Modular Confinement and the Universal Hadronic Spectrum: A Prediction from ℤ₆ Substrate Topology

The emergence of exotic hadrons—such as the compact hexaquark d^* (2380) and the doubly charmed tetraquark T₂₂^+ (3875) —challenges conventional quark-model taxonomy. This research demonstrates that these states, alongside the conventional meson spectrum, obey a universal confinement pattern dictated by the discrete Z₆ topology of the vacuum. Building on the Modular Substrate Theory (MST), this framework derives the dimensionless constants R₅ₔ₍₃ = 2/ (6 3) and =3/4 directly from the Standard Model gauge group quotient (SU (3) C SU (2) L U (1) Y) /Z₆ and holographic entropy bounds. By introducing a strict modular zero-sum rule (rᵢ 0 6), the theory generates a natural, predictive taxonomy distinguishing between primons (permanently confined constituents), duplexes (mesons), hexaquarks, and tetraquarks. Furthermore, the model demonstrates that for linearly confined systems, where the non-relativistic Schrödinger equation reduces to the Airy equation, the level spacings undergo a dimensional projection compressed by the topological factor = 3/4. This yields a theoretical compression factor of ₌ₒₓ = 0. 75. This prediction is robustly validated against the Particle Data Group (PDG) 2024 mass values for the Bottomonium and Charmonium families, yielding an observed factor of ₎₁ₒ = 0. 7257 0. 0042 (p = 0. 152, showing a 96. 8% match). Finally, the framework maps known exotics to specific Airy nodes (n=1 for d^*, n=1. 5 for T₂₂^+) and provides precise, falsifiable mass predictions for their first radial excitations (d^ at 3619 40 MeV and T₂₂^ at 4778 50 MeV). Strikingly, it also reveals a flavor-blind mass quantization principle, where different quark sectors condense at the same topological stability nodes—evidenced by the predicted d^ mass aligning within 0. 07% of the experimental ₂₂^++ baryon mass. Uploaded File Inventory MSTModularConfinementₐndₜheUniversalHadronicSpectrum. pdf The complete compiled manuscript in PDF format. It includes all theoretical derivations, experimental data tables, statistical analyses, and bibliographic references. MSTModularConfinementₐndₜheUniversalHadronicSpectrum. tex The pristine LaTeX source code of the manuscript. This ensures full typographic reproducibility and allows other researchers to examine the exact structural layout. MSTModularConfinementₐndₜheUniversalHadronicSpectrum. ipynb The core interactive Jupyter (Google Colab) Notebook written in Python 3. It contains the complete, self-contained computational workflow required to reproduce every numerical output and statistical claim in the paper (including Airy root extraction, modular zero-sum rule validation, Student's t-test hypothesis testing, inverse-variance error propagation, and exotic mass scaling calculations). MST-ModularConfinementₐndₜheUniversalHadronicSpectrum. ipynb - Colab. pdf A static PDF snapshot of the pre-executed Jupyter Notebook. It preserves the exact runtime terminal outputs, printed tables, and embedded plots directly from the Google Colab environment for quick empirical verification. mstₕadronicₚredictions. png The high-resolution analytic graphic generated by the notebook. It features a unified side-by-side plot showcasing the dimensional compression analysis in conventional mesons (left panel) and the quantitative mass predictions for undiscovered exotic states (right panel). License and Metadata License: Creative Commons Attribution 4. 0 International (CC BY 4. 0). Sharing, adapting, and building upon this code and data is permitted, provided proper credit is given to the original author. Keywords: Hadronic Spectroscopy, Exotic Hadrons, Tetraquarks, Hexaquarks, Modular Substrate Theory, Airy Function, Particle Data Group, Standard Model Topology, Dimensional Compression.