Constants Derivation Planck Time Inner Nonneg

The positivity lemma

The Planck time is a unit of time built from the three constants that anchor modern physics: the speed of light c, the reduced Planck constant ħ, and Newton's gravitational constant G. Its standard definition is t_P = sqrt(ħG/c⁵). In the International System of units, with c = 299792458 m/s exactly, ħ = 1.054571817×10⁻³⁴ J·s, and G = 6.67430×10⁻¹¹ m³/(kg·s²), this works out to about 5.39×10⁻⁴⁴ seconds. That is the time it takes light to cross one Planck length, roughly 1.6×10⁻³⁵ meters, a scale where quantum effects of gravity are expected to become strong.

The declaration planck_time_inner_nonneg is a lemma inside a machine-checked library of formal theorems. It proves that the expression inside the square root in the Planck time formula, namely ħG/c⁵, is greater than or equal to zero. Because the library has already shown that ħ, G, and c are all positive numbers, and that c⁵ is positive as well, the lemma follows directly: a product of positives divided by a positive is positive. The lemma's conclusion is 0 ≤ ħG/c⁵, and its proof is a single step from the positivity of the numerator and denominator.

This is a small, technical result, but it earns its place in the library because it makes the definition of the Planck time well-founded. A square root of a negative number would not be a real number, so the lemma guarantees that the Planck time as defined is a genuine real quantity. In the same file, a companion lemma proves the stronger statement that the Planck time itself is strictly positive, 0 < t_P, using the same ingredients.

In Recognition Science, the framework models physical constants as derived from a discrete ledger of recognition events rather than as free parameters. This lemma is part of that derivation's bookkeeping: it does not by itself establish any physical law, but it certifies that the derived quantity is mathematically sound. The library's derivation_status string records that the relation between the framework's base time unit tau0 and the Planck time is proven, along with other consistency checks, with no proof holes.

What the lemma does not claim is just as important. It does not assert that the Planck time is the smallest possible time interval, nor that time is discrete. It does not prove that the formula for the Planck time is physically correct; that is a definitional choice, not a theorem. And it does not derive the fine-structure constant α, which remains an open target in the framework. The lemma is a guardrail, not a discovery: it ensures the mathematics is sane before the physics proceeds.

lemma planck_time_inner_nonneg : 0 ≤ hbar_codata * G_codata / c_codata ^ 5 :=
  le_of_lt (div_pos (mul_pos hbar_codata_pos G_codata_pos) (pow_pos c_codata_pos 5))

lemma planck_time_pos : 0 < planck_time := by
  unfold planck_time
  exact sqrt_pos.mpr (div_pos (mul_pos hbar_codata_pos G_codata_pos) (pow_pos c_codata_pos 5))

def derivation_status : String :=
  "✓ tau0_sq_eq PROVEN\n" ++
  "✓ planck_relation_satisfied PROVEN\n" ++
  "✓ G_relation_satisfied PROVEN\n" ++
  "✓ tau0_planck_relation PROVEN\n" ++
  "✓ units_self_consistent PROVEN\n" ++
  "✓ NO PROOF HOLES"

What this page does not claim

This lemma does not assert that the Planck time is the smallest possible time interval or that time is discrete. This lemma does not prove that the formula for the Planck time is physically correct; that is a definitional choice, not a theorem. This lemma does not derive the fine-structure constant α, which remains an open target in the framework.

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

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Derived articles

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