Constants Consistency

Constants consistency

Constants consistency is a formal audit within Recognition Science (RS). It checks that the framework's derived constants, which emerge from a single forcing chain, do not contradict each other or the measured values of physical quantities. The audit lives in a machine-checked library of formal theorems, meaning every equality is verified by a computer rather than asserted by hand.

The framework works in its own natural units where the fundamental tick of time τ₀ equals 1 and the speed of light c equals 1 voxel per tick. These are dimensionless choices for internal reasoning. To compare against laboratory experiments, the module defines an SI-calibrated tick duration, τ₀_SI, using CODATA values for the reduced Planck constant, gravitational constant, and speed of light. The formula is √(ħG/(πc³))/c, which yields a duration in seconds.

The central result is a theorem stating that this SI-calibrated value exactly equals the framework's internally derived τ₀. In plain language: the tick duration predicted from RS's own constants matches the one computed from standard physics constants, with no free parameters. A companion lemma proves this duration is positive. The module also defines the octave duration as eight ticks, matching the framework's eight-tick recognition cycle, and proves it positive as well.

A second theorem verifies that the golden ratio φ is defined identically across different modules of the library, preventing subtle drift between separate definitions. The consistency check is summarized in a string literal that lists each verified equality. Its final note clarifies the unit convention: RS-native units are dimensionless, while SI values serve only for external calibration.

What this establishes is that RS's derived constants form a coherent system. The tick, the octave, and φ all agree across modules, and the tick's SI value matches experiment-derived constants. For a reader, this means the framework's internal bookkeeping is self-consistent and anchored to measurable physics, not merely an abstract formalism.

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

/-- τ₀_SI is positive. -/
lemma tau0_SI_pos : 0 < tau0_SI := by
  rw [tau0_SI_eq_derivation]
  exact Derivation.tau0_pos

/-- Octave duration is positive. -/
lemma octave_SI_pos : 0 < octave_SI := by
  unfold octave_SI
  exact mul_pos (by norm_num : (0 : ℝ) < 8) tau0_SI_pos

/-- The golden ratio φ is defined consistently across modules. -/
theorem phi_consistency :
    Constants.phi = IndisputableMonolith.Constants.phi := rfl

What this page does not claim

This module does not derive the SI values of ħ, G, or c; it uses them as inputs. The consistency check does not prove that RS's constants match experiment, only that internal definitions agree with each other and with the SI-calibrated formula. No claim is made that the tick duration has been directly measured.

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.

Derived articles

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