scaledIonization
scaledIonization computes the dimensionless φ-scaled ionization energy for atomic number Z by raising phi to twice the period and multiplying by the normalized proxy. Modelers deriving periodic ionization trends from the Recognition Science ladder would cite this when generating predictions before eV anchoring. The definition is a direct algebraic product of the rail factor and the position factor.
claim$I_1(Z) = phi^{2p} · (ionizationProxy(Z) / periodLength(Z))$ where $p$ denotes the noble-gas-bounded period of $Z$.
background
The module derives first ionization energies from φ-ladder scaling. RS predicts a sawtooth pattern forced by φ-rail scaling as φ^{2n} for period n, a position factor that increases toward period closure, and fixed φ-packing offsets for subshells. Key predictions are that alkali metals reach minimum ionization per period while noble gases reach the maximum, all without data fitting.
proof idea
The definition is a direct computation. It calls periodOf Z to obtain the integer period, forms the rail factor via Real.rpow on Constants.phi raised to twice that period, then multiplies by the result of normalizedIonization Z.
why it matters in Recognition Science
scaledIonization supplies the scaling factor consumed by predictedI1_eV to produce the anchored eV prediction. It implements the φ-rail component of the ionization model in the module documentation, enabling the subsequent sawtooth ordering theorems. The construction rests on the phi fixed point and ladder scaling that appear throughout the Recognition framework.
scope and limits
- Does not incorporate experimental ionization values for calibration.
- Does not apply relativistic corrections for high-Z elements.
- Does not predict higher-order ionization energies I2 or I3.
- Does not extend beyond the first ionization energy proxy.
Lean usage
predictedI1_eV 11formal statement (Lean)
47def scaledIonization (Z : ℕ) : ℝ :=
proof body
Definition body.
48 let period := periodOf Z
49 let rf := Real.rpow Constants.phi ((2 : ℝ) * (period : ℝ))
50 rf * normalizedIonization Z
51
52/-- Predicted ionization energy in eV (display seam).
53 Uses E_coh as the universal energy anchor. -/