THE RECOGNITION HILBERT SPACE: CANONICAL QUANTIZATION OF THE FOUR-DIMENSIONAL RECOGNITION PHASE SPACE

The companion papers on the σ-Noether charge (element 72) and the (Z, Θ) Heisenberg principle (element 73) close the symplectic structure of Recognition Science (RS) ledger phase space: two canonical-conjugate pairs (J, σ) and (Z, Θ) with action quanta ℏ_R = E_coh · τ₀ and ℏ_C = ℏ_R/D²(D + 2) = ℏ_R/45 at D = 3. This paper performs the canonical quantization. The recognition Hilbert space H_RS factors as a tensor product H_(J,σ) ⊗ H_(Z,Θ) because the ratio ℏ_C/ℏ_R = 1/45 is rational with purely combinatorial denominator. An irrational ratio would produce a noncommutative product; the integer D²(D + 2) prevents that. The symplectic volume quantum is ℏ_R · ℏ_C = ℏ²/45. Both cosmic Noether charges (the σ-charge from element 72 and the Z-Θ charge from element 73) equal φ in RS-natural units. The small-deviation limit reproduces the quadratic emergence J(1 + ε) = ε²/2 + O(ε³) that gives rise to the Schrödinger Hamiltonian. The Lean module Foundation.RecognitionHilbertSpace formalizes the construction in 455 lines, 0 sorry, 0 axioms. With H_RS in hand, every undetermined quantity in RS (gauge boson masses, the α_s boundary condition, the BIT δw(z) kernel) becomes an eigenvalue or expectation value of one operator.