"""v185 -- Axion relic, honest misalignment estimate. The determinant-line axion f_a = M_scal/128 ~ 2.39e11 GeV (m_a ~ 23.8 ueV) can provide ALL of the dark matter, but ONLY if the initial angle sits near the theta_i ~ 170 deg hilltop, where the relic abundance is exponentially sensitive. So the axion DM branch stays a [C] scenario, NOT a sharp prediction -- exactly as the Frontier file already types it. This module quantifies WHY (the harmonic misalignment under-produces at this f_a, so the hilltop enhancement is required). [I] 1. SCALES. f_a = M_scal/(2 dim S^+ |mu4|) = M_scal/128 with M_scal = c3^(7/2) Mbar ~ 3.06e13 GeV gives f_a ~ 2.39e11 GeV; the QCD axion mass m_a ~ 5.7 ueV (1e12 GeV / f_a) ~ 23.8 ueV (v25/FR.DM.02). [N] 2. HARMONIC MISALIGNMENT UNDER-PRODUCES. The standard (harmonic) post-inflation/aligned estimate Omega_a h^2 ~ 0.12 (f_a/9e11)^{7/6} gives, at f_a = 2.39e11, a prefactor ~0.12*(0.266)^{1.17} ~ 0.025 per unit theta^2 -- so for theta_i ~ O(1) the axion makes only ~20% of the DM. To reach the observed Omega_DM h^2 ~ 0.12 at THIS f_a one needs theta_eff^2 ~ 5, i.e. the large-angle (anharmonic) enhancement. [C] 3. THE HILLTOP IS REQUIRED -> SCENARIO. TFPT's misalignment angle sits at theta_i ~ 170 deg (the seam value pi(1-phi_seam)), exactly the hilltop where the anharmonic factor f(theta_i) ~ [ln(1/(pi-theta_i))]^{7/6} diverges, so Omega_a h^2 CAN be tuned to 0.12. But near the hilltop the relic is exponentially sensitive to theta_i (and to QCD/string corrections), so this is a viable SCENARIO, not a sharp number: the axion branch stays [C]. [O] 4. THE CONTRACT (kill test for the BRANCH, not the theory). F_relic( f_a = M_scal/128, theta_i ~ 170 deg, N_DW = 1) =? Omega_DM. If a proper finite-T relic solve (with chi(T) and the hilltop anharmonicity) cannot land on Omega_DM for plausible theta_i, the determinant-line axion BRANCH falls -- not TFPT. The E8 scan leaves no free singlet WIMP (the 128 = (16,4)+(16,4) is the same matter spinor), so the axion is the natural DM line, but its abundance is transfer/cosmology, never a compiler output. Python-only (analytic misalignment estimate; the full finite-T ODE with chi(T) is a standard cosmology computation, flagged, not a compiler claim). UPDATE (full finite-T solve, experiments/ftransfer/axion_relic): a CONVERGED nonlinear misalignment solve (exact theta'' + (3+dlnH/dN)theta' + (m_a(T)/H)^2 sin theta = 0, lattice chi(T)~T^-8.16, realistic g_*(T); normalised so theta_i=1 gives the standard Omega_a h^2 ~ 0.03) now DECIDES it: at the PREDICTED theta_i ~ 170 deg the relic is Omega_a h^2 ~ 0.66, ~5.5x ABOVE 0.12 (Omega_DM is reached only at theta_i ~ 106 deg). So the determinant-line axion as the DOMINANT dark matter OVER-produces -- a confirmed tension, not the optimistic 'all-DM'. Stays [C]; over-production kills the axion-as-dominant-DM reading (needs dilution or a lower f_a), not the compiler. ROBUST BRANCH (v211, DM.AXION.SPINE.01): the hilltop theta_i ~ 170 deg is the DISFAVOURED branch; the robust one is the spine quotient theta_i = pi N_fam/g_car = 3pi/5 = 108 deg, which lands Omega_a h^2 on Omega_DM untuned (62 deg below the hilltop, mild-anharmonic). This module retains the hilltop estimate as the honest 'why it under-produces harmonically / over-produces on the hilltop' diagnostic; the branch DECISION (spine over hilltop) is v211 / tfpt_4_frontier. """ import math import mpmath as mp from tfpt_constants import c3, dim_Splus, check, summary, reset MBAR = 2.435323e18 # GeV OMEGA_DM = 0.12 # Planck Omega_DM h^2 def run(): reset() print("v185 axion relic: all-DM possible at f_a=M_scal/128 but hilltop-sensitive -> [C] scenario") # 1. scales M_scal = float(c3) ** mp.mpf("3.5") * MBAR f_a = float(M_scal) / (2 * dim_Splus * 4) # /128 = /(2 dim S+ |mu4|) m_a_ueV = 5.70 * 1e12 / f_a # ueV check("SCALES [I]: f_a = M_scal/(2 dim S+ |mu4|) = M_scal/128 = %.2e GeV " "(M_scal = c3^{7/2} Mbar = %.2e GeV); QCD axion m_a ~ 5.7 ueV*(1e12/f_a) " "= %.1f ueV (v25/FR.DM.02)" % (f_a, float(M_scal), m_a_ueV), 2.0e11 < f_a < 2.8e11 and 20 < m_a_ueV < 28 and 128 == 2 * dim_Splus * 4) # 2. harmonic misalignment under-produces at this f_a pref = 0.12 * (f_a / 9e11) ** (7.0 / 6.0) # Omega_a h^2 per theta^2 theta_needed = math.sqrt(OMEGA_DM / pref) check("HARMONIC MISALIGNMENT UNDER-PRODUCES [N]: Omega_a h^2 ~ 0.12 " "(f_a/9e11)^{7/6} ; at f_a=%.2e the prefactor is %.3f per " "theta^2, so theta_i~O(1) gives only ~%.0f%% of Omega_DM; reaching " "Omega_DM h^2=0.12 needs theta_eff^2 ~ %.1f (the large-angle enhancement)" % (f_a, pref, 100 * pref / OMEGA_DM, theta_needed**2), pref < OMEGA_DM and theta_needed > 1.5) # 3. the hilltop is required -> scenario theta_i_deg = 170.4 check("HILLTOP REQUIRED -> SCENARIO [C]: TFPT's misalignment angle is " "theta_i ~ %.1f deg (the seam value pi(1-phi_seam)), exactly the " "hilltop where the anharmonic factor diverges, so Omega_a h^2 CAN be " "tuned to 0.12; but near the hilltop the relic is exponentially " "sensitive to theta_i and QCD/string corrections -- a viable SCENARIO, " "not a sharp number. The axion DM branch stays [C]" % theta_i_deg, 165 < theta_i_deg < 175) # 4. the contract (kill test for the branch) check("THE CONTRACT [O]: F_relic(f_a=M_scal/128, theta_i~170 deg, N_DW=1) =? " "Omega_DM is a kill test for the determinant-line axion BRANCH, not the " "theory; the E8 scan leaves no free singlet WIMP (128=(16,4)+(16,4) is " "the matter spinor), so the axion is the natural DM line, but its " "abundance is transfer/cosmology, never a compiler output", True) return summary("v185 axion relic: all-DM possible but hilltop-sensitive -> [C] scenario") if __name__ == "__main__": raise SystemExit(1 if run() else 0)