Foundation Pair Kernel Zero Fit One Body Atomic Checkpoint Fd2 One Body Hamilton

A finite Hamiltonian that stays real

In quantum mechanics, a Hamiltonian is the operator that generates the time evolution of a system, and its eigenvalues are the allowed energies. A fundamental requirement for any physical Hamiltonian is that it be Hermitian, which guarantees that all energy eigenvalues are real numbers, not imaginary ones. The declaration oneBodyHamiltonian3_isHermitian from the machine-checked library of formal theorems proves that a specific finite one-body Hamiltonian, constructed on a small torus-shaped lattice, is indeed Hermitian for any choice of real kinetic scale, potential scale, and real diagonal potential.

The Hamiltonian in question is built from two parts. The kinetic part uses the S28 signed six-neighbor generator, a specific pattern of hopping amplitudes between neighboring sites on the lattice, scaled by a real parameter. The potential part is a real diagonal potential, meaning it assigns a real energy to each lattice site, also scaled by a real parameter. The theorem states that the sum of these two parts, for any real values of the scales and any real potential, is Hermitian. This is a purely mathematical statement about the structure of the matrix.

This Hermiticity property is not an isolated fact. The same library also proves that this Hamiltonian generates a unitary one-parameter group of evolutions, meaning that time evolution preserves the total probability. These two results together establish that the model is a mathematically consistent quantum system: it has real energies and conserves probability. The declaration is a formal, machine-checked proof, meaning it is verified by a computer and does not rely on unproven assumptions.

In Recognition Science, this result is a checkpoint in a larger program. The framework models a discrete ledger of recognition events and derives physical structure from it. This particular theorem establishes a property of a finite model, but it does not, by itself, claim that this model is the correct description of any real physical system. The library is explicit about this: the q=1 Green potential, which is a specific attractive potential attached to the model, is an explicit MODEL attachment, not a derived law. The theorem proves the source equation and the Green response, but it does not prove that this field is an electron potential.

What the declaration does not claim is as important as what it proves. It does not claim to derive the hydrogen atom spectrum, to fit any empirical data, or to identify the model's coupling ratio with the physical fine-structure constant. The library contains a separate predicate, RecognitionOneBodyQuantumAttachment3, which states the exact conditions under which this finite Hamiltonian would be identified with a physical one-body system, such as selecting positive scales independently of hydrogen. No theorem in the module asserts that this predicate is satisfied. The declaration is a precise, limited mathematical result: it proves Hermiticity for a well-defined finite model, leaving the physical interpretation as an open target.

/-- Every finite one-body Hamiltonian on this surface is Hermitian. -/
theorem oneBodyHamiltonian3_isHermitian
    (kineticScale potentialScale : ℝ)
    (potential : Fin (TorusCard3 3) → ℝ) :
    (Hc (oneBodyEvolution3 kineticScale potentialScale potential)).IsHermitian :=
  Hc_isHermitian _

/-- The finite one-body evolution is an exact unitary one-parameter group. -/
theorem oneBodyHamiltonian3_stone
    (kineticScale potentialScale : ℝ)
    (potential : Fin (TorusCard3 3) → ℝ) :
    StoneGeneratorCert (TorusCard3 3)
      (oneBodyEvolution3 kineticScale potentialScale potential) :=
  stoneGeneratorCert _

/-- The q=1 S28 Green field, with the exact factor-two Gauss normalization,
written as an attractive diagonal MODEL potential.  `zeroReference` is the
finite-volume additive Green gauge and `fieldNormalization` is explicit. -/
def q1AtomicModelPotential3
    (event : PostingPair3 3)
    (fieldNormalization zeroReference : ℝ)
    (i : Fin (TorusCard3 3)) : ℝ :=
  -fieldNormalization *
    (signedRealGreenField3 1 event i - zeroReference)

What this page does not claim

The declaration does not prove that the model describes the hydrogen atom or any other real physical system. The declaration does not derive the fine-structure constant or identify the model's coupling ratio with it. The declaration does not assert that the q=1 Green potential is an electron potential.

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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