Mathematics Number Theory From Rs Rsi Count Five

The five identities

The golden ratio φ is the number that satisfies φ² = φ + 1, about 1.618. A machine-checked library of formal theorems, part of the Recognition Science framework, certifies five identities that connect φ to other numbers. The first is the defining property itself. The second is φ⁵ = 5φ + 3, a Fibonacci-style relation. The third is φ⁸ = 21φ + 13, another Fibonacci relation. The fourth is that 45, called gap45, equals 3² × (3 + 2), which is 9 × 5. The fifth is the count: the number of these key identities is 5.

The declaration rsi_count_five establishes that fifth item. It states that rsKeyIdentityCount, a definition that names the number of key identities, equals 5. This is not a deep theorem about primes; it is a bookkeeping fact. The library defines the count as 5, and the declaration confirms that definition is correct. The other four identities are proved theorems, each with its own formal proof. The count declaration simply ties them together under one label.

In Recognition Science, these identities are part of a larger story. The framework models reality as maintaining a ledger, a discrete record of recognition events, and derives constants like the golden ratio from a forced cost function. Here, the number 5 appears as the count of canonical identities, and 45 appears as a gap tied to the number of spatial dimensions in the framework, D = 3. The library's proofs use only the standard axioms of its type theory, with no framework-specific assumptions.

The declaration does not claim that these five identities are the only important number-theoretic facts in the framework, nor that the count 5 has a deep structural meaning beyond being the number of listed identities. It does not claim that φ itself is prime, or that 45 is prime. It establishes a precise, narrow fact: the library's own definition of the key identity count is 5, and this is verified by the machine checker.

theorem rsi_count_five : rsKeyIdentityCount = 5 := rfl

/-- φ^5 = 5φ+3 (Fibonacci). -/
theorem phi5_fibonacci : phi ^ 5 = 5 * phi + 3 := by
  have h2 := phi_sq_eq
  have h3 : phi ^ 3 = 2 * phi + 1 := by nlinarith
  have h4 : phi ^ 4 = 3 * phi + 2 := by nlinarith
  nlinarith

/-- φ^8 = 21φ+13 (Fibonacci). -/
theorem phi8_fibonacci : phi ^ 8 = 21 * phi + 13 := by
  have h2 := phi_sq_eq
  have h3 : phi ^ 3 = 2 * phi + 1 := by nlinarith
  have h4 : phi ^ 4 = 3 * phi + 2 := by nlinarith
  nlinarith [sq_nonneg (phi ^ 4)]

/-- gap45 = D²(D+2) at D=3. -/
theorem gap45_from_D : 3 ^ 2 * (3 + 2) = 45 := by decide

What this page does not claim

The count 5 has a deep structural meaning beyond being the number of listed identities. The golden ratio φ is a prime number. The declaration proves any new property of the number 5 itself.

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/Mathematics/NumberTheoryFromRS.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: