Chemistry Periodic Blocks Block Capacity

Block capacity

The periodic table's familiar blocks, s, p, d, and f, record which kind of orbital holds the newest electron. Chemists count these blocks by the azimuthal quantum number, 0 for s, 1 for p, 2 for d, 3 for f. The capacity of a shell, the number of electrons it can hold, follows the pattern 2, 6, 10, 14, which is 2, 6, 10, 14, or 4 times 0.5, 1.5, 2.5, 3.5. This pattern is standard textbook material, and it is the classical fact any model of block structure must reproduce.

In Recognition Science, the framework's machine-checked library of formal theorems defines a quantity it calls block_capacity. The definition is a model, a choice, not a proof. It sets the capacity of the n-th shell to phi raised to the power 2n, where phi is the golden ratio, about 1.618. So shell 0 has capacity phi^0, which is 1; shell 1 has capacity phi^2, about 2.618; shell 2 has capacity phi^4, about 6.854. These numbers do not match the classical electron counts 2, 8, 18. The definition also introduces a shell scale, written as E_coh times block_capacity, and proves by reflexivity that this shell scale equals E_coh times the capacity. That identity is a theorem in the library, but it is a tautology: it restates the definition, since shell is defined as E_coh times block_capacity.

The framework models the shell scale as an energy-like quantity, and the docstring calls the whole construction a proxy for periodic table block structure from phi-packing of orbitals. The word proxy matters. The definition does not derive the periodic table from first principles. It does not predict the 2, 8, 18, 32 shell capacities, and it does not explain why the blocks have their observed widths. It supplies a dimensionless capacity for each shell and an energy scale built from it, and the library records the identity between them. The value of E_coh, the coherence energy, is a separate constant in the framework, and this file does not fix it.

What the declaration establishes is narrow. It defines a model object and proves one trivial identity about it. The identity, shell n equals E_coh times block_capacity n, holds by definition, not by physical argument. A reader should not take the golden-ratio powers as measured electron capacities; they are a chosen bookkeeping device within one framework's account of chemistry. The classical periodic table, with its 2, 6, 10, 14 pattern, remains the empirical target that any such model would need to meet, and this file does not claim to meet it.

noncomputable def block_capacity (n : Nat) : ℝ := Constants.phi ^ (2 * n)

/-- Identity: shell scale equals `E_coh` times capacity at each n. -/
@[simp] theorem blocks_holds (n : Nat) : shell n = Constants.E_coh * block_capacity n := by
  rfl

What this page does not claim

This answer does not claim that block_capacity predicts the observed electron shell capacities 2, 8, 18, 32. This answer does not claim that the golden ratio powers are measured physical quantities. This answer does not claim that the identity shell n = E_coh * block_capacity n is a physical law; it is a definitional tautology.

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

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