Informational Unification of Fundamental Interactions: An Operator-Based Framework with Emergent Spectral Hierarchies

We present an exploratory, operator-based framework in which the variety of fundamental interactions is approached from an informational perspective. The framework is built around a single functional acting on a relational substrate, combining local, non-local, and hierarchical contributions together with an irreversible cost term. Rather than treating gauge symmetries as fundamental inputs, the approach investigates whether structured interaction sectors can be associated with distinct stability properties of this operator. The physical content of the proposal is examined through the spectral properties of the Hessian of the informational functional. In this setting, the organisation of eigenvalues is used as a diagnostic tool to characterise the presence or absence of hierarchical structures that may be interpreted as interaction sectors. No attempt is made to derive specific gauge groups or coupling constants. A sequence of computational toy models is introduced to probe the internal consistency of the framework. These models are analysed through systematic comparisons with carefully defined null constructions, using spectral entropy, gap-based measures, and concentration indicators. The results show that non-trivial spectral hierarchies arise only when the full informational structure is present, and are suppressed when key terms are removed. Robustness tests across scales, system sizes, and coupling regimes suggest that these features are not solely attributable to fine-tuned parameter choices. The framework is intended as a pre-unificatory, information-theoretic investigation rather than a complete unification scheme. Its purpose is to clarify how interaction-like structures might emerge from a single operator, and to provide a transparent computational setting in which such ideas can be tested and refined.我们提出了一个探索性的、基于算符（operator）的框架，从信息论视角来探讨各类基本相互作用。该框架以一个作用于关系性基底的单一泛函（functional）为核心，融合了局域、非局域与层级性贡献项，以及不可逆代价项。本研究未将规范对称性（gauge symmetry）作为基础输入条件，而是通过该框架探究：结构化相互作用扇区是否可与该算符的独特稳定性性质建立对应关联。 本提案的物理内涵可通过信息论泛函的黑塞矩阵（Hessian）的谱性质进行检验。在此框架下，本征值的分布模式被用作诊断工具，以表征可被解释为相互作用扇区的层级性结构是否存在。本研究未尝试推导具体的规范群或耦合常数。 我们引入了一系列计算玩具模型，以探究该框架的内在自洽性。通过系统对比精心定义的零构造模型，并结合谱熵、能隙相关度量与集中度指标对这些模型进行分析。结果表明，仅当完整的信息论结构完整存在时，才会出现非平庸的谱层级结构；若移除关键项，则此类结构会被抑制。针对不同尺度、系统规模与耦合参数区间的鲁棒性测试显示，这些特征并非仅能通过精细调谐的参数选取来实现。 本框架旨在作为一项预统一的信息论研究，而非一套完整的统一理论方案。其核心目标在于阐明：类相互作用结构如何可由单一算符衍生而来，并提供一个透明的计算实验环境，以供此类设想得到检验与完善。