Mathematics Optimization Theory From Rs Global Minimum

The global minimum

In classical optimization theory, a global minimum is the point where a function attains its lowest possible value. For a function f(x), a point x* is a global minimum if f(x*) ≤ f(x) for every x in the domain. The Recognition Science framework, which treats recognition as a forced cost, defines its cost function J(x) = (x + 1/x)/2 − 1. The framework's machine-checked library of formal theorems proves that J(1) = 0, meaning the cost reaches its lowest value at x = 1.

This result is the recognition equilibrium, the state where recognition costs nothing. The theorem global_minimum in the library's OptimizationTheoryFromRS file establishes that the cost function's global minimum is exactly zero at the input 1. The same library proves that for any positive input other than 1, the cost is strictly greater than zero. Together these two results characterize the cost function completely: it has one and only one minimum, at x = 1, and that minimum value is 0.

The framework connects this to the five canonical optimization problem types: linear, nonlinear, combinatorial, convex, and stochastic. Its library defines these five types and proves there are exactly five of them. In the framework's account, all optimization is J-cost minimization, so the global minimum at J = 0 corresponds to the recognition equilibrium, while local minima occur at J > 0 where the cost is positive but locally optimal.

What the declaration does not claim is equally important. It does not claim that the cost function J is the only possible cost function; that uniqueness is a separate theorem requiring five conditions. It does not claim that x = 1 is a minimum in any domain other than the positive reals, since the function is not defined at zero and the theorem's hypotheses specify 0 < r. It does not claim that the five optimization problem types are exhaustive in any mathematical sense beyond the framework's own definition. The declaration is a precise statement about a specific function, not a universal claim about all optimization problems.

/-- Global minimum: J = 0. -/
theorem global_minimum : Jcost 1 = 0 := Jcost_unit0

/-- Local minimum: J > 0. -/
theorem local_minimum {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < Jcost r := Jcost_pos_of_ne_one r hr hne

theorem optimizationProblemTypeCount : Fintype.card OptimizationProblemType = 5 := by decide

What this page does not claim

The uniqueness of the cost function J is not established by global_minimum alone. The declaration does not apply to non-positive inputs, where the cost function is not defined. The five problem types are exhaustive only within the framework's own definition, not in general mathematics.

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

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