Chemistry Sodium Ionization Rs Na Ionization Rs

The formal skeleton

Sodium's first ionization energy is 5.139 electronvolts, the energy needed to strip one outer electron from a neutral sodium atom. The Recognition Science library contains a declaration named NaIonizationRS, and the first thing to know is what that declaration actually is: a small structure with three fields, each field a proved statement about a generic cost function, not about sodium.

The three fields are general facts. The first says the cost function J, defined as J(x) = (x + 1/x)/2 - 1, vanishes when its input equals 1. The second says J is nonnegative for positive inputs. The third says the number phi - 3/2 is positive, where phi is the golden ratio. These are proved in the machine-checked library of formal theorems. They hold for any positive real numbers, which is why the declaration's own docstring states plainly that it proves nothing specific to sodium.

The declaration takes its name from a research note attached to the module, which records an intended numerical match: phi^4 times 0.75 equals 5.14 electronvolts, close to the measured 5.139. That note is not a theorem. It is a stated aspiration, and the library's own documentation flags it as such. The formal structure NaIonizationRS does not define the electron mass m or the electron charge e in sodium's terms; it only defines domainCost as J(m/e) without reference to one.

What the declaration does establish, then, is a template. It shows that the cost function J has the three listed properties, and it packages those properties into a named structure with a certificate of existence. The certificate is a ledger, a discrete record of formal facts, but the ledger has no sodium-specific entries. The gap between the research note and the formal content is the gap between a hoped-for identification and a proved result.

For a reader, the practical takeaway is simple. The library proves three general inequalities and one positivity statement. It does not prove that sodium's ionization energy equals phi^4 times 0.75. That numerical claim remains a research note, not a theorem, and the library's own text says so in the same breath as it records the idea.

structure NaIonizationRS where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The declaration does not prove that sodium's ionization energy equals 5.14 electronvolts. The declaration does not define the electron mass or charge in physical terms. The numerical match in the research note is not a theorem, only a stated aspiration.

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/Chemistry/Sodium_Ionization_RS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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