normalizedIonization
The definition computes a dimensionless ionization value for atomic number Z by dividing the valence proxy by the current period length. Chemists working on periodic trends within the Recognition Science framework cite it to obtain normalized scores where noble gases reach 1 and alkali metals reach 1 over period length. It is realized as a guarded division that returns zero when the period length vanishes.
claimFor atomic number $Z$, the normalized ionization equals the valence-electron proxy divided by the period length when that length is positive, and equals zero otherwise. Noble-gas elements therefore evaluate to 1 while alkali metals evaluate to $1/n$ for period length $n$.
background
The module derives first ionization energies from phi-ladder scaling. It combines a rail factor that grows as phi to the power of twice the period index with a position factor that increases from the start to the end of each period. The sawtooth pattern arises directly from these two contributions without empirical fitting. Upstream, ionizationProxy supplies the valence count that runs from 1 for alkali metals to the full period length for noble gases, capturing the cost of breaking into a forming shell. periodLength returns the number of elements between successive shell closures. The Z anchors from the mass and policy modules supply the integer atomic numbers used throughout.
proof idea
This is a direct definition that applies the division of ionizationProxy by periodLength, guarded by an equality check against zero.
why it matters in Recognition Science
The definition supplies the position factor required by the downstream scaledIonization, which multiplies it by the phi-rail term phi to the power 2 times period. It therefore realizes the second ingredient of the phi-ladder scaling stated in the module doc-comment and enables the predicted ordering alkali minimum to noble-gas maximum within each period. The construction sits inside the broader Recognition Science program that forces constants and mass ladders from the J-uniqueness relation and the eight-tick octave.
scope and limits
- Does not incorporate the phi-rail scaling factor.
- Does not output energies in electron-volts.
- Does not apply relativistic or many-body corrections.
- Does not extend to non-integer atomic numbers.
- Does not compute higher ionization stages.
Lean usage
scaledIonization 11formal statement (Lean)
40def normalizedIonization (Z : ℕ) : ℝ :=
proof body
Definition body.
41 if periodLength Z = 0 then 0
42 else (ionizationProxy Z : ℝ) / (periodLength Z : ℝ)
43
44/-- φ-scaled ionization energy (dimensionless).
45 Combines rail factor with position factor.
46 I_scaled(Z) = φ^{2·period} × (valence / periodLength) -/