Foundation Jcost Convexity In Log Space Same Symmetry

The shared symmetry

The recognition cost function, a measure of the forced price of a recognition event, is defined on positive numbers. The declaration same_symmetry concerns what happens when you view that cost through a logarithmic lens. Instead of looking at the cost as a function of the original value x, you look at it as a function of t = ln(x), the logarithm of x. This change of coordinates is a standard mathematical move that often turns multiplicative relationships into additive ones.

In these log coordinates, the theorem establishes two functions with a matching shape. The first is g(t), the cost function expressed in the new variable. The second is h(t) = t²/2, the simplest quadratic function in that same variable. The declaration proves that both are even functions: g(t) = g(-t) and h(t) = h(-t) for every t. An even function is symmetric about the vertical axis, meaning its value at any distance to the right of zero is identical to its value at the same distance to the left. The theorem also confirms both functions share a fixed point at t = 0, where both equal zero, and that both are positive everywhere else.

The shared symmetry means the cost function and its quadratic approximation are indistinguishable from the perspective of this reflection property. Near t = 0, the cost function g(t) is approximately t²/2, so the quadratic h(t) is its local model. The theorem certifies that the two agree on this structural feature: the same zero point, the same evenness, the same sign pattern. It is a machine-checked statement, part of a library of formal theorems, that the framework's cost function and the log-ratio form belong to the same symmetry family.

What the declaration does not claim is that the two functions are equal, or that the cost function is globally convex in log space. It establishes a shared symmetry and fixed point, not an identity. The quadratic h(t) is an approximation valid near the origin, not a description of the cost function everywhere. The theorem also says nothing about the physical meaning of the cost function or about any control system; it is a purely mathematical statement about two real-valued functions of a real variable.

In Recognition Science, this symmetry result supports the framework's closed-loop control analysis. The fact that the cost function and its quadratic model share this reflection property means that arguments relying on the symmetry of the quadratic carry over to the actual cost function. The declaration gives the framework a precise, verified handle on the shape of its central cost object in the coordinate system where that shape is simplest to reason about.

/-- Both g and h are even functions. -/
theorem same_symmetry : ∀ t, g t = g (-t) ∧ h t = h (-t) :=
  fun t => ⟨g_even t, h_even t⟩

/-- g and h share the same fixed point at t = 0. -/
theorem same_fixed_point : g 0 = 0 ∧ h 0 = 0 := ⟨g_at_zero, h_at_zero⟩

/-- g(0) = J(e⁰) = J(1) = 0. -/
theorem g_at_zero : g 0 = 0 := by
  unfold g
  simp [Jcost_unit0]

What this page does not claim

The declaration does not claim that the cost function equals the quadratic function h(t) for all t. The declaration does not claim that the cost function is convex in log space. The declaration does not claim any physical or control-theoretic consequence on its own.

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

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