Gravity Macroscopic Ledger Macroscopic Shift Map Smul

The macroscopic shift

In quantum mechanics, a superposition is a sum of possible states, each with its own complex-number amplitude. A linear operator respects that sum: applying it to the whole is the same as applying it to each part and then adding. The theorem MacroscopicShift_map_smul establishes one half of that property for a large-scale update rule in the Recognition Science framework. It states that multiplying a multi-site configuration by a complex number c, and then applying the macroscopic shift, gives the same result as applying the shift first and then multiplying by c.

The framework's ledger, a discrete record of recognition events, assigns each site an eight-dimensional signal space. The single-site update, a cyclic shift through those eight ticks, was already known to be linear. The new theorem extends that linearity to a finite tensor product of many sites. If a configuration is a pure tensor of site states, the macroscopic shift acts on each factor independently. The scalar-multiplication theorem is the second of two linearity clauses, paired with an additivity theorem, and together they certify that the multi-site update is a genuine linear map.

This matters because it upgrades the macroscopic ledger from a definition to a structural theorem. The framework can now treat a finite collection of sites as a single Hilbert space in which superpositions are preserved by the recognition update. The theorem is machine-checked, with no unproved axioms beyond the standard logical ones. It is a formal result about the algebra of the update, not a claim about the physical force of gravity.

What the theorem does not claim is equally precise. It does not identify the macroscopic shift with the gravitational response operator; that identification remains an open target. It does not say that any particular multi-site configuration is physically realized, only that the mathematical structure supports such superpositions. The scalar-multiplication property is a necessary condition for linearity, but the theorem alone does not establish the full physical content of the macroscopic ledger.

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

/-- 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

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

What this page does not claim

The theorem does not identify the macroscopic shift with the gravitational response operator. The theorem does not assert that any particular multi-site configuration is physically realized. Scalar multiplication alone does not establish full linearity without the additivity clause.

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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