ml_is_phi_power'
plain-language theorem explainer
If the recognition cost differential Δδ equals an integer multiple n of the elementary ledger bit cost ln φ, then the equilibrium stellar mass-to-light ratio equals φ to the power n. Astrophysicists modeling stellar populations would cite this to place observed M/L ratios on discrete φ-ladder rungs. The proof is a direct term reduction that substitutes the hypothesis into the exponential definition of ml_from_cost_diff and invokes the real exponentiation identity exp(n log φ) = φ^n.
Claim. If the cost differential satisfies $Δδ = n · ln φ$ for integer $n$, then the equilibrium mass-to-light ratio equals $φ^n$.
background
Stellar assembly minimizes total ledger cost during collapse. Photon emission carries recognition cost δ_emit = J(r_emit) while bound mass storage carries δ_store = J(r_store). The differential Δδ = δ_emit - δ_store fixes the equilibrium partition. The module defines J_bit as the elementary bit cost ln φ and sets ml_from_cost_diff(Δδ) := exp(Δδ), so that Δδ = n · J_bit immediately yields M/L = φ^n (MODULE_DOC).
proof idea
The term proof first simplifies the definitions of ml_from_cost_diff and J_bit. It rewrites using the hypothesis Δδ = n · J_bit. With φ positive (Constants.phi_pos), it applies Real.rpow_intCast and Real.rpow_def_of_pos to obtain exp(n log φ) = φ^n, then normalizes by ring.
why it matters
This theorem supplies the definitional claim that stellar M/L is a power of φ, completing the core derivation of the StellarAssembly module. It supports the hypothesis that the characteristic mass-to-light ratio is fixed by the eight-tick partition. In the framework it instantiates the phi-ladder mass formula at astrophysical scales, with integer steps n constrained by the octave structure (T7) and the recognition cost J.
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