IndisputableMonolith.Cost.FixedPoint
Cost.FixedPoint isolates the algebraic fixed point whose positive solution is the golden ratio φ. Researchers building the phi-ladder mass formulas or the self-similar structure in Recognition Science cite this result. The argument reduces x = 1 + 1/x to the quadratic x² - x - 1 = 0 and matches its positive root to the standard definition of φ.
claimThe positive real number satisfying the equation $x = 1 + 1/x$ is the golden ratio φ.
background
Recognition Science derives all structure from a single functional equation whose self-similar fixed point appears in the cost domain. The upstream Constants module supplies the fundamental RS time quantum (RS-native) τ₀ = 1 tick. This module extracts the fixed-point relation that forces φ in the T6 step of the unified forcing chain.
proof idea
The module contains the single lemma phi_is_cost_fixed_point. Its proof solves the quadratic obtained by clearing the denominator in x = 1 + 1/x and identifies the positive root with the golden ratio.
why it matters in Recognition Science
This lemma supplies the fixed point required by T6 in the unified forcing chain. It anchors the phi-ladder used in the mass formula yardstick * φ^(rung - 8 + gap(Z)) and the eight-tick octave. The result closes the algebraic step that forces φ from self-similarity.
scope and limits
- Does not compute numerical approximations to φ.
- Does not embed the fixed point into the full J-cost function.
- Does not prove uniqueness outside the positive reals.
- Does not address the Berry creation threshold or Z_cf.