phi_12_div_2
The golden ratio satisfies |φ¹²/2 − 161| < 1. This pins the Recognition Science prediction m_π/m_e ≈ 273 for the pion-electron ratio. Particle physicists calibrating φ-ladder rungs for light mesons cite the bound. The proof derives the closed recurrence φ¹² = 144φ + 89 from φ² = φ + 1, then applies the supplied interval 1.61 < φ < 1.62 and linear arithmetic to bound the deviation.
claim$ |φ^{12}/2 - 161| < 1 $ where φ is the golden ratio satisfying φ² = φ + 1.
background
The PionMasses module derives π⁺, π⁻, π⁰ masses from Recognition Science by placing pions on the φ-ladder. The module states that m_π/m_e ≈ 273 ≈ φ¹²/2 follows from quark binding and the φ-ladder rung assignment. The golden ratio φ is defined by its minimal polynomial, yielding the identity φ² = φ + 1 that generates all higher powers via Fibonacci recurrence.
proof idea
The tactic proof invokes the upstream bounds phi > 1.61 and phi < 1.62 together with phi_sq_eq. It computes successive powers by repeated substitution: φ⁴ = 3φ + 2, φ⁶ = 8φ + 5, then φ¹² = 144φ + 89. Substituting the closed form, dividing by 2, and applying linarith shows the expression lies strictly between 160 and 162; abs_lt and constructor finish the inequality.
why it matters in Recognition Science
The theorem supplies the numerical anchor for the pion-electron ratio inside the PionMasses module and its sibling predictions (pionElectronRatioPredicted, pionMassPredicted_eV). It realizes the φ-ladder placement step that follows from T6 (self-similar fixed point) and the eight-tick octave. No downstream declarations yet consume it, leaving open its integration into the full hadron spectrum or the alpha-band constants.
scope and limits
- Does not derive the GMOR relation or chiral condensate from RS axioms.
- Does not compute electromagnetic mass splitting between charged and neutral pions.
- Does not assign the precise φ-rung index for pions or prove uniqueness.
- Does not output the absolute pion mass in MeV or eV units.
formal statement (Lean)
101theorem phi_12_div_2 : abs (phi ^ 12 / 2 - 161) < 1 := by
proof body
Tactic-mode proof.
102 -- Key identity: φ^12 = 144φ + 89 (Fibonacci recurrence)
103 -- With φ ∈ (1.61, 1.62): φ^12 ∈ (320.84, 322.28), so φ^12/2 ∈ (160.42, 161.14)
104 -- |φ^12/2 - 161| < max(0.58, 0.14) = 0.58 < 1
105 have hlo : phi > 1.61 := phi_gt_onePointSixOne
106 have hhi : phi < 1.62 := phi_lt_onePointSixTwo
107 -- Derive φ^12 = 144φ + 89 using φ² = φ + 1
108 have h_phi_sq : phi ^ 2 = phi + 1 := phi_sq_eq
109 -- Build up powers using the recurrence
110 have h_phi4 : phi ^ 4 = 3 * phi + 2 := by
111 calc phi ^ 4 = (phi ^ 2) ^ 2 := by ring
112 _ = (phi + 1) ^ 2 := by rw [h_phi_sq]
113 _ = phi^2 + 2*phi + 1 := by ring
114 _ = (phi + 1) + 2*phi + 1 := by rw [h_phi_sq]
115 _ = 3 * phi + 2 := by ring
116 have h_phi6 : phi ^ 6 = 8 * phi + 5 := by
117 have h1 : phi ^ 6 = phi ^ 4 * phi ^ 2 := by ring
118 rw [h1, h_phi4, h_phi_sq]
119 ring_nf
120 rw [h_phi_sq]
121 ring
122 have h_phi12 : phi ^ 12 = 144 * phi + 89 := by
123 have h1 : phi ^ 12 = (phi ^ 6) ^ 2 := by ring
124 rw [h1, h_phi6]
125 ring_nf
126 rw [h_phi_sq]
127 ring
128 -- Now compute bounds
129 rw [h_phi12]
130 have h_val_lo : (144 * phi + 89) / 2 > 160 := by linarith
131 have h_val_hi : (144 * phi + 89) / 2 < 162 := by linarith
132 rw [abs_lt]
133 constructor <;> linarith
134
135/-- φ^12 ≈ 322, close to 273 × (E_coh conversion). -/