Constants Kdisplay K Gate Units Invariant

Unit invariance of the K gate

The K gate is a dimensionless number that Recognition Science uses to check whether two independent measurements agree. One measurement produces a time interval, the other a length, and the ratio of each to its own unit scale should come out the same. The declaration K_gate_units_invariant proves that this ratio does not change when you rescale both unit sizes by the same positive factor. If you multiply the unit of time and the unit of length by α, the ratio of the displayed time to the unit time stays exactly the same as before. The proof is a formal theorem in the machine-checked library of formal theorems, not an assumption or a numerical coincidence.

The theorem states it precisely. For any unit system U with a positive time unit, and any positive rescaling factor α, the ratio (tau_rec_display U') / U'.tau0 equals (tau_rec_display U) / U.tau0, where U' is the rescaled system. The proof rewrites both ratios to the same constant K_gate_ratio, which depends only on the structure of the framework, not on the chosen units. This is the kind of statement that makes a dimensionless observable trustworthy: it cannot be an artifact of picking meters versus feet, or seconds versus hours.

What the theorem does not claim is just as important. It does not claim that the K gate ratio has any particular numerical value, nor that any specific measurement already matches it. The framework defines a validation predicate, validateKGate, which checks whether two measured values fall within a tolerance computed from their uncertainties, but that predicate is a definition, not a theorem. The theorem only guarantees scale invariance of the ratio, not that any experimental apparatus has actually passed the gate. A separate structure, KGateMeasurement, exists to hold such a measurement, and a falsifier predicate, falsifier_K_gate_mismatch, names the condition under which a measurement would fail the check. None of that is proved to occur or not occur; it is the protocol for a future empirical test.

The practical consequence is that the framework's dimensionless bridge ratio is a stable target for experiment. Since the ratio cannot be shifted by choosing different units, any mismatch between the two measured values must come from physics, not from convention. That is what the invariance theorem buys: a clean separation between unit choice and physical content, so that a future experiment either confirms the ratio or falsifies it, with no ambiguity about the measuring stick.

/-- K-gate is independent of units rescaling -/
theorem K_gate_units_invariant (U : RSUnits) (α : ℝ) (hα : 0 < α) (hτ : 0 < U.tau0) :
  let U' : RSUnits := { tau0 := α * U.tau0, ell0 := α * U.ell0, c := U.c,
                        c_ell0_tau0 := by calc U.c * (α * U.tau0) = α * (U.c * U.tau0) := by ring
                                              _ = α * U.ell0 := by rw [U.c_ell0_tau0] }
  (tau_rec_display U') / U'.tau0 = (tau_rec_display U) / U.tau0 := by
  intro U'
  have hα' : α ≠ 0 := ne_of_gt hα
  have hτ' : U.tau0 ≠ 0 := ne_of_gt hτ
  rw [tau_rec_display_ratio U hτ', tau_rec_display_ratio U' (mul_ne_zero hα' hτ')]

/-- K-gate is independent of units rescaling -/
theorem K_gate_units_invariant (U : RSUnits) (α : ℝ) (hα : 0 < α) (hτ : 0 < U.tau0) :
  let U' : RSUnits := { tau0 := α * U.tau0, ell0 := α * U.ell0, c := U.c,
                        c_ell0_tau0 := by calc U.c * (α * U.tau0) = α * (U.c * U.tau0) := by ring
                                              _ = α * U.ell0 := by rw [U.c_ell0_tau0] }
  (tau_rec_display U') / U'.tau0 = (tau_rec_display U) / U.tau0 := by
  intro U'
  have hα' : α ≠ 0 := ne_of_gt hα
  have hτ' : U.tau0 ≠ 0 := ne_of_gt hτ
  rw [tau_rec_display_ratio U hτ', tau_rec_display_ratio U' (mul_ne_zero hα' hτ')]

What this page does not claim

The K gate ratio has a specific numerical value in this theorem. Any physical measurement has already been validated against the K gate. The invariance theorem applies to anything other than the ratio of displayed time to unit time.

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

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