Gravity No Graviton Unit Bridge Bmv Phase Rate Si Eq Kappa Alpha Factored

The unit bridge

The declaration bmvPhaseRateSI_eq_kappa_alpha_factored is a proved identity inside the Recognition Science framework's machine-checked library of formal theorems. It states that a certain phase rate, the rate at which two masses accumulate an entangling phase in a proposed experiment, can be written as a product of three factors: a unit conversion constant, a dimensionless coupling, and a geometric factor depending only on the distances between the masses. The identity itself is a rearrangement of definitions, but the content is that the physical rate is not a new free parameter; it is built from the same dimensionless coupling that governs gravity in the framework.

The classical physics context is the BMV proposal, named after Bose, Marletto, and Vedral, for testing whether gravity is quantum. The idea is to put two masses in superposition, let them interact gravitationally, and look for an entangling phase. The rate of that phase is proportional to G m1 m2 / ℏ times a combination of inverse distances. In the framework's native units, where ℏ = φ⁻⁵ and G = φ⁵/π, the ratio G/ℏ becomes the closed-form quantity φ¹⁰/π, fixed entirely by the golden ratio φ. The theorem bmv_phase_rate_native_eq proves this closed form for the native rate.

The SI version, bmvPhaseRateSI_eq_kappa_alpha_factored, composes that native rate with a unit conversion. The conversion is not derived; it is an input, an inhabitant of an external calibration structure. The theorem is therefore conditional: it proves the algebraic structure of the SI rate, but the numerical value in SI units depends on a calibration that the framework has not yet discharged. The docstring marks this explicitly as a conditional theorem with the calibration as input, living at an open frontier called Foundation.DimensionalBridgeStructural.

What the declaration does not claim is just as important. It does not prove that the BMV experiment will work, that gravity is quantum, or that the proposed geometry is physically realizable. It does not derive the unit conversion from first principles. It does not even assert that the dimensionless coupling κ_rs equals any particular measured number; it only proves the algebraic identity that κ_rs times another constant equals G/ℏ. The theorem is a bridge blueprint, not a bridge that has been built to the shore of SI units.

/-- **T4 master closed form.** Under a unit-bridge input, the SI BMV
phase rate equals `Uconv · κ_rs · α_RS · m₁ m₂ · g`. -/
theorem bmvPhaseRateSI_eq_kappa_alpha_factored (U : UnitBridgeInput) :
    bmvPhaseRateSI U
      = U.Uconv *
          (ZeroParameterGravity.kappa_rs * alphaRS *
            U.m1 * U.m2 *
            bmvGeometryFactor U.r_LL U.r_LR U.r_RL U.r_RR) := by
  unfold bmvPhaseRateSI BMVPhaseRateNative
  rw [show G * U.m1 * U.m2 / hbar
            = (G / hbar) * U.m1 * U.m2 by ring,
      ← kappa_rs_alphaRS_eq_G_over_hbar]

/-- **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

What this page does not claim

The theorem does not prove that the BMV experiment will produce a measurable phase. It does not derive the unit conversion from first principles. It does not assert that the dimensionless coupling equals any particular measured value.

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)

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: