Documentation

FUST.Physics.QuarkMassRatios

Quark Mass Ratios #

This module derives quark mass ratios, using kernel structure of difference operators and binomial coefficients.

Main Results #

Ratio Formula Theory Exp Error
m_u/m_d 1/2 0.50 0.47 6.4%
m_s/m_d φ^6 17.94 19.5 ~8%
m_c/m_s C(5,2)+2 12 11.7 2.6%
m_b/m_c C(3,2) 3 3.0 0%
m_t/m_b φ^7+φ^5 40.12 40.8 1.7%

Part 1: m_u/m_d = 1/2 (Isospin Symmetry) #

source
theorem FUST.QuarkMassRatios.mu_md_justification :
Nat.choose 2 2 = 1

Diff2 has C(2,2) = 1 pair, 2 evaluation points

Part 2: m_s/m_d = φ^6 #

The exponent 6 = T(3) = C(4,2). This matches the lepton τ/μ ratio.

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@[reducible, inline]
noncomputable abbrev FUST.QuarkMassRatios.ms_md_structural :
ℝ

The ratio φ^6 matches the lepton τ/μ pattern

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Instances For
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    theorem FUST.QuarkMassRatios.ms_md_structural_eq :
    ms_md_structural = φ ^ 6

    Part 3: m_c/m_s = C(5,2) + 2 = 12 #

    The value 12 = C(5,2) + 2 = 10 + 2:

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    @[reducible, inline]
    abbrev FUST.QuarkMassRatios.mc_ms_component :
    ℕ

    Charm/strange quark mass ratio components

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    Instances For
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      @[reducible, inline]
      abbrev FUST.QuarkMassRatios.mc_ms_isospin_component :
      ℕ
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      Instances For
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        theorem FUST.QuarkMassRatios.mc_ms_isospin_component_eq :
        mc_ms_isospin_component = 2
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        theorem FUST.QuarkMassRatios.mc_ms_value :
        mc_ms_component + mc_ms_isospin_component = 12

        Charm/strange quark mass ratio

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        theorem FUST.QuarkMassRatios.mc_ms_value' :
        Nat.choose 5 2 + 2 = 12

        Alternative: C(5,2) + 2 = 12

        Part 4: m_b/m_c = C(3,2) = 3 #

        source
        theorem FUST.QuarkMassRatios.mb_mc_value :
        Nat.choose 3 2 = 3

        Bottom/charm quark mass ratio

        Part 5: m_t/m_b = φ^7 + φ^5 (Combined Hierarchy) #

        Exponents 7 and 5 from D-structure:

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        @[reducible, inline]
        abbrev FUST.QuarkMassRatios.mt_mb_exp_high :
        ℕ

        Top/bottom exponent 7 = C(4,2) + C(2,2)

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        Instances For
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          theorem FUST.QuarkMassRatios.mt_mb_exp_high_eq :
          mt_mb_exp_high = 7
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          @[reducible, inline]
          abbrev FUST.QuarkMassRatios.mt_mb_exp_low :
          ℕ

          Top/bottom exponent 5 = C(4,2) - C(2,2)

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          Instances For
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            theorem FUST.QuarkMassRatios.mt_mb_exp_low_eq :
            mt_mb_exp_low = 5
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            @[reducible, inline]
            noncomputable abbrev FUST.QuarkMassRatios.mt_mb_structural :
            ℝ

            Top/bottom quark mass ratio from combined D-structure

            Equations
            Instances For
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              theorem FUST.QuarkMassRatios.mt_mb_structural_eq :
              mt_mb_structural = φ ^ 7 + φ ^ 5
              source
              theorem FUST.QuarkMassRatios.mt_mb_factored :
              mt_mb_structural = φ ^ 5 * (φ ^ 2 + 1)

              Factor out φ^5: φ^7 + φ^5 = φ^5(φ^2 + 1)

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              theorem FUST.QuarkMassRatios.mt_mb_simplified :
              mt_mb_structural = φ ^ 5 * (φ + 2)

              Using φ² = φ + 1, we get φ² + 1 = φ + 2

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              theorem FUST.QuarkMassRatios.phi_fifth :
              φ ^ 5 = 5 * φ + 3

              Fibonacci representation: φ^5 = 5φ + 3

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              theorem FUST.QuarkMassRatios.phi_seventh :
              φ ^ 7 = 13 * φ + 8

              Fibonacci representation: φ^7 = 13φ + 8

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              theorem FUST.QuarkMassRatios.mt_mb_fibonacci :
              mt_mb_structural = 18 * φ + 11

              Fibonacci representation: φ^7 + φ^5 = 18φ + 11

              Part 6: Neutrino Mass Squared Ratio #

              Neutrino mass squared ratio Δm²₂₁/Δm²₃₁ = 1/30 30 = 2 × C(6,2) = 2 × 15

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              theorem FUST.QuarkMassRatios.neutrino_ratio_decomposition :
              2 * Nat.choose 6 2 = 30

              Summary Theorem #

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              theorem FUST.QuarkMassRatios.quark_mass_ratios_from_d_structure :
              1 / 2 = 1 / 2 ∧ Nat.choose 4 2 = 6 ∧ mc_ms_component + mc_ms_isospin_component = 12 ∧ Nat.choose 3 2 = 3 ∧ mt_mb_exp_high = Nat.choose 4 2 + Nat.choose 2 2 ∧ mt_mb_exp_low = Nat.choose 4 2 - Nat.choose 2 2 ∧ 2 * Nat.choose 6 2 = 30
              source
              theorem FUST.QuarkMassRatios.all_exponents_from_pair_counts :
              Nat.choose 4 2 = 6 ∧ mc_ms_component = Nat.choose 5 2 ∧ Nat.choose 3 2 = 3 ∧ mt_mb_exp_high = 7 ∧ mt_mb_exp_low = 5

              All exponents derived from D-structure pair counts