Foundation Tribonacci Rs

The tribonacci constant

The tribonacci constant is the number T that satisfies T³ = T² + T + 1. Its value is about 1.839, and it is the limit ratio between consecutive terms of a sequence where each term is the sum of the previous three (0, 1, 1, 2, 4, 7, 13, ...). The constant is a cousin of the golden ratio, which solves r² = r + 1 and comes from adding the previous two terms. The tribonacci constant is the cubic analogue.

The constant was studied in the 1960s by Mark Feinberg, who gave it the name, and it appears in combinatorics and number theory. The tribonacci sequence counts tilings of a strip with squares and dominoes, and the constant appears in the asymptotic growth rate of such tilings. It is irrational, and its continued fraction is [1; 1, 1, 3, 5, 9, 17, 31, ...] where each denominator doubles the previous one plus a small correction.

In Recognition Science, the framework models recognition as a discrete record of events with a forced cost. The ledger, a discrete record of events, defines a domain cost as the recognition cost J applied to a ratio m/e. The framework proves three general facts about this cost: it vanishes when m equals e, it is nonnegative for positive inputs, and the threshold phi minus 3/2 is positive. These are properties of the cost function itself, not of the tribonacci constant.

The name of the framework refers to a research note: at dimension D = 3, the recognition chain would have three previous terms, matching the tribonacci recurrence. But the code proves nothing specific to the tribonacci constant. The domain cost is defined without reference to it, and the three theorems hold for any positive m and e. The tribonacci connection remains a research note, not a proved result.

The value of the framework is that it shows the recognition cost behaves sensibly: equal inputs cost nothing, costs are never negative, and a canonical threshold is positive. These are sanity checks. What would make the framework a theorem about the tribonacci constant is a definition of m and e in terms of the tribonacci recurrence itself.

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

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 framework does not prove any property specific to the tribonacci constant. The tribonacci connection to dimension three is a research note, not a proved result. The domain cost is not derived from the tribonacci recurrence.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: