Cost Jcost Weak Triangle False

A triangle inequality that fails

The cost function J(x) = (x + 1/x)/2 - 1 measures the price of a multiplicative change by a factor x, where x = 1 costs nothing. For positive x, this cost is always at least zero, it is symmetric under x → 1/x, and it grows as x moves away from 1 in either direction. One might hope that the cost of a combined change x·y is bounded by the sum of the individual costs, as happens for ordinary distances. The declaration Jcost_weak_triangle_FALSE proves this hope is wrong.

The theorem states that a specific weak triangle inequality is false: it is not true that for all positive x and y, J(x·y) ≤ 2(J(x) + J(y)) + 2√(J(x)·J(y)). The factor 2 and the square-root term make this a mild, weakened form of a triangle inequality. The proof exhibits a counterexample, so the failure is not a matter of a missing proof but a genuine property of J. This is a theorem, a statement proved in the machine-checked library of formal theorems, not a conjecture or a numerical observation.

What does survive is a different, stronger-looking bound. The library proves J(x·y) ≤ 2(J(x) + J(y)) + 2·J(x)·J(y), which holds for all positive x and y. The difference between the two inequalities is the last term: the true bound has the product J(x)·J(y), while the false one has the geometric mean √(J(x)·J(y)). Since the product can be much smaller than the geometric mean when one cost is tiny, the true bound is actually weaker, which is why it can hold while the other fails.

The failure matters because it shows that J does not behave like a metric on the multiplicative group of positive reals. A metric would satisfy a triangle inequality; J does not. This is a deliberate feature of the framework, not a defect. The cost function is forced by the five plain conditions of Recognition Science, and the failure of a triangle inequality is a consequence of that forcing. It tells a reader that the cost of a combined change is not simply the sum of parts, and that the framework's notion of cost is genuinely different from a distance.

The declaration does not claim that all triangle-like inequalities fail for J, nor that J is a metric, nor that the framework is wrong. It establishes exactly one negative result about one specific inequality. The positive bound J(x·y) ≤ 2(J(x) + J(y)) + 2·J(x)·J(y) remains true and is part of the same library. The reader can take away a precise fact: the cost function resists a natural geometric analogy, and the resistance is proved.

/-- **DEPRECATED**: The "weak triangle" J(xy) ≤ 2(J(x)+J(y)) + 2√(J(x)J(y)) is FALSE.

    Counterexample: x = y = 10
    - J(100) ≈ 49.005
    - 2(J(10) + J(10)) + 2√(J(10)·J(10)) = 2(4.05 + 4.05) + 2·4.05 ≈ 24.3
    - 49.005 > 24.3

    Use `Jcost_submult` instead: J(xy) ≤ 2J(x) + 2J(y) + 2J(x)J(y), which IS proved. -/
theorem Jcost_weak_triangle_FALSE :
    ¬ (∀ x y : ℝ, 0 < x → 0 < y →
      Jcost (x * y) ≤ 2 * (Jcost x + Jcost y) + 2 * Real.sqrt (Jcost x * Jcost y)) := by
  intro h
  -- Counterexample: x = y = 10, J(100) > 2(J(10)+J(10)) + 2*sqrt(J(10)*J(10))
  -- J(100) = (100 + 1/100)/2 - 1 = 49.005
  -- J(10) = (10 + 0.1)/2 - 1 = 4.05
  -- RHS = 2*(4.05 + 4.05) + 2*sqrt(4.05*4.05) = 16.2 + 8.1 = 24.3
  -- 49.005 > 24.3 is TRUE, not ≤, so the inequality fails
  have hc := h 10 10 (by norm_num) (by norm_num)
  -- The claim asserts ≤ but the counterexample shows >
  -- J(100) = 49.005, RHS = 24.3, so 49.005 > 24.3
  simp only [Jcost] at hc
  nlinarith [sq_nonneg (10 : ℝ), Real.sqrt_nonneg (Jcost 10 * Jcost 10)]

/-- From d'Alembert: J(xy) ≤ 2J(x) + 2J(y) + 2J(x)J(y) since J(x/y) ≥ 0 -/
lemma Jcost_submult {x y : ℝ} (hx : 0 < x) (hy : 0 < y) :
    Jcost (x * y) ≤ 2 * Jcost x + 2 * Jcost y + 2 * Jcost x * Jcost y := by
  have h := dalembert_identity hx hy
  have hnonneg : 0 ≤ Jcost (x / y) := Jcost_nonneg (div_pos hx hy)
  linarith

What this page does not claim

This answer does not claim that Jcost_weak_triangle_FALSE shows the cost function is inconsistent or that the framework's axioms are contradictory. This answer does not claim that all triangle inequalities fail for J; only the specific weak form is proved false.

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

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