Cosmology Siconversion Planck Time Si Pos

The Planck time in SI units

The Planck time is a unit of time built from fundamental constants of nature. In the International System of Units (SI), it is defined as the time it takes light to travel one Planck length, and it measures roughly 5.391247 × 10⁻⁴⁴ seconds. This is an extraordinarily small duration: about 10⁻²⁰ times the time it takes light to cross a proton. Physicists use it as a natural scale where quantum effects of gravity become significant, and it appears in theories of the very early universe.

The Planck time was first discussed in the late 19th century by Max Planck, who noticed that combining the speed of light, Newton's gravitational constant, and the reduced Planck constant produces a unique time scale. The value is not a prediction of any single theory; it is a derived constant from known physics. In modern metrology, the Planck time is not a defined unit like the second, but a computed quantity from measured constants, and its numerical value carries the uncertainty of those measurements.

In Recognition Science (RS), the framework works in its own native units where the speed of light, a reference length, and a reference time are all set to one. To express RS predictions in human units, the framework needs a calibration anchor, and it uses the Planck scale for this purpose. The declaration planck_time_SI_pos in the framework's machine-checked library of formal theorems establishes a simple but important fact: the SI value of the Planck time is a positive number. This positivity is proved from the definition, which sets the value to 5.391247 × 10⁻⁴⁴ seconds.

The theorem itself is deliberately modest. It does not derive the Planck time from RS principles, nor does it claim that the SI value is a prediction of the framework. The docstring is explicit: the SI numerical values are CODATA-sourced experimental numbers, not RS predictions. They form the calibration seam through which RS predictions are expressed in human units. The theoretical content of RS lies in the ratios of observables to the Planck scale, not in the SI values themselves.

What the theorem does support is the consistency of the calibration. Along with a companion theorem for the Planck length, it feeds into a calibration certificate that checks the ratio of Planck length to speed of light against the Planck time, ensuring the two scales are mutually consistent within one percent. This certificate is what lets the framework convert its native-unit predictions into seconds and meters, and it is the bridge that makes cosmological comparisons possible.

theorem planck_time_SI_pos : 0 < planck_time_SI := by
  unfold planck_time_SI; norm_num

/-- Planck time in seconds (CODATA 2018).
    t_P = ℓ_P/c = 5.391247 × 10⁻⁴⁴ s. -/
def planck_time_SI : ℝ := 5.391247e-44

/-- Convert a time from Planck times to seconds. -/
def planck_to_seconds (t_planck : ℝ) : ℝ := t_planck * planck_time_SI

theorem si_calibration_cert : SICalibrationCert where
  planck_length_positive := planck_length_SI_pos
  planck_time_positive := planck_time_SI_pos
  consistency := by unfold planck_length_SI c_SI planck_time_SI; norm_num
  consistency2 := by unfold planck_length_SI c_SI planck_time_SI; norm_num

What this page does not claim

The theorem does not derive the numerical value of the Planck time from RS principles. The theorem does not claim the SI value of the Planck time is an RS prediction. The theorem does not establish the physical significance of the Planck time.

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/Cosmology/SIConversion.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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