Verification Phi Non Degenerate Cert

The non-degeneracy certificate

The golden ratio φ is the positive solution to r² = r + 1, about 1.618. A non-degeneracy certificate is a formal record that φ is not zero and not one. In Recognition Science, a framework that derives physical constants from a forced cost of recognition, this certificate is a small but load-bearing audit: it guarantees that expressions like φ⁻¹ and ratios involving φ are well-defined, and that φ is not accidentally the trivial point where the cost function vanishes.

The two properties are direct consequences of the already-proved fact that 1 < φ. Since φ is greater than one, it is automatically neither zero nor one. The certificate packages these two inequalities as explicit statements, φ ≠ 0 and φ ≠ 1, in a machine-checked library of formal theorems. The proof is a two-line construction: each inequality follows from the same prior theorem, and the library verifies both in one step.

Why does this matter? Division by φ appears throughout the framework, for example in φ⁻¹ and in the J-cost function J(x) = (x + 1/x)/2 − 1. Knowing φ ≠ 0 makes those divisions safe. The second property, φ ≠ 1, is more subtle: the cost minimum sits at J(1) = 0, so if φ equaled one, it would be a trivial, uninteresting point. The certificate rules that out, confirming that φ is a genuine, non-trivial constant in the framework's structure.

The certificate is a verification artifact, not a new discovery. It does not prove that φ has any particular value; it only records that φ is not degenerate. This is the kind of check that keeps a formal library honest: before building larger results on φ, the framework confirms the ground is solid.

structure PhiNonDegenerateCert where
  deriving Repr

structure PhiNonDegenerateCert where
  deriving Repr

What this page does not claim

The certificate does not prove that φ is the golden ratio or that it equals 1.618. The certificate does not establish that φ is the unique self-similar scaling; that is a separate theorem. The certificate does not derive any physical constant; it only records two inequalities about φ.

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/Verification/PhiNonDegenerateCert.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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