Gravity Macroscopic Ledger Macroscopic Shift Finite Sum

Superposition across sites

In quantum mechanics, a system can exist in a combination of states at once, and the mathematics of that combination is linear: adding two states and then evolving them gives the same result as evolving each and adding the outcomes. The declaration MacroscopicShift_finite_sum proves that this same linear behavior holds when the framework's single-site recognition update, a discrete record of events called a ledger, is extended to a finite collection of sites. Concretely, for any finite set of locations, any finite list of complex amplitudes, and any finite family of multi-site configurations, the update commutes with the sum: applying the update to the combined configuration equals the sum of the updates applied to each part.

The proof works by building a larger carrier space from the single-site one. Each site carries an eight-tick signal, and the multi-site carrier is the tensor product of those single-site spaces. The update acts on each factor independently, and the theorem PiTensorProduct.map guarantees the result is linear. The finite-sum theorem then follows by induction over the finite set of terms, using additivity and scalar homogeneity of the update at each step. This is a structural result: it establishes that the macroscopic update is a well-behaved linear operator on the tensor-product space, preserving superpositions exactly as the single-site update does.

In Recognition Science, this theorem upgrades the macroscopic ledger Hilbert carrier from a conditional statement to a structural theorem in the framework's machine-checked library of formal theorems. The library verifies the claim with no unproved assumptions beyond the standard axioms of the underlying type theory. The practical consequence is that multi-site ledger configurations, superpositions of states across many locations, are mathematically coherent objects that evolve consistently under the recognition update.

The theorem does not claim that this macroscopic update is physically identical to the gravitational response operator. That identification remains an open target, tracked separately in the framework's master plan. The finite-sum theorem concerns the amplitude-linear level only: it says the mathematics of superposition works, not that the resulting operator is the one that governs gravity. The physical bridge from this algebraic structure to a gravitational channel is not established by this declaration.

/-- **Macroscopic ledger superposition.** For a finite family of
macroscopic ledger configurations `Ψ : κ → MacroscopicLedger ι` and
amplitudes `c : κ → ℂ` indexed by a finite set, the macroscopic
recognition update commutes with the finite linear combination. This is
the explicit superposition principle for multi-site ledger
configurations. -/
theorem MacroscopicShift_finite_sum
    {κ : Type*} (s : Finset κ) (c : κ → ℂ) (Ψ : κ → MacroscopicLedger ι) :
    MacroscopicShift (∑ α ∈ s, c α • Ψ α) =
      ∑ α ∈ s, c α • MacroscopicShift (Ψ α) := by
  classical
  induction s using Finset.induction_on with
  | empty => simp
  | @insert α s hα ih =>
    rw [Finset.sum_insert hα, Finset.sum_insert hα,
        MacroscopicShift_map_add, MacroscopicShift_map_smul, ih]

/-- Action of the macroscopic recognition update on a pure tensor
configuration: `R̂_macro (⨂ᵢ ψᵢ) = ⨂ᵢ R̂ ψᵢ`. -/
theorem MacroscopicShift_tprod (ψ : ι → Signal8) :
    MacroscopicShift (PiTensorProduct.tprod ℂ ψ) =
      PiTensorProduct.tprod ℂ (fun i => cyclic_shift (ψ i)) := by
  unfold MacroscopicShift
  rw [PiTensorProduct.map_tprod]
  rfl

/-- **Linearity.** The macroscopic recognition update is `ℂ`-linear
(by construction, since `PiTensorProduct.map` returns a `LinearMap`).
This restates the universal property explicitly. -/
theorem MacroscopicShift_map_add (Ψ Φ : MacroscopicLedger ι) :
    MacroscopicShift (Ψ + Φ) = MacroscopicShift Ψ + MacroscopicShift Φ :=
  MacroscopicShift.map_add Ψ Φ

theorem MacroscopicShift_map_smul (c : ℂ) (Ψ : MacroscopicLedger ι) :
    MacroscopicShift (c • Ψ) = c • MacroscopicShift Ψ :=
  MacroscopicShift.map_smul c Ψ

theorem macroscopicLedgerTheorem_inhabited
    (ι : Type) [Fintype ι] [DecidableEq ι] :
    Nonempty (MacroscopicLedgerTheorem ι) :=
  ⟨macroscopicLedgerTheorem ι⟩

What this page does not claim

The macroscopic update is not claimed to be physically identical to the gravitational response operator. The theorem does not establish any empirical content about gravity; it is a purely algebraic statement about linear operators. The finite-sum result does not address infinite superpositions or continuous families of configurations.

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

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Derived articles

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