The Spectral Existence Theorem for Developmental Geometry
Abstract
Paper E of the Developmental Geometry program. Proves the Spectral Existence Theorem: a developmental manifold (M,gDev)(M, g_\mathrm{Dev}) (M,gDev) exists whose minimal spectral subspace has spectrum {1,2,3,…}\{1,2,3,\ldots\} {1,2,3,…} matching the Collatz packet-size function k(n)=v2(3n+1)k(n) = v_2(3n+1) k(n)=v2(3n+1). Construction uses inverse Sturm-Liouville theory (Gel'fand-Levitan). Closes the last foundational gap in the DG program. The Collatz conjecture is equivalent to the Unified Density Conjecture unconditionally within the DG framework.
Fields of Study & Subjects
Benford’s Law and Fraud Detection
Probability and Statistical Research
Mathematics
Computing
and Information Processing
Publication details
- Journal / source
- Zenodo (CERN European Organization for Nuclear Research)
- Publication date
- 24 March 2026
- Language
- EN
- Document type
- Preprint