A Tau-Based Framework for Spectral Structure, Renormalization, and Critical Flow

Three-Volume Set: τ-Spectral Structure, Renormalization Mapping, and Spectral Critical Flow

This repository contains a unified three-volume research set developing a τ-based framework for organizing fermion structure and clarifying how physical meaning emerges under deformation, renormalization, and representation change. Each volume is logically self-contained, while the full set forms a coherent narrative arc from local eigenstructure to global mechanism.

Volumes

• Volume I — “Recursive Eigenvalue Problem for Fermion Mass Hierarchies”
  Introduces a τ-dependent spectral formulation that organizes fermion hierarchies through a recursive eigenvalue structure. This volume establishes the local spectral picture and motivates τ as an organizing parameter.
• Volume II — “Renormalization Mapping from Tau-Spectral Eigenmodes to Standard Model Fermion Masses”
  Develops the mapping from τ-spectral structure to Standard Model observables, emphasizing renormalization group (RG) effects and representation dependence. This volume clarifies why naïve eigenvalue-to-mass identification is generally invalid, especially in strongly coupled sectors.
• Volume III — “Spectral Critical Flow and Persistent Branches in τ-Deformed Operator Systems”
  Provides the mechanism-level completion of the framework by defining Spectral Critical Flow, in which τ parametrizes a continuous deformation of operator structure punctuated by critical transitions. Physical entities are identified with persistent branches—structures that remain stable across representational reorganization. Masses are treated as regime-dependent branch diagnostics rather than universal invariants. This volume explicitly integrates Onto-Physical Entanglement as the criterion linking ontology, dynamics, and representation.

What’s new in this set

• Mechanism-first framing: τ is treated as a deformation/flow parameter, not a physical ruler or mass generator.
• Global completion: eigenstates remain valid local descriptions, while globally meaningful structure is captured by persistent branches.
• Renormalization-consistent interpretation: RG flow is treated as a dominant component of the overall deformation dynamics in relevant sectors.
• Invariants reframed: emphasis shifts from fragile numerical constants to scale-/scheme-independent universality classes and flow topology.

Software and reproducibility note

This set discusses TauSolver Prism™ strictly at the level of functional role as a numerical probe of τ-dependent spectral structure. Proprietary implementation details are intentionally not disclosed and are covered by existing and/or pending intellectual property protections.

How to cite

If you use or reference this work, please cite the Zenodo record for the full three-volume set. If referencing a specific argument or mechanism, cite the relevant volume by title within the set.

Keywords: Spectral flow; critical phenomena; renormalization group; fermion hierarchy; operator deformation; universality classes; τ-framework; persistent branches; Onto-Physical Entanglement