TRPG Physics

The master equation

One action holds the whole framework. Most terms are the established physics of Einstein, Yang and Mills, Dirac and Higgs. TRPG adds the containment field, the scale layer, and a relational form for the masses.

STRPG = ∫ d⁴x √(−g) [ (c⁴/16πG) R − ρΛ − ½(∂P)² − (λ/4) v⁴ (P² − 1)² − (∂σ)² − ¼ F² + ψ̄ iγμDμ ψ − gP P ψ̄qψq − (√2/vH) ℓ̄ Sℓ² ℓ H + quark terms ]

S_TRPG = ∫ d⁴x √(−g) × [ the seven terms below ]

  1. Gravity: (c⁴/16πG) R. In TRPG this is not put in by hand. It emerges from area-law entanglement of the medium (Jacobson; M14, M16, P4), with G ≈ 3.5 a² in units of the medium's spacing a.

  2. Cosmic baseline resistance: −ρ_Λ, the dark-energy baseline, about 3.3 × 10⁻⁴² MeV/fm³ (energy scale 2.24 meV; M8).

  3. Containment field P: −½(∂P)² − (λ/4) v⁴ (P² − 1)², with λ ≈ 1160 to 1210 and v ≈ 35 MeV. The depth B = 60 (57 to 64) MeV/fm³ comes from the proton and Delta masses (Q3; earlier taken from the neutron-star window 62 to 70), and the stiffness from the 1.7 GeV glueball (M1). Quarks feel it through −g_P P ψ̄_qψ_q, which confines them when g_P is large.

  4. Scale layer σ (no privileged scale): −(∂σ)² with m = m₀e^{2σ}, so the distance between scales is S = ½ ln(m₁/m₂) (M3). The curvature of the layer space is dynamic, set by wave superposition, and averages to flat (M5).

  5. Forces: −¼ F² for U(1) × SU(2) × SU(3). The groups follow the division algebras (M18, P3). Given 3 colours (measured), anomaly cancellation forces the charges +2/3, −1/3, −1 and 0 (M16, Q2).

  6. Matter: ψ̄ iγ^μD_μ ψ for the three generations (Dirac).

  7. Mass couplings through the Higgs: −(√2/v_H) ℓ̄ S_ℓ² ℓ H. The lepton square-root-mass matrix is S_ℓ = M₀ × circulant(1, e^{iδ}/√2, e^{−iδ}/√2) with δ = 2/9 and 3M₀² = 941.6 MeV (M2, P5). The neutrinos use the same rule (normal ordering, sum 0.0592 eV). The quark couplings are measured inputs (P2).

Count of measured inputs: G (or a), ρ_Λ, B, the glueball mass, one lepton scale M₀, two neutrino splittings, the three force strengths, the Higgs mass and v_H, six quark masses and four quark mixing numbers. Add the three neutrino mixing angles and their phase, and the strong-force CP angle, which the first count left out. Corrected total (Q2): about 25, compared with about 28 for the Standard Model with neutrino masses, gravity and dark energy, if B and the glueball mass are counted as consequences of the strong force (B is now set by the proton, Q3). The saving of 3 is in the leptons: two charged-lepton masses and the lightest neutrino mass.

Every number above is computed by a script you can download and run on the Data & code page.