Foundation Coupled Recognition Cores Finite Dimensional Exact Embedding

The exact embedding

In quantum mechanics, a physical system is often described by a state vector, a list of complex numbers giving the probability amplitudes for each possible outcome. A system with d possible outcomes lives in a d-dimensional complex vector space. The Recognition Science framework's machine-checked library of formal theorems proves that any such finite-dimensional state space can be placed exactly, without loss or approximation, into a larger space built from a collection of four-state subsystems.

The construction starts with a single four-state system, called a ququart carrier, whose states are indexed by the numbers 0, 1, 2, 3. Combining N such carriers gives a space of dimension 4^N, since each carrier contributes four possibilities. The theorem, named finite_dimensional_exact_embedding, states that for any dimension d, provided d is no larger than 4^N, the d-dimensional space can be embedded as a subspace of the N-carrier space. The embedding is exact: it is a linear map that preserves the inner product structure, so probabilities and overlaps between states are unchanged.

The library proves this by constructing an explicit map, embedState, and showing it has the required properties. It also establishes the supporting arithmetic: the number of distinct configurations of N four-state carriers is exactly 4^N, and the basic operations on the four-state carriers, such as shifting and adding indices, behave correctly. These results are verified in a proof assistant, meaning the proofs are checked by a computer and contain no gaps.

In Recognition Science, this result acts as a structural bridge. The framework models recognition events as discrete records, and a single recognition event is represented by a four-state carrier. The theorem shows that any finite quantum-like system, regardless of its dimension, can be built from a finite number of these fundamental four-state building blocks. This is a statement about mathematical structure, not about which physical systems exist in nature.

What the theorem does not claim is equally important. It does not say that every quantum system is physically composed of four-state subsystems. It does not assert that the embedding is unique, nor that it respects any particular dynamics or Hamiltonian. It is a purely mathematical statement about vector spaces and their subspaces. It also does not claim that the number of carriers N is determined by the dimension d; any N with 4^N at least d will do, and the theorem holds for all such N.

/-- Zero-pad a finite-dimensional state into a coupled-core carrier. -/
def embedState {d N : ℕ} (_h : d ≤ Fintype.card (CoupledCoreIndex N)) :
    (Fin d → ℂ) →ₗ[ℂ] CoupledCoreSpace N where
  toFun := fun v s =>
    let i := coupledCoreEquivFin N s
    if hi : i.val < d then v ⟨i.val, hi⟩ else 0
  map_add' := by
    intro v w
    ext s
    dsimp [coupledCoreEquivFin]
    split_ifs <;> simp
  map_smul' := by
    intro z v
    ext s
    dsimp [coupledCoreEquivFin]
    split_ifs <;> simp

/-- The dimension of the coupled-core index space is exactly `4^N`. -/
theorem coupledCoreIndex_card (N : ℕ) :
    Fintype.card (CoupledCoreIndex N) = 4 ^ N := by
  simp [CoupledCoreIndex]

/-- Zero-pad a finite-dimensional state into a coupled-core carrier. -/
def embedState {d N : ℕ} (_h : d ≤ Fintype.card (CoupledCoreIndex N)) :
    (Fin d → ℂ) →ₗ[ℂ] CoupledCoreSpace N where
  toFun := fun v s =>
    let i := coupledCoreEquivFin N s
    if hi : i.val < d then v ⟨i.val, hi⟩ else 0
  map_add' := by
    intro v w
    ext s
    dsimp [coupledCoreEquivFin]
    split_ifs <;> simp
  map_smul' := by
    intro z v
    ext s
    dsimp [coupledCoreEquivFin]
    split_ifs <;> simp

What this page does not claim

The theorem does not claim that every quantum system is physically composed of four-state subsystems. The theorem does not specify a unique embedding or determine the number of carriers N from the dimension d. The theorem does not address dynamics, Hamiltonians, or any time evolution of the embedded states.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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