{"nbformat":4,"nbformat_minor":0,"metadata":{"colab":{"provenance":[{"file_id":"1xjV1ay9IUnOLByqyRTIbtVzs_tGK-gWZ","timestamp":1774513300741}],"authorship_tag":"ABX9TyMU18210QSM2Rue92/mNa8j"},"kernelspec":{"name":"python3","display_name":"Python 3"},"language_info":{"name":"python"}},"cells":[{"cell_type":"code","execution_count":null,"metadata":{"id":"zJbmr5yZOFBP"},"outputs":[],"source":[]},{"cell_type":"markdown","metadata":{"id":"bd826434"},"source":["## RFT Prediction #1: SN Ia Hubble Diagram 'Ringing'\n","\n","### 1. The Falsifiable Signature\n","Rendered Frame Theory predicts that the distance modulus residuals (relative to standard Λ-CDM) are not smooth, but contain a coherent oscillatory signal. This 'ringing' is the direct observational fingerprint of the discrete rendering updated at the fundamental frequency $\\Omega_f = 212.76$.\n","\n","### 2. Physical Origin of the Nodes\n","The harmonic nodes ($z \\approx 0.15, 0.45, 1.2, 2.5$) arise from the zeros and extrema of the **Non-Standard Topological Gradient (\\u2207̃)**. Because the expansion $H_{\\mathrm{RFT}}(z)$ is modulated by this gradient, the resulting luminosity distance $d_L(z)$ carries a 'phase-lag' that manifests as small, periodic fluctuations in the perceived brightness of standard candles like Type Ia Supernovae.\n","\n","### 3. Falsifiability and Future Testing\n","This prediction is highly falsifiable. Standard hierarchical models predict a smooth residual curve once systematic errors are removed. To test this RFT signature:\n","*   **Target Surveys:** Future high-cadence surveys like the **Nancy Grace Roman Space Telescope** and **LSST (Vera Rubin)** will provide the necessary sample size at $z > 1$.\n","*   **Method:** Non-parametric reconstructions of the Hubble diagram (e.g., using Gaussian Processes) should reveal the oscillatory structure clustered around the RFT nodes if the rendering lattice is physical."]},{"cell_type":"markdown","source":[],"metadata":{"id":"b3dd9gYJOgjh"}},{"cell_type":"code","metadata":{"id":"38eca773","colab":{"base_uri":"https://localhost:8080/","height":565},"executionInfo":{"status":"ok","timestamp":1774513109866,"user_tz":0,"elapsed":1510,"user":{"displayName":"RFTsystems4Ai “Gold Standard” Verification First","userId":"07478342073546913366"}},"outputId":"92ab3be7-3bd9-4790-f0f2-661ee0fdd939"},"source":["import numpy as np\n","import matplotlib.pyplot as plt\n","from scipy.integrate import simpson\n","\n","# --- 1. MASTER RFT ENGINE (Fully Self-Contained) ---\n","def H_RFT_UNIFIED(z):\n","    # Frozen Parameters\n","    p1_e, p2_e = -0.5976, 4.8900\n","    p1_l, p2_l = -3.1239, 3.1852\n","    nex_c, t_z, H0 = 0.0631, 2.5, 70.0\n","\n","    s = 1.0 / (1.0 + np.exp(-5.0 * (z - t_z)))\n","    p1 = p1_e * s + p1_l * (1.0 - s)\n","    p2 = p2_e * s + p2_l * (1.0 - s)\n","    tau = 1.0 * (1 + p1 * z + p2 * np.log(1.0 + z))\n","    # Hyper-Gradient (∇̃)\n","    h_sum = sum(np.sin(z * n * 212.76) / n for n in range(4, 12))\n","    return np.maximum(H0 * (1.0 - (tau - 1.0)) * (1.0 + nex_c * h_sum), 1.0)\n","\n","def H_LCDM_STD(z):\n","    return 67.4 * np.sqrt(0.315 * (1 + z)**3 + 0.685)\n","\n","# --- 2. RESIDUAL CALCULATOR ---\n","z_grid = np.linspace(0.0001, 2.6, 2000)\n","c_light = 299792.458\n","\n","# Integrate distances\n","inv_h_rft = np.array([1.0 / H_RFT_UNIFIED(z) for z in z_grid])\n","inv_h_lcdm = np.array([1.0 / H_LCDM_STD(z) for z in z_grid])\n","\n","chi_rft = np.array([simpson(inv_h_rft[:i], x=z_grid[:i]) for i in range(2, 2001)])\n","chi_rft = np.insert(chi_rft, 0, 0.0)\n","chi_lcdm = np.array([simpson(inv_h_lcdm[:i], x=z_grid[:i]) for i in range(2, 2001)])\n","chi_lcdm = np.insert(chi_lcdm, 0, 0.0)\n","\n","# Compute Modulus residuals\n","mu_rft = 5 * np.log10((1 + z_grid) * c_light * chi_rft + 1e-10) + 25\n","mu_lcdm = 5 * np.log10((1 + z_grid) * c_light * chi_lcdm + 1e-10) + 25\n","\n","# --- 3. FINAL VISUALIZATION ---\n","plt.figure(figsize=(12, 6))\n","plt.plot(z_grid, mu_rft - mu_lcdm, 'm-', linewidth=2, label='RFT Falsifiable Signal')\n","nodes = [0.15, 0.45, 1.2, 2.5]\n","for node in nodes:\n","    plt.axvline(node, color='cyan', linestyle='--', alpha=0.4)\n","plt.title(\"RFT Falsifiable Prediction: SN Ia Residual Ringing\", fontsize=14)\n","plt.xlabel(\"Redshift z\"); plt.ylabel(\"Δμ (RFT - ΛCDM)\")\n","plt.legend(); plt.grid(True, alpha=0.2)\n","plt.show()"],"execution_count":null,"outputs":[{"output_type":"display_data","data":{"text/plain":["<Figure size 1200x600 with 1 Axes>"],"image/png":"[encoded data, 46404 characters]\n"},"metadata":{}}]}]}