======================================================================== SOLAR P-MODE EIGENVALUE ANALYSIS Data: BiSON Davies et al. (2014), 22-year dataset ======================================================================== ── TEST 1: Large Frequency Separation Δν ── l n→n+1 Δν (μHz) Δν/Δν_mean --------------------------------------------- 0 6→7 145.378 1.03331 0 7→8 145.205 1.03208 0 8→9 144.274 1.02547 0 9→10 140.864 1.00123 0 10→11 138.258 0.98271 0 11→12 135.608 0.96387 0 12→13 135.250 0.96133 mean Δν(l=0) = 140.691 μHz 1 7→8 144.031 1.02955 1 8→9 143.204 1.02364 1 9→10 139.885 0.99991 1 10→11 136.561 0.97615 1 11→12 135.804 0.97074 mean Δν(l=1) = 139.897 μHz 2 8→9 141.164 1.02453 2 9→10 138.685 1.00654 2 10→11 135.770 0.98538 2 11→12 135.516 0.98354 mean Δν(l=2) = 137.784 μHz 3 9→10 137.552 1.00497 3 10→11 136.192 0.99503 mean Δν(l=3) = 136.872 μHz ── TEST 2: Frequency Ratios vs Golden Ratio φ ── φ = 1.6180339887 1/φ = 0.6180339887 φ² = 2.6180339887 l = 0: n₂/n₁ ν₂/ν₁ closest φ^p p error -------------------------------------------------- 6/7 1.149471 1.127838 0.25 +0.021633 7/8 1.129880 1.127838 0.25 +0.002042 8/9 1.114213 1.127838 0.25 -0.013625 9/10 1.100083 1.127838 0.25 -0.027755 10/11 1.089295 1.127838 0.25 -0.038544 11/12 1.080403 1.127838 0.25 -0.047435 12/13 1.074223 1.127838 0.25 -0.053615 l = 1: n₂/n₁ ν₂/ν₁ closest φ^p p error -------------------------------------------------- 7/8 1.121483 1.127838 0.25 -0.006355 8/9 1.107702 1.127838 0.25 -0.020137 9/10 1.094976 1.127838 0.25 -0.032862 10/11 1.084677 1.127838 0.25 -0.043161 11/12 1.077634 1.127838 0.25 -0.050204 l = 2: n₂/n₁ ν₂/ν₁ closest φ^p p error -------------------------------------------------- 8/9 1.101215 1.127838 0.25 -0.026623 9/10 1.090298 1.127838 0.25 -0.037540 10/11 1.081079 1.127838 0.25 -0.046759 11/12 1.074858 1.127838 0.25 -0.052981 l = 3: n₂/n₁ ν₂/ν₁ closest φ^p p error -------------------------------------------------- 9/10 1.086427 1.127838 0.25 -0.041411 10/11 1.078765 1.127838 0.25 -0.049073 ── TEST 3: Second Differences vs A_N Eigenvalue Pattern ── The second difference δ²ν = ν(n+1) - 2ν(n) + ν(n-1) probes the tridiagonal structure. For A_N: δ² is related to the discrete Laplacian eigenvalues. l = 0: second differences δ²ν(n) n ν(n) μHz δ²ν (μHz) -------------------------------- 7 1117.993 -0.173 8 1263.198 -0.931 9 1407.472 -3.410 10 1548.336 -2.606 11 1686.594 -2.650 12 1822.202 -0.358 A_6 fit (N=6): RMS residual = 0.4667 Normalized data: ['1.000', '0.766', '0.000', '0.248', '0.235', '0.943'] A_6 eigenvals: ['1.000', '0.846', '0.623', '0.377', '0.154', '0.000'] l = 1: second differences δ²ν(n) n ν(n) μHz δ²ν (μHz) -------------------------------- 8 1329.635 -0.827 9 1472.839 -3.319 10 1612.724 -3.324 11 1749.285 -0.757 A_4 fit (N=4): RMS residual = 0.6267 Normalized data: ['0.973', '0.002', '0.000', '1.000'] A_4 eigenvals: ['1.000', '0.691', '0.309', '0.000'] l = 2: second differences δ²ν(n) n ν(n) μHz δ²ν (μHz) -------------------------------- 9 1535.853 -2.479 10 1674.538 -2.915 11 1810.308 -0.254 A_3 fit (N=3): RMS residual = 0.8061 Normalized data: ['0.164', '0.000', '1.000'] A_3 eigenvals: ['1.000', '0.500', '0.000'] l = 3: second differences δ²ν(n) n ν(n) μHz δ²ν (μHz) -------------------------------- 10 1729.088 -1.360 ── TEST 4: Small Separations d₀₂ = ν(n,0) − ν(n−1,2) ── These probe the deep interior / tachocline region. n= 9: d₀₂ = 1407.472 - 1394.689 = +12.783 μHz n=10: d₀₂ = 1548.336 - 1535.853 = +12.483 μHz n=11: d₀₂ = 1686.594 - 1674.538 = +12.056 μHz n=12: d₀₂ = 1822.202 - 1810.308 = +11.894 μHz n=13: d₀₂ = 1957.452 - 1945.824 = +11.628 μHz d₀₂(n=10)/d₀₂(n=9) = 0.976531 (φ⁻¹ = 0.618034, diff = +0.358497) d₀₂(n=11)/d₀₂(n=10) = 0.965793 (φ⁻¹ = 0.618034, diff = +0.347759) d₀₂(n=12)/d₀₂(n=11) = 0.986563 (φ⁻¹ = 0.618034, diff = +0.368529) d₀₂(n=13)/d₀₂(n=12) = 0.977636 (φ⁻¹ = 0.618034, diff = +0.359602) ── TEST 5: Large Separation vs Fundamental Constants ── Mean Δν(l=0) = 140.691 μHz Δν / α⁻¹ = 1.026672 Δν / φ = 86.951820 Δν × φ = 227.642820 Δν / (φ × α⁻¹)= 0.634518 Δν / 135 = 1.042156 (canonical Δν ≈ 135 μHz) ── TEST 6: Direct A_N Eigenvalue Overlay ── Scaling A_N eigenvalues to the observed frequency range. l = 0, N = 8 modes k A_N eig scaled (μHz) observed Δ (μHz) ------------------------------------------------------- 1 +1.87939 1957.452 972.615 +984.837 2 +1.53209 1866.457 1117.993 +748.464 3 +1.00000 1727.044 1263.198 +463.846 4 +0.34730 1556.029 1407.472 +148.557 5 -0.34730 1374.038 1548.336 -174.298 6 -1.00000 1203.023 1686.594 -483.571 7 -1.53209 1063.610 1822.202 -758.592 8 -1.87939 972.615 1957.452 -984.837 RMS residual: 668.666 μHz Frequency range: 984.837 μHz Relative RMS: 67.90% l = 1, N = 6 modes k A_N eig scaled (μHz) observed Δ (μHz) ------------------------------------------------------- 1 +1.80194 1885.089 1185.604 +699.485 2 +1.24698 1777.376 1329.635 +447.741 3 +0.44504 1621.726 1472.839 +148.887 4 -0.44504 1448.967 1612.724 -163.757 5 -1.24698 1293.317 1749.285 -455.968 6 -1.80194 1185.604 1885.089 -699.485 RMS residual: 489.204 μHz Frequency range: 699.485 μHz Relative RMS: 69.94% ── TEST 7: Consecutive Spacing Ratios ── For A_N: Δλ_k / Δλ_{k-1} follows a specific sine pattern. l = 0: k Δν_obs ratio_obs ratio_A_N diff -------------------------------------------------- 2 145.205 0.99881 1.53209 -0.53328 3 144.274 0.99359 1.22668 -0.23309 4 140.864 0.97636 1.06418 -0.08781 5 138.258 0.98150 0.93969 +0.04181 6 135.608 0.98083 0.81521 +0.16563 7 135.250 0.99736 0.65270 +0.34466 l = 1: k Δν_obs ratio_obs ratio_A_N diff -------------------------------------------------- 2 143.204 0.99426 1.44504 -0.45078 3 139.885 0.97682 1.10992 -0.13309 4 136.561 0.97624 0.90097 +0.07527 5 135.804 0.99446 0.69202 +0.30244 l = 2: k Δν_obs ratio_obs ratio_A_N diff -------------------------------------------------- 2 138.685 0.98244 1.36603 -0.38359 3 135.770 0.97898 1.00000 -0.02102 4 135.516 0.99813 0.73205 +0.26608 ======================================================================== ANALYSIS COMPLETE ========================================================================