SpiralHelios: Spectral Geometric Energy of Symbols and Numbers in a Finite Hilbert Space

This work documents the continued development of SpiralHelios, a fully deterministic, executable, and offline alphabet–geometric processing system. The present version preserves full continuity with the previously published SpiralHelios Core and extends the work by formally introducing the ROM architectural layer, which defines a deterministic and auditable framework for the canonical encoding of established scientific knowledge. This record is a continuation of the previously published SpiralHelios Core andSpiralHelios ROM works. It contains the main research article“SpiralHelios: Spectral Geometric Energy of Symbols and Numbers in a Finite Hilbert Space”,together with its Supplementary Figure S1 and Supplementary Note S1. The work presents experimental validation of spectral geometric energy associatedwith symbols and numbers in a finite real Hilbert space of dimension D = 1024.A fully deterministic system maps symbolic and numeric sequences to unit-normgeometric states, enabling observable spectral analysis through self-adjointcirculant Laplacian operators. All experiments were executed fully offline on consumer mobile hardware,including execution under extreme resource constraints, with invariant spectralstructure and deterministic behavior. The supplementary materials provide experimental evidence of multilingualdeterministic execution and the complete axiomatic mathematical foundationof the spectral geometric energy framework. All components form a unified deterministic geometric system and arepublished under a single DOI as a coherent scientific record. Author:Christos Baltas (PH-GAIOS) ORCID:https://orcid.org/0009-0000-9743-3678 Contact:info@spiral-code.com Zenodo Community:https://zenodo.org/communities/spiral-code-energy/