Density and Goldstone emission of an RP³ ≃ SO(3) global-string network relative to U(1), and a massive-bath systematic of the emission estimator

We simulate networks of global Z₂ strings whose vacuum manifold is RP³ ≃ SO(3), produced by a real symmetric-tensor order parameter, and compare them with U(1) axion-string networks evolved in the same code, with the same Goldstone decay constant and core mass, at 576³ (U(1) control also at 768³, where it passes preregistered scaling gates). The string-density ratio is R_ξ ≃ 1.6 for every protocol tested: sparse and dense starts, with and without draining the initial massive radiation, and networks started late and dense that approach the attractor from above, which bracket it to R_ξ ∈ [1.57, 1.68] at ℓ = ln(m_r/H) = 5.61. We identify a systematic of the standard kinetic Goldstone-emission estimator: the conversion of massive radiation left by the initial state into Goldstones is counted as string emission. It is 19% (RP³) and 28% (U(1)) of the measured emission for sparse starts and more than half for dense ones; for U(1) two independent methods, a drain of the massive modes and a direct measurement of the radial-to-Goldstone transfer, agree. With the bath drained, the emission ratio is R_Γ = 1.56 ± 0.09 (stat) ± 0.08 (sampling), independent of the initial density within ≈7%, and the emission per unit string length is within 13% of U(1) from both sides of the attractor. The spectral slope of the RP³ emission at ℓ = 5 is q = 0.87 ± 0.06 with our estimator, but an independent re-implementation shows that the convention used to map the physical fitting window onto lattice momenta shifts it by up to 0.07; we therefore quote q as compatible with 1. All verdict-bearing analyses were preregistered and are reproducible from the released data. Code, network summaries, preregistered specifications, the independent audit and the figure scripts: https://github.com/indigenica/whalefall-rp3-string-networks (code GPL-3.0-or-later; data CC BY-NC 4.0). A script in that repository recomputes every number quoted in the paper from the stored results and checks it against the text.