Two Lenses, One Invariant: Empirical Confirmation That ρ Is Basis-Independent

The Latent Number \(\rho\) — the analyticity parameter that governs computational complexity in smooth systems — can be measured through at least two independent methods: spectral decay (fitting the exponential decline of expansion coefficients in an orthonormal basis) and grade norm ratios (measuring the Taylor/grade structure of the system's defining equations). We present six experiments confirming that these two "lenses" detect the same underlying invariant and that \(\rho\) serves as a universal computational complexity diagnostic across dynamical regimes.

For five analytic functions with known singularity structure, the Chebyshev and Taylor lenses both recover the exact singularity location (§2). For the SIR epidemiological ODE, Chebyshev and Legendre expansions yield Pearson \(r = 0.9999\) across eight parameter regimes, confirming basis independence (§3). Sweeping \(R_0\) from 0.3 to 15 reveals that the SIR epidemic threshold is a \(\rho\)-phase transition (§4). Applying trajectory-based grade and spectral lenses to the Van der Pol oscillator — the first non-epidemiological, polynomial vector field test — yields perfect Spearman rank correlation (\(\rho_s = 1.0\)), with both diagnostics decreasing as relaxation oscillations sharpen (§5). The Lorenz system demonstrates that the onset of deterministic chaos at \(r \approx 24.74\) corresponds to \(\rho\) dropping from 1.16 (stable) to 1.01 (fully chaotic), with the required spectral terms \(N^\) increasing from 63 to over 1000 (§6). Finally, convergence curves for the SIR model verify the predicted truncation error rate \(\|\cdot\|_\infty \sim \rho^{-N}\) and the formula \(N^ = \lceil \log(1/\varepsilon)/\log\rho \rceil\) on real ODE solutions (§7).

The invariant is the singularity structure; the lens determines the coordinate system, not the value.

Keywords: Latent Number, analyticity parameter, basis independence, grade method, spectral decay, SIR model, Van der Pol, Lorenz, phase transition, convergence rate

MSC 2020: 65M70, 41A25, 37M05, 92D30, 37D45