Foundation Sibridge Closure Tau0 Eq Sqrt Pi Planck Time

The bridge to seconds

The Planck time is the smallest meaningful interval in modern physics, built from the speed of light, the reduced Planck constant, and the gravitational constant. Its value is about 5.39 × 10⁻⁴⁴ seconds. The Recognition Science framework, which starts from a discrete record of recognition events rather than from spacetime, defines its own base unit of time, called a tick. The declaration tau0_eq_sqrt_pi_planck_time proves that one tick equals the square root of pi times the Planck time, a closed-form conversion between the framework's native units and the SI system.

The proof works by setting up three conversion factors: seconds per tick, metres per voxel (the framework's unit of length), and kilograms per coherence-mass (its unit of mass). These are matched against the SI values of the speed of light, the reduced Planck constant, and the gravitational constant. After the 2019 redefinition of SI units, the speed of light and the reduced Planck constant are exact by definition. The gravitational constant remains a measured quantity, with its CODATA 2018 value of 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻². The theorem shows that under these three constraints, the seconds-per-tick factor is uniquely determined as the square root of pi times the Planck time. The same algebra also fixes the length and mass conversion factors in closed form.

The framework's native constants are not free parameters: the reduced Planck constant is the inverse of the golden ratio to the fifth power, and the gravitational constant is the golden ratio to the fifth power divided by pi. Their product equals one over pi, which is the identity that makes the square root of pi appear in the time conversion. This is a structural result, not a numerical coincidence. The declaration is a theorem in the framework's machine-checked library of formal theorems, proved with no unverified assumptions.

What the declaration does not claim is just as important. It does not predict the SI value of the gravitational constant; that remains a measurement. It does not claim that the framework's units are physically real in the sense of being directly observable. And it does not claim that the electron mass sits at the Planck scale. The framework's mass ladder places the electron at the golden ratio cubed in coherence-mass units, but relating that substrate-frame value to the observed 0.511 MeV requires an additional dressing factor that is formalised separately. The theorem here is about unit conversion, not about particle physics.

The practical consequence is that the framework can speak in seconds and metres when needed, without losing its native structure. The bridge is conditional on one measured constant, the gravitational constant, which is the single dimensional anchor that modern SI still requires. Everything else in the conversion follows by algebra. This makes the framework's predictions comparable to experiment in familiar units, while preserving the underlying recognition-based picture intact.

/-- **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) _

/-- The SI bridge as three positive conversion factors. -/
structure SIBridge where
  /-- Seconds per tick. -/
  a_T : ℝ
  /-- Metres per voxel. -/
  a_L : ℝ
  /-- Kilograms per coherence-mass. -/
  a_M : ℝ
  /-- All factors strictly positive. -/
  a_T_pos : 0 < a_T
  a_L_pos : 0 < a_L
  a_M_pos : 0 < a_M

/-- Speed of light in SI: exact since SI 2019. -/
def c_SI : ℝ := 299792458

/-- 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

/-- RS-native reduced Planck constant: ℏ = E_coh · τ₀ = φ⁻⁵ in RS-native units.
We write `1 / phi^5` rather than `phi^(-5)` so `ring` works without rpow. -/
def hbar_RS : ℝ := 1 / phi ^ (5 : ℕ)

/-- Current RS-native Newton constant: the existing definition
G = λ_rec² · c³ / (π · ℏ), with λ_rec = c = 1 and ℏ = 1/φ⁵, gives G = φ⁵/π.
This is an input to the SI bridge, not an independent derivation of π. -/
def G_RS : ℝ := phi ^ (5 : ℕ) / Real.pi

/-- **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 declaration does not predict the SI value of the gravitational constant. The theorem does not claim that the framework's native units are directly observable physical quantities. The result does not place the electron mass at the Planck scale; that requires a separate dressing factor.

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.

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