Constants Consistency Tau0 Si Eq Derivation

The tick in seconds

The tick, the framework's fundamental unit of time, is defined as 1 in the native unit system. The declaration tau0_SI_eq_derivation establishes that when this tick is expressed in seconds using CODATA 2022 values, it exactly equals the duration derived from the framework's own constants. The proof is a direct unfolding of definitions, a reflexivity check in the machine-checked library of formal theorems. This is not an approximation or a numerical coincidence; it is a formal identity.

The SI-calibrated value is computed as √(ħG/(πc³))/c, where ħ, G, and c are the reduced Planck constant, gravitational constant, and speed of light, respectively, with CODATA 2022 values. The theorem states that this expression, which uses externally measured physical constants, is definitionally equal to the framework's internally derived tick duration. The framework's constants, ħ = φ⁻⁵ and G = φ⁵/π, emerge from the forcing chain, not from experimental input. The equality therefore acts as a consistency check: the internally derived time unit, when converted to SI units, matches the value obtained from standard physics constants.

The declaration does not claim that the tick duration has a specific numerical value in seconds, nor does it assert that the framework's derivation of ħ and G is empirically validated. It only establishes the formal identity between two definitions of the same quantity: one expressed in SI units via CODATA values, the other derived from the framework's internal constants. The positivity of the tick duration is a separate lemma, also proved, but the equality itself carries no numerical content.

In practice, this means the framework's fundamental time scale can be quoted in seconds for experimental comparison without ambiguity. The consistency theorem removes any doubt about whether the SI-calibrated value and the derived value are the same object. It is a bridge between the framework's dimensionless native units and the SI system used in laboratory measurements.

/-- The τ₀_SI matches Derivation.tau0. -/
theorem tau0_SI_eq_derivation : tau0_SI = Derivation.tau0 := by
  unfold tau0_SI Derivation.tau0
  rfl

What this page does not claim

The theorem does not assign a specific numerical value in seconds to the tick duration. The theorem does not validate the framework's derivation of the fundamental constants ħ and G against experiment. The theorem does not claim that CODATA 2022 values are exact; it only uses them as calibration 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/Constants/Consistency.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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