Chemistry Periodic Table From Phi Ladder Electron Block Count

The five block types

Chemists group the periodic table into blocks named for the atomic orbital an element's outermost electrons occupy. The s-block holds 2 elements per period, the p-block 6, the d-block 10, and the f-block 14, for a total of 32 across the four known block types. A fifth block, the g-block, is predicted to appear in period 8 with 18 elements, though no g-block element has been synthesized.

The standard electron shell capacities follow the formula 2n², where n is the principal quantum number: 2 for the first shell, 8 for the second, 18 for the third, and 32 for the fourth. These numbers are not new; they appear in any inorganic chemistry textbook. The Recognition Science framework's machine-checked library of formal theorems records these facts as definitions and proofs, confirming that the shell capacities are 2, 8, 18, and 32, and that the five block types s, p, d, f, and predicted g exhaust the list.

In Recognition Science, the declaration electronBlockCount establishes that the number of electron block types is exactly five. The framework models the block types as a finite set with five members, and a theorem proves the count by direct computation. This is not a derivation of the periodic table's structure from first principles; it is a formal record of a standard chemistry fact, checked by a computer.

The framework's contribution here is modest but precise. It does not claim that the number five follows from the golden ratio, nor that the phi-ladder pattern generates the block structure. The docstring explicitly states that the shell capacities are not phi-ladder numbers. What the declaration provides is a clean, machine-verified statement of a familiar counting fact, useful as a foundation stone for further formal work in chemistry.

theorem electronBlockCount : Fintype.card ElectronBlock = 5 := by decide

theorem shellCapacity_1 : shellCapacity 1 = 2 := by decide

theorem shellCapacity_2 : shellCapacity 2 = 8 := by decide

theorem shellCapacity_3 : shellCapacity 3 = 18 := by decide

theorem shellCapacity_4 : shellCapacity 4 = 32 := by decide

What this page does not claim

The number five is not derived from the golden ratio or any phi-ladder scaling. The declaration does not prove that g-block elements exist; it only records the block as a predicted type. The shell capacities 2, 8, 18, 32 are not claimed to be phi-ladder numbers.

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/PeriodicTableFromPhiLadder.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.

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