Constants Consistency Octave Si

The octave in seconds

An octave in music spans eight notes. In Recognition Science, an octave is the framework's fundamental recognition cycle, a span of eight ticks (discrete time steps in the framework's ledger of events). The declaration octave_SI gives this cycle a duration in ordinary seconds, so the framework's internal time can be compared with laboratory measurements.

The definition is simple: octave_SI = 8 * tau0_SI, where tau0_SI is the framework's fundamental tick duration expressed in seconds. That tick duration comes from a formula involving the framework's derived values for the reduced Planck constant, the gravitational constant, and the speed of light, evaluated with CODATA values. The framework proves that this SI-calibrated tick is positive, and therefore the octave duration is positive as well.

In Recognition Science, the ledger (a discrete record of recognition events) works in its own native units where one tick is simply 1 by definition. The SI-calibrated values exist only for external calibration, to connect the framework's dimensionless internal time to the seconds used in physical experiments. The declaration octave_SI is a consistency check: it confirms that the framework's eight-tick cycle has a well-defined, positive duration in real-world units.

What octave_SI does not claim is any particular numerical value for that duration. The declaration does not assert that an octave lasts, say, 10^-43 seconds or any other specific amount. It only establishes the relationship between the framework's internal time unit and SI seconds, given the CODATA input values. The actual number depends on those inputs and is not itself a derived constant of the framework.

/-- Octave duration in SI units (8 × τ₀_SI). -/
noncomputable def octave_SI : ℝ := 8 * tau0_SI

/-- 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

What this page does not claim

octave_SI does not assign a specific numerical duration in seconds to the octave. The declaration does not establish any physical claim about the nature of time. It does not claim the octave duration is a fundamental constant independent of CODATA 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)

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