Reasoning. (1) Catalysis rate: f_p = Integral n_M v_rel sigma rho_p N_A d^3x with sigma = sigma_0/beta_rel^2; assuming monopoles settle at the solar center (rho_p = 50 g/cm^3, beta_rel = thermal = 1.7 x 10^-3) gives f_p = 1.7 x 10^6 (sigma_0/1mb) N_M decays/s. (2) Neutrino budget: each decay yields 3 neutrinos through pi+ → mu+ nu_mu, mu+ → e+ nu_e nu_mu-bar, with nu_e fraction f_nu_e = f_pi+ (1 - a_pi+), f_pi+ ~ 0.5 (GUT-model branching to pi+ + anything), a_pi+ = 0.2 pion absorption at the solar center; so f_p = 4 pi d^2 I0 / 3 / f_nu_e at Earth-Sun distance d = 1.5 x 10^13 cm. (3) Capture: N_M = pi R_sun^2 (1 + (beta_esc/beta_M)^2) epsilon(M_M, beta_M) 4 pi F_M t_sun, with beta_esc = 2 x 10^-3, t_sun = 4.6 x 10^9 yr, and capture efficiency epsilon from monopole energy-loss calculations — epsilon falls below 1 for M_M > 10^17 GeV, which bounds the limit’s mass validity. Model dependences carried explicitly: sigma_0 and f_pi+ appear as scaling factors in the quoted bound; assumed central accumulation; monopole-antimonopole annihilation neglected. The nontrivial content is that a null counting experiment on Earth bounds the product (ambient monopole flux) x (catalysis cross section) — the same “nature’s own long-exposure experiment” logic as the compact-star bounds on stable black holes.
Step 6 — validity verdict
approved / checked. Reconstruction — the chain: captured monopoles N_M proportional to ambient flux F_M times 4.6-Gyr solar exposure (with capture efficiency epsilon(M_M, beta_M)); catalyzed proton decay f_p proportional to N_M sigma_0; solar low-energy nu_e flux at Earth proportional to f_p times branching/absorption factors. Load-bearing step: every link is monotone increasing in the product F_M sigma_0, so Super-K’s null flux limit I90 inverts to an upper bound on that product — a valid inversion, and the bound is correctly reported as a bound on the product with (sigma_0/1mb) and (f_pi+/0.5) carried as explicit scaling factors rather than hidden assumptions, with the mass-validity edge (epsilon < 1 above ~1e17 GeV) stated. I traced the algebraic structure and dimensional consistency of the three links as recorded; the numerical coefficients (central-settling density, beta_rel, pion absorption) enter as premises whose truth step 8 prices. The “nature’s long-exposure experiment” framing adds nothing beyond the chain and smuggles nothing. Checked.