Astrophysics Stellar Assembly Tick Partition

The tick partition

A tick, in the Recognition Science framework, is a discrete unit of recognition events, the framework's term for the basic counting of physical processes. The declaration tick_partition is a theorem in the framework's machine-checked library of formal theorems. It states that the number of ticks assigned to mass storage, five, plus the number assigned to light emission, three, equals the total number of ticks in the framework's eight-tick cycle. The proof is a definitional equality, meaning it holds by the way the numbers are defined, not by any physical measurement.

The partition is part of a wider derivation in the framework's stellar assembly module. The framework models the mass-to-light ratio of a star as a function of the recognition cost difference between photon emission and mass storage. The cost function, J(x) = (x + 1/x)/2 - 1, is proved unique in the framework. The theorem ml_is_phi_power shows that if this cost difference equals an integer multiple of the framework's unit cost, then the mass-to-light ratio equals the golden ratio raised to that integer. The tick_partition theorem fixes the integer in question: the framework's eight-tick structure, split as five mass ticks and three light ticks, yields a tick ratio of 5/3, which the framework interprets as a characteristic tier of one.

What the declaration does not do is assert that real stars have a mass-to-light ratio of exactly the golden ratio, about 1.618 solar units. The framework's own hypothesis, H_StellarML, states that the stellar mass-to-light ratio equals the golden ratio, and the theorem ml_falsifiable shows this hypothesis is falsifiable: if the ratio were not the golden ratio, the hypothesis would be false. The framework's docstring notes that this value falls within the observed range of stellar mass-to-light ratios, roughly 0.5 to 5 solar units, but that comparison is an empirical check, not a theorem. The tick_partition declaration itself is a statement about the framework's internal arithmetic, not about any telescope measurement.

theorem tick_partition : mass_ticks + light_ticks = total_ticks := rfl

theorem tick_partition : mass_ticks + light_ticks = total_ticks := rfl

/-- When Δδ = n · J_bit = n · log(φ), we get M/L = φ^n -/
theorem ml_is_phi_power (n : ℤ) (Δδ : ℝ) (h : Δδ = n * J_bit) :
    ml_from_cost_diff Δδ = φ ^ n := by
  simp only [ml_from_cost_diff, J_bit] at *
  rw [h]
  -- exp(n * log(φ)) = φ^n by definition of zpow for positive reals
  have hφ : 0 < φ := Constants.phi_pos
  rw [← Real.rpow_intCast φ n]
  rw [Real.rpow_def_of_pos hφ]
  ring

/--- SCAFFOLD: M/L falsifiability check. -/
theorem ml_falsifiable (h : H_StellarML) :
    ml_stellar ≠ φ → False := by
  intro h_neq
  exact h_neq h

What this page does not claim

The tick_partition theorem does not claim that real stars have a mass-to-light ratio of exactly the golden ratio. The framework does not claim that the five-plus-three tick split is derived from physical measurement rather than definition. The framework does not claim that its golden-ratio mass-to-light value is a measured astrophysical constant.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Astrophysics/StellarAssembly.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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