Cosmology Siconversion Si Calibration Cert

The calibration seam

The Planck scale is the natural bridge between the abstract units of a physical theory and the human-made units of the laboratory. The Planck length, about 1.616255 × 10⁻³⁵ meters, and the Planck time, about 5.391247 × 10⁻⁴⁴ seconds, are combinations of the fundamental constants that set the scale where quantum effects and gravity meet. The Recognition Science framework derives physics in its own native units, where the speed of light, a reference length, and a reference time are all set to 1. To report its predictions in meters and seconds, it needs one fixed point of contact with the human unit system.

The declaration si_calibration_cert provides that contact point. It is a machine-checked certificate, a formal object in the framework's library of theorems, that packages three facts about the SI values of the Planck scale. First, the Planck length and Planck time are positive numbers. Second, the ratio of the Planck length to the speed of light is consistent with the Planck time to within one percent. The certificate proves these statements by direct computation from the defined constants, with no additional assumptions.

The epistemic content of the certificate is deliberately narrow. The SI values for the Planck length and time are not predictions of the framework; they are external facts, sourced from CODATA, that depend on the human definitions of the meter and the second. The framework's theoretical content lives in the ratios it computes, such as the age of the universe divided by the Planck time, not in the absolute SI numbers. The certificate exists to make those ratios expressible in familiar units, not to derive the length of a meter from first principles.

In Recognition Science, this calibration seam is what allows cosmological observables, like the Hubble constant or the age of the universe, to be stated in kilometers per second per megaparsec or in gigayears. The framework proves relationships in its native units; the certificate and its companion conversion functions translate those results into the units a telescope reports. The certificate is a piece of bookkeeping infrastructure, not a new physical law.

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

theorem planck_length_SI_pos : 0 < planck_length_SI := by
  unfold planck_length_SI; norm_num

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

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

/-- Planck length in meters (CODATA 2018).
    ℓ_P = √(ℏG/c³) = 1.616255 × 10⁻³⁵ m.
    Uncertainty: ±0.000018 × 10⁻³⁵ m (relative: 1.1 × 10⁻⁵). -/
def planck_length_SI : ℝ := 1.616255e-35

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

What this page does not claim

The certificate does not derive the numerical values of the meter or the second from the framework's axioms. The certificate does not assert that the Planck scale is the true fundamental scale of nature, only that it serves as the chosen calibration anchor. The certificate does not validate the CODATA measurements themselves; it treats them as given inputs.

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)

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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