Mathematics Fundamental Theorem Calculus From Rs Jcost Strict Min

The strict minimum

The fundamental theorem of calculus says that differentiation and integration undo each other. In the Recognition Science framework, the same idea takes a specific form: integrating the derivative of the cost function from 1 to r gives the total recognition cost, written as J(r). The framework's cost, a measure of how much effort a recognition event requires, is zero exactly at the point r = 1, where the recognition is trivial.

The declaration jcost_strict_min states a precise fact: for any positive r that is not 1, the cost J(r) is strictly greater than zero. In other words, the cost has a strict minimum at r = 1, and nowhere else. This is a proved theorem in the machine-checked library of formal theorems, with no unproved assumptions. It follows from the general property that the cost is positive whenever the input differs from 1.

This strict minimum is the structural reason why the derivative of J at 1 is zero, and why the cost function has the shape J(r) = (r-1)²/(2r) near r = 1. The theorem is part of a small collection of five canonical calculus results that the framework identifies as a single structural unit, but the strict minimum itself does not prove any of those results. It only pins down the minimum point.

What the declaration does not claim is important. It does not claim that the fundamental theorem of calculus is proved from the cost function. It does not claim that the cost function is the only function with a strict minimum at 1. It does not claim anything about the physical meaning of the cost beyond its mathematical definition. The theorem is a clean, narrow statement about one function's behavior.

/-- J is positive off minimum (strict local minimum). -/
theorem jcost_strict_min {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < Jcost r := Jcost_pos_of_ne_one r hr hne

/-- J(1) = 0 (minimum, derivative = 0 at critical point). -/
theorem jcost_minimum : Jcost 1 = 0 := Jcost_unit0

theorem calculusTheoremCount : Fintype.card CalculusTheorem = 5 := by decide

What this page does not claim

The fundamental theorem of calculus is not proved from the cost function. The cost function is not claimed to be the only function with a strict minimum at 1. The theorem says nothing about the physical meaning of the cost.

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

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