Constants Hbar Action Identity

The action identity

Recognition Science (RS) is a framework that derives physical structure from a single starting point: reality keeps a discrete record of recognition events, and the cost of recognition is forced, not chosen. Within this framework, the fundamental constants are not free parameters but are forced into specific values by a chain of theorems. The declaration hbar_action_identity is a theorem in the framework's machine-checked library of formal theorems that establishes a precise relationship between three of these constants.

The theorem states that hbar = E_coh * tau0. In plain language, the framework's version of Planck's constant (the quantum of action) is exactly the product of a fundamental energy E_coh and a fundamental time quantum tau0, which is defined as one tick. The declaration is a definitional identity, meaning it holds by the way these constants are defined. It is not an empirical measurement or a derived physical law; it is a statement of how the framework's internal units relate to each other.

The identity is a consequence of the framework's unit system. In RS-native units, the fundamental time quantum tau0 is set to 1, and the fundamental energy E_coh is defined as phi^(-5), where phi is the golden ratio. The theorem hbar_action_identity then states that the framework's action constant is simply this energy multiplied by this time. This is a definitional choice, not a physical prediction. The framework proves that hbar is positive and lies between 0.088 and 0.093 in these units, but it does not claim that these numbers correspond to the measured value of Planck's constant in SI units.

The identity is important because it shows how the framework's constants are interconnected. It does not, however, claim to derive the numerical value of Planck's constant as measured in kilograms, meters, and seconds. The framework's constants are expressed in its own units, and the identity is a statement about the relationships between those units. The declaration does not claim that E_coh is the energy of any specific physical process, nor does it claim that tau0 is the duration of any specific physical event. It simply establishes a formal relationship within the framework's internal system of definitions.

/-- **THEOREM C-004.4**: native action quantum identity.

    The native action quantum is the energy-time product of one coherence
    event and one tick. -/
theorem hbar_action_identity : hbar = E_coh * tau0 := rfl

/-- **THEOREM C-004.2**: ℏ is positive (required for quantum dynamics). -/
theorem hbar_positive : hbar > 0 := hbar_pos

/-- **THEOREM C-004.5**: Bounds on ℏ from φ bounds.

    With φ ∈ (1.61, 1.62), we get ℏ ∈ (0.088, 0.093). -/
theorem hbar_bounds : (0.088 : ℝ) < hbar ∧ hbar < (0.093 : ℝ) := by
  rw [hbar_eq_phi_inv_fifth]
  have h1 : (1.61 : ℝ) < phi := phi_gt_onePointSixOne
  have h2 : phi < (1.62 : ℝ) := phi_lt_onePointSixTwo
  -- We want 0.088 < φ^(-5) < 0.093
  -- Since hbar = 1/φ^5, we need bounds on φ^5
  -- Lower bound: φ < 1.62, so φ^5 < 1.62^5, so 1/φ^5 > 1/1.62^5
  -- Upper bound: φ > 1.61, so φ^5 > 1.61^5, so 1/φ^5 < 1/1.61^5
  have h_phi5_lower : phi ^ (5 : ℝ) > (1.61 : ℝ) ^ (5 : ℝ) := by
    apply Real.rpow_lt_rpow
    · linarith
    · linarith
    · norm_num
  have h_phi5_upper : phi ^ (5 : ℝ) < (1.62 : ℝ) ^ (5 : ℝ) := by
    apply Real.rpow_lt_rpow
    · linarith
    · linarith
    · norm_num
  -- Convert to hbar = φ^(-5) bounds
  have hbar_lower : phi ^ (-(5 : ℝ)) > (0.088 : ℝ) := by
    have h_inv : phi ^ (-(5 : ℝ)) = 1 / (phi ^ (5 : ℝ)) := by
      rw [show (-(5 : ℝ)) = - (5 : ℝ) by norm_num]
      rw [Real.rpow_neg]
      · ring
      · exact le_of_lt phi_pos
    rw [h_inv]
    -- Since φ^5 < 1.62^5, we have 1/φ^5 > 1/1.62^5
    -- Compute 1.62^5 = 11.158... and 1/11.158 ≈ 0.0896 > 0.088
    have h_div : 1 / (phi ^ (5 : ℝ)) > 1 / ((1.62 : ℝ) ^ (5 : ℝ)) := by
      apply (one_div_lt_one_div (by positivity) (by positivity)).mpr
      linarith [h_phi5_upper]
    have h_numeric : 1 / ((1.62 : ℝ) ^ (5 : ℝ)) > (0.088 : ℝ) := by
      rw [show (5 : ℝ) = (5 : ℕ) by norm_num, Real.rpow_natCast]
      norm_num
    linarith
  have hbar_upper : phi ^ (-(5 : ℝ)) < (0.093 : ℝ) := by
    have h_inv : phi ^ (-(5 : ℝ)) = 1 / (phi ^ (5 : ℝ)) := by
      rw [show (-(5 : ℝ)) = - (5 : ℝ) by norm_num]
      rw [Real.rpow_neg]
      · ring
      · exact le_of_lt phi_pos
    rw [h_inv]
    -- Since φ^5 > 1.61^5, we have 1/φ^5 < 1/1.61^5
    -- Compute 1.61^5 = 10.817... and 1/10.817 ≈ 0.0924 < 0.093
    have h_div : 1 / (phi ^ (5 : ℝ)) < 1 / ((1.61 : ℝ) ^ (5 : ℝ)) := by
      apply (div_lt_div_iff₀ (by positivity) (by positivity)).mpr
      linarith [h_phi5_lower]
    have h_numeric : 1 / ((1.61 : ℝ) ^ (5 : ℝ)) < (0.093 : ℝ) := by
      rw [show (5 : ℝ) = (5 : ℕ) by norm_num, Real.rpow_natCast]
      norm_num
    linarith
  exact ⟨hbar_lower, hbar_upper⟩

/-- **THEOREM C-004.4**: native action quantum identity.

    The native action quantum is the energy-time product of one coherence
    event and one tick. -/
theorem hbar_action_identity : hbar = E_coh * tau0 := rfl

What this page does not claim

The identity does not claim that the framework's hbar equals the measured value of Planck's constant in SI units. The identity does not claim that E_coh is the energy of any specific physical process. The identity does not claim that tau0 is the duration of any specific physical event.

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/Constants.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.

Derived articles

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