Foundation Tribonacci Rs Tribonacci Cert

The certificate's scope

The Tribonacci constant is the number T ≈ 1.839 that solves T³ = T² + T + 1, the cubic cousin of the golden ratio. In the Recognition Science framework's machine-checked library of formal theorems, a declaration named TribonacciCert exists, but it does not establish anything about that constant. The certificate is a small package of three proved facts about a cost function, a mathematical device that assigns a penalty to a comparison between two quantities.

The three facts are general and modest. First, when the two quantities in the cost function are equal, the cost is zero. Second, for positive quantities, the cost is never negative. Third, a certain threshold value, defined as the golden ratio minus 1.5, is positive. The certificate bundles these three statements into a single object and proves that such an object exists. These are the only claims the certificate makes.

The name is aspirational, not descriptive. The library's own documentation says the Tribonacci idea was where the research was meant to go, but the formal proof never reaches it. The cost function is defined as J(m/e), a ratio of two real numbers, with no definition of what m and e mean for the Tribonacci sequence. The three proved facts hold for any positive numbers, not for anything specific to the Tribonacci recurrence.

In Recognition Science, this certificate is a template. The same three facts are stated once, universally, in a shared module that 2,383 sibling modules copy verbatim. What would turn this certificate into a theorem about its subject is a definition of m and e in the Tribonacci setting. Until that definition exists, the certificate proves the shape of a result, not the result itself.

structure TribonacciCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

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]

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

What this page does not claim

The certificate does not prove any property of the Tribonacci constant itself. The certificate does not define what m and e mean for the Tribonacci sequence. The certificate does not establish that the Tribonacci ratio is phi^2 or any other specific value.

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/Tribonacci_RS.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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