Foundation Sibridge Closure Si Bridge Closure Cert Inhabited

The SI Bridge Certificate

The declaration siBridgeClosureCert_inhabited is a formal proof that a certain mathematical object, the SI bridge certificate, is not empty. In plain terms, it establishes that a consistent set of three positive conversion factors can exist, which translate the framework's native units of time, length, and mass into seconds, metres, and kilograms. This is a structural result: it shows the conversion is possible, not that a specific numerical value is predicted.

The framework's native constants are the speed of light c_RS = 1, the reduced Planck constant ħ_RS = φ⁻⁵, and the gravitational constant G_RS = φ⁵/π, all in its own system of units. The certificate proves that, given the SI definitions of c and ħ (which are exact by definition since 2019) and the measured value of G, there is a unique way to assign the conversion factors. The key theorem a_T_sq_eq shows that the square of the time conversion factor, a_T², equals π·ħ_SI·G_SI / c_SI⁵. This leads to the closed form τ₀ = √π · τ_Planck, meaning one native tick equals the square root of π times the Planck time.

The declaration does not predict the SI value of G. It takes G_SI as a given input, a CODATA measurement, and proves the algebra of conversion. It also does not derive the value of π; the π in G_RS is inherited from the framework's constants. The certificate is conditional: it proves uniqueness and existence of the bridge, given the supplied dimensional anchor. This is a structural closure, not a new physical prediction.

In Recognition Science, this certificate is the final step in connecting its internal dimensionless predictions to the familiar units of experimental physics. It reduces the problem of matching the framework's numbers to SI to a single measured input, G. The result is a machine-checked guarantee that the translation is consistent and unique, which is a necessary condition for any empirical comparison of the framework's predictions with measured quantities.

theorem siBridgeClosureCert_inhabited : Nonempty SIBridgeClosureCert :=
  ⟨siBridgeClosureCert⟩

/-- **MAIN ALGEBRAIC IDENTITY**: under c + ℏ + G constraints,
`a_T² = π · ℏ_SI · G_SI / c_SI⁵`.

Proof: multiply `(a_M · a_T) · (a_T / a_M) = a_T²` using the helper
identities, and use `ℏ_RS · G_RS = 1/π`. -/
theorem a_T_sq_eq (b : SIBridge) (hC : IsClosedBridge b) :
    b.a_T ^ 2 = Real.pi * hbar_SI * G_SI / c_SI ^ 5 := by
  obtain ⟨hC_c, hC_h, hC_G⟩ := hC
  have h_aMaT := aM_aT_eq_of_c_hbar b hC_c hC_h
  have h_aTaM := aT_aM_eq_of_c_G b hC_c hC_G
  have hM_ne : b.a_M ≠ 0 := ne_of_gt b.a_M_pos
  -- (a_M · a_T) · (a_T / a_M) = a_T²
  have h_prod : (b.a_M * b.a_T) * (b.a_T / b.a_M) = b.a_T ^ 2 := by
    rw [show (b.a_M * b.a_T) * (b.a_T / b.a_M)
          = (b.a_M / b.a_M) * (b.a_T * b.a_T) from by ring]
    rw [div_self hM_ne, one_mul, sq]
  -- Substitute the helper identities and simplify using ℏ_RS · G_RS = 1/π
  have h_hG : hbar_RS * G_RS = 1 / Real.pi := hbar_RS_mul_G_RS
  have hbar_RS_ne : hbar_RS ≠ 0 := hbar_RS_pos.ne'
  have G_RS_ne : G_RS ≠ 0 := G_RS_pos.ne'
  have c_SI_ne : c_SI ≠ 0 := c_SI_pos.ne'
  have hpi_ne : Real.pi ≠ 0 := Real.pi_pos.ne'
  have c_SI2_ne : c_SI ^ 2 ≠ 0 := pow_ne_zero _ c_SI_ne
  have c_SI3_ne : c_SI ^ 3 ≠ 0 := pow_ne_zero _ c_SI_ne
  have c_SI5_ne : c_SI ^ 5 ≠ 0 := pow_ne_zero _ c_SI_ne
  -- Compute (ℏ_SI / (ℏ_RS · c²)) · (G_SI / (G_RS · c³)) = π · ℏ_SI · G_SI / c⁵
  have h_target : (hbar_SI / (hbar_RS * c_SI ^ 2)) * (G_SI / (G_RS * c_SI ^ 3))
      = Real.pi * hbar_SI * G_SI / c_SI ^ 5 := by
    -- Combine fractions: numerator product over denominator product.
    have h_combine : (hbar_SI / (hbar_RS * c_SI ^ 2)) * (G_SI / (G_RS * c_SI ^ 3))
        = hbar_SI * G_SI / (hbar_RS * G_RS * c_SI ^ 5) := by
      rw [div_mul_div_comm]
      congr 1
      ring
    rw [h_combine, h_hG]
    -- Goal: ℏ_SI · G_SI / ((1/π) · c⁵) = π · ℏ_SI · G_SI / c⁵
    rw [show (1 / Real.pi) * c_SI ^ 5 = c_SI ^ 5 / Real.pi from by ring]
    rw [div_div_eq_mul_div]
    rw [show hbar_SI * G_SI * Real.pi = Real.pi * hbar_SI * G_SI from by ring]
  rw [← h_prod, h_aMaT, h_aTaM, h_target]

/-- Newton's gravitational constant in SI (CODATA 2018 recommended value).
This is the SINGLE remaining dimensional measurement after SI 2019. -/
def G_SI : ℝ := 6.67430e-11

What this page does not claim

The declaration does not predict the numerical value of the gravitational constant G. The declaration does not derive the value of π from the framework's axioms. The declaration does not establish that the framework's native units correspond to any physical reality, only that a consistent conversion 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/Foundation/SIBridgeClosure.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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