Mathematics Fundamental Theorem Calculus From Rs Calculus Theorem Count

The five calculus theorems

The fundamental theorem of calculus is the bridge between two operations: differentiation, which measures how fast a quantity changes, and integration, which accumulates those changes. In its simplest form, the theorem says that if you integrate the derivative of a function from a starting point to an end point, you get the net change in the function itself. The notation is compact: ∫₁ʳ J'(x) dx = J(r) - J(1). This is not a new discovery; it is the standard result taught in every calculus course, and it holds for any differentiable function with a continuous derivative.

In the Recognition Science framework, this classical theorem takes a specific form. The framework models a ledger, a discrete record of recognition events, and assigns a cost to each recognition. That cost is the function J(r) = (r + 1/r)/2 - 1, which measures how far a ratio r is from unity. The framework's library, a machine-checked collection of formal theorems, proves that the integral of the derivative of J from 1 to r equals J(r) itself, since J(1) = 0. The derivative J'(1) = 0 marks a minimum: the cost is zero when the ratio is exactly 1, and positive otherwise.

The declaration calculusTheoremCount does something narrower. It counts five canonical calculus theorems: the two parts of the fundamental theorem, the mean value theorem, the intermediate value theorem, and L'Hôpital's rule. The machine-checked proof shows this count is exactly 5, no more and no less. The framework associates this number with a structural dimension, but the count itself is a simple combinatorial fact about a list of five named theorems, not a claim about the physical world.

What the count does not do is prove the fundamental theorem of calculus from scratch. The theorem is assumed as a standard result of real analysis; the framework's contribution is to observe that, for its specific cost function, the classical theorem holds and that five standard theorems form a natural set. The declaration also does not claim that these five theorems are the only possible ones, nor that the number 5 has any physical significance beyond this particular enumeration. It is a bookkeeping fact about a chosen list, verified by a proof assistant, and nothing more.

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

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

/-- 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 declaration does not prove the fundamental theorem of calculus from first principles. The count of five does not imply that these are the only calculus theorems that exist. The number 5 carries no demonstrated physical meaning beyond this specific enumeration.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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