Cosmology Siconversion Planck Length Si Pos

The SI anchor

The Planck length is the scale at which quantum effects of gravity are expected to become strong, about 1.616255 × 10⁻³⁵ meters. It is constructed from three fundamental constants: the reduced Planck constant ħ, Newton's gravitational constant G, and the speed of light c, combined as ℓ_P = √(ħG/c³). This is a standard physics definition, not something unique to any one theory. The number is staggeringly small: a proton is roughly 10²⁰ times larger.

The measured value comes from CODATA, the international committee that publishes recommended values of physical constants. The 2018 CODATA adjustment gives ℓ_P ≈ 1.616255 × 10⁻³⁵ m, with the meter defined by the distance light travels in a fixed time interval. This is an experimental input, not a theoretical prediction. The framework's ledger, a discrete record of recognition events from which it derives physics, does not determine the length of a human-defined meter.

In Recognition Science, the declaration planck_length_SI_pos states that this SI value is positive: 0 < 1.616255e-35. The proof is trivial, a direct computation from the definition. Its purpose is not to discover the value but to certify it as a usable anchor. The framework derives physics in native units where c = ℓ₀ = τ₀ = 1, and needs a bridge to human units for comparison with observations. This positivity theorem is part of that bridge, ensuring the conversion factor is mathematically sound.

The framework's library also proves that in its native units, the Planck length equals 1/√π, a result from its internal derivation. The SI value, however, is explicitly external. The theoretical content lies in ratios: an observed radius divided by ℓ_P, or an age divided by the Planck time. These ratios are dimensionless and comparable across unit systems. The SI constants themselves carry no predictive weight.

What planck_length_SI_pos does not claim is more important than what it proves. It does not assert that the Planck length is the smallest possible length, a claim physics has not established. It does not claim the framework predicts the meter's length. It does not even claim the CODATA value is exact; it only certifies positivity. The declaration is a calibration seam, a place where theory meets measurement, not a statement about the nature of spacetime at the Planck scale.

/-- 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 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

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

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

What this page does not claim

The Planck length is the smallest possible length in physics. The framework predicts the SI value of the meter. The CODATA value is exact rather than a measured approximation.

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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