Disclosure Geometry III The Spectral Sphere
Year: 2026
Type: preprint
Abstract: This disclosure report develops the spectral completion of the circle model. In the first circle disclosure, the circle was interpreted physically: the center as particle or localized closure, the area as wave or resonant mediation, and the circumference as field or boundary of possibility. In the second circle disclosure, the same geometry was interpreted ontologically: the center as the coherence seed, the area as the Euler phase seed, the circumference as the scale seed, and the whole circle as the identity seed. The present report extends that sequence into spectral geometry. The central claim is that spectral shell harmonics require the sphere. The circle is sufficient for disclosing center, mediation, boundary, and whole. It gives the minimal closure object. But spectral decomposition requires an orthogonal angular domain, and this is supplied by the sphere. The sphere is therefore not an arbitrary addition to the circle model. It is the circle’s orthogonal completion under spectral requirements.The report develops a unit-ball derivation in which the null center r = 0 supplies the coherence-origin or regularity condition, the unit boundary r = 1 supplies the scale-closure condition, the spherical interior 0 < r < 1 supplies the Euler phase-domain, and the resonance operator supplies identity-selection. Under these conditions, the continuous spherical domain decomposes into admissible eigenmodes. These eigenmodes appear as spectral harmonic shells. The resulting thesis is compact: spectral shell harmonics arise when the null center and unit boundary constrain a spherical phase-domain under an identity-selecting resonance operator. Keywords spectral sphere; disclosure geometry; spherical harmonics; spectral shells; eigenmodes; unit ball; null center; unit boundary; closure geometry; resonance operator; seed tetrad; UCCF; harmonic shell decomposition
Author: Lilien, Philip
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7162496751
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