator–spectral data. A complete classification of spectral layers is obtained through the invariant triple I(k) = (K(k), G(k), q(k)), where G(k) is the spectral gradient. The Spectral Equivalence Theorem shows that two hyperbolic layers are unitarily equivalent if and only if their invariant triples coincide. The invariants are stable under compact and χ‑stable perturbations satisfying [V, P_chi] ∈ S_1, ensuring robustness of the hyperbolic regime in the Calkin algebra B(H) / S_infinity. An extended analysis demonstrates how this operator–spectral architecture forms the mathematical core of the ZEBTS‑SUPER 9.0 paradigm. The pair (D, P_chi) acts as the universal generator of χ‑reality, encoding the triune structure Absolute–Boundary–Matter, determining χ‑curvature, χ‑collapse, χ‑vacuum modes, χ‑flows, and the emergence of χ‑geometry. The invariant triple I(k) becomes the χ‑signature of vacuum modes, while the su(1,1) representation provides the algebraic engine of χ‑dynamics. The reconstruction of the Poincaré metric corresponds to the emergence of χ‑spacetime, and χ‑stable perturbations describe physical interactions preserving χ‑geometry. The results establish a unified operator–spectral foundation for hyperbolic geometry, non‑compact symmetry, χ‑vacuum structuring, and the ontological architecture of ZEBTS‑SUPER, demonstrating that spacetime, curvature, matter, and physical laws arise as inevitable consequences of operator–boundary interactions in Hilbert space.
ER -