Encyclopedia Constants Constants Hbar Lt One
ARTICLE 4 claims 3 theorems 1 model
Constants Hbar Lt One
A theorem in a machine-checked library proves a fundamental unit of action is less than one, and the proof is a matter of definition.
The small quantum
In physics, an action is a quantity with units of energy multiplied by time, and the reduced Planck constant, written ℏ, is the tiny quantum of action that sets the scale of quantum effects. Its measured value in SI units is about 1.054 × 10⁻³⁴ joule-seconds, a number so small that quantum behavior is hidden in everyday life. Recognition Science, a framework that derives physical structure from a ledger of recognition events, works in its own native units, where the speed of light and a fundamental length are both set to one. In those units, the framework defines its own ℏ, not as a free parameter but as a product of a locking constant and a fundamental time tick.
The declaration hbar_lt_one is a theorem in the framework's machine-checked library of formal theorems, proving that this native ℏ is less than 1. The proof is not a numerical approximation; it follows from the definitions. The framework defines the locking constant as the golden ratio raised to the power −5, and the golden ratio is about 1.618, so that constant is about 0.09017. The fundamental time tick is defined to be exactly 1, and the product of 0.09017 and 1 is indeed less than 1. A companion theorem proves ℏ is positive, and a bounds theorem pins it between 0.088 and 0.093. The framework's ℏ is thus a small positive number in its own units, consistent with the familiar idea that the quantum of action is small.
In Recognition Science, this smallness is not an accident. The framework models the cost of a recognition event, and its central theorem forces a specific cost function. From that function, the golden ratio emerges as a unique self-similar scaling, and the framework derives the constants of nature, including this ℏ, rather than taking them as inputs. The theorem hbar_lt_one is a small but necessary step in that chain: it confirms that the derived quantum of action is in the expected range, less than one in the framework's units. It also connects to the framework's identity that ℏ equals the product of a coherence energy and the time tick, a relation that mirrors the classical action-energy-time link.
What the theorem does not claim is just as important as what it proves. It does not prove that the framework's ℏ equals the measured Planck constant in SI units; that comparison is an empirical check, not a theorem. It does not prove that the golden ratio itself is fundamental to physics; that is a choice the framework makes in its definitions. And it does not prove that the framework's units are the same as conventional physics units; the theorem is internal to the framework's own system of units. The declaration is a precise, definition-driven statement about a number the framework constructs, not a claim about the measured world.
THEOREM hbar_lt_one · IndisputableMonolith/Constants.lean
/-- **THEOREM C-004.3**: ℏ < 1 (the action quantum is small compared to natural units).
Proof: φ > 1 ⟹ φ⁵ > 1 ⟹ φ⁻⁵ < 1. -/
theorem hbar_lt_one : hbar < 1 := by
rw [hbar_eq_phi_inv_fifth]
have h1 : phi ^ (5 : ℝ) > 1 := by
have hphi : phi > 1 := one_lt_phi
have hexp : (5 : ℝ) > 0 := by norm_num
have h1_lt : (1 : ℝ) < phi ^ (5 : ℝ) := by
rw [← Real.one_rpow (5 : ℝ)]
apply Real.rpow_lt_rpow
· norm_num
· linarith
· norm_num
linarith
have h2 : 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 [h2]
have h3 : phi ^ (5 : ℝ) > 0 := by positivity
apply (div_lt_iff₀ h3).mpr
linarith
MODEL cLagLock · IndisputableMonolith/Constants.lean
/-- Canonical locked C_lag constant: C_lock = φ^{−5}. -/
@[simp] noncomputable def cLagLock : ℝ := phi ^ (-(5 : ℝ))
THEOREM hbar_positive · hbar_bounds · IndisputableMonolith/Constants.lean
/-- **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 hbar_lt_one · IndisputableMonolith/Constants.lean
/-- **THEOREM C-004.3**: ℏ < 1 (the action quantum is small compared to natural units).
Proof: φ > 1 ⟹ φ⁵ > 1 ⟹ φ⁻⁵ < 1. -/
theorem hbar_lt_one : hbar < 1 := by
rw [hbar_eq_phi_inv_fifth]
have h1 : phi ^ (5 : ℝ) > 1 := by
have hphi : phi > 1 := one_lt_phi
have hexp : (5 : ℝ) > 0 := by norm_num
have h1_lt : (1 : ℝ) < phi ^ (5 : ℝ) := by
rw [← Real.one_rpow (5 : ℝ)]
apply Real.rpow_lt_rpow
· norm_num
· linarith
· norm_num
linarith
have h2 : 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 [h2]
have h3 : phi ^ (5 : ℝ) > 0 := by positivity
apply (div_lt_iff₀ h3).mpr
linarith
What this page does not claim
The theorem does not prove the framework's ℏ equals the measured Planck constant in SI units. The theorem does not prove the golden ratio is fundamental to physics; that is a definitional choice. The theorem does not claim the framework's units are the same as conventional physics units.
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
This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red:
- What is the empirical comparison between the framework's ℏ and the measured Planck constant?
- How does the framework derive the golden ratio from its cost function?
- What is the physical interpretation of the framework's native units?
MACHINE LAYER · GROUNDED CLAIM TABLE · CLICK TO EXPAND
THEOREM hbar_lt_one · IndisputableMonolith/Constants.lean
/-- **THEOREM C-004.3**: ℏ < 1 (the action quantum is small compared to natural units). Proof: φ > 1 ⟹ φ⁵ > 1 ⟹ φ⁻⁵ < 1. -/ theorem hbar_lt_one : hbar < 1 := by rw [hbar_eq_phi_inv_fifth] have h1 : phi ^ (5 : ℝ) > 1 := by have hphi : phi > 1 := one_lt_phi have hexp : (5 : ℝ) > 0 := by norm_num have h1_lt : (1 : ℝ) < phi ^ (5 : ℝ) := by rw [← Real.one_rpow (5 : ℝ)] apply Real.rpow_lt_rpow · norm_num · linarith · norm_num linarith have h2 : 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 [h2] have h3 : phi ^ (5 : ℝ) > 0 := by positivity apply (div_lt_iff₀ h3).mpr linarithThe declaration hbar_lt_one is a theorem in the framework's machine-checked library of formal theorems, proving that this native ℏ is less than 1. hbar_lt_one · IndisputableMonolith/Constants.leanMODEL cLagLock · IndisputableMonolith/Constants.lean
/-- Canonical locked C_lag constant: C_lock = φ^{−5}. -/ @[simp] noncomputable def cLagLock : ℝ := phi ^ (-(5 : ℝ))The framework defines the locking constant as the golden ratio raised to the power −5, and the golden ratio is about 1.618, so that constant is about 0.09017. cLagLock · IndisputableMonolith/Constants.leanTHEOREM hbar_positive · hbar_bounds · IndisputableMonolith/Constants.lean
/-- **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⟩A companion theorem proves ℏ is positive, and a bounds theorem pins it between 0.088 and 0.093. hbar_positive · hbar_bounds · IndisputableMonolith/Constants.leanTHEOREM hbar_lt_one · IndisputableMonolith/Constants.lean
/-- **THEOREM C-004.3**: ℏ < 1 (the action quantum is small compared to natural units). Proof: φ > 1 ⟹ φ⁵ > 1 ⟹ φ⁻⁵ < 1. -/ theorem hbar_lt_one : hbar < 1 := by rw [hbar_eq_phi_inv_fifth] have h1 : phi ^ (5 : ℝ) > 1 := by have hphi : phi > 1 := one_lt_phi have hexp : (5 : ℝ) > 0 := by norm_num have h1_lt : (1 : ℝ) < phi ^ (5 : ℝ) := by rw [← Real.one_rpow (5 : ℝ)] apply Real.rpow_lt_rpow · norm_num · linarith · norm_num linarith have h2 : 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 [h2] have h3 : phi ^ (5 : ℝ) > 0 := by positivity apply (div_lt_iff₀ h3).mpr linarithThe framework's ℏ is thus a small positive number in its own units, consistent with the familiar idea that the quantum of action is small. hbar_lt_one · IndisputableMonolith/Constants.lean