Foundation Sibridge Closure A L Eq Of C Constraint

The length conversion

The declaration aL_eq_of_c_constraint establishes a single, simple relationship between two of the framework's fundamental units: the length of a tick (a discrete unit of time) and the length of a voxel (a discrete unit of space). It proves that, given the framework's own definition that the speed of light in native units is exactly 1, the length of a voxel in metres is always equal to the speed of light in metres per second multiplied by the length of a tick in seconds. In other words, the conversion factor between the framework's space and time units is not chosen freely; it is forced by the constancy of the speed of light.

This is a purely algebraic consequence of the framework's setup. The framework defines a bridge between its native units and the SI system using three positive conversion factors: seconds per tick, metres per voxel, and kilograms per coherence-mass. The speed-of-light constraint states that the SI value of c (299,792,458 metres per second) must equal the native value (1) times the ratio of the length factor to the time factor. The theorem aL_eq_of_c_constraint simply rearranges this constraint to solve for the length factor directly.

The theorem does not, by itself, determine the numerical value of either the tick or the voxel. It only fixes their ratio. To pin down the actual size of a tick in seconds, the framework must bring in two more constraints, one from the reduced Planck constant and one from the gravitational constant. Only when all three constraints are applied together does the framework derive a unique value for the tick, which it identifies as the square root of pi times the Planck time.

In Recognition Science, this result is part of a larger structural claim: that the framework's internal units are not arbitrary but are uniquely calibrated to the SI system once the dimensional anchor is supplied. The framework's library of machine-checked theorems proves the algebra of this calibration, but it does not predict the measured value of the gravitational constant. That value remains an input, a CODATA measurement, not a derivation.

The practical consequence is that the framework's unit conversions are not a matter of convention or choice. Once the framework commits to the speed of light being exactly 1 in its native units, the relationship between its space and time units is locked in. This is a small but necessary step in the framework's broader project of deriving physical constants from a single cost function, and it shows how a purely formal system can connect to the measured world through a minimal set of anchors.

/-- From c-constraint, `a_L = c_SI · a_T`. -/
theorem aL_eq_of_c_constraint (b : SIBridge) (hC : c_constraint b) :
    b.a_L = c_SI * b.a_T := by
  unfold c_constraint c_RS at hC
  have hT_ne : b.a_T ≠ 0 := ne_of_gt b.a_T_pos
  -- hC : c_SI = 1 * (a_L / a_T)
  rw [one_mul] at hC
  -- hC : c_SI = a_L / a_T
  rw [eq_div_iff hT_ne] at hC
  -- hC : c_SI * a_T = a_L
  linarith

/-- From c-constraint, `a_L = c_SI · a_T`. -/
theorem aL_eq_of_c_constraint (b : SIBridge) (hC : c_constraint b) :
    b.a_L = c_SI * b.a_T := by
  unfold c_constraint c_RS at hC
  have hT_ne : b.a_T ≠ 0 := ne_of_gt b.a_T_pos
  -- hC : c_SI = 1 * (a_L / a_T)
  rw [one_mul] at hC
  -- hC : c_SI = a_L / a_T
  rw [eq_div_iff hT_ne] at hC
  -- hC : c_SI * a_T = a_L
  linarith

/-- **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) _

What this page does not claim

The theorem does not predict the SI value of the gravitational constant. The theorem does not determine the absolute size of the tick or voxel, only their ratio. The theorem does not establish the physical existence of discrete space-time voxels.

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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