Quadratic Condensation and Relational Local Time in a Many-Wave Complete Relational System — From Relational Closure without Spacetime as Fundamental Variables to the Kepler Equation and the Inverse-Square Central Force

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We report an unexpected conservation structure found in a preregistered 99-run survey (4096 steps each) of a nonlinear many-wave complete relational system extended from our minimal closed system [K1]. Although the anticipated relaxation-type evolution (floor, onset, saturation) did not occur - 5 runs are exact fixed points and 94 depart immediately - the complex quadratic sum of all relational waves is exactly conserved in every run: sum z_m^2 = C^2. This follows rigorously because each simultaneous update is a real orthogonal transformation generated by a real antisymmetric kernel. We read this as quadratic condensation: many internal waves are exactly represented by one complex quadratic quantity, and the closure form is invariant under hierarchical coarse-graining, sum_n C_n^2 = sumn,m zₙₘ^2. The simultaneous conservation of H and C^2 fixes the Gram matrix of the real and imaginary parts, with eigenvalues (H+-|C^2|)/2 and shape index chi=|C^2|/H in [0,1]; we prove chi=1 iff K=0 under connectivity, matching exactly the five frozen runs, and give a closed form chi(L,m_a,m_b) that reproduces all 99 measured values to machine precision. An antisymmetric quadratic flux J_nm=-J_mn governs exchange between condensates, and a single-valued relational phase U_AB with U_BA=U_AB⁻¹ and composition U_AB U_BC = U_AC defines directed local clocks on a double cover, with the two lifts +-i carrying the two time directions. Finally, squaring the fixed Gram 2-frame maps it to an ellipse with one focus at the origin whose eccentricity equals chi exactly; a three-component zero-closure embedding realizes any nondegenerate eccentricity 0<=e<1; and under a single explicit clock-readout axiom dt=r dtau, the Kepler equation, the area law, the inverse-square central force, and the third law n^2A^3=mu are reconstructed without assuming physical time or spatial coordinates as fundamental variables. Japanese and English versions (Markdown, LaTeX, PDF) are included.