On the Exponential Mass Hierarchy of the Standard Model of Particles

The empirical mass spectrum of Standard Model of particles reveals an exponential structure. When the logarithm of particle’s mass is plotted against the integer exponent of an exponential equation, a straight line is produced. This straight‐line structure—confirms the exponential mass‐variation structure. This structure extends from the electron, muon, tau to the nucleons, light mesons, heavy quarkonia, and the electroweak bosons, as well as characteristic photon energies. We derive the exponential equation that reproduces this straight‐line structure—or the exponential mass‐variation structure—from first principles. The geometrical derivation is rooted in the Balmer– Rydberg formula and Planck’s constant, and sectional area of a very specific geometric object: a horn–torus. We show how a single physical input from hydrogen spectroscopy caliberates the overall scale of our exponential equation. This shows the equation is scale‐invariant and suggests a possible common geometric origin for matter and radiation. The geometric origin of the exponential structure also provides a unified way to categorize the known particle and radiation masses. To calibrate our exponential equation we noted the energy density vary exponentially along the radial direction of the atom. That lead to derivation of an exponential mass equation, that only contained (h,c,α) and an index n, a geometric index. This mass equation reproduced the entire empirical particle mass spectrum. Masses of leptons, baryons, mesons, quark scales, antiparticles and photon energies, as a function of only a few universal constants. A plot of (ln m) versus (n), shows all known particles of the standard model of particles, fall on a single straight line without per–particle tuning. The alignment is structural: one slope, one prefactor, and one geometric index governs the entire spectrum. The straight-line relationship also remains exact across Lyman, Balmer, Paschen, Brackett, Pfund lines. It also predicts the masses for empty ”n”-slots that may constitute concrete targets for unobserved resonances. Our analysis reveals a unified geometric picture in which mass and radiation are governed by a single exponential structure of spacetime, suggesting their emergent nature and common origin. The framework is algebraic, dimensionally consistent, and empirically anchored, offering a new geometric interpretation of some of the oldest data in quantum physics