Foundation Pair Kernel Coherence Scaled Event Operator Evolution Hamiltonian Eq

The common scale

In quantum mechanics, the Hamiltonian is the operator that generates the time evolution of a system, and its structure determines the energy spectrum. A common simplification is to write the Hamiltonian as a kinetic part plus a potential part, each with its own coefficient. The declaration evolution_hamiltonian_eq_common_scale establishes that, for a specific constructed operator called CoherenceScaledEventOperator3, both of those coefficients are equal to a single shared energy scale, denoted eventGeneratorEnergyScale. In symbols, the theorem states that the Hamiltonian matrix element at row i and column j equals this common scale multiplied by the corresponding element of a dimensionless evolution operator. This means the ratio of the kinetic to the potential coupling is fixed to one, a deliberate and restrictive choice.

The operator itself is built from a discrete record of events, called a ledger, on a three-dimensional torus. It uses a signed graph Laplacian for the kinetic part and a mean-zero Green response for the source part, both derived from a single unit posting. The carrier has no free real coefficients; the only scale is the common one. The declaration is a theorem in the machine-checked library of formal theorems, meaning it is proved from the definitions without additional axioms. It also comes with companion theorems: the operator is Hermitian, it satisfies Stone's theorem for unitary evolution, and it witnesses a recognition-based one-body quantum attachment.

What this declaration does not claim is as important as what it proves. The operator is a new physical structure, a model, and the framework's own premises do not prove that this model is the molecular Hamiltonian of any real system. The definition fixes the dimensionless coupling ratio to one, but this is a modeling choice, not a derived fact. The module does not prove the physical correctness of the emitted spectrum, nor does it establish a remaining physical equality or source-action duality. It also does not use any hydrogen or CODATA input, so the common scale is not fitted to experiment. The theorem is a statement about the internal consistency of a proposed operator, not a statement about the world.

The practical consequence is that a deterministic executable can now be built around this exact object. The next step is to put it in front of atomic reality and compare its predictions with measurements. The declaration clears the mathematical ground, but the empirical test remains open.

/-- The native matrix is one common energy scale times the dimensionless
event operator, entry by entry. -/
theorem evolution_hamiltonian_eq_common_scale
    (op : CoherenceScaledEventOperator3)
    (i j : Fin (TorusCard3 3)) :
    op.evolution.hamiltonian i j =
      eventGeneratorEnergyScale *
        op.dimensionlessEvolution.hamiltonian i j := by
  unfold CoherenceScaledEventOperator3.evolution
    CoherenceScaledEventOperator3.dimensionlessEvolution
    oneBodyEvolution3
  by_cases hij : i = j
  · simp [hij]
    ring
  · simp [hij]

theorem coherenceScaledEventOperator_isHermitian
    (op : CoherenceScaledEventOperator3) :
    (Hc op.evolution).IsHermitian :=
  oneBodyHamiltonian3_isHermitian _ _ _

theorem coherenceScaledEventOperator_stone
    (op : CoherenceScaledEventOperator3) :
    StoneGeneratorCert (TorusCard3 3) op.evolution :=
  oneBodyHamiltonian3_stone _ _ _

/-- The carrier is a concrete instance of the old attachment predicate. This
states compatibility only; that predicate does not force physical identity. -/
theorem coherenceScaledEventOperator_witnesses_model_attachment
    (op : CoherenceScaledEventOperator3) :
    RecognitionOneBodyQuantumAttachment3 op.evolution := by
  refine
    ⟨op.event, op.event_realized,
      eventGeneratorEnergyScale, eventGeneratorEnergyScale,
      1, 0, eventGeneratorEnergyScale_pos,
      eventGeneratorEnergyScale_pos, by norm_num, ?_⟩
  rw [CoherenceScaledEventOperator3.evolution,
    unitPostingPotential3_eq_q1_model]

/-- One realized primitive posting, with no Hamiltonian coefficient fields. -/
structure CoherenceScaledEventOperator3 where
  event : PostingPair3 3
  event_realized : event ∈ realizedPrimitivePostingPairs3 3

/-- This MODEL emits dimensionless source/kinetic ratio one. The theorem is
an audit of the constructor, not a physical-selection theorem. -/
theorem dimensionless_couplingRatio_eq_one :
    oneBodyCouplingRatio
        eventGeneratorEnergyScale eventGeneratorEnergyScale =
      1 := by
  unfold oneBodyCouplingRatio
  exact div_self (ne_of_gt eventGeneratorEnergyScale_pos)

What this page does not claim

This declaration does not prove that the operator is the actual Hamiltonian of any physical molecule. This declaration does not derive the value of the common scale from first principles; it is defined as hbar over tau0. This declaration does not establish that the emitted spectrum matches any experimental data.

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/PairKernelCoherenceScaledEventOperator.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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