phi
plain-language theorem explainer
The golden ratio φ is defined as the real number (1 + sqrt(5))/2. Galaxy cluster modelers cite this definition when showing that observed mass-to-light ratios lie between φ^10 and φ^13. The definition is a direct algebraic assignment with no computation or lemmas required.
Claim. Let $φ$ denote the golden ratio, defined by the equation $φ = (1 + √5)/2$.
background
This definition sits inside the module that closes Gap 10 by deriving mass-to-light ratios from the ledger rather than from external galaxy data. The module states that M/L emerges from recognition-weighted stellar assembly, ledger budget constraints, and curvature partition across the 8-tick cycle, with observed values matching powers of φ between 100 and 500. Upstream structures supply the J-cost function from PhysicsComplexityStructure, the ledger L from Recognition, and the phi-tier organization of nuclear densities from NucleosynthesisTiers.
proof idea
The definition is a direct algebraic assignment of the standard closed-form expression for the golden ratio. No lemmas or tactics are invoked; it serves as a primitive constant for the sibling derivations that follow.
why it matters
The definition supplies the base constant for all subsequent M/L derivations in the module, allowing the claim that observed ratios equal φ^n for integer n near 10-13. It anchors the local theoretical setting that all dimensionless ratios in Recognition Science are algebraic in φ, consistent with the self-similar fixed point forced in the unified chain. Sibling declarations such as phi_power and ml_is_phi_power apply it directly to match the 100-500 observational window.
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