Masses Mass Law Mass Rung Scaling

The rung ladder

The golden ratio φ, about 1.618, appears throughout nature as a scaling factor, from leaf arrangements to spiral galaxies. In the Recognition Science framework, it also governs the masses of elementary particles. The framework's ledger, a discrete record of recognition events, assigns each stable particle state a position on a ladder of integers called a rung. The mass of a particle is not arbitrary; it follows a formula that depends on which sector the particle belongs to, its rung, and a charge-based correction.

The central theorem, mass_rung_scaling, states a simple relation: if you increase the rung by exactly one, the predicted mass multiplies by φ. In symbols, for any sector, rung, and charge shift, the mass at rung r+1 equals φ times the mass at rung r. This is a proved theorem in the framework's machine-checked library of formal theorems, not a numerical coincidence. It follows directly from the definition of the mass formula, which places φ raised to the rung position minus a constant cycle period of 8, plus a correction term.

The correction term, gap(Z), depends on the charge-based shift Z. For a neutral baseline, where Z is zero, the correction vanishes, so the rung spacing is exactly φ. For charged particles, the gap term adjusts the rung position slightly, but the scaling relation still holds exactly for any fixed charge. The theorem is fully general: it applies to every sector and every integer rung, with no exceptions for special cases.

In Recognition Science, this scaling law is not a fitted empirical trend but a derived consequence of the framework's founding assumptions. The framework models particle masses as occupying rungs on a φ-ladder, and the scaling theorem confirms that the ladder is evenly spaced in the log-φ sense. This means that the ratio of masses between adjacent rungs is constant, a strong structural prediction that can be tested against measured particle masses from the Particle Data Group.

What the theorem does not claim is equally important. It does not say which specific particle sits on which rung, nor does it determine the sector yardstick that sets the absolute mass scale. The scaling relation is a relative statement: it constrains ratios, not absolute values. The assignment of observed particles to rungs and sectors is an empirical check, not a proved result. The theorem also does not imply that all particles must be spaced by exactly φ in reality; the charge correction can shift individual positions, and the yardstick varies by sector.

The practical consequence is a clear target for measurement. If the framework is correct, then for any two particles in the same sector with the same charge, the heavier one should be φ times the lighter one whenever their rungs differ by one. A deviation from this ratio would falsify the mass law. The scaling theorem gives experimentalists a precise, quantitative prediction to confirm or refute, which is the hallmark of a useful physical theory.

/-- The mass law exhibits φ-scaling: increasing rung by 1 scales mass by φ. -/
theorem mass_rung_scaling (s : Sector) (r : ℤ) (Z_val : ℤ) :
    predict_mass s (r + 1) Z_val = phi * predict_mass s r Z_val := by
  unfold predict_mass
  -- φ^(r+1-8+gap) = φ^1 * φ^(r-8+gap)
  set gap := gap_correction Z_val
  have h_add : (((r + 1 : ℤ) : ℝ) - 8 + gap) = 1 + (((r : ℤ) : ℝ) - 8 + gap) := by
    push_cast
    ring
  rw [h_add, Real.rpow_add phi_pos]
  rw [Real.rpow_one]
  ring

/-- The "gap" term corrects for the charge-based shift.
    When Z=0 (neutral sector baseline), gap(0) = 0. -/
theorem gap_zero_neutral : gap_correction 0 = 0 := by
  unfold gap_correction
  simp only [Int.cast_zero, zero_div, add_zero, Real.log_one, zero_div]

/-- The mass law exhibits φ-scaling: increasing rung by 1 scales mass by φ. -/
theorem mass_rung_scaling (s : Sector) (r : ℤ) (Z_val : ℤ) :
    predict_mass s (r + 1) Z_val = phi * predict_mass s r Z_val := by
  unfold predict_mass
  -- φ^(r+1-8+gap) = φ^1 * φ^(r-8+gap)
  set gap := gap_correction Z_val
  have h_add : (((r + 1 : ℤ) : ℝ) - 8 + gap) = 1 + (((r : ℤ) : ℝ) - 8 + gap) := by
    push_cast
    ring
  rw [h_add, Real.rpow_add phi_pos]
  rw [Real.rpow_one]
  ring

What this page does not claim

The theorem does not identify which specific particle occupies which rung. The theorem does not fix the absolute mass scale or the sector yardstick. The theorem does not assert that all particles in nature are spaced by φ without charge corrections.

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/Masses/MassLaw.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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