Mathematics Fundamental Theorem Calculus From Rs Jcost Minimum

The cost minimum

The fundamental theorem of calculus says that differentiation and integration undo each other. In the Recognition Science framework, the same pair of operations appears with a cost attached. The framework models recognition as a ledger, a discrete record of events, and assigns a cost to each recognition event. The cost function J(x) = (x + 1/x)/2 - 1 measures how far a recognition ratio x is from the unit ratio 1. At x = 1, the cost is zero: J(1) = 0, and this point is a strict local minimum, meaning J(x) is positive for every x other than 1.

The theorem jcost_minimum in the framework's machine-checked library of formal theorems establishes exactly that J(1) = 0. It is a structural fact about the cost function, not a numerical accident. The derivative J'(1) equals 0, so the point r = 1 is a critical point and the minimum of the cost. The framework then uses this minimum to build an integral from 1 to r of J'(x) dx, which equals J(r) - J(1) = J(r). This is the framework's version of the fundamental theorem of calculus: the integral of the cost derivative from 1 to r is the total recognition cost.

The framework also counts five canonical calculus theorems, FTC-1, FTC-2, mean value theorem, intermediate value theorem, and L'Hôpital's rule, as a set of size 5. This count is a theorem in the library, proved by deciding the finite type. The minimum at 1 and the strict positivity elsewhere are packaged into a single certificate structure, CalculusCert, which bundles the five-theorem count, the minimum at 1, and the strict minimum property.

What jcost_minimum does not claim is broader. It does not prove the fundamental theorem of calculus in the classical sense. The framework's integral identity is a derivation within its own cost model, not a new proof of the classical theorem. The theorem also does not claim that the cost function is the only possible cost function; that uniqueness is a separate result. And the count of five theorems is a definitional choice about which theorems to include, not a claim about the number of theorems in all of mathematics.

/-- 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

/-- 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

What this page does not claim

The theorem does not prove the classical fundamental theorem of calculus. The uniqueness of the cost function is a separate result, not established by jcost_minimum. The five-theorem count is a definitional choice, not a claim about all of 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/FundamentalTheoremCalculusFromRS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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