This deposit contains a self-contained Wolfram Language verification script and its printable PDF rendering, supporting a manuscript on a unified description of representative meson and baryon masses within the Fermion–Boson Duality (FBD) framework.

Central formula. Each hadron mass is reconstructed as

m_H^FBD = T_q(E_H) Q_H⁽⁰⁾ + T_g(E_H) G_H,

where Q_H⁽⁰⁾ is a quark-baseline contribution, G_H = Σ_f n_f^(H) M_f^(g) is a flavor-dependent effective-gluon contribution, and T_q, T_g are sigmoid (logistic) transition functions of the hadron energy E_H. For fitted hadrons E_H is set equal to the observed PDG mass; for unmeasured doubly or triply heavy baryons, the same equation is solved self-consistently or evaluated in its saturated limit (T_q, T_g → 1).

Inputs. The script uses three flavor-resolved mass scales: PDG current quark masses (used for baryonic Q_H⁽⁰⁾), meson effective basis masses (used for mesonic Q_H⁽⁰⁾), and common effective-gluon masses (used in G_H for both sectors). The calibration set consists of 12 mesons (π⁺, K⁺, K⁰, η, η′, ρ, K^*, φ, D^*, B^*, J/ψ, Υ) and 17 baryons (the spin-1/2 SU(3) octet plus Ω⁻, together with the single-charm and single-bottom baryons Λ_c⁺, Ξ_c^+,0, Ω_c⁰, Λ_b⁰, Ξ_b^0,−, Ω_b⁻).

Numerical results. With four shared transition-function parameters (β_q = 0.0162091 MeV⁻¹, μ_q = 0 MeV, β_g = 0.0147425 MeV⁻¹, μ_g = 498.236 MeV), the script reproduces:

• Mesons: R² = 0.999296, relative RMSE 2.16%, max |error| 3.32%.
• Light baryons: relative RMSE 3.38%, max |error| 5.00%.
• Heavy baryons: relative RMSE 0.57%, max |error| 1.27%.
• All 17 baryons: R² = 0.999648, relative RMSE 2.49%.
• All 29 observed hadrons combined: R² = 0.999450, relative RMSE 2.36%.

As an independent consistency check that does not enter the calibration, the saturated-limit prediction for the doubly-charmed baryon Ξ_cc⁺⁺ agrees with its PDG value to within −1.28% (predicted 3575.3 MeV vs. observed 3621.6 MeV). Predictions for the as-yet-unmeasured doubly heavy baryons (Ξ_cc⁺, Ω_cc⁺, Ξ_cb^+,0, Ω_cb⁰, Ξ_bb^0,−, Ω_bb⁻) and triply heavy baryons (Ω_ccc⁺⁺, Ω_bbb⁻) are also tabulated for future experimental validation.

What the script produces. When executed, the script prints the full meson and baryon fit tables (observed mass, FBD reconstruction, relative error, and the explicit T_qQ_H⁽⁰⁾ and T_gG_H component decomposition), exports CSV tables (meson results, baryon results, heavy-baryon estimates), and exports the four PDF figures that accompany the manuscript: the transition functions, meson residuals, baryon residuals, and the unified meson–baryon scatter plot.

Reproducibility. The script is intentionally written as a single readable file with no manuscript-specific section numbering, so that it can be inspected and rerun as standalone supplementary material. It depends only on a standard Wolfram Language installation; no external packages are required.

Files in this deposit:

FBD_DoubleTransition_Probability_SU2.wl — the main verification script (Wolfram Language source) reproducing the unified meson and baryon mass formula.
FBD_DoubleTransition_Probability_SU2.pdf — a printable rendering of the main script with executed output (numerical tables and figures).
hadron_mass_unified.nb — a supplementary Mathematica notebook verifying the mathematical properties of the bosonic gamma matrices (omega^mu) underlying the FBD framework, including the Clifford algebra of gamma and omega matrices, the chiral relations involving gamma_5, the transverse projection identity (omega^mu A_mu)^2 = -A_1^2 - A_2^2, the localization of the transverse Proca term A^mu A_mu - (omega.A)^2 = A_0^2 - A_3^2 = 2 A^+ A^-, and the identification of omega^0 = omega^3 with the light-cone gamma matrix gamma^+/sqrt(2). This notebook supports the discussion of the FBD-QCD extended Lagrangian and the "leftover 2-DOF mass generation" picture in the manuscript Appendix B.
hadron_mass_unified.pdf — a printable rendering of the supplementary notebook with all executed output (matrix anticommutators, the symbolic identity for the transverse Proca term, and the light-cone gamma matrix identification).

Version 8

Section / appendix label updates. The manuscript structure has been revised to separate the main-text Appendix from a dedicated Supplementary Material (SM) file. Notebook filenames and section/appendix labels are realigned with the new structure:

Previous Appendix A, B, C, D, E -> Supplementary Material SM1 ... SM5 Previous Appendix F -> Main-text Appendix A Previous Appendix G -> Main-text Appendix B

The main-text Section numbering (Sec. 1 - Sec. 9) is unchanged.

Files in v8.

FBD_00_CommonSetup.wl shared Table 1 parameters (loaded by 01, 03, 04, 05)

FBD_01_MainFigures.nb Main text Sec. 3 - Sec. 7

FBD_02_OmegaMatrix_SM2_7.nb SM Sec. SM2.7 (was AppB7 in v7)

FBD_03_StatMech_Sec8_SM4.nb Main Sec. 8 + SM Sec. SM4 (was Sec 8 + AppD in v7)

FBD_04_ComponentTables_SM1_SM2.nb SM Sec. SM1, SM2 (was AppA, AppB in v7)

FBD_05_Summary_Verification.nb overall RMSE cross-check

FBD_06_BetaQ_AppB.nb Main-text Appendix B (was AppG in v7)

FBD_07_StatAnalysis_AppA.nb Main-text Appendix A (was AppF in v7)

Unchanged. All 13 free parameters, all meson / baryon / heavy-baryon numerical results, and all calculations in notebooks 02, 06, 07 are identical to v7.