Gravity Ledger Superposition Ledger Superposition Theorem Inhabited

Superposition in the ledger

In quantum mechanics, a particle can be in a superposition, a weighted combination of two or more definite states at the same time. The classic picture is Schrödinger's cat, neither simply alive nor simply dead until measured. Recognition Science, a framework that derives physics from a discrete record of recognition events, asks whether its own ledger, the record of events, can hold such superpositions. The declaration ledgerSuperpositionTheorem_inhabited proves that it can: the space of recognition states is a complex vector space, and the one-tick update that advances the ledger is a linear map that preserves the inner product. In plain terms, any weighted sum of ledger configurations is itself a valid physical state, and the update acts on the sum by acting on each part separately.

The proof is built from three properties. First, the update respects addition and scalar multiplication, so shifting a sum of states equals the sum of the shifted states. Second, this holds for finite sums of any length. Third, the update preserves the inner product, the measure of overlap between two states, and therefore preserves the norm, the length of a state. Together these mean the update is a unitary operator, the quantum-mechanical way of saying a transformation does not change probabilities. The theorem is unconditional in the framework's machine-checked library, a collection of formal theorems verified by computer, and it reuses earlier results on the Schrödinger equation and complex structure without adding new axioms.

What the theorem does not claim is just as important. It does not say that any particular superposition is realized in nature, only that the mathematics allows it. It does not claim to have observed a superposition in a laboratory. The physical interpretation, that these superpositions correspond to actual gravitational or quantum states, is a modeling choice, not a proved fact. The theorem establishes a necessary condition for a quantum theory of gravity within the framework, not a sufficient one.

theorem ledgerSuperpositionTheorem_inhabited :
    Nonempty LedgerSuperpositionTheorem :=
  ⟨ledgerSuperpositionTheorem⟩

/-- **T1.i (Linearity).** The recognition update is `ℂ`-linear: it maps
linear combinations of ledger configurations to the corresponding linear
combinations of evolved configurations. This is
`SchrodingerDerivation.schrodinger_linear` repackaged in the ledger
superposition language. -/
theorem ledger_superposition_preserved
    (ψ φ : Signal8) (a b : ℂ) :
    cyclic_shift (a • ψ + b • φ) =
      a • cyclic_shift ψ + b • cyclic_shift φ :=
  IndisputableMonolith.Foundation.SchrodingerDerivation.schrodinger_linear ψ φ a b

/-- **T1.i (Finite-sum form).** For a finite family of definite ledger
configurations `L : ι → Signal8` and amplitudes `c : ι → ℂ` indexed by a
finite set, the recognition update commutes with the finite linear
combination. This is the explicit superposition principle for ledger
configurations: `R̂ Σ c_α |L_α⟩ = Σ c_α R̂ |L_α⟩`. -/
theorem ledger_superposition_finite_sum
    {ι : Type*} (s : Finset ι) (c : ι → ℂ) (L : ι → Signal8) :
    cyclic_shift (∑ α ∈ s, c α • L α) =
      ∑ α ∈ s, c α • cyclic_shift (L α) := by
  classical
  induction s using Finset.induction_on with
  | empty =>
      ext t
      simp [cyclic_shift]
  | @insert α s hα ih =>
      have hsum :
          (∑ β ∈ insert α s, c β • L β)
            = c α • L α + ∑ β ∈ s, c β • L β := by
        simp [Finset.sum_insert hα]
      rw [hsum,
          IndisputableMonolith.Foundation.SchrodingerDerivation.cyclic_shift_add,
          IndisputableMonolith.Foundation.SchrodingerDerivation.cyclic_shift_smul,
          ih]
      simp [Finset.sum_insert hα]

/-- **T1.iii (Inner-product preservation).** The recognition update
preserves the canonical `inner8` Hermitian inner product. The proof is
direct from the definitions of `inner8` and `cyclic_shift`: the cyclic
shift permutes the eight summands without changing the value of the
sum. -/
theorem recognition_update_inner_preserved (f g : Signal8) :
    inner8 (cyclic_shift f) (cyclic_shift g) = inner8 f g := by
  show IndisputableMonolith.Foundation.ComplexStructureForcing.inner8
      (cyclic_shift f) (cyclic_shift g)
      = IndisputableMonolith.Foundation.ComplexStructureForcing.inner8 f g
  unfold IndisputableMonolith.Foundation.ComplexStructureForcing.inner8 cyclic_shift
  rw [Fin.sum_univ_eight, Fin.sum_univ_eight]
  simp
  ring

What this page does not claim

The theorem does not claim that any particular superposition is physically realized. The theorem does not claim to have observed a superposition in a laboratory. The physical interpretation of the superpositions as gravitational states is a modeling choice, not a proved fact.

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

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