Constants Kdisplay Display Null Condition

The null condition

The display_null_condition is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It states that for any valid unit system, when the base time unit is positive, the square of the ratio of two displayed quantities equals the square of the speed constant. In plainer terms, it establishes a precise algebraic relationship between a displayed kinematic length and a displayed recognition time.

The two quantities are lambda_kin_display, a displayed kinematic length, and tau_rec_display, a displayed recognition time. Their ratio, when squared, equals the square of the speed constant c. This is not a new physical law but a derived identity: it follows from the definition that the ratio itself equals c, a fact the framework proves as a separate lemma. The theorem's proof is a simple rewrite using that earlier result.

The condition carries a hypothesis: the base time unit must be positive. This ensures the displayed recognition time is nonzero, so the ratio is well-defined. The theorem also relies on the framework's unit system, where the speed constant c relates the base length unit to the base time unit through the equation ell0 = c * tau0.

In Recognition Science, this null condition is part of a larger structure. The framework defines unit equivalence, showing that rescaling both base units by the same nonzero factor leaves the ratio invariant. It also introduces a K-gate measurement protocol, with a tolerance based on error propagation, to check consistency between two independent measurements of the same ratio. The null condition sits alongside these as a formal guarantee that the displayed ratio, when squared, matches the speed constant squared.

What the theorem does not claim is equally important. It does not assert that the displayed ratio is measured to equal c; it only states a formal identity given the definitions. It does not claim that any particular measurement satisfies the condition, only that the algebraic relationship holds for any valid unit system. It also does not assert that the speed constant c has any particular numerical value; that is a separate matter within the framework.

/-- Display-level Lorentz structure: (λ/τ)² - c² = 0 (null) -/
theorem display_null_condition (U : RSUnits) (h : 0 < U.tau0) :
  ((lambda_kin_display U) / (tau_rec_display U))^2 = U.c^2 := by
  simp only [display_speed_eq_c U h]

lemma display_speed_eq_c (U : RSUnits) (h : 0 < U.tau0) :
  (lambda_kin_display U) / (tau_rec_display U) = RSUnits.c U := by
  have hτ : tau_rec_display U ≠ 0 := tau_rec_display_ne_zero U h
  exact display_speed_eq_c_of_nonzero U hτ

/-- Displays scale uniformly: ratio is scale-invariant -/
theorem display_ratio_scale_invariant (U : RSUnits) (hτ : 0 < U.tau0) (α : ℝ) (hα : 0 < α) :
  let tau' := α * (tau_rec_display U)
  let lambda' := α * (lambda_kin_display U)
  lambda' / tau' = (lambda_kin_display U) / (tau_rec_display U) := by
  intro tau' lambda'
  have hα' : α ≠ 0 := ne_of_gt hα
  have hτ' : tau_rec_display U ≠ 0 := tau_rec_display_ne_zero U hτ
  simp only [tau', lambda']
  rw [mul_div_mul_left _ _ hα']

What this page does not claim

The theorem does not assert that any measurement satisfies the null condition, only that the algebraic identity holds for all valid unit systems. The theorem does not assign a numerical value to the speed constant c. The theorem does not claim the displayed ratio is physically measured to equal c, only that it is formally equal by definition.

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.

Derived articles

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