Foundation Pair Kernel Zero Fit One Body Atomic Checkpoint Fd2

The finite atomic checkpoint

A quantum system needs a state space, a rule for how the state changes, and a way to compute energies. The foundation pair kernel zero fit one body atomic checkpoint fd2 is a machine-checked module that assembles the smallest finite version of all three, small enough that a computer can verify every step. It works on a three-by-three grid wrapped into a torus, so the state is a function from nine points to complex numbers. The kinetic part comes from a fixed signed generator, and a real diagonal potential can be attached to the same Hamiltonian. The module proves the resulting matrix is Hermitian and generates an exact unitary evolution, the finite analogue of quantum time development.

The checkpoint is called zero fit because it contains no hydrogen answer, no empirical normalization, no orbital exponent, and no cutoff. The potential is a specific model attachment: the q=1 Green field, written as an attractive diagonal potential with an explicit normalization and a zero reference. The module proves the source equation and the Green response, but it does not prove that this field is an electron potential, does not select the kinetic-to-field coupling ratio, and does not identify the current alpha expression with the physical infrared coupling. A predicate called RecognitionOneBodyQuantumAttachment3 states that missing semantic arrow, and no theorem asserts it.

What the module does establish is the exact structure of the energy. The scalar energy has a line expansion: the response to a variation is a linear term plus a quadratic remainder, both descending from the same scalar energy. A common rescaling of the kinetic and potential scales multiplies the whole energy by that common factor, and the dimensionless coupling ratio survives. The module also exports the dimensionless hydrogen ground energy ratio as minus alpha squared over two, and the Bohr-to-reduced-Compton length ratio as the inverse alpha. These are definitions, not derivations; their physical interpretation still depends on the attachment predicate and the infrared alpha identification.

In plain language, this checkpoint is a promise kept and a boundary drawn. The promise is that a finite quantum system can be built from the framework's ingredients with no fitted constants and no unproved axioms. The boundary is that the step from this mathematical object to a physical hydrogen atom remains open, stated as a predicate rather than assumed as a theorem. The module shows exactly how far the framework's own proofs reach, and where the empirical identification would have to begin.

/-- 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 _

/-- Exact finite variational identity.  The linear response and quadratic
remainder both descend from `oneBodyScalarEnergy3`. -/
theorem oneBodyScalarEnergy3_line_expansion
    (kineticScale potentialScale : ℝ)
    (potential state variation : Fin (TorusCard3 3) → ℝ)
    (t : ℝ) :
    oneBodyScalarEnergy3 kineticScale potentialScale potential
        (fun i => state i + t * variation i) =
      oneBodyScalarEnergy3 kineticScale potentialScale potential state +
        t * oneBodyResponsePairing3
          kineticScale potentialScale potential state variation +
        t ^ 2 *
          oneBodyScalarEnergy3 kineticScale potentialScale potential variation := by
  have hpotential :
      (∑ i, potential i * (state i + t * variation i) ^ 2) =
        (∑ i, potential i * (state i) ^ 2) +
          2 * t * (∑ i, potential i * state i * variation i) +
          t ^ 2 * (∑ i, potential i * (variation i) ^ 2) := by
    have hpoint : ∀ i,
        potential i * (state i + t * variation i) ^ 2 =
          potential i * (state i) ^ 2 +
            2 * t * (potential i * state i * variation i) +
            t ^ 2 * (potential i * (variation i) ^ 2) := by
      intro i
      ring
    simp_rw [hpoint]
    simp only [Finset.sum_add_distrib, ← Finset.mul_sum]
  unfold oneBodyScalarEnergy3 oneBodyResponsePairing3
  rw [action_line_expansion, hpotential]
  ring

/-- A common energy-unit rescaling multiplies the entire scalar energy. -/
theorem oneBodyScalarEnergy3_common_rescale
    (c kineticScale potentialScale : ℝ)
    (potential state : Fin (TorusCard3 3) → ℝ) :
    oneBodyScalarEnergy3 (c * kineticScale) (c * potentialScale)
        potential state =
      c * oneBodyScalarEnergy3 kineticScale potentialScale potential state := by
  unfold oneBodyScalarEnergy3
  ring

theorem oneBodyCouplingRatio_common_rescale
    (c kineticScale potentialScale : ℝ)
    (hc : c ≠ 0) (hk : kineticScale ≠ 0) :
    oneBodyCouplingRatio (c * kineticScale) (c * potentialScale) =
      oneBodyCouplingRatio kineticScale potentialScale := by
  unfold oneBodyCouplingRatio
  field_simp

/-- Dimensionless infinite-nucleus hydrogen energy ratio exported for the
frozen external checkpoint.  Its physical interpretation still depends on
`RecognitionOneBodyQuantumAttachment3` and the infrared alpha identification. -/
def recognitionHydrogenGroundEnergyRatio : ℝ :=
  -(Constants.alpha ^ 2) / 2

/-- Dimensionless Bohr-to-reduced-Compton length ratio exported for the same
checkpoint. -/
def recognitionHydrogenLengthRatio : ℝ :=
  Constants.alphaInv

What this page does not claim

The module does not prove that the q=1 Green field is an electron potential. The module does not select the kinetic-to-field coupling ratio or identify alpha with the physical coupling. The module contains no hydrogen answer, empirical normalization, orbital exponent, or cutoff.

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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