Foundation Coupled Recognition Cores

Coupled recognition cores

A recognition core is the smallest unit of recognition, the framework's term for a discrete record of an event. The core is modeled as a ququart, a quantum system with four distinguishable states. The standard basis states are written |0⟩, |1⟩, |2⟩, and |3⟩, and a general state is a complex-valued function on those four states.

The central object is a coupled collection of N such cores. A configuration of the whole system is an assignment of one of the four states to each core, and the space of all configurations has exactly 4^N elements. The theorem coupledCoreIndex_card establishes that the number of configurations of N cores is 4^N. This exponential growth is the first structural fact about how recognition scales with the number of cores.

On a single core, two operations move between basis states. The shift operation adds one to a state, wrapping around from 3 back to 0. The phase operation multiplies a state by a power of the imaginary unit i. These two operations satisfy the defining relation of a Weyl pair: applying phase after shift gives i times the result of applying shift after phase. This relation is proved as ququartWeyl_relation_apply.

The same two operations lift to the full coupled space, acting independently on each core. They form an orthogonal family: any two distinct operators of this kind have zero inner product, while each operator has inner product 4 with itself. This orthogonality is what makes the operators a usable basis for the space of linear maps on the coupled system.

In Recognition Science, the framework proves that an eight-tick recognition cycle forces 2^3 as a structural constant, and the ququart is the natural carrier for the local quarter-turn of that cycle. The coupled space is the setting in which multiple recognition cores act together, and the theorems guarantee that the local operations on each core remain well-defined and independent when combined. No particular physical system is claimed to realize these cores; the mathematical structure that a physical realization would have to respect is what is established.

/-- 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]

/-- Pointwise Weyl relation `ZX ψ = i XZ ψ`. -/
lemma ququartWeyl_relation_apply (ψ : QuquartState) :
    ququartZ (ququartX ψ) = (Complex.I : ℂ) • ququartX (ququartZ ψ) := by
  ext m
  fin_cases m
  · simp [ququartZ, ququartX, prev4]
    symm
    have hI : Complex.I * Complex.I = (-1 : ℂ) := by
      have h := Complex.I_sq; rw [sq] at h; exact h
    calc
      -(Complex.I * (Complex.I * ψ 3)) = -((Complex.I * Complex.I) * ψ 3) := by ring
      _ = -((-1) * ψ 3) := by rw [hI]
      _ = ψ 3 := by ring
  · simp [ququartZ, ququartX, prev4]
  · simp [ququartZ, ququartX, prev4]
    have hI : Complex.I * Complex.I = (-1 : ℂ) := by
      have h := Complex.I_sq; rw [sq] at h; exact h
    calc
      -ψ 1 = (-1 : ℂ) * ψ 1 := by ring
      _ = (Complex.I * Complex.I) * ψ 1 := by rw [hI]
      _ = Complex.I * (Complex.I * ψ 1) := by ring
  · simp [ququartZ, ququartX, prev4, Complex.I_sq]

/-- 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

/-- Each local Weyl monomial has Hilbert-Schmidt norm squared `4`. -/
theorem localWeylMonomial_self_inner (a b : Fin 4) :
    localOperatorInner (localWeylMonomial a b) (localWeylMonomial a b) = 4 := by
  unfold localOperatorInner
  have hs : ∀ s : Fin 4,
      ∑ t : Fin 4,
        star ((localWeylMonomial a b (basisKet s)) t) *
          (localWeylMonomial a b (basisKet s)) t = 1 := by
    intro s
    have hphase : (-Complex.I) ^ (b.val * (add4 a s).val) *
        Complex.I ^ (b.val * (add4 a s).val) = 1 := by
      simpa using I_pow_star_mul_self (b.val * (add4 a s).val)
    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)
    calc
      ∑ 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
                simpa [localWeylMonomial_basisKet] using hinner
      _ = 1 := hphase
  rw [Fin.sum_univ_four]
  rw [hs 0, hs 1, hs 2, hs 3]
  norm_num

What this page does not claim

No physical system is claimed to realize a recognition core. The module does not derive the eight-tick cycle; it assumes the ququart as a carrier for it. The orthogonality of the Weyl family does not imply completeness of that family as a basis.

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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