#!/usr/bin/env python3 """ EXECUTION --------- Dependencies: pip install numpy matplotlib sympy Run: python3 mft_neutrino_hierarchy.py Runtime: ~5 seconds Outputs: mft_neutrino_hierarchy.png + console THE δ⁴ − 1 NEUTRINO HIERARCHY PREDICTION ========================================== Monistic Field Theory predicts the ratio of atmospheric to solar neutrino mass-squared differences: Δm²₃₂ / Δm²₂₁ = δ⁴ − 1 = 16 + 12√2 ≈ 32.97 Observed (PDG 2024): 32.58 ± 0.3 Error: 1.21% This is a PARAMETER-FREE prediction from the silver ratio δ = 1 + √2 alone. No fitting. No new parameters. DERIVATION ---------- The MFT sextic potential V₆(φ) = m²φ²/2 − λ₄φ⁴/4 + λ₆φ⁶/6 has three critical points: φ = 0 (linear vacuum) V(0) = 0 φ = φ_b (barrier) V(φ_b) = 1/(3δ) φ = φ_v (nonlinear vacuum) V(φ_v) = −δ/3 The Family-of-Three Theorem associates one neutrino with each critical point. The neutrino mass-squared splitting at each critical point is proportional to V(φ)² — the square of the potential energy — arising from second-order gravitational backreaction through the F(φ)R coupling in the MFT action. The hierarchy ratio is: [V²(φ_v) − V²(φ_b)] / [V²(φ_b) − V²(0)] = [(δ/3)² − (1/(3δ))²] / [(1/(3δ))² − 0] = [δ²/9 − 1/(9δ²)] / [1/(9δ²)] = δ⁴ − 1 = (1+√2)⁴ − 1 = 16 + 12√2 ≈ 32.97 This is the 13th manifestation of the silver ratio in MFT. Author: Dale Wahl / MFT research programme, April 2026 """ import numpy as np import matplotlib matplotlib.use('Agg') import matplotlib.pyplot as plt from matplotlib.patches import FancyArrowPatch import os _SCRIPT_DIR = os.path.dirname(os.path.abspath(__file__)) def _out(fn): return os.path.join(_SCRIPT_DIR, fn) # ══════════════════════════════════════════════════════════════════ # CONSTANTS # ══════════════════════════════════════════════════════════════════ # MFT potential parameters (normalised) M2 = 1.0 LAM4 = 2.0 LAM6 = 0.5 # Silver ratio DELTA = 1 + np.sqrt(2) SQRT2 = np.sqrt(2) # Critical points PHI_B = np.sqrt(2 - SQRT2) # barrier PHI_V = np.sqrt(2 + SQRT2) # nonlinear vacuum # Observed neutrino mass-squared differences (PDG 2024) DM2_21_OBS = 7.53e-5 # eV² (solar) DM2_32_OBS = 2.453e-3 # eV² (atmospheric, normal ordering) RATIO_OBS = DM2_32_OBS / DM2_21_OBS # ══════════════════════════════════════════════════════════════════ # POTENTIAL # ══════════════════════════════════════════════════════════════════ def V(phi): """MFT sextic potential V₆(φ) = m²φ²/2 − λ₄φ⁴/4 + λ₆φ⁶/6.""" return M2*phi**2/2 - LAM4*phi**4/4 + LAM6*phi**6/6 # ══════════════════════════════════════════════════════════════════ # STEP-BY-STEP DERIVATION # ══════════════════════════════════════════════════════════════════ print("=" * 70) print("THE δ⁴ − 1 NEUTRINO HIERARCHY PREDICTION") print("=" * 70) print("\nSTEP 1: The three critical points of V₆(φ)") print("-" * 50) print(f" φ = 0 (linear vacuum) V(0) = {V(0):.6f}") print(f" φ = φ_b (barrier) V(φ_b) = {V(PHI_B):.6f}") print(f" φ = φ_v (nonlinear vacuum) V(φ_v) = {V(PHI_V):.6f}") print(f"\n φ_b = √(2−√2) = {PHI_B:.6f}") print(f" φ_v = √(2+√2) = {PHI_V:.6f}") print("\nSTEP 2: Exact potential values in terms of δ") print("-" * 50) V_0 = 0 V_b = 1/(3*DELTA) V_v = DELTA/3 print(f" V(0) = 0") print(f" V(φ_b) = 1/(3δ) = {V_b:.8f}") print(f" V(φ_v) = −δ/3 = {-V_v:.8f}") print(f"\n Verify: V(φ_b) from formula = {V_b:.8f}") print(f" V(φ_b) computed = {V(PHI_B):.8f}") print(f" Match: {abs(V_b - V(PHI_B)) < 1e-10} ✓") print(f" Verify: V(φ_v) from formula = {-V_v:.8f}") print(f" V(φ_v) computed = {V(PHI_V):.8f}") print(f" Match: {abs(-V_v - V(PHI_V)) < 1e-10} ✓") print("\nSTEP 3: Square the potential values") print("-" * 50) V2_0 = V_0**2 V2_b = V_b**2 V2_v = V_v**2 print(f" V²(0) = 0") print(f" V²(φ_b) = 1/(9δ²) = {V2_b:.10f}") print(f" V²(φ_v) = δ²/9 = {V2_v:.10f}") print("\nSTEP 4: Compute the mass-squared differences") print("-" * 50) dV2_21 = V2_b - V2_0 # ν₂ − ν₁ dV2_32 = V2_v - V2_b # ν₃ − ν₂ print(f" Δ(V²)₂₁ = V²(φ_b) − V²(0) = 1/(9δ²) = {dV2_21:.10f}") print(f" Δ(V²)₃₂ = V²(φ_v) − V²(φ_b) = δ²/9 − 1/(9δ²) = {dV2_32:.10f}") print("\nSTEP 5: Take the ratio") print("-" * 50) ratio_pred = dV2_32 / dV2_21 print(f" Δ(V²)₃₂ / Δ(V²)₂₁ = [δ²/9 − 1/(9δ²)] / [1/(9δ²)]") print(f" = [δ² − 1/δ²] × δ²") print(f" = δ⁴ − 1") print(f" = (1+√2)⁴ − 1") print(f" = 16 + 12√2") print(f" = {ratio_pred:.6f}") # Verify algebraically delta4_minus1 = 16 + 12*SQRT2 print(f"\n Verify: 16 + 12√2 = {delta4_minus1:.6f}") print(f" Verify: δ⁴ − 1 = {DELTA**4 - 1:.6f}") print(f" Match: {abs(ratio_pred - delta4_minus1) < 1e-10} ✓") print("\nSTEP 6: Compare to observation") print("-" * 50) print(f" Predicted: δ⁴ − 1 = 16 + 12√2 = {ratio_pred:.4f}") print(f" Observed: Δm²₃₂/Δm²₂₁ = {RATIO_OBS:.4f}") print(f" Error: {abs(ratio_pred - RATIO_OBS)/RATIO_OBS * 100:.2f}%") print("\n" + "=" * 70) print("SUMMARY") print("=" * 70) print(f""" The MFT sextic potential, whose shape is derived from the Symmetric Back-Reaction Theorem (λ₄² = 8m²λ₆), predicts: Δm²₃₂ / Δm²₂₁ = δ⁴ − 1 = 16 + 12√2 ≈ {ratio_pred:.2f} This uses: • V(φ_b) = 1/(3δ) and V(φ_v) = −δ/3 (exact, from the potential) • The V² splitting mechanism (second-order gravitational backreaction) • The Family-of-Three theorem (one neutrino per critical point) • Nothing else — no free parameters, no fitting Observed: {RATIO_OBS:.2f} ± ~0.3 Error: {abs(ratio_pred - RATIO_OBS)/RATIO_OBS * 100:.2f}% This is the 13th manifestation of the silver ratio δ = 1+√2 in MFT, and the first involving δ⁴. """) # ══════════════════════════════════════════════════════════════════ # FIGURE # ══════════════════════════════════════════════════════════════════ fig = plt.figure(figsize=(14, 10)) gs = fig.add_gridspec(2, 2, hspace=0.35, wspace=0.3) # ── Panel 1: The sextic potential with critical points ──────────── ax1 = fig.add_subplot(gs[0, 0]) phi = np.linspace(-0.2, 2.2, 500) ax1.plot(phi, V(phi), 'k-', lw=2) ax1.axhline(0, color='gray', ls='-', lw=0.5) # Mark the three critical points for phi_c, label, color, va in [(0, r'$\varphi=0$', 'blue', 'bottom'), (PHI_B, r'$\varphi_b$', 'orange', 'bottom'), (PHI_V, r'$\varphi_v$', 'red', 'top')]: ax1.plot(phi_c, V(phi_c), 'o', color=color, ms=10, zorder=5) offset = 0.03 if va == 'bottom' else -0.03 ax1.annotate(f'{label}\nV={V(phi_c):.3f}', xy=(phi_c, V(phi_c)), xytext=(phi_c + 0.05, V(phi_c) + offset + (0.1 if va == 'bottom' else -0.15)), fontsize=9, color=color, fontweight='bold', arrowprops=dict(arrowstyle='->', color=color, lw=1.5)) ax1.set_xlabel(r'$\varphi$', fontsize=12) ax1.set_ylabel(r'$V_6(\varphi)$', fontsize=12) ax1.set_title('MFT sextic potential with\nthree critical points', fontsize=11) ax1.set_ylim(-1.0, 0.25) ax1.grid(True, alpha=0.3) # ── Panel 2: V² at the critical points ─────────────────────────── ax2 = fig.add_subplot(gs[0, 1]) labels = [r'$\nu_1$' + '\n' + r'($\varphi=0$)', r'$\nu_2$' + '\n' + r'($\varphi_b$)', r'$\nu_3$' + '\n' + r'($\varphi_v$)'] V2_vals = [V2_0, V2_b, V2_v] colors = ['blue', 'orange', 'red'] x_pos = [0, 1, 2] bars = ax2.bar(x_pos, V2_vals, color=colors, alpha=0.7, width=0.6, edgecolor='black', linewidth=1.5) # Annotate the values for x, v2, col in zip(x_pos, V2_vals, colors): if v2 > 0.001: ax2.text(x, v2 + 0.01, f'{v2:.4f}', ha='center', fontsize=10, fontweight='bold') else: ax2.text(x, v2 + 0.01, f'{v2:.1e}', ha='center', fontsize=9) # Draw the Δ(V²) arrows ax2.annotate('', xy=(1, V2_b), xytext=(0, V2_0), arrowprops=dict(arrowstyle='<->', color='green', lw=2)) ax2.text(0.5, V2_b/2 + 0.005, r'$\Delta(V^2)_{21}$' + f'\n= 1/(9δ²)\n= {dV2_21:.4f}', ha='center', fontsize=9, color='green', fontweight='bold', bbox=dict(boxstyle='round,pad=0.3', fc='lightyellow', alpha=0.9)) ax2.annotate('', xy=(2, V2_v), xytext=(1, V2_b), arrowprops=dict(arrowstyle='<->', color='purple', lw=2)) ax2.text(1.5, (V2_v + V2_b)/2, r'$\Delta(V^2)_{32}$' + f'\n= {dV2_32:.4f}', ha='center', fontsize=9, color='purple', fontweight='bold', bbox=dict(boxstyle='round,pad=0.3', fc='lavender', alpha=0.9)) ax2.set_xticks(x_pos) ax2.set_xticklabels(labels, fontsize=10) ax2.set_ylabel(r'$V^2(\varphi_{\rm core})$', fontsize=12) ax2.set_title(r'$V^2$ at each critical point', fontsize=11) ax2.grid(True, alpha=0.3, axis='y') # ── Panel 3: The derivation chain ──────────────────────────────── ax3 = fig.add_subplot(gs[1, 0]) ax3.axis('off') derivation_text = ( r"$\bf{Derivation:}$" + "\n\n" r"$V(0) = 0$" + "\n" r"$V(\varphi_b) = \frac{1}{3\delta}$" + "\n" r"$V(\varphi_v) = -\frac{\delta}{3}$" + "\n\n" r"$V^2(\varphi_v) = \frac{\delta^2}{9}, \quad V^2(\varphi_b) = \frac{1}{9\delta^2}$" + "\n\n" r"$\frac{\Delta m^2_{32}}{\Delta m^2_{21}} = \frac{V^2(\varphi_v) - V^2(\varphi_b)}{V^2(\varphi_b) - V^2(0)}$" + "\n\n" r"$= \frac{\delta^2/9 - 1/(9\delta^2)}{1/(9\delta^2)} = \delta^4 - 1$" + "\n\n" r"$= (1+\sqrt{2})^4 - 1 = 16 + 12\sqrt{2}$" + "\n\n" r"$\approx 32.97$" ) ax3.text(0.05, 0.95, derivation_text, transform=ax3.transAxes, fontsize=13, va='top', family='serif', bbox=dict(boxstyle='round,pad=0.5', fc='lightyellow', alpha=0.9)) # ── Panel 4: Comparison with observation ────────────────────────── ax4 = fig.add_subplot(gs[1, 1]) # Bar chart: predicted vs observed x = [0, 1] vals = [ratio_pred, RATIO_OBS] clrs = ['#2196F3', '#4CAF50'] lbls = [f'MFT: δ⁴−1\n= {ratio_pred:.2f}', f'Observed\n= {RATIO_OBS:.2f}'] bars = ax4.bar(x, vals, color=clrs, alpha=0.8, width=0.5, edgecolor='black', linewidth=1.5) for xi, vi in zip(x, vals): ax4.text(xi, vi + 0.5, f'{vi:.2f}', ha='center', fontsize=14, fontweight='bold') ax4.set_xticks(x) ax4.set_xticklabels(lbls, fontsize=11) ax4.set_ylabel(r'$\Delta m^2_{32} \,/\, \Delta m^2_{21}$', fontsize=13) ax4.set_title(f'Neutrino hierarchy ratio\n(error: {abs(ratio_pred-RATIO_OBS)/RATIO_OBS*100:.2f}%)', fontsize=11) ax4.set_ylim(0, 38) ax4.grid(True, alpha=0.3, axis='y') # Add the error annotation err_pct = abs(ratio_pred - RATIO_OBS)/RATIO_OBS * 100 ax4.annotate(f'Error: {err_pct:.2f}%', xy=(0.5, max(vals) + 2), fontsize=14, ha='center', fontweight='bold', color='darkred', bbox=dict(boxstyle='round,pad=0.4', fc='mistyrose', alpha=0.9)) fig.suptitle(r'MFT Neutrino Hierarchy: $\Delta m^2_{32}/\Delta m^2_{21} = \delta^4 - 1 = 16 + 12\sqrt{2}$', fontsize=14, fontweight='bold', y=0.98) plt.savefig(_out('mft_neutrino_hierarchy.png'), dpi=150, bbox_inches='tight') print(f"Figure saved: {_out('mft_neutrino_hierarchy.png')}") # ══════════════════════════════════════════════════════════════════ # VERDICT # ══════════════════════════════════════════════════════════════════ print("\n" + "=" * 70) print("VERDICT") print("=" * 70) print(f"\n Predicted: δ⁴ − 1 = 16 + 12√2 = {ratio_pred:.4f}") print(f" Observed: Δm²₃₂/Δm²₂₁ = {RATIO_OBS:.4f}") print(f" Error: {err_pct:.2f}%") print(f" Status: {'PASS' if err_pct < 5 else 'FAIL'} (threshold: 5%)")