Gravity No Graviton Unit Bridge Unit Bridge Theorem

The unit bridge

The Recognition Science library's UnitBridgeTheorem is a formal statement about how a dimensionless number from its internal framework can be converted into a quantity that a tabletop experiment could in principle measure. The theorem itself is a structure, a bundle of three proved facts, not a single equation. It assembles, in one place, the pieces needed to take the framework's gravity constant and express it in standard physical units.

The core result is an identity. In the framework's own units, the ratio of the gravitational constant G to the reduced Planck constant ħ is a closed-form number built only from the golden ratio φ: G/ħ = φ¹⁰/π. The theorem proves this identity and then uses it to rewrite the rate at which two masses would entangle, a quantity from the framework's quantum channel model, into a factored form. That form separates the framework's dimensionless coupling constant, a geometric factor depending on the distances between the masses, and a conversion factor that carries the units.

The conversion factor is where the honest boundary lies. The theorem does not supply the numerical value of that conversion factor. It is parameterized by an input structure that represents an external calibration between the framework's native units and the SI system. The theorem proves that if such a calibration exists, then the phase rate in SI units follows. The existence of that calibration is declared an open frontier in the framework's own dimensional bridge module. Until that frontier is discharged, the theorem is conditional: it is a proof of a relationship, not a prediction of a number.

What the theorem does not claim is as important as what it proves. It does not claim that the framework's gravity constant equals any measured value. It does not claim that the entangling phase rate has been observed. It does not even claim that the calibration input exists. The theorem's achievement is narrower and cleaner: it shows that the framework's internal gravity physics, if its unit system is connected to ours, has a definite and computable consequence for a laboratory-scale observable. The bridge is built, but the road on the far side is not yet open.

/-- **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

/-- **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]

/-- A unit-bridge input parameterizes the conversion of the
RS-native BMV phase rate to SI. It records:
1. The RS-native algebraic identity `G/ℏ = (φ⁵/π) · φ⁵` (already a
   theorem, but bundled for clean propagation);
2. A scale factor `Uconv : ℝ` representing the dimensional
   `seconds_per_tick × meters_per_voxel⁻¹ × ...` combination supplied by
   `Constants.RSNativeUnits.ExternalCalibration`;
3. The test-mass parameters `(m₁, m₂, {r_ab})` in SI units.

This is a deliberately simple `Prop`-valued structure: the actual
`ExternalCalibration` instance lives in
`Foundation.DimensionalBridgeStructural`'s named open frontier. -/
structure UnitBridgeInput where
  /-- SI conversion scale (positive). -/
  Uconv : ℝ
  /-- Conversion is positive. -/
  Uconv_pos : 0 < Uconv
  /-- Mass 1 (SI). -/
  m1 : ℝ
  /-- Mass 1 positivity. -/
  m1_pos : 0 < m1
  /-- Mass 2 (SI). -/
  m2 : ℝ
  /-- Mass 2 positivity. -/
  m2_pos : 0 < m2
  /-- Branch separation r_LL. -/
  r_LL : ℝ
  /-- Nonzero. -/
  r_LL_ne : r_LL ≠ 0
  /-- Branch separation r_LR. -/
  r_LR : ℝ
  /-- Nonzero. -/
  r_LR_ne : r_LR ≠ 0
  /-- Branch separation r_RL. -/
  r_RL : ℝ
  /-- Nonzero. -/
  r_RL_ne : r_RL ≠ 0
  /-- Branch separation r_RR. -/
  r_RR : ℝ
  /-- Nonzero. -/
  r_RR_ne : r_RR ≠ 0

What this page does not claim

The theorem does not claim that the framework's gravity constant matches any measured value of G. The theorem does not claim that the entangling phase rate has been observed in any experiment. The theorem does not claim that the external calibration between framework units and SI units exists.

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

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