Hearing Shape from Spectrum: Connes’ Non-Commutative Geometry, Kac’s Inverse Spectral Problem, and the Highway Linking Geometry and Inverse Problems.
Two lineages run through the corpus — the geometric mainstream (Riemann…Hopf, Paper 91) and the integral-brain / inverse problem (Papers 89/99). This paper links the two for the FIRST time by a single road, via the question “can shape be heard from the spectrum?” Mark Kac’s “Can one hear the shape of a drum?” (1966) — recover a drum’s shape from the eigenvalues (spectrum) of its Laplacian — is an INVERSE problem, and the answer is in general NO (isospectral non-isometric shapes exist = non-unique). Meanwhile Connes’ NON-COMMUTATIVE GEOMETRY rewrites geometry in the spectrum of the Dirac operator D (spectral triple (A,H,D)), defining even distance as d(x,y)=sup{|f(x)-f(y)| : ||[D,f]||<=1} using the operator alone (Connes’ reconstruction theorem: a commutative spectral triple IS a Riemannian manifold). Two disciplines apply: (1) DISTINCT ROOTS — “distance from the metric” (Riemann 91) and “distance from the spectrum” (Connes) coincide on a manifold yet are distinct roots; and the family of inverse problems — scattering (GLM, 89) / boundary-value (Calderón, 99, unique but ill-posed) / spectral (Kac, non-unique) — share the stance but are distinct mathematics. (2) HONEST NUANCE — the Connes-Chamseddine non-commutative Standard Model predicted a Higgs mass that was ruled out and then adjusted by adding a field, so even a rigorous program brushes the cliff of Paper 101 (Eddington’s caution). What is audible from the spectrum: dimension, volume, total scalar curvature (Weyl’s law, heat kernel) — spectral dimension is a new distinct instance of “dimension” (69/94). The keystone of the bridge is Paper 70 (index theorem = home of the Dirac operator), where geometry (91) and inverse problems (99) meet on the spectrum of D. IMPORTANT (scope): a Layer-2 structural interpretation of established mathematics (NCG, inverse spectral problem); NCG kept to a structural introduction, NOT exaggerated into “deriving the Standard Model”; no new mathematics and no new prediction. The Universe OS is a Layer-3 non-science lens.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). Hearing Shape from Spectrum [Paper 102]. Zenodo.
DOI: 10.5281/zenodo.21836735
ORCID: 0009-0006-8587-3450