A New Constant from Ramanujan's π and Primorial Compression: Numerical Experiments and Anomalies

We present a numerical experiment in experimental mathematics. Starting from Ramanujan's well-known approximation for π, P = (355/113)·(1 − 0.0003/3553) ≈ 3.1415926536, we construct a dynamical system by successively compressing this constant. Factorial compression yields the classical product P·e, where e is Euler's number. Replacing factorials with primorials (products of consecutive primes) introduces a pulsating, stepwise dynamics and leads to a new constant R = Σ P/(p_n#) ≈ 5.3571385829..., which we find absent from the OEIS database. Unexpectedly, this constant shows remarkably accurate connections with the golden ratio φ, Euler's number e, the Euler-Mascheroni constant γ, and √π. A continued fraction expansion reveals an anomalous spike at 1193, whose properties point to a hidden geometric symmetry. Removing the first three primes (2, 3, 5) destroys this structure, confirming their role as a structural skeleton. This work presents a new constant, previously unknown, and a set of numerical anomalies that invite further analytical investigation.