Cost Real Trace Root Real Trace Root Sq Sub Four Nonneg

The square root's domain

The declaration realTraceRoot is a definition from the framework's machine-checked library of formal theorems. It names the principal root of the quadratic equation X² − tX + 1 = 0, for any real number t at least 2. The formula is (t + √(t² − 4)) / 2, taking the larger of the two roots. The theorem realTraceRoot_sq_sub_four_nonneg proves that, whenever t ≥ 2, the expression t² − 4 is nonnegative, so the square root in the formula is always a real number.

This is a technical precondition, not a grand result. It says the domain of the definition is well-formed: the input range t ≥ 2 is exactly where the discriminant t² − 4 does not go negative. If t were less than 2, the square root would be imaginary, and the formula would not produce a real-valued cost. The theorem is proved by a routine algebraic inequality, using only the assumption that 2 ≤ t.

In the broader framework, this object appears in the study of recognition costs, where the number t is a trace value tied to a composition law. The cost function J(x) = (x + 1/x)/2 − 1, which the framework proves to be forced by five plain conditions, has the same quadratic structure. Here, the lemma ensures that the construction stays within real arithmetic, a necessary step before any further properties are derived.

What this declaration does not claim is any physical content. It does not assert that t represents a measurable quantity, nor that the root has a physical interpretation. It is purely a statement about real numbers and the domain of a definition. The theorem's scope is the algebraic guarantee that the formula is real-valued on its intended input range.

theorem realTraceRoot_sq_sub_four_nonneg {t : ℝ} (ht : 2 ≤ t) :
    0 ≤ t ^ 2 - 4 := by nlinarith

/-- The principal (≥ 1) root of `X² - t X + 1 = 0`, for `t ≥ 2`. -/
noncomputable def realTraceRoot (t : ℝ) : ℝ :=
  (t + Real.sqrt (t ^ 2 - 4)) / 2

What this page does not claim

The theorem does not assign any physical meaning to the number t or the root. It does not assert that the framework's cost function is derived from this lemma alone. It does not claim that t² − 4 is nonnegative for any t below 2.

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

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