IndisputableMonolith.PhiSupport.Alternatives
The PhiSupport.Alternatives module demonstrates that sqrt(2), sqrt(3), sqrt(5), pi and e fail the phi selection criterion from PhiSelectionCore. Researchers confirming the uniqueness of the golden ratio within Recognition Science would cite these explicit counterexamples. Proofs proceed by direct algebraic verification that each candidate squared differs from the candidate plus one.
claim$\sqrt{2}$ fails the criterion $x^2 = x + 1$ with $x > 0$, as do $\sqrt{3}$, $\sqrt{5}$, $\pi$ and $e$.
background
The module imports the phi selection criterion requiring a positive real satisfying $x^2 = x + 1$ from PhiSelectionCore. It also draws on the RS time quantum $\tau_0 = 1$ tick defined in Constants. The module provides lemmas establishing that common constants do not satisfy this equation.
The theoretical setting is the Recognition Science framework in which phi is forced as the self-similar fixed point. Upstream results supply the exact criterion and base time unit used to frame the failures.
proof idea
The module consists of separate lemmas, one per constant. Each performs a direct algebraic check that the candidate squared is not equal to the candidate plus one, following the pattern shown for sqrt(2) in the module doc comment. No complex tactics or upstream lemmas beyond the definitions are applied.
why it matters in Recognition Science
These failure results support the phi selection criterion in PhiSelectionCore by eliminating common alternatives. They contribute to the uniqueness arguments and the forcing of phi as the self-similar fixed point in the broader Recognition Science chain.
scope and limits
- Does not prove that phi is the unique positive solution.
- Does not analyze constants outside the listed set.
- Does not address physical interpretations of the constants.
- Does not extend to negative reals or complex numbers.