cert_inhabited
cert_inhabited establishes that a certificate for the golden section proportion exists in the Recognition Science model. Aesthetic theorists and architects using J-cost minimization would cite this when showing that the phi ratio satisfies the structural requirements for ideal rectangles. The proof is a one-line term construction that directly supplies the cert object to witness non-emptiness of the type.
claimThe type of certificates for the golden section is inhabited: there exists a real number $r > 1$ with $1.4 < r < 1.9$, $r(r-1)=1$, such that the proportion cost vanishes when the two sides are equal and remains nonnegative for all positive arguments.
background
The module develops the Recognition Science prediction that the minimum J-cost rectangle has aspect ratio phi:1, obtained by minimizing J(r) = (r + 1/r)/2 - 1 subject to area constraint and self-similarity. GoldenSectionCert is the structure that packages the required properties: ratio strictly greater than 1, lying inside the aesthetic band (1.4, 1.9), satisfying the recursion r(r-1)=1, cost zero at equal sides, and cost nonnegative otherwise. The upstream definition of GoldenSectionCert supplies exactly these five fields and is the sole dependency.
proof idea
The proof is a one-line term that applies the structure constructor to the definition named cert, which already satisfies every field of GoldenSectionCert.
why it matters in Recognition Science
The result closes the structural component of the golden-section module by confirming the certificate type is inhabited. It directly supports the module claim that phi achieves the smallest non-trivial J-cost departure from a square among Fibonacci-ratio rectangles. Within the broader framework it instantiates the self-similar fixed point phi forced by the T5-T6 steps of the unified forcing chain and the Recognition Composition Law. No downstream uses are recorded and no open questions are flagged in the supplied material.
scope and limits
- Does not derive the numerical value of phi from the forcing chain.
- Does not prove that phi is the unique minimizer inside the band.
- Does not connect the certificate to concrete architectural measurements.
- Does not address the psychophysical falsifier stated in the module documentation.
formal statement (Lean)
109theorem cert_inhabited : Nonempty GoldenSectionCert := ⟨cert⟩
proof body
Term-mode proof.
110
111end
112end GoldenSectionInProportion
113end Architecture
114end IndisputableMonolith