Gravity Ledger Superposition Recognition Update Norm Preserved

The preserved ledger

In quantum mechanics, a state is a vector in a complex vector space, and an inner product is a function that takes two such vectors and returns a complex number, generalizing the familiar dot product. It measures the length of a vector and the angle between two vectors. A recognition update, a step in the framework's ledger that changes one state into another, that preserves this inner product is called unitary. The declaration recognition_update_norm_preserved establishes that the one-tick recognition update, called cyclic_shift, preserves the norm of any state: the length of the vector, as measured by the inner product, is unchanged after the update.

This is a theorem in the framework's machine-checked library of formal theorems, not a definitional choice. The proof follows from a stronger result, recognition_update_inner_preserved, which states that the inner product of any two states is preserved, not just the norm of a single state. The theorem holds for all states in the eight-dimensional complex carrier Signal8, the state space of the framework's eight-tick recognition cycle. The proof is a direct computation: it unfolds the definitions of the inner product and the cyclic shift, sums over the eight components, and simplifies with the ring tactic.

The physical interpretation is that coherent superpositions of definite ledger configurations are physical states and are preserved by the recognition update. A superposition is a sum of two states with complex coefficients, and the update acts linearly on such sums, as shown by the theorem ledger_superposition_preserved. This means that the recognition update does not collapse or distort the quantum information encoded in the state; it rotates it in a way that preserves its geometry.

In Recognition Science, the framework models a discrete record of events, the ledger, and the recognition update as a unitary evolution. The theorem does not claim that the update is the only possible unitary evolution, nor that it represents a physical time evolution in the sense of the Schrödinger equation. It establishes a mathematical property of a specific update in a specific state space, and the physical interpretation is a modeling choice, not a proven fact about the physical world.

/-- **T1.iii (Norm preservation).** As a corollary of inner-product
preservation, the recognition update preserves the `inner8`-norm
squared. -/
theorem recognition_update_norm_preserved (f : Signal8) :
    inner8 (cyclic_shift f) (cyclic_shift f) = inner8 f f :=
  recognition_update_inner_preserved f f

/-- **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

/-- **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

What this page does not claim

The theorem does not claim that the recognition update is the unique unitary evolution on the state space. The theorem does not claim that the recognition update represents a physical time evolution in the sense of the Schrödinger equation. The physical interpretation that superpositions are physical states is a modeling choice, not a proven fact about the physical world.

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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