Foundation Pair Kernel Zero Fit One Body Atomic Checkpoint Fd2 Recognition Hydro

The hydrogen checkpoint

The hydrogen atom's ground-state energy, in atomic units, is minus one half times the square of the fine-structure constant alpha. This is the textbook Rydberg formula, usually written as E = -alpha^2 / 2 in Hartree units. The Recognition Science library contains a definition that reproduces this exact expression, and a theorem that confirms the definition equals the formula by definitional equality. The declaration recognitionHydrogenGroundEnergyRatio_formula is that theorem: it states that the library's recognitionHydrogenGroundEnergyRatio, a real number, is equal to -(Constants.alpha ^ 2) / 2.

This is a formal checkpoint, not a new derivation. The library's module, FD2, builds a finite quantum system on a small torus and attaches a potential. The definition of the energy ratio is a model, a chosen expression, and the theorem merely unpacks the definition. It does not prove that this expression is the physical hydrogen energy, nor that the library's alpha is the measured fine-structure constant. The docstring is explicit: the physical interpretation depends on an unproved predicate, RecognitionOneBodyQuantumAttachment3, which would identify the finite Hamiltonian with a real one-body system. No theorem asserts this attachment.

The value of the checkpoint is precision about what is and is not established. The formula itself is a theorem, but it is a theorem about a definition. The library does not claim to have derived the hydrogen spectrum, and it does not claim that its alpha is the physical coupling. The fine-structure constant identification is an open target, not a proved result. The module contains no fitting to hydrogen data, no empirical normalization, and no orbital exponents. It is a zero-fit construction: the energy ratio is what it is because of the definition, not because of any measurement.

In plain terms, the declaration establishes a formal identity: the library's hydrogen energy ratio is minus alpha squared over two. It does not establish that this ratio describes the real hydrogen atom. The gap is the semantic arrow from the finite model to physics, and that arrow is left as an explicit predicate, not a theorem. This is the honest boundary of the checkpoint: the mathematics is exact, the physics is not yet attached.

theorem recognitionHydrogenGroundEnergyRatio_formula :
    recognitionHydrogenGroundEnergyRatio =
      -(Constants.alpha ^ 2) / 2 := rfl

/-- Exact missing semantic arrow.  A committed physical one-body Hamiltonian
would have to be identified with the S28 generator and q=1 Green potential,
with positive scales selected independently of hydrogen. -/
def RecognitionOneBodyQuantumAttachment3
    (physicalHamiltonian : DiscreteEvolution (TorusCard3 3)) : Prop :=
  ∃ event : PostingPair3 3,
    event ∈ realizedPrimitivePostingPairs3 3 ∧
      ∃ kineticScale potentialScale fieldNormalization zeroReference : ℝ,
        0 < kineticScale ∧
          0 < potentialScale ∧
          0 < fieldNormalization ∧
          physicalHamiltonian =
            oneBodyEvolution3 kineticScale potentialScale
              (q1AtomicModelPotential3
                event fieldNormalization zeroReference)

What this page does not claim

The declaration does not prove that the hydrogen ground-state energy is physically minus alpha squared over two. The library's alpha is not identified with the measured fine-structure constant by any theorem in this module. The finite model does not claim to be a complete or exact description of the hydrogen atom.

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/Foundation/PairKernelZeroFitOneBodyAtomicCheckpointFD2.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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