Exact Global Bound HL(5) = 2 for the Classical Liénard Equation of Degree 5 via Hilbert-Fourier Variety Analysis

We establish an exact and rigorous global proof that the maximum number of limit cyclesfor the classical planar Liénard differential system of degree 5, x_ = y 􀀀 F(x), y_ = 􀀀x, isstrictly two: HL(5) = 2. By embedding the dynamics into an infinite-dimensional algebraicvariety in the Hilbert space `2 under a natural parity-scaling hierarchy—assigning to evendissipative terms and 2 to odd dissipative terms—we completely decouple the amplitudecorrections and frequency shifts order by order. We prove that all odd-order corrections tothe fundamental amplitude vanish identically (X1;2m+1 0) due to exact temporal parity.At order 2, the fundamental energy balance collapses to a quadratic polynomial in thebasal energy u = X21;0, which admits at most two positive real roots. Furthermore, wedemonstrate that the vanishing of the Jacobian polynomial Q(X1;0) = 0 corresponds strictlyto a saddle-node (fold) bifurcation of limit cycles, where two hyperbolic orbits coalesce andannihilate without increasing the cycle capacity. Finally, we show that the complementarybranch X1;0 = 0 is blocked by a hyperbolic focus barrier unless the linear dissipation vanishes(a1 = 0), in which case it reduces to a weak focus governed by the Blows–Lloyd theorem(b(5 􀀀 1)=2c = 2). These results definitively confirm the validity of the Lins–de Melo–Pughconjecture for degree 5.