{ "cells": [ { "cell_type": "markdown", "id": "76d63f79", "metadata": {}, "source": [ "# DCT Laboratory — Volume I, Chapter 16\n", "## Synthesis, Future Directions, and Transition to Volume II\n", "**Seed `26116`** · Companion to the chapter and AXIOM Module **AXIOM-16**\n", "\n", "**The capstone: one integrated run reusing the volume's own instruments.**\n", "The GEOP's allocation (Ch. 15) is recalled and re-verified; the unified dynamics\n", "(Chs. 13–14) run under an invariant gate; a **digital twin** (Def., §16.10)\n", "tracks the \"actual\" enterprise and a drift monitor catches their divergence; and\n", "the **Optimization Readiness Theorem** is scored as an executable checklist.\n", "Mirrored in `DCT_V1_Ch16_Lab.xlsx`." ] }, { "cell_type": "code", "execution_count": null, "id": "051611af", "metadata": { "execution": { "iopub.execute_input": "2026-07-13T23:27:21.671256Z", "iopub.status.busy": "2026-07-13T23:27:21.671035Z", "iopub.status.idle": "2026-07-13T23:27:22.536281Z", "shell.execute_reply": "2026-07-13T23:27:22.535168Z" } }, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "plt.rcParams['figure.dpi']=110\n", "\n", "import numpy as np\n", "SEED = 26116\n", "# --- Ch. 15 recall: the GEOP's corner optimum ---\n", "K0c, DELTAc, ALPHAc = np.array([40.,30.,20.]), np.array([.02,.06,.10]), np.array([.3,.4,.3])\n", "BASEc = (1-DELTAc)*K0c\n", "def Y(I): return float(10*np.prod((BASEc+np.asarray(I))**ALPHAc))\n", "S2 = BASEc[1]+BASEc[2]+6.0\n", "Kp = ALPHAc[1:]/ALPHAc[1:].sum()*S2\n", "I_STAR = np.array([0.0, Kp[0]-BASEc[1], Kp[1]-BASEc[2]])\n", "\n", "# --- Chs. 13-14 recall: integrated (K, p) dynamics, g = 0.06 ---\n", "A = np.array([[0.90,0.20],[0.06,0.80]]); Bv = np.array([4.0,2.0])\n", "Z0, N, FLOOR = np.array([40.0,50.0]), 16, 25.0\n", "def twin_path(n=N):\n", " z=np.empty((n+1,2)); z[0]=Z0\n", " for k in range(n): z[k+1]=A@z[k]+Bv\n", " return z\n", "\n", "# --- Ch. 16: the digital twin vs the 'actual' enterprise ---\n", "NOISE_P = np.array([1.1735, -2.1226, 2.1835, -0.1833, -2.0614, 2.2657,\n", " 1.0514, 0.6913, 1.2287, -0.3311, -0.9874, 0.7519,\n", " -0.0573, 0.7441, -1.55, 1.855]) # frozen: rng(26116).normal(0,1.5,16) # frozen seed-26116 draws, sigma=1.5\n", "def actual_path(n=N):\n", " z=np.empty((n+1,2)); z[0]=Z0\n", " for k in range(n):\n", " z[k+1]=A@z[k]+Bv\n", " z[k+1,1]+=NOISE_P[k]\n", " return z\n", "\n", "def reference_values():\n", " tw, ac = twin_path(), actual_path()\n", " diff = ac[1:,1]-tw[1:,1]\n", " rmse = float(np.sqrt(np.mean(diff**2)))\n", " drift = int(np.argmax(np.abs(diff)>3.0))+1 if (np.abs(diff)>3.0).any() else 0\n", " rho = float(max(abs(np.linalg.eigvals(A))))\n", " minK = float(tw[:,0].min())\n", " viol = int((tw[:,0]=FLOOR)+int(abs(ALPHAc.sum()-1)<1e-9)+int(abs(marg-1)<1e-6)\n", " return {\n", " \"Y_star_recall\": round(Y(I_STAR),4),\n", " \"rho_integrated\": round(rho,4),\n", " \"minK_16\": round(minK,4),\n", " \"invariant_violations\": viol,\n", " \"twin_rmse\": round(rmse,4),\n", " \"drift_quarter\": drift,\n", " \"max_abs_gap\": round(float(np.abs(diff).max()),4),\n", " \"readiness_score\": ready,\n", " }\n", "if __name__ == \"__main__\":\n", " [print(f\"{k:22s} {v}\") for k,v in reference_values().items()]" ] }, { "cell_type": "markdown", "id": "49b96dd3", "metadata": {}, "source": [ "## Panel 1 — The readiness checklist, executable\n", "The Enterprise Optimization Readiness Theorem asks four questions of any\n", "enterprise claiming to be ready for Volume II. Each is a computation the volume\n", "built: dynamics certified ($\\rho < 1$, Ch. 14), invariants held (Ch. 13's gate),\n", "objective declared (weights sum to 1, Ch. 11), optimality conditions verified\n", "(marginal ratio 1, Ch. 15). Score: **4 of 4** — this enterprise may proceed." ] }, { "cell_type": "code", "execution_count": null, "id": "292dd6b9", "metadata": { "execution": { "iopub.execute_input": "2026-07-13T23:27:22.539030Z", "iopub.status.busy": "2026-07-13T23:27:22.538169Z", "iopub.status.idle": "2026-07-13T23:27:22.547006Z", "shell.execute_reply": "2026-07-13T23:27:22.545900Z" } }, "outputs": [], "source": [ "tw = twin_path()\n", "rho = float(max(abs(np.linalg.eigvals(A))))\n", "minK = tw[:,0].min()\n", "marg = (ALPHAc[1]/(BASEc[1]+I_STAR[1]))/(ALPHAc[2]/(BASEc[2]+I_STAR[2]))\n", "checks = [\n", " (\"dynamics certified: rho < 1 (Ch. 14)\", rho, rho < 1),\n", " (\"invariant held: min K >= 25 (Ch. 13)\", minK, minK >= FLOOR),\n", " (\"objective declared: sum(alpha) = 1 (Ch. 11)\", ALPHAc.sum(), abs(ALPHAc.sum()-1)<1e-9),\n", " (\"optimality verified: marginal ratio = 1 (Ch. 15)\", marg, abs(marg-1)<1e-6)]\n", "for name, val, ok in checks:\n", " print(f\"{'READY' if ok else 'NOT READY':>9s} {name:46s} value = {val:.4f}\")\n", "print(f\"\\nreadiness score: {sum(ok for _,_,ok in checks)} / 4\")\n", "print(f\"GEOP value recalled and re-verified: Y* = {Y(I_STAR):.4f}\")" ] }, { "cell_type": "markdown", "id": "411117e9", "metadata": {}, "source": [ "## Panel 2 — The digital twin and its drift monitor\n", "The **twin** is the deterministic core; the **actual** enterprise is the same\n", "system with frozen seed-`26116` disturbances on performance. The twin tracks\n", "well (RMSE 1.68 over 16 quarters) until quarter **9**, when the gap first\n", "exceeds the declared tolerance of 3.0 — the drift monitor fires, and §16.10's\n", "lifecycle says: re-estimate, re-validate, re-deploy. A model is an instrument\n", "under maintenance (Ch. 4), and the twin is that sentence industrialized." ] }, { "cell_type": "code", "execution_count": null, "id": "744b0c0e", "metadata": { "execution": { "iopub.execute_input": "2026-07-13T23:27:22.549567Z", "iopub.status.busy": "2026-07-13T23:27:22.548611Z", "iopub.status.idle": "2026-07-13T23:27:23.072491Z", "shell.execute_reply": "2026-07-13T23:27:23.071159Z" } }, "outputs": [], "source": [ "tw, ac = twin_path(), actual_path()\n", "t = np.arange(N+1)\n", "diff = ac[1:,1]-tw[1:,1]\n", "drift = int(np.argmax(np.abs(diff)>3.0))+1\n", "fig, axes = plt.subplots(1,2, figsize=(10.2,4.0))\n", "axes[0].plot(t, tw[:,1], c=\"#0B3D2E\", lw=2.2, label=\"digital twin p\")\n", "axes[0].plot(t, ac[:,1], \"o-\", c=\"#C8A24B\", lw=1.6, ms=4, label=\"actual p (seed 26116)\")\n", "axes[0].axvline(drift, c=\"#B0532F\", ls=\":\", lw=1.4)\n", "axes[0].set(xlabel=\"quarter\", title=\"Twin vs actual\"); axes[0].legend(frameon=False, fontsize=9); axes[0].grid(alpha=.25)\n", "axes[1].plot(range(1,N+1), np.abs(diff), \"o-\", c=\"#1B6B52\", lw=2, ms=4)\n", "axes[1].axhline(3.0, c=\"#B0532F\", ls=\"--\", lw=1.2, label=\"tolerance 3.0\")\n", "axes[1].axvline(drift, c=\"#B0532F\", ls=\":\", lw=1.4, label=f\"drift fires: q{drift}\")\n", "axes[1].set(xlabel=\"quarter\", ylabel=\"|gap|\", title=\"The drift monitor\")\n", "axes[1].legend(frameon=False, fontsize=9); axes[1].grid(alpha=.25)\n", "plt.tight_layout(); plt.show()\n", "print(f\"twin RMSE: {np.sqrt(np.mean(diff**2)):.4f} max |gap|: {np.abs(diff).max():.4f} drift quarter: {drift}\")" ] }, { "cell_type": "markdown", "id": "bf10b986", "metadata": {}, "source": [ "## Panel 3 — The volume in one run\n", "Sixteen quarters of the integrated enterprise, with every layer's instrument\n", "annotated where it acts: the GEOP chose the allocation, the certified dynamics\n", "carried it, the invariant gate stayed green (zero violations, min K = 40), and\n", "the twin watched. This panel is Volume I's table of contents, executed." ] }, { "cell_type": "code", "execution_count": null, "id": "b189483a", "metadata": { "execution": { "iopub.execute_input": "2026-07-13T23:27:23.074159Z", "iopub.status.busy": "2026-07-13T23:27:23.073966Z", "iopub.status.idle": "2026-07-13T23:27:23.326064Z", "shell.execute_reply": "2026-07-13T23:27:23.325323Z" } }, "outputs": [], "source": [ "fig, ax = plt.subplots(figsize=(8.6,4.4))\n", "ax.plot(t, tw[:,0], c=\"#0B3D2E\", lw=2.4, label=\"capital K\")\n", "ax.plot(t, tw[:,1], c=\"#C8A24B\", lw=2.4, label=\"performance p\")\n", "ax.axhline(FLOOR, c=\"#B0532F\", ls=\"--\", lw=1.2, label=\"invariant K ≥ 25 (Ch. 13-14)\")\n", "ax.annotate(\"GEOP allocation applied (Ch. 15)\", (0.2, tw[0,0]+3), fontsize=9, color=\"#0B3D2E\")\n", "ax.annotate(f\"drift monitor fires q{drift} (Ch. 16)\", (drift+0.2, 30), fontsize=9, color=\"#B0532F\")\n", "ax.set(xlabel=\"quarter\", ylabel=\"level\", title=\"One integrated run — seeds 26101→26116 behind it\")\n", "ax.legend(frameon=False, fontsize=9); ax.grid(alpha=.25); plt.tight_layout(); plt.show()\n", "print(f\"invariant violations: {int((tw[:,0]