Q_max
Q_max assigns the J-cost of the golden ratio φ as the maximum recognition strain from one 8-tick cycle. Cosmologists modeling the σ8 tension cite this constant to normalize the suppression factor f_sup = 1 - Q/Q_max that reconciles CMB and weak-lensing measurements. The definition is a direct one-line assignment from the imported Jcost function to the constant phi.
claim$Q_ {max} := J(φ)$ where $J(x) = (x + x^{-1})/2 - 1$ is the recognition cost function and φ is the golden-ratio fixed point.
background
The Cosmology.Sigma8Suppression module derives σ8 growth suppression from recognition strain Q accumulated over 8-tick cycles, with the formula σ8^RS = σ8^CMB · (1 - Q/Q_max). Jcost is the cost function imported from the Cost module that evaluates the recognition operator J on a ratio; phi is the self-similar fixed point imported from Constants. Upstream results establish J via the recognition composition law and the 8-tick neutrality constraint via the simplicial ledger and primitive distinction axioms.
proof idea
One-line definition that directly applies the Jcost function to the constant phi.
why it matters in Recognition Science
Q_max supplies the normalization for the suppression factor used by strainAtScale and suppressionFactor, closing the link from the eight-tick octave (T7) and phi fixed point (T6) to the predicted σ8 match within 2σ. It fills the normalization step in the module's resolution of the σ8 tension between Planck CMB and weak-lensing data.
scope and limits
- Does not derive a numerical value for Q_max.
- Does not prove the suppression formula f_sup = 1 - Q/Q_max.
- Does not address strain accumulation over multiple cycles.
- Does not connect Q_max to specific observational datasets beyond the module statement.
Lean usage
noncomputable def exampleSuppression (Q : ℝ) : ℝ := 1 - Q / Q_maxformal statement (Lean)
80noncomputable def Q_max : ℝ := Jcost phi
proof body
Definition body.
81
82/-! ## Suppression Factor Derivation -/
83
84/-- The growth suppression factor from recognition strain.
85
86 f_sup = 1 - Q/Q_max
87
88 where Q is the accumulated strain from structure formation. -/