Foundation Sibridge Closure Hbar Rs Mul G Rs

The native product

The declaration hbar_RS_mul_G_RS is a formal theorem inside the Recognition Science framework's machine-checked library of formal theorems. It states a plain algebraic fact about two constants the framework defines in its own native units. In those units, the reduced Planck constant is the reciprocal of the golden ratio raised to the fifth power, and the gravitational constant is the golden ratio raised to the fifth power divided by pi. The theorem proves that their product is exactly one over pi. The golden ratio itself is the number φ satisfying φ² = φ + 1, approximately 1.618.

The proof is a short algebraic cancellation. Since the reduced Planck constant is 1/φ⁵ and the gravitational constant is φ⁵/π, multiplying them gives (1/φ⁵) × (φ⁵/π). The φ⁵ terms cancel, leaving exactly 1/π. The theorem is tagged as a structural theorem, meaning it is proved with zero axioms specific to the framework, relying only on the standard logical foundations of the machine-checked library. The declaration is part of a larger module, the SI Bridge Closure, which uses this identity as one input to a calibration map.

In Recognition Science, this identity is not a prediction about the physical world. It is a definitional consequence of how the framework chooses its native units. The framework models physical constants as pure numbers derived from its forcing chain; the golden ratio appears as a fundamental scaling, and pi enters through the gravitational constant. The product identity is a consistency check that these definitions align with each other. It does not, by itself, say anything about the measured values of the Planck constant or the gravitational constant in SI units.

What the declaration does not claim is equally precise. It does not predict the numerical value of Newton's gravitational constant G in kilograms, metres, and seconds. That value, 6.67430 × 10⁻¹¹, is a CODATA measurement used as an input to the SI bridge, not a theorem output. The identity also does not derive the fine-structure constant, which remains an open target in the framework. Finally, the theorem does not establish any physical law; it is a pure algebraic statement about the framework's own definitions, a necessary but small step in the larger project of connecting native units to SI.

/-- The product `ℏ_RS · G_RS = 1/π` (Planck identity in RS-native). -/
theorem hbar_RS_mul_G_RS : hbar_RS * G_RS = 1 / Real.pi := by
  unfold hbar_RS G_RS
  have hpi_ne : Real.pi ≠ 0 := Real.pi_pos.ne'
  have hphi5_ne : phi ^ (5 : ℕ) ≠ 0 := phi_pow_5_pos.ne'
  -- (1/φ⁵) · (φ⁵/π) = φ⁵/(φ⁵·π) = 1/π
  rw [div_mul_div_comm, one_mul,
      div_eq_div_iff (mul_ne_zero hphi5_ne hpi_ne) hpi_ne]
  ring

/-- RS-native reduced Planck constant: ℏ = E_coh · τ₀ = φ⁻⁵ in RS-native units.
We write `1 / phi^5` rather than `phi^(-5)` so `ring` works without rpow. -/
def hbar_RS : ℝ := 1 / phi ^ (5 : ℕ)

/-- Current RS-native Newton constant: the existing definition
G = λ_rec² · c³ / (π · ℏ), with λ_rec = c = 1 and ℏ = 1/φ⁵, gives G = φ⁵/π.
This is an input to the SI bridge, not an independent derivation of π. -/
def G_RS : ℝ := phi ^ (5 : ℕ) / Real.pi

/-- The product `ℏ_RS · G_RS = 1/π` (Planck identity in RS-native). -/
theorem hbar_RS_mul_G_RS : hbar_RS * G_RS = 1 / Real.pi := by
  unfold hbar_RS G_RS
  have hpi_ne : Real.pi ≠ 0 := Real.pi_pos.ne'
  have hphi5_ne : phi ^ (5 : ℕ) ≠ 0 := phi_pow_5_pos.ne'
  -- (1/φ⁵) · (φ⁵/π) = φ⁵/(φ⁵·π) = 1/π
  rw [div_mul_div_comm, one_mul,
      div_eq_div_iff (mul_ne_zero hphi5_ne hpi_ne) hpi_ne]
  ring

What this page does not claim

The theorem does not predict the measured value of Newton's gravitational constant in SI units. The theorem does not derive the fine-structure constant, which remains an open target. The theorem does not establish any physical law beyond the framework's own definitions.

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

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Derived articles

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