On the original Ulam’s problem and its quantization

The paper proves, under the pp,qq-resonance π/(2)·T/(AB)=p/q (Definition 3.1), that the energy at N periods is exactly a quadratic polynomial E(N)=aN^2+bN+c with a≥0 (Theorem 2), using: (i) a “stop-the-wall” change of variables and time reparametrization that convert the moving boundary to a Schrödinger equation with two δ-kicks per period (eqs. (24)–(27)), and (ii) a finite-dimensional, unitary Floquet reduction RS at resonance (Proposition 3.2), whose eigenphases ξj(x) yield the N^2 term after differentiating with respect to x (formulas (40)–(42)) . By contrast, the candidate solution asserts a resonance-free unipotent/Heisenberg argument that hinges on an incorrect factor in the kick strengths and an incorrect scaling law for a linear observable. In the correct kicked formulation, the quadratic kicks have exponents ±iJx^2 with J1=B(A−B)/(2T), J2=A(A−B)/(2T) (hence momentum shifts ±kℓ rather than ±2kℓ), so the one-period Heisenberg matrix is not the unipotent form used in the model; moreover, the paper explicitly requires resonance to reduce the Floquet dynamics to q×q matrices and to define the energy via a q-fold average (eq. (37))—a structural step the model bypasses . The paper’s proof is internally consistent and matches standard Floquet/kicked-system techniques; the model’s derivation contains a factor-of-two error in the kicks, a sign error in the key linear observable, and an unjustified claim that resonance is unnecessary.