Gravity Ledger Superposition Recognition Update Inner Preserved

The preserved ledger

In quantum mechanics, the inner product of two state vectors is the overlap that determines probabilities: how much one state resembles another, and how likely a measurement is to find one when the system is in the other. The declaration recognition_update_inner_preserved proves that this overlap is untouched by the basic step of the framework's ledger update. The ledger, a discrete record of recognition events, advances by a cyclic shift, a rotation through its eight possible states. The theorem states that for any two states f and g in the eight-dimensional signal space, the inner product after the shift equals the inner product before it.

This is a unitarity result: the recognition update preserves the geometry of the state space, so probabilities computed before the update remain valid after it. The proof is direct and computational, expanding the inner product over the eight components and simplifying with ring arithmetic. The theorem also implies that the norm of any single state, its total probability, is conserved by the update. This is the formal backbone of the ledger superposition theorem, which states that coherent superpositions of definite ledger configurations are physical states and remain physical under the update.

In Recognition Science, the framework models the cost of recognition as forced, and this declaration is part of the chain that gives the framework its quantum-mechanical character. The theorem is proved in the machine-checked library of formal theorems, with no new axioms beyond the standard three. The mathematical statement is unconditional; the physical interpretation, that this is how gravity emerges from the ledger, is a model choice.

What the declaration does not claim is just as important. It does not say that the update preserves the ledger's classical content, only its quantum overlap. It does not derive the inner product itself, nor does it say anything about what happens to states outside the eight-dimensional signal space. The theorem is a precise, narrow fact about one specific update on one specific space, and its power comes from that precision.

/-- **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.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

/-- The canonical inhabitant of `LedgerSuperpositionTheorem`. -/
def ledgerSuperpositionTheorem : LedgerSuperpositionTheorem where
  linearity := ledger_superposition_preserved
  linearity_sum := fun s c L => ledger_superposition_finite_sum s c L
  inner_preserved := recognition_update_inner_preserved
  norm_preserved := recognition_update_norm_preserved

What this page does not claim

The declaration does not say that the update preserves the ledger's classical content, only its quantum overlap. The declaration does not derive the inner product itself. The declaration says nothing about states outside the eight-dimensional signal space.

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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