Foundation Coupled Recognition Cores Tensor Weyl Monomial Basis Image Orthogonal

The tensor Weyl monomial

A recognition core, in the Recognition Science framework, is a discrete record of events where each event is one of four states. The framework models a single core as a ququart, a quantum system with four basis states, and builds larger systems by coupling many cores together. The declaration tensorWeylMonomial defines a specific kind of operator on these coupled systems: it shifts the state at each position by an amount determined by one index, and multiplies by a phase factor determined by another index. This is the tensor product of the local Weyl operators, the standard shift-and-phase building blocks of quantum information.

The key theorem, tensorWeylMonomial_basis_image_orthogonal, proves that these operators form an orthogonal basis for the space of all linear maps on the coupled system. Two operators with different shift indices are orthogonal, meaning their inner product is zero, and two with the same shift but different phase indices are also orthogonal. The proof builds on a chain of lemmas: the local operators are shown to be orthogonal, the phase characters are shown to be orthogonal for distinct indices, and the tensor product of orthogonal families is orthogonal. The result is that the 16 operators on a single core, and the 4^N operators on N coupled cores, are linearly independent and span the full operator space.

This is a structural result, not a physical prediction. It does not say that any particular physical system realizes these operators, nor that the recognition cores themselves exist in nature. It establishes a mathematical fact about the framework's own construction: the operators defined by the framework form a complete, orthogonal set. This is a necessary step for any later claim that these operators can be used to decompose or measure states, but the framework does not yet claim that such measurements are physically realized.

The theorem also does not claim that the operators are unitary, nor that they form a group under composition. The proof establishes orthogonality with respect to a specific inner product, the Hilbert-Schmidt-like inner product defined in the framework. It does not establish any property about the eigenvalues or eigenvectors of these operators. The result is purely about the linear-algebraic structure of the operator family.

/-- The tensor-Weyl monomial on the concrete coupled-core carrier. -/
def tensorWeylMonomial {N : ℕ} (a b : CoupledCoreIndex N) :
    CoupledCoreSpace N →ₗ[ℂ] CoupledCoreSpace N where
  toFun := fun ψ s => phaseCharacter b s * ψ (shiftedConfig a s)
  map_add' := by
    intro ψ χ
    ext s
    simp [phaseCharacter]
    ring
  map_smul' := by
    intro z ψ
    ext s
    simp [phaseCharacter]
    ring

/-- Distinct shift labels force Hilbert-Schmidt orthogonality. -/
theorem localWeylMonomial_shift_orthogonal {a a' : Fin 4} (h : a ≠ a') (b b' : Fin 4) :
    localOperatorInner (localWeylMonomial a b) (localWeylMonomial a' b') = 0 := by
  unfold localOperatorInner
  apply Finset.sum_eq_zero
  intro s hs
  rw [localWeylMonomial_basisKet, localWeylMonomial_basisKet]
  have hidx : add4 a s ≠ add4 a' s := by
    intro hEq
    exact h ((add4_eq_add4_iff_left a a' s).mp hEq)
  have hinner :=
    scaled_basisKet_inner
      (Complex.I ^ (b.val * (add4 a s).val))
      (Complex.I ^ (b'.val * (add4 a' s).val))
      (add4 a s) (add4 a' s)
  simpa [localWeylMonomial_basisKet, hidx] using hinner

set_option maxHeartbeats 800000

/-- Distinct shift labels force Hilbert-Schmidt orthogonality. -/
theorem localWeylMonomial_shift_orthogonal {a a' : Fin 4} (h : a ≠ a') (b b' : Fin 4) :
    localOperatorInner (localWeylMonomial a b) (localWeylMonomial a' b') = 0 := by
  unfold localOperatorInner
  apply Finset.sum_eq_zero
  intro s hs
  rw [localWeylMonomial_basisKet, localWeylMonomial_basisKet]
  have hidx : add4 a s ≠ add4 a' s := by
    intro hEq
    exact h ((add4_eq_add4_iff_left a a' s).mp hEq)
  have hinner :=
    scaled_basisKet_inner
      (Complex.I ^ (b.val * (add4 a s).val))
      (Complex.I ^ (b'.val * (add4 a' s).val))
      (add4 a s) (add4 a' s)
  simpa [localWeylMonomial_basisKet, hidx] using hinner

set_option maxHeartbeats 800000

/-- Equal shifts but distinct phase labels are orthogonal in the one-core Weyl family. -/
theorem localWeylMonomial_phase_orthogonal (a : Fin 4) {b b' : Fin 4} (h : b ≠ b') :
    localOperatorInner (localWeylMonomial a b) (localWeylMonomial a b') = 0 := by
  unfold localOperatorInner
  have hs : ∀ s : Fin 4,
      ∑ t : Fin 4,
        star ((localWeylMonomial a b (basisKet s)) t) *
          (localWeylMonomial a b' (basisKet s)) t =
        (-Complex.I) ^ (b.val * (add4 a s).val) *
          Complex.I ^ (b'.val * (add4 a s).val) := by
    intro s
    have hinner :=
      scaled_basisKet_inner
        (Complex.I ^ (b.val * (add4 a s).val))
        (Complex.I ^ (b'.val * (add4 a s).val))
        (add4 a s) (add4 a s)
    simpa [localWeylMonomial_basisKet] using hinner
  rw [Fin.sum_univ_four]
  rw [hs 0, hs 1, hs 2, hs 3]
  fin_cases a <;> fin_cases b <;> fin_cases b' <;>
    simp at h <;>
    simp [add4, neg_I_pow, Complex.I_pow_eq_pow_mod] <;>
    ring_nf <;> simp [I_pow_five]
set_option maxHeartbeats 200000

What this page does not claim

The theorem does not claim that the operators are unitary or form a group. The theorem does not claim that any physical system realizes these operators. The theorem does not establish any property about eigenvalues or eigenvectors of the operators.

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)

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