Foundation Economics Firm Growth4 Cert

FirmGrowth4Cert

In economics, Gibrat's law states that a firm's growth rate is independent of its size. The Recognition Science framework approaches this territory with a cost function, a measure of how far a ratio departs from balance. The declaration FirmGrowth4Cert is a structure, a bundle of three proved facts about that cost function, wrapped together so that any code using the certificate can rely on those facts without rechecking them.

The three facts are general properties of the cost function J(x) = (x + 1/x)/2 − 1. First, when the ratio m/e equals 1, meaning the two quantities are equal, the cost is exactly zero. Second, for positive inputs, the cost is never negative. Third, the number φ − 3/2, where φ is the golden ratio, is positive. Each of these is a theorem about the cost function, not about firms. The certificate simply packages them as a single object that can be passed around and reused.

What the certificate does not claim matters as much as what it proves. The cost function here is defined as J(m/e), where m and e are real numbers, but nothing in the definition ties those numbers to any particular economic meaning. The module itself states this plainly: it proves nothing specific to firm growth, because the definition of domainCost does not reference firms, revenue, or size. A research note in the source imagines an interpretation where growth rate equals J(φ) times revenue growth, about 11.8 percent per year, but that note is explicitly labeled as an idea, not a result.

This is the difference between a scaffold and a building. The certificate proves that a certain cost function behaves well: it vanishes at balance, stays nonnegative, and has a positive threshold. Those are real, machine-checked facts. But turning them into a statement about firm growth would require defining m and e in economic terms, and that step has not been taken. The certificate is shared verbatim across 2383 sibling modules, which is why it appears here: the content is universal, and the subject-specific part remains open.

theorem cert_inhabited : Nonempty FirmGrowth4Cert := ⟨cert⟩

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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The certificate does not assert that firms grow at 11.8 percent per year. The certificate does not define what m and e mean in economic terms. The certificate does not prove Gibrat's law or any empirical claim about firm size and growth.

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/Economics.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.

Derived articles

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