hbar_positive

formal source

 315  simp
 316
 317/-- **THEOREM C-004.2**: ℏ is positive (required for quantum dynamics). -/
 318theorem hbar_positive : hbar > 0 := hbar_pos
 319
 320/-- **THEOREM C-004.3**: ℏ < 1 (the action quantum is small compared to natural units).
 321
 322    Proof: φ > 1 ⟹ φ⁵ > 1 ⟹ φ⁻⁵ < 1. -/
 323theorem hbar_lt_one : hbar < 1 := by
 324  rw [hbar_eq_phi_inv_fifth]
 325  have h1 : phi ^ (5 : ℝ) > 1 := by
 326    have hphi : phi > 1 := one_lt_phi
 327    have hexp : (5 : ℝ) > 0 := by norm_num
 328    have h1_lt : (1 : ℝ) < phi ^ (5 : ℝ) := by
 329      rw [← Real.one_rpow (5 : ℝ)]
 330      apply Real.rpow_lt_rpow
 331      · norm_num
 332      · linarith
 333      · norm_num
 334    linarith
 335  have h2 : phi ^ (-(5 : ℝ)) = 1 / (phi ^ (5 : ℝ)) := by
 336    rw [show (-(5 : ℝ)) = - (5 : ℝ) by norm_num]
 337    rw [Real.rpow_neg]
 338    · ring
 339    · exact le_of_lt phi_pos
 340  rw [h2]
 341  have h3 : phi ^ (5 : ℝ) > 0 := by positivity
 342  apply (div_lt_iff₀ h3).mpr
 343  linarith
 344
 345/-- **THEOREM C-004.4**: ℏ = E_coh · τ₀ (the action quantum identity).
 346
 347    This is the fundamental physical interpretation: Planck's constant
 348    is the minimal action (energy × time) for a recognition event. -/