What does this research mean for the field?

The Orbital Space O, constructed from the equation e^{P(x)(az+b)} = f(x), has a complete theory that includes a metric structure, symmetry group, and classification into 2^{n+1} classes, revealing intrinsic geometric properties of closed orbits. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.ESTABLISHES_NEW_DIRECTION.

What question did this study set out to answer?

The aim is to develop a comprehensive theory of the Orbital Space related to analytic functions.

March 12, 2026Open Access

Orbital Functional Geometry: Orbits, Spectra, and the Geometry of Analytic Functions

Key Points

The aim is to develop a comprehensive theory of the Orbital Space related to analytic functions.
Constructed Orbital Space O from the equation e^{P(x)(az+b)} = f(x)
Analyzed closed orbits as trajectories of solutions z
Developed metric structures including Hausdorff distance and foliation
Explored symmetry group D∞ and classification into 2^{n+1} classes
Established principles of orbital spectroscopy and calculus
Determined a continuous spectrum λ = 2/a² for the orbital Laplacian
Showed that the intrinsic geometry is flat
Encoded zeros and poles as readable from orbits

Abstract

We introduce the Orbital Space O, constructed from the equation e^P (x) (az+b) = f (x). A point of O is a closed orbit — the trajectory of the solution z as x makes one full turn around a root of P in ℂ. We develop the complete theory: metric structure (Hausdorff distance), foliation, symmetry group (D∞), classification into 2^n+1 classes, encoding theorem (zeros and poles readable from orbits), orbital spectroscopy, differential and integral calculus in three directions, orbital Laplacian with continuous spectrum λ = 2/a², and flatness of the intrinsic geometry.

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Judicael Brindel

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Orbital Functional Geometry: Orbits, Spectra, and the Geometry of Analytic Functions

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Abstract

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