Foundation Phi Square Identity Phi Sq Ident Cert

The certification record

The golden ratio φ is the number whose square equals itself plus one: φ² = φ + 1. Solving the quadratic gives φ = (1 + √5)/2 ≈ 1.618. This identity is the defining algebraic property of φ, and it leads directly to the Fibonacci recurrence, where each term is the sum of the two before it. The classical story is well known: Euclid described the ratio in his Elements as the extreme and mean ratio, and it appears throughout geometry, from the pentagon to the golden spiral.

In Recognition Science, the framework models recognition events through a ledger, a discrete record of events, and assigns each event a cost, a forced penalty for recognition. The framework's library, a machine-checked collection of formal theorems, contains a file named Phi_Square_Identity.lean. Despite the name, the file does not prove φ² = φ + 1. Instead, it defines a cost function J(x) = (x + 1/x)/2 - 1 and proves three general facts about it: the cost vanishes when the two inputs are equal, the cost is never negative for positive inputs, and the quantity φ - 3/2 is positive.

The declaration PhiSqIdentCert is a structure that bundles these three facts into a single certificate. It contains the theorem that domainCost r r = 0 for any nonzero r, the theorem that domainCost m e ≥ 0 when both m and e are positive, and the theorem that canonicalThreshold = φ - 3/2 is positive. The framework then constructs a certificate cert satisfying all three conditions and proves that such a certificate exists. The file's docstring explicitly notes that this body is shared verbatim with 2383 sibling modules, and that the real subject-specific content would require defining m and e in that subject's own terms.

What the declaration does not claim is the headline its filename suggests. It does not prove the golden ratio identity, nor does it establish any connection between the cost function and the Fibonacci recurrence. The paragraph above the Lean code, which mentions φ² = φ + 1 and the Fibonacci recognition recursion, is marked as a research note recording where the idea was meant to go, not as a result. The machine-checked theorems prove only the three general cost facts, universally quantified, with no reference to any specific domain.

This distinction matters for anyone reading the framework's library. A name like PhiSqIdentCert invites a reader to expect a proof of the golden ratio property, but the actual content is a generic template. The framework's honest accounting separates what is proved, the three cost facts, from what is intended, the application to a specific subject. The certificate is real, but it is a skeleton awaiting a definition of m and e to give it flesh.

structure PhiSqIdentCert 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]

structure PhiSqIdentCert 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

What this page does not claim

The golden ratio identity φ² = φ + 1 is not proved in this file. The Fibonacci recurrence is not derived from the cost function here. The certificate does not apply to any particular physical or mathematical domain.

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/Phi_Square_Identity.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.

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