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The axion misalignment branch

Original ledger classification: Physical [C] (spine angle = structural rational / NOT observable post-inflationary; relic from f_a + N_DW=1; reframed by FR.DM.02)

Recorded statement: AXION DM SPINE-ANGLE BRANCH theta_i = pi N_fam/g_car = 3 pi/5 = 108 deg (2026-06-15): an ALTERNATIVE, more ROBUST misalignment angle than the determinant-line hilltop theta_i ~ 170 deg (FR.DM.03/v185, which over-produces Omega_a h^2 ~ 0.66…

Statement, assumptions and sources

Paper context: Frontier and open physical interfaces

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DM.AXION.SPINE.01

Original ledger classification: Physical [C] (spine angle = structural rational / NOT observable post-inflationary; relic from f_a + N_DW=1; reframed by FR.DM.02)

Original claim text

AXION DM SPINE-ANGLE BRANCH theta_i = pi N_fam/g_car = 3 pi/5 = 108 deg (2026-06-15): an ALTERNATIVE, more ROBUST misalignment angle than the determinant-line hilltop theta_i ~ 170 deg (FR.DM.03/v185, which over-produces Omega_a h^2 ~ 0.66 ~ 5.5x). (1) [I] RATIONAL MOTIVE: theta_i = pi N_fam/g_car = 3 pi/5 = 108 deg EXACTLY (the central spine quotient 3/5 = N_fam/g_car, a rational multiple of pi, NOT tuned to the relic). (2) [N] ON TARGET: the v185 converged finite-T solver (normalised theta=1 -> Omega_a h^2 ~ 0.03) reaches Omega_DM h^2 = 0.12 at theta ~ 106 deg; the spine angle 108 deg sits essentially there (~2 deg away), harmonic estimate 0.03 theta^2 = 0.107 confirms the order, a mild anharmonic factor lifts to ~0.12. (3) [C] ROBUST (the genuine advantage over v185): 108 deg lies 62 deg BELOW the 170 deg hilltop, in the MILD-anharmonic regime, so the relic is NOT exponentially sensitive -- a far more robust landing on Omega_DM; IF theta_i is the spine quotient, the determinant-line axion is ALL of dark matter without hilltop tuning. (4) [O] HONEST -- AN ALTERNATIVE ANSATZ: theta_i = 3 pi/5 (spine motive) and theta_i = pi(1-phi_seam) ~ 170 deg (seam-value motive, v185) are MUTUALLY EXCLUSIVE readings of the same angle; neither is forced -- the full finite-T solver decides. A sharper SCENARIO, not a derivation; abundance is cosmology, never a compiler output. (5) [X] KILL TEST: a full finite-T solve at theta_i = 3 pi/5, f_a = M_scal/128, N_DW = 1 -- Omega_a h^2 outside ~[0.08,0.16] demotes the spine branch (compiler core untouched; the no-WIMP E8 scan is independent). Python-only. POST-INFLATIONARY REFRAMING [C] (2026-07-10, FR.DM.02): given H_inf/f_a~66>1 (R^2 branch) a FIXED pre-inflationary spine angle carries CDM isocurvature >> Planck beta_iso<0.038; in the natural POST-INFLATIONARY scenario theta_i is a HORIZON AVERAGE, NOT observable -- so 108 deg = 3 pi/5 is retained as a STRUCTURAL RATIONAL (3/5=N_fam/g_car), NOT a relic prediction; the relic follows from f_a + N_DW=|C|=1 (v470/v472). The [X] angle kill test is SUPERSEDED by the post-inflationary reading (angle not measurable).

Original classification detail

Physical [C]: theta_i = pi N_fam/g_car = 3 pi/5 = 108 deg (spine quotient) lands on Omega_DM ~106 deg ROBUSTLY (mild-anharmonic, far below the 170 deg hilltop) -- a more robust DM branch than v185; an alternative ansatz, the full solver decides; kill test outside ~[0.08,0.16] demotes the branch

Dependencies as recorded

FR.DM.01; FR.DM.02; FR.DM.03; v185_axion_relic_solver.py

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Registered verification modules

  • v211_axion_spine_angle.pyRegistered verification module · frontier
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    axion DM, the spine-angle branch θ_i = π N_fam/g_car = 3π/5 (108°): an ALTERNATIVE, more robust misalignment angle than the determinant-line hilltop θ_i≈170° (v185, which over-produces Ω_a h²≈0.66). [I] rational motive θ_i=π N_fam/g_car=3π/5=108° exactly (spine quotient 3/5, no fit); [N] the v185 finite-T solver (θ=1→0.03) reaches Ω_DM at θ≈106°, so 108° is essentially on target (harmonic 0.03θ²=0.107); [C] ROBUST: 108° is 62° below the 170° hilltop, in the mild-anharmonic regime — not exponentially sensitive, a far more robust landing than v185; [O] HONEST: an alternative ansatz to θ_i=π(1−φ_seam)≈170° (mutually exclusive, neither forced — the full solver decides, DM.AXION.SPINE.01); [X] kill test: full finite-T solve, Ω_a h² outside ~[0.08,0.16] demotes the branch. A [C] kill-test branch, NOT a closure.

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