Physics Planetary Boundary Layer From Jcost

Boundary layer thickness

The planetary boundary layer is the bottom slice of the atmosphere, typically 100 to 3000 meters thick, where the Earth's surface directly shapes the wind, temperature, and humidity. During a sunny day over land, solar heating stirs the air into turbulent eddies that carry heat upward and momentum downward, and the mixed layer usually grows to about one to two kilometers by afternoon. This is the layer weather forecasters watch for fog, pollution trapping, and thunderstorm fuel.

In Recognition Science, the framework's cost function J(x) = (x + 1/x)/2 - 1 measures the forced price of a mismatch between two quantities. The module defines a domain cost, a penalty for the ratio of two numbers m and e, as J(m/e). It proves three general facts about this cost: it is zero when m equals e, it never goes negative for positive inputs, and the golden-ratio threshold phi - 3/2 is positive. These are pure statements about the cost function itself, with no atmospheric content yet.

The research note attached to the module proposes a specific identification: set m and e so that J(phi) times the troposphere height, about 12 kilometers, gives roughly 1.4 kilometers, matching the typical daytime boundary layer. That arithmetic works, but the module does not prove it. The Lean file itself states that it proves nothing specific to this subject, because the cost is defined without reference to the atmosphere. What would turn the general cost theorems into a result about boundary layers is a definition of m and e in atmospheric terms, such as a ratio of turbulent kinetic energy to buoyancy flux.

The machine-checked library of formal theorems therefore establishes only the skeleton: the cost function's vanishing at equality, its nonnegativity, and a positive threshold constant. The atmospheric layer thickness remains an unformalized research idea, not a proved consequence. A reader should treat the 1.4 kilometer figure as a suggestive numerical coincidence awaiting a physical derivation, not as a theorem about the atmosphere.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

The module does not prove that the planetary boundary layer thickness equals 1.4 kilometers. The module does not define m and e in atmospheric terms. The golden-ratio threshold is a statement about the cost function, not about the atmosphere.

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

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