Gravity Macroscopic Ledger Cyclic Shift Linear Map Smul

The shift's linearity

In quantum mechanics, states combine by addition and scale by complex numbers. A physical operation is called linear when it respects both moves: the result of acting on a sum is the sum of the results, and the result of acting on a scaled state is the scaled result. The theorem cyclicShiftLinear_map_smul proves the second half of that property for one specific operation, the recognition update, which is a fixed rule that advances a system's discrete record of events by one step.

The operation in question is the cyclic shift on an eight-tick signal carrier, a vector space with eight basis states. The theorem states that for any complex number c and any state ψ, shifting c·ψ gives the same result as c times the shifted ψ. In symbols: cyclicShiftLinear (c • ψ) = c • cyclicShiftLinear ψ. This scalar homogeneity, together with its additive counterpart, makes the shift a proper linear map rather than a merely formal rule.

Linearity matters because it is the bridge from one site to many. The framework builds a macroscopic ledger, a tensor product of many single-site carriers, and defines the macroscopic update as the shift applied factor by factor. The scalar property proved here is one of the five clauses in the certificate that the macroscopic update is a genuine linear extension. Without it, the whole construction would not qualify as a linear map, and the superposition principle for multi-site configurations would fail.

In Recognition Science, this theorem upgrades a paper claim to a structural theorem in the machine-checked library of formal theorems. It does not, however, identify the macroscopic update with any gravitational response operator; that physical identification remains an open target. The theorem is about the algebraic structure of the shift, not about what the shift means physically.

theorem cyclicShiftLinear_map_smul (c : ℂ) (ψ : Signal8) :
    cyclicShiftLinear (c • ψ) = c • cyclicShiftLinear ψ :=
  cyclicShiftLinear.map_smul c ψ

/-- The macroscopic ledger theorem is verified. -/
noncomputable def macroscopicLedgerTheorem
    (ι : Type) [Fintype ι] [DecidableEq ι] :
    MacroscopicLedgerTheorem ι where
  single_site_linear ψ φ a b := by
    rw [cyclicShiftLinear.map_add, cyclicShiftLinear.map_smul,
        cyclicShiftLinear.map_smul]
  tensor_action := MacroscopicShift_tprod
  additive := MacroscopicShift_map_add
  scalar_homogeneous := MacroscopicShift_map_smul
  finite_superposition := MacroscopicShift_finite_sum

What this page does not claim

The theorem does not identify the macroscopic recognition update with any gravitational response operator. The theorem does not establish that the cyclic shift is the only possible recognition update. The theorem does not prove that the macroscopic ledger construction is physically realized.

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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