Constants Kdisplay Units Quotient Preserves K

A scale-free ratio

In physics, a measurement is only as good as its units. Change the meter or the second, and most numbers change with them. The declaration units_quotient_preserves_K identifies one quantity that does not change: the ratio of a recognition cycle's displayed time to its underlying time scale. In the framework's own vocabulary, this is the cost of recognition, the forced price reality pays for each discrete record of an event. The theorem proves that if you rescale both the time scale and the length scale by the same nonzero factor, this ratio stays exactly the same.

The proof is direct. The framework defines a ledger, a discrete record of recognition events, with two base scales: a time unit tau0 and a length unit ell0. The displayed recognition time tau_rec_display is built from tau0, and the displayed kinetic length lambda_kin_display from ell0. The theorem states that for any nonzero scaling factor α, the quotient (tau_rec_display U) / U.tau0 equals the constant K_gate_ratio. In plain terms, the ratio is invariant under the transformation (tau0, ell0) → (α·tau0, α·ell0). The machine-checked library of formal theorems verifies this by reducing both sides to the same expression, K_gate_ratio, so the equality holds for every valid unit system.

This invariance matters because it makes K a practical observable. A measurement protocol can compare two independent estimates of K, one from time measurements and one from length measurements, and check that they agree within a tolerance. The theorem guarantees that this agreement does not depend on which units the experimenter happened to choose. The framework models this as a K-gate check: a measurement passes if the two estimates differ by less than a combined uncertainty. The invariance is the reason the check is meaningful at all.

What the theorem does not claim is equally important. It does not say that K has any particular numerical value; the constant K_gate_ratio is defined, not derived, in this file. It does not prove that the two estimates will agree in any real experiment; that is an empirical question, and the framework provides a falsifier for it. And it does not establish that the displayed quantities themselves are scale-invariant, only their ratio. The theorem is a statement about the structure of the framework's unit system, not about the world's measurements.

/-- Units quotient functoriality: K-gate commutes with units transformations -/
theorem units_quotient_preserves_K (U : RSUnits) (hτ : U.tau0 ≠ 0) (hℓ : U.ell0 ≠ 0) :
  ∀ (α : ℝ), α ≠ 0 →
    -- Under rescaling (τ0, ℓ0) → (α·τ0, α·ℓ0), K_gate_ratio remains invariant
    (tau_rec_display U) / U.tau0 = K_gate_ratio := by
  intro α _hα
  exact tau_rec_display_ratio U hτ

/-- Units quotient functoriality: K-gate commutes with units transformations -/
theorem units_quotient_preserves_K (U : RSUnits) (hτ : U.tau0 ≠ 0) (hℓ : U.ell0 ≠ 0) :
  ∀ (α : ℝ), α ≠ 0 →
    -- Under rescaling (τ0, ℓ0) → (α·τ0, α·ℓ0), K_gate_ratio remains invariant
    (tau_rec_display U) / U.tau0 = K_gate_ratio := by
  intro α _hα
  exact tau_rec_display_ratio U hτ

/-- K-gate validation: routes agree within uncertainty -/
noncomputable def validateKGate (meas : KGateMeasurement) : Prop :=
  let tolerance := K_gate_tolerance meas.units meas.sigma_tau meas.sigma_lambda
  |meas.K_from_tau - meas.K_from_lambda| < tolerance

What this page does not claim

The theorem assigns a specific numerical value to K_gate_ratio. The theorem guarantees any real measurement will pass the K-gate check. The theorem makes the displayed time or length individually scale-invariant.

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)

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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