Transient Phase Atlas: Short-Orbit Modular Fingerprints of Constants

We introduce a reproducible computational framework for studying the transient modular behavior of numerical constants under two complementary dynamical projections: (i) raw-power orbits $x_n = {\alpha^n}$ and (ii) log-relative orbits $x_n = {n\rho}$ where $\rho = \log\alpha / \log\beta$. The raw-power view provides an empirical transient fingerprint sensitive to algebraic structure; the log-relative view converts multiplicative comparisons into additive irrational rotations amenable to Diophantine analysis via continued fractions. We combine visualization with quantitative diagnostics from discrepancy theory (star discrepancy, histogram deviation, entropy, gap statistics) and with modulus-robustness tests comparing a composite modulus (990) against nearby primes (997, 1009). For $N = 30$ we report representative diagnostics and case studies, including (a) a continued-fraction explanation for why $e$ relative to $\pi$ exhibits an 8-sector skeleton linked to the convergent $7/8$, and (b) algebraic mechanisms producing raw-power structure for $\varphi$ (Pisot collapse) and $\sqrt{2}$ (exact parity locking). As an arithmetic control, we verify that prime residues on the same composite wheel flatten toward uniformity, weakening the objection that the modular lens manufactures structure by itself. We emphasize conservative interpretation: observed transient geometry is documented as computational output; structural explanations are stated only where supported by classical theory.