Foundation Sibridge Closure Si Bridge Closed Under Three Constraints

The unique calibration

Every system of units faces the same problem: how do you translate its internal scale into the seconds, metres, and kilograms of everyday physics? The Recognition Science framework defines its own base units: a tick (a discrete unit of time), a voxel (a unit of length), and a coherence-mass (a unit of mass). The declaration si_bridge_closed_under_three_constraints is a theorem in the framework's machine-checked library of formal theorems. It proves that the conversion factors from these native units to SI units are uniquely determined, provided three physical constraints are satisfied.

The three constraints are the measured values of the speed of light, the reduced Planck constant, and Newton's gravitational constant. The framework's native values for these constants are not arbitrary: it sets the speed of light to 1, the reduced Planck constant to phi^-5 (where phi is the golden ratio), and the gravitational constant to phi^5 divided by pi. The theorem shows that if a bridge factor for time, one for length, and one for mass satisfy all three matching equations simultaneously, then the time factor is forced to be exactly sqrt(pi) times the Planck time. This is the unique solution; no other choice of positive conversion factors can satisfy all three constraints at once.

The result is a structural theorem, not a new measurement. It does not predict the value of the gravitational constant; it takes the measured SI value as an input. What it proves is that the conversion algebra closes: given the framework's dimensionless constants and one dimensional anchor (here, the measured G), the entire map from native units to SI units is fixed. The framework's claim is conditional on that anchor. It also reduces the number of free dimensional parameters compared to the Standard Model, which has 19 or more, down to one.

The theorem leaves a known frontier open. If the electron's mass in coherence-mass units is phi^3, the framework's direct conversion places it at the Planck scale, not at the observed 0.511 MeV. The framework attributes this gap to a separate cosmic-Z dressing scale, a structural factor shared by all Standard Model masses. That hierarchy is formalised elsewhere and is not part of this closure theorem.

/-- **MASTER STATEMENT**: under the c, ℏ, G calibration constraints, the SI
bridge is uniquely determined and `a_T = √π · τ_Planck`. This closes the
conversion-map problem conditional on the dimensional anchor. -/
theorem si_bridge_closed_under_three_constraints :
    ∀ b : SIBridge, IsClosedBridge b →
      b.a_T = Real.sqrt Real.pi * tau_Planck := tau0_eq_sqrt_pi_planck_time

/-- **HEADLINE THEOREM**: τ₀ = √π · τ_Planck under the calibrated bridge. -/
theorem tau0_eq_sqrt_pi_planck_time (b : SIBridge) (hC : IsClosedBridge b) :
    b.a_T = Real.sqrt Real.pi * tau_Planck := by
  rw [a_T_eq b hC]
  unfold tau_Planck
  rw [show Real.pi * hbar_SI * G_SI / c_SI ^ 5 =
      Real.pi * (hbar_SI * G_SI / c_SI ^ 5) from by ring]
  exact Real.sqrt_mul (le_of_lt Real.pi_pos) _

/-- **MASTER STATEMENT**: under the c, ℏ, G calibration constraints, the SI
bridge is uniquely determined and `a_T = √π · τ_Planck`. This closes the
conversion-map problem conditional on the dimensional anchor. -/
theorem si_bridge_closed_under_three_constraints :
    ∀ b : SIBridge, IsClosedBridge b →
      b.a_T = Real.sqrt Real.pi * tau_Planck := tau0_eq_sqrt_pi_planck_time

What this page does not claim

This theorem does not predict the numerical value of Newton's gravitational constant G. This theorem does not prove that the framework's native constants are the only possible ones. This theorem does not explain the cosmic-Z hierarchy that separates the Planck scale from the electroweak scale.

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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