Foundation Closed Observable Framework Strict Convexity From Closure

The convexity obligation

Strict convexity is a property of a curve or function: it means the line segment connecting any two points on the graph lies strictly above the graph itself. For a function of one variable, this is equivalent to having a positive second derivative wherever the function is smooth. In plain terms, a strictly convex cost function has no flat spots and no valleys that are not the unique lowest point.

In Recognition Science, the declaration StrictConvexityFromClosure is a formal structure, a named container in the framework's machine-checked library of formal theorems, that packages exactly one requirement: the function J, which measures the cost of a recognition event, must be strictly convex on the positive real numbers. The declaration does not establish that J is convex. It states the condition as an obligation, one of three separate requirements that a complete reconstruction of the framework's ledger must satisfy.

The other two obligations are continuity and calibration. Together the three form a split version of what the library calls the regularity seam: the set of assumptions needed to turn the framework's axioms into a working comparison mechanism. The declaration's role is to make the convexity requirement auditable on its own, so that a proof or a counterexample can target it directly rather than argue about a broad, undifferentiated hypothesis.

What the declaration does not claim is that closure alone forces strict convexity. The name might suggest that the framework's closure condition, the requirement that a system has no external input, logically implies convexity. The formal structure does not assert that implication. It merely records the convexity condition as a separate premise that a reconstruction must supply. The distinction matters: a theorem in the library that uses this declaration is only as strong as the obligation it assumes.

The practical consequence is that the framework's central result, the unique cost function J(x) = (x + 1/x)/2 - 1, depends on this convexity assumption as an input, not as a derived conclusion. The declaration makes that dependence explicit and local. A reader who wants to challenge the framework can now ask a precise question: is strict convexity actually forced by the closure axioms, or is it an extra assumption wearing a theorem's clothes?

/-- Strict-convexity obligation extracted from closure/no-arbitrage. -/
structure StrictConvexityFromClosure (J : ℝ → ℝ) : Prop where
  strict_convex : StrictConvexOn ℝ (Set.Ioi 0) J

/-- Strict-convexity obligation extracted from closure/no-arbitrage. -/
structure StrictConvexityFromClosure (J : ℝ → ℝ) : Prop where
  strict_convex : StrictConvexOn ℝ (Set.Ioi 0) J

/-- Explicit split version of the regularity seam.

Instead of a single broad `RegularityCert`, the reconstruction theorem now
tracks continuity, convexity, and calibration as independently auditable
obligations. -/
structure FiniteDescriptionRegularity (J : ℝ → ℝ) : Prop where
  continuity : ContinuityFromFiniteDescription J
  convexity : StrictConvexityFromClosure J
  calibration : CalibrationFromUnitChoice J

What this page does not claim

The declaration does not establish that closure forces strict convexity. The declaration does not establish the uniqueness of the cost function J. The declaration does not assert that any actual physical system satisfies the convexity obligation.

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

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