Gravity No Graviton Unit Bridge G Over Hbar Rs Native

A fixed ratio for gravity

Newton's gravitational constant G and Planck's constant hbar are usually measured independently, their ratio a contingent number. In the Recognition Science framework, both constants are derived from a single forcing chain, so their ratio is fixed. The declaration G_over_hbar_RS_native proves that, in the framework's native units, G divided by hbar equals phi to the tenth power divided by pi, where phi is the golden ratio (1 + sqrt(5))/2. The proof unfolds the definitions of G and hbar and simplifies, leaving a closed form with no free parameters.

The ratio appears in the framework's model of gravity as an entangling phase rate. The BMV phase rate, the rate at which two masses acquire a quantum phase from their gravitational interaction, takes the form (G m1 m2 / hbar) times a geometry factor that depends on the distances between the masses. Substituting the proved ratio gives the phase rate as (phi^10 / pi) m1 m2 g. The theorem bmv_phase_rate_native_eq states this closed form exactly.

In Recognition Science, the declaration is one step in a larger bridge. The framework also defines a dimensionless coupling kappa_rs = 8 phi^5 and an alphaRS = phi^5 / (8 pi), and proves that kappa_rs times alphaRS equals G / hbar. This identity links the dimensionless coupling to the dimensionful ratio. The bridge to SI units, however, is not yet complete. The conversion to tabletop SI values requires an external calibration input, which lives at an open frontier in the framework's dimensional bridge. Until that calibration is discharged, the SI-valued phase rate is a conditional theorem, not an unconditional one.

What the declaration does not claim: it does not prove that the framework's G and hbar match the measured SI values. It only fixes their ratio in native units. It does not derive the fine-structure constant or any other coupling from first principles. The bridge to SI remains open, so the framework does not yet produce a number a laboratory can check.

/-- **Helper: G/ℏ in RS-native units.**
We compute `G/ℏ = (φ⁵/π) · φ⁵ = φ¹⁰/π` directly by unfolding the
RS-native definitions:
* `G = λ_rec² c³ / (π ℏ)` with `λ_rec = c = 1` and `ℏ = φ⁻⁵`,
* so `G = 1/(π · φ⁻⁵) = φ⁵/π`,
* and `G/ℏ = (φ⁵/π)/φ⁻⁵ = φ¹⁰/π`. -/
theorem G_over_hbar_RS_native :
    G / hbar = phi ^ (5 : ℝ) / Real.pi * phi ^ (5 : ℝ) := by
  unfold G hbar cLagLock lambda_rec ell0 c tau0 tick
  have hpi : Real.pi ≠ 0 := Real.pi_ne_zero
  have hphi_ne : phi ^ (-(5 : ℝ)) ≠ 0 :=
    ne_of_gt (Real.rpow_pos_of_pos phi_pos _)
  -- After the unfold, we have (1^2 * 1^3) / (π * (φ⁻⁵ * 1)) / (φ⁻⁵ * 1)
  simp only [one_pow, mul_one, div_one]
  -- Goal: 1 / (π * φ⁻⁵) / φ⁻⁵ = φ⁵/π * φ⁵
  rw [Real.rpow_neg phi_pos.le]
  field_simp

/-- BMV entangling phase rate in RS-native units. By T3 the entangling
invariant is `(G m₁ m₂ T / ℏ) · g`; the per-time rate is
`(G m₁ m₂ / ℏ) · g`. -/
noncomputable def BMVPhaseRateNative
    (m1 m2 r_LL r_LR r_RL r_RR : ℝ) : ℝ :=
  (G * m1 * m2 / hbar) * bmvGeometryFactor r_LL r_LR r_RL r_RR

/-- **Closed form for the RS-native BMV phase rate.** -/
theorem bmv_phase_rate_native_eq
    (m1 m2 r_LL r_LR r_RL r_RR : ℝ) :
    BMVPhaseRateNative m1 m2 r_LL r_LR r_RL r_RR
      = (phi ^ (5 : ℝ) / Real.pi * phi ^ (5 : ℝ))
          * m1 * m2 * bmvGeometryFactor r_LL r_LR r_RL r_RR := by
  unfold BMVPhaseRateNative
  rw [show G * m1 * m2 / hbar = (G / hbar) * m1 * m2 by ring,
      G_over_hbar_RS_native]

/-- **κ_rs · α_RS = G/ℏ in RS-native units.** -/
theorem kappa_rs_alphaRS_eq_G_over_hbar :
    ZeroParameterGravity.kappa_rs * alphaRS = G / hbar := by
  unfold ZeroParameterGravity.kappa_rs alphaRS
  rw [G_over_hbar_RS_native]
  -- (8 · φ⁵) · (φ⁵ / (8π)) = φ⁵/π · φ⁵
  -- The LHS uses a Nat exponent (from `kappa_rs` and `alphaRS` definitions
  -- as `phi ^ (5 : ℕ)` via `^`), the RHS from `G_over_hbar_RS_native` uses
  -- `phi ^ (5 : ℝ)`. We bridge with `Real.rpow_natCast`.
  have hbridge : phi ^ (5 : ℕ) = phi ^ ((5 : ℕ) : ℝ) := by
    rw [Real.rpow_natCast]
  have hpi : Real.pi ≠ 0 := Real.pi_ne_zero
  have h5 : ((5 : ℕ) : ℝ) = (5 : ℝ) := by norm_num
  rw [hbridge, h5]
  field_simp

What this page does not claim

The framework's G and hbar match the measured SI values. The fine-structure constant is derived from first principles. The SI-valued phase rate is unconditional; it requires an external calibration.

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

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