Foundation Ckmhierarchy From Phi Ladder Mass Ratio Top Up Above 30000

The top-to-up ratio

The standard model of particle physics has six quarks, the building blocks of protons and neutrons. Their masses span an enormous range: the top quark weighs about 172.7 GeV, while the up quark weighs about 2.16 MeV. That makes the top quark roughly 80,000 times heavier than the up quark. No accepted theory explains why the gap is so large. The CKM matrix, which describes how quarks change flavor, shows a similar hierarchy, and its entries are fitted from experiment rather than predicted.

In Recognition Science (RS), quark masses sit on a ladder of integer steps. The framework models the mass of a quark at rung k as m_unit · φ^k, where φ is the golden ratio, about 1.618, and m_unit is a base mass. The six quarks occupy rungs 8 through 30: up at 8, down at 9, strange at 14, charm at 17, bottom at 22, and top at 30. Because adjacent rungs differ by exactly φ, the ratio of the top mass to the up mass is φ^(30-8) = φ^22, which is approximately 39,089.

The declaration mass_ratio_top_up_above_30000 proves a precise lower bound: 30000 < φ^22. This is a machine-checked theorem in the framework's library of formal theorems. It is a structural statement about the ladder itself, not a measurement. The framework's library also proves that the ratio is positive, that masses strictly increase with rung number, and that the six-quark count and rung positions are fixed. These are all consequences of the definition of the ladder, not empirical inputs.

The empirical ratio m_t / m_u is about 80,000, which is within a factor of 2 of the φ^22 prediction. The framework ascribes the discrepancy to a scale-running correction at the gap-45 energy scale. This comparison is a check, not a proof: the measured masses come from experiment, and the ladder is a model. The theorem itself only establishes the lower bound for the ratio φ^22, not that any real quark mass equals this value.

What the declaration does not claim is equally important. It does not say that measured quark masses match φ^22 exactly; the empirical ratio is about 80,000, not 39,089. It does not derive the CKM matrix elements from the ladder. And it does not claim that the rung assignments themselves are forced by the framework's axioms; they are chosen to fit the gauge structure. The theorem is a clean statement about the golden ratio ladder, and the comparison to nature is a separate, weaker claim.

/-- Numerical lower bound: `φ^22 > 30,000` (within a factor 3 of
empirical 80,000 top-to-up mass ratio). We use that `1.61^22 > 30000`
via piecewise computation. -/
theorem mass_ratio_top_up_above_30000 : 30000 < mass_ratio_top_up := by
  unfold mass_ratio_top_up
  have h_phi : 1.61 < phi := phi_gt_onePointSixOne
  have h_pos : (0 : ℝ) ≤ 1.61 := by norm_num
  have h_pow : (1.61 : ℝ) ^ 22 ≤ phi ^ 22 :=
    pow_le_pow_left₀ h_pos (le_of_lt h_phi) 22
  -- (1.61)^22 = (1.61)^11 · (1.61)^11; (1.61)^11 ≈ 187.4
  -- (1.61)^11 > 175
  have h_11 : (175 : ℝ) < (1.61 : ℝ) ^ 11 := by
    have : (1.61 : ℝ) ^ 11 = 1.61 * 1.61 * 1.61 * 1.61 * 1.61 * 1.61 *
                              1.61 * 1.61 * 1.61 * 1.61 * 1.61 := by
      ring
    rw [this]; norm_num
  -- (1.61)^22 = ((1.61)^11)^2 > 175^2 = 30625
  have h_22 : (1.61 : ℝ) ^ 22 = ((1.61 : ℝ) ^ 11) ^ 2 := by ring
  have h_compute : (30000 : ℝ) < ((1.61 : ℝ) ^ 11) ^ 2 := by
    have h_11_pos : (0 : ℝ) < (1.61 : ℝ) ^ 11 := by positivity
    have h_sq_lt : (175 : ℝ)^2 ≤ ((1.61 : ℝ) ^ 11) ^ 2 := by
      have h_175_pos : (0 : ℝ) ≤ 175 := by norm_num
      exact pow_le_pow_left₀ h_175_pos (le_of_lt h_11) 2
    have h_175_sq : (175 : ℝ) ^ 2 = 30625 := by norm_num
    linarith
  rw [← h_22] at h_compute
  linarith

/-- Mass at rung `k`, parameterised by base mass unit. -/
def mass_at_rung (m_unit : ℝ) (k : ℕ) : ℝ := m_unit * phi ^ k

/-- Up-quark rung (lightest, first-generation up-type). -/
def up_rung : ℕ := 8

/-- Top-quark rung (heaviest, third-generation up-type, scale-
saturating). -/
def top_rung : ℕ := 30

/-- Adjacent rungs differ by exactly `φ`. -/
theorem mass_geometric (m_unit : ℝ) (k : ℕ) :
    mass_at_rung m_unit (k + 1) = mass_at_rung m_unit k * phi := by
  unfold mass_at_rung
  rw [pow_succ]
  ring

/-- The top-to-up mass ratio: `φ^(top_rung - up_rung) = φ^22`. -/
def mass_ratio_top_up : ℝ := phi ^ 22

What this page does not claim

Measured quark masses equal φ^22 exactly; the empirical ratio is about 80,000, not 39,089. The rung assignments are forced by the framework's axioms; they are a modeling choice. The CKM matrix elements are derived from the ladder.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/CKMHierarchyFromPhiLadder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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