Constants Consistency Phi Consistency

The golden ratio's consistency check

The golden ratio φ, approximately 1.618, is the number that satisfies φ² = φ + 1. It appears throughout mathematics, from the regular pentagon to the Fibonacci sequence, where successive terms approach it as a limit. In the Recognition Science framework, φ also emerges as the unique self-similar scaling forced by the framework's cost function, a result proved in the machine-checked library of formal theorems.

The declaration phi_consistency is a consistency check, not a new derivation. It states, as a theorem proved by reflexivity, that the golden ratio defined in one module is exactly equal to the golden ratio defined in another module. The theorem reads: Constants.phi = IndisputableMonolith.Constants.phi. In plain terms, this confirms that two different parts of the framework's codebase refer to the same number when they write φ.

This matters because a large formal library can drift. One module might define φ as (1 + √5)/2, another as the positive root of x² = x + 1, and a third as the limit of Fibonacci ratios. These definitions are mathematically equivalent, but a machine-checked proof must verify the equality explicitly. The phi_consistency theorem closes that gap, ensuring that any result proved about φ in one module applies to φ in another.

The theorem does not claim that φ is physically real, that the golden ratio appears in nature by necessity, or that the framework's derivation of φ from its cost function is experimentally confirmed. It is a bookkeeping statement about the internal coherence of a formal system. The framework's larger claims about φ, such as its role in the forcing chain that leads to three spatial dimensions, rest on separate theorems with their own proofs and empirical checks.

/-- The golden ratio φ is defined consistently across modules. -/
theorem phi_consistency :
    Constants.phi = IndisputableMonolith.Constants.phi := rfl

/-- The golden ratio φ is defined consistently across modules. -/
theorem phi_consistency :
    Constants.phi = IndisputableMonolith.Constants.phi := rfl

What this page does not claim

The theorem does not prove that φ appears in nature or that the golden ratio is physically fundamental. The theorem does not confirm the framework's derivation of φ from the cost function; that is a separate result. The theorem does not establish any experimental or empirical validity for φ's role in particle physics.

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/Constants/Consistency.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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