IndisputableMonolith.Cost.FixedPoint
The Cost.FixedPoint module isolates the fixed-point equation satisfied by the golden ratio in the Recognition Science cost setting. It states that φ is the positive solution to x = 1 + 1/x. Researchers tracing the self-similar step in the forcing chain cite this result. The module consists of one canonical lemma that rests directly on the imported Constants definition of the time quantum.
claimThe golden ratio satisfies the equation $x = 1 + 1/x$ and is its unique positive real solution.
background
The module sits in the Cost domain and imports the Constants module. The upstream Constants module defines the fundamental RS time quantum τ₀ = 1 tick. The local theoretical setting is the derivation of all physics from one functional equation, with the fixed-point property of φ arising inside the T0-to-T8 forcing chain.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the fixed-point property required at step T6 of the forcing chain, where φ is forced as the self-similar fixed point. It therefore supports the Recognition Composition Law and the subsequent extraction of RS-native constants such as ħ = φ^{-5}. No direct downstream declarations are recorded in the current graph.
scope and limits
- Does not prove uniqueness outside the positive reals.
- Does not connect the fixed point to the J-cost function or defect distance.
- Does not reference the eight-tick octave or spatial dimension D = 3.
- Does not derive the numerical value of φ or the alpha band.