Mathematics Calculus Variations From Rs Jcost Off Minimum

The minimum at rest

The calculus of variations is the branch of mathematics that finds the curve, surface, or path that minimizes a quantity called a functional. The classic problems are named: the brachistochrone finds the fastest slide between two points under gravity, the geodesic finds the shortest path on a curved surface, the minimal surface finds the shape with least area for a given boundary, and Fermat's principle finds the path light takes through a medium. Each problem sets up a functional, then applies the Euler-Lagrange equation to find where that functional reaches an extremum, usually a minimum.

In the Recognition Science framework, the functional is the recognition cost, a discrete record of how much effort reality spends to recognize a state. The framework's cost function is J(r) = (r + 1/r)/2 - 1, where r is a positive real number representing a scale or ratio. The framework proves, in a machine-checked library of formal theorems, that this cost has exactly one minimum: at r = 1, the cost is zero. The declaration jcost_off_minimum states the other half: for any positive r not equal to 1, the cost is strictly greater than zero. In plain language, the system rests only at unity, and any departure from rest carries a positive price.

The framework's library also counts five canonical variational problems (brachistochrone, geodesic, minimal surface, Fermat, and J-cost minimization) and proves that number is exactly five. This count is a definitional choice about which problems to include, not a theorem about all possible variational problems. The Euler-Lagrange equation for J is written in the framework's documentation, and the minimum at r = 1 is the equilibrium point where the first derivative of J vanishes.

What jcost_off_minimum does not claim is important. It does not claim that the cost function is the only possible cost function; that uniqueness is a separate theorem. It does not claim that r = 1 is the only point where the derivative is zero; the theorem only states the cost is positive away from unity. It does not claim anything about which of the five variational problems has a solution, only that the framework recognizes five of them. The theorem is a local statement about one function's behavior, not a global statement about all of calculus of variations.

The consequence for a reader is concrete: in this framework, equilibrium is not a choice but a forced point. Any state that is not the rest state pays a positive cost, and the cost grows as r moves away from 1. This makes the framework's dynamics directional: systems are pushed toward unity, not merely allowed to sit anywhere. The minimum at rest is the anchor that makes the framework's other results, like the golden ratio and the eight-tick cycle, hang together.

/-- J-cost is strictly below off-equilibrium: J > 0 for r ≠ 1. -/
theorem jcost_off_minimum {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < Jcost r := Jcost_pos_of_ne_one r hr hne

/-- J-cost minimum: J = 0 at r = 1. -/
theorem jcost_variational_minimum : Jcost 1 = 0 := Jcost_unit0

theorem variationalProblemCount : Fintype.card VariationalProblem = 5 := by decide

What this page does not claim

The theorem does not claim uniqueness of the cost function. The theorem does not claim that r = 1 is the only critical point of J. The theorem does not claim that any of the five variational problems has a solution.

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

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