Constants Kdisplay Single Inequality Audit

The single inequality audit

In physics, a ratio between two measured quantities should not depend on the units you happen to use. The Recognition Science framework encodes this expectation in a formal object called RSUnits, a choice of base units for time and length. The declaration single_inequality_audit is a theorem in the framework's machine-checked library of formal theorems. It states that for any such unit choice, as long as the base time and length are nonzero, a certain derived ratio is always less than or equal to another derived ratio. In plainer terms: if you measure the same physical rate two different ways, one from a time-based definition and one from a length-based definition, the time-based number never exceeds the length-based number.

The two ratios come from display quantities, which are the framework's way of converting internal unit choices into numbers a physicist could write down. The theorem itself is a formal inequality: (tau_rec_display U) / U.tau0 ≤ (lambda_kin_display U) / U.ell0. It is proved by rewriting both sides using a prior lemma, K_gate_eqK, which shows each ratio equals the same constant K_gate_ratio. Because both sides equal the same number, the inequality collapses to equality. The proof is short and entirely mechanical; the Lean kernel checks every step.

What the theorem does not claim is just as important. It does not say that the two measurement methods agree to within experimental error. That would be a statement about real instruments, with tolerances and noise. The framework handles that separately in a definition called validateKGate, which checks whether the absolute difference between two measured values falls below a tolerance computed from measurement uncertainties. The theorem is about the ideal, noiseless ratios; the validation is about real data. The theorem also does not say that the constant K_gate_ratio has a particular numerical value. It only says the two ratios are equal, whatever that common value may be. Finally, the theorem does not establish that the framework's units are physically realized; it assumes a unit structure and proves a property of it.

For a reader, the practical upshot is a consistency check. If you ever compute a rate using the framework's time-based display and another using its length-based display, you can trust that the two will not contradict each other in the ideal case. The inequality is a small but load-bearing piece of the framework's claim that its internal definitions are coherent, independent of the arbitrary choice of base units.

/-- Single-inequality audit: checking one route inequality suffices (routes equal).

    Since `(tau_rec_display U)/τ0 = (lambda_kin_display U)/ℓ0` by `K_gate_eqK`,
    the inequality direction is immediate. -/
theorem single_inequality_audit (U : RSUnits) (hτ : U.tau0 ≠ 0) (hℓ : U.ell0 ≠ 0) :
  (tau_rec_display U) / U.tau0 ≤ (lambda_kin_display U) / U.ell0 := by
  have h := K_gate_eqK U hτ hℓ
  rw [h.1, h.2]

/-- Single-inequality audit: checking one route inequality suffices (routes equal).

    Since `(tau_rec_display U)/τ0 = (lambda_kin_display U)/ℓ0` by `K_gate_eqK`,
    the inequality direction is immediate. -/
theorem single_inequality_audit (U : RSUnits) (hτ : U.tau0 ≠ 0) (hℓ : U.ell0 ≠ 0) :
  (tau_rec_display U) / U.tau0 ≤ (lambda_kin_display U) / U.ell0 := by
  have h := K_gate_eqK U hτ hℓ
  rw [h.1, h.2]

What this page does not claim

The theorem does not claim that measured values from real instruments will agree within experimental error. The theorem does not claim that K_gate_ratio has a specific numerical value. The theorem does not claim that the framework's unit structure is physically realized.

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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Derived articles

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