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 $T_{cc}^+(3875)$—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 $\mathbb{Z}_6$ topology of the vacuum. Building on the Modular Substrate Theory (MST), this framework derives the dimensionless constants $R_{\text{fund}} = \ln 2/(6\ln 3)$ and $\beta=3/4$ directly from the Standard Model gauge group quotient $(SU(3)_C \times SU(2)_L \times U(1)_Y)/\mathbb{Z}_6$ and holographic entropy bounds. By introducing a strict modular zero-sum rule ($\sum r_i \equiv 0 \pmod{6}$), 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 $\beta = 3/4$. This yields a theoretical compression factor of $\Gamma_{\text{MST}} = 0.75$. 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 $\Gamma_{\text{obs}} = 0.7257 \pm 0.0042$ ($p = 0.152$, showing a 96.8% match). Finally, the framework maps known exotics to specific Airy nodes ($n=1$ for $d^*$, $n=1.5$ for $T_{cc}^+$) and provides precise, falsifiable mass predictions for their first radial excitations ($d^{}$ at $3619 \pm 40$ MeV and $T_{cc}^{}$ at $4778 \pm 50$ 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 $\Xi_{cc}^{++}$ baryon mass. Uploaded File Inventory MST_Modular_Confinement_and_the_Universal_Hadronic_Spectrum.pdf The complete compiled manuscript in PDF format. It includes all theoretical derivations, experimental data tables, statistical analyses, and bibliographic references. MST_Modular_Confinement_and_the_Universal_Hadronic_Spectrum.tex The pristine LaTeX source code of the manuscript. This ensures full typographic reproducibility and allows other researchers to examine the exact structural layout. MST_Modular_Confinement_and_the_Universal_Hadronic_Spectrum.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-Modular_Confinement_and_the_Universal_Hadronic_Spectrum.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. mst_hadronic_predictions.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.