Cost Classical Results Spherical Cap Measure Bounds

Spherical cap measure bounds

A spherical cap is the part of a sphere cut off by a plane. If the sphere has radius 1 and the cap's half-angle is θ, measured from the north pole down to the cutting plane, its surface area is 2π(1 − cos θ). For any angle between 0 and π/2, the cosine lies between 0 and 1, so the area is always zero or positive. The declaration spherical_cap_measure_bounds states exactly this: for every θ in that range, the expression 2π(1 − cos θ) is nonnegative.

The result belongs to a module of classical mathematical facts that a machine-checked library of formal theorems records as axioms pending full formalization. These are textbook results from real analysis, justified by standard references, not new physical assumptions. The theorem itself is proved from the basic inequality cos θ ≤ 1, so its content is elementary. Its role is to supply a needed bound in a larger derivation, likely one involving angles and areas on a sphere.

In Recognition Science, the framework's library uses such classical facts as building blocks. The declaration does not claim that spherical caps are the only shapes with nonnegative area, nor does it say anything about caps on spheres of other radii, about volumes, or about angles outside the stated range. It is a narrow, exact statement: for a unit sphere and a half-angle up to a right angle, the cap's area does not go negative.

What matters for a reader is that this is a sanity bound, not a discovery. It guarantees that a quantity appearing in a framework derivation stays physically sensible. The theorem does not derive the area formula; it assumes it and proves the sign. That distinction keeps the framework honest: the formula itself is a classical given, and the bound is a checked consequence.

theorem spherical_cap_measure_bounds :
    ∀ θmin ∈ Set.Icc (0 : ℝ) (Real.pi/2),
      0 ≤ (2 * Real.pi * (1 - Real.cos θmin)) :=
  spherical_cap_pos

What this page does not claim

The declaration does not derive the spherical cap area formula itself. It does not claim anything about volumes, spheres of radius other than 1, or angles outside the stated range. It does not assert that spherical caps are the only surfaces with nonnegative area.

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

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