The Square Root of Mass

I. The mass dimension

Linear quantities in physics — field values, amplitudes — interfere and transform; their squares — energies, probabilities — are positive and additive. Mass behaves like the latter: it is variance-like. The natural linear coordinate underneath it is therefore χ ≡ √m, in exact analogy with the standard deviation underlying a variance. The framework promotes χ to a physical coordinate of a fifth dimension, with mass as its additive, variance-like norm.

The proposed interpretation maps the decomposition onto the Standard Model: Higgs-type masses are displacements of a condensate mean (χ̄), QCD masses are fluctuation variances (Var χ) [D], but this is not a Standard-Model theorem. The measured fermion spectrum is consistent with a log-uniform distribution — the fixed-point measure under m ↔ χ — so the radial measure cannot distinguish the two descriptions; the question must be addressed by angular and dynamical structure [B].

II. The Wesson test and its double exclusion

The five-dimensional dynamics builds on the Space-Time-Matter program of Wesson, differing in exactly one place: the dictionary between the extra coordinate and mass. In the canonical 5D metric the √m dictionary doubles Wesson's predicted secular mass drift. Both the differential and the universal drift scenarios are then excluded by clock and ephemeris data — by factors of 2×10⁶ and 10³ respectively [B]. What survives is a sharp requirement:

III. The horn barrier and radial stabilization

The canonical five-dimensional warp makes every finite spatial separation collapse at ξ → 0. With no independent SQUS radial scale, self-similarity gives a physical circumferential radius f ∝ ξ²: the quadratic horn is derived rather than postulated [A/D]. The revised dynamics distinguishes a turning point from a circular orbit. A constant-ξ orbit requires both zero radial velocity and zero radial acceleration, equivalently h ≡ d ln f/d ln ξ = 1; linear stability additionally requires h′ < 0 [A].

The pure quadratic horn has h = 2 everywhere. It creates an inner centrifugal barrier and reflection from the tip, but no constant-radius orbit. A completed profile that crosses h = 1 with negative slope, or a separate radial potential V_rad, is therefore required. The companion equipartition potential fixes family direction but has an exact radial flat direction and cannot set the absolute mass scale by itself.

IV. The winding spectrum and the U(1) holonomy

Given a stabilizing radial closure, treating the three neutrinos as consecutive winding modes ℓ = 1, 2, 3 runs into a second no-go theorem: in the stated separable radial problem the level ratios ε₂/ε₁ and ε₃/ε₁ are bounded by 4 and 9 — while the equipartition-cone spectrum demands 4.878 ± 0.044 and 11.737 ± 0.182 [A/B]. A minimal one-parameter resolution within this ansatz is a U(1) holonomy shift; an identical stationary spectrum can also arise from a rotating background:

It places a candidate ℓ = 4 level at 166.7 meV and formal ℓ = 0, −1, −2 levels near 0.7 µeV, 1.47 meV, and 17.3 meV. These are conditional level locations, not complete experimental predictions, until mixing and population rules are derived [B/D].

V. The Koide relation as equipartition

The charged-lepton masses satisfy Koide's Q = 2/3 to a relative accuracy of 3×10⁻⁶ (Q = 0.6666645 ± 0.0000051), without an accepted explanation for four decades. In χ = √m variables the relation is exact equipartition of |χ⃗|² between the singlet and doublet irreducible representations of the family group S₃; the unique S₃-invariant potential of degree ≤ 4 enforcing it at every scale is V_eq = λ(n₁ − n₂)² [A]. Q is a renormalization-group quasi-invariant, exact on-shell with a closed-form O(α) drift in MS̄ [A/B]. Interpreting interactions as covariances predicts that |Q − 2/3| grows with the coupling across fermion triplets — confirmed over five orders of magnitude, with signs reproduced by a level-repulsion formula [B].

Extended to neutrinos through the hyperoctahedral group B₃, the equipartition cone intersects exactly one sign orbit — a single reflection on the lightest mode. For normal ordering this fixes (m₁, m₂, m₃) = (0.364, 8.662, 50.13) meV, Σm_ν = 59.16 ± 0.24 meV, m_β = 8.85 ± 0.10 meV and m_ββ = 1.25–3.97 meV — sharply falsifiable by JUNO, cosmology, and 0νββ searches [B]. The local statistical significance is ≈ 4.3σ, robust under look-elsewhere accounting, and independently supported by the trials-free 1981 out-of-sample prediction of the τ mass (1776.97 MeV, then measured 1784.2 ± 3.2 — later data landed on the prediction).

VI. One measure, three measurements

Three independent measurements exhibit one and the same scale-invariant fixed-point measure along the mass axis: the log-uniformity of the fermion mass spectrum, the renormalization-group freezing of the Koide ratio, and the electron's spectral dressing profile, whose plateau equals the QED anomalous dimension exactly [B/C]. A single scale-invariant structure seen three ways is the framework's strongest internal cross-check.

VII. Origins — from SQUS to the mass dimension

This program began with the Single Quantum Uncertainty Sphere (SQUS) hypothesis — a perspective on the cosmological constant integrating quantum mechanics and general relativity, in which vacuum energy within a universe-encompassing uncertainty sphere reproduces the observed Λ without dark energy. The 2026 papers grew out of that line of work and supersede it as the framework's core: the emphasis moved from the uncertainty sphere to the geometry of the mass dimension itself. The original paper remains available on ResearchGate ↗.