Jobs Leadership Shadow Engine – Attractor & Bifurcation Math (Notebook)

jobs_shadow_math.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Jobs Leadership Shadow Engine – Attractor & Bifurcation Math\n",
        "\n",
        "Let:\n",
        "\n",
        "- $ \\mathbf{L} \\in \\mathbb{R}^{21} $ = vector of the 21 leadership dimensions.\n",
        "- $ M \\in \\mathbb{R} $ = meta-coherence (D22: Vision Attractor Coherence).\n",
        "- $ \\mathbf{S} \\in \\mathbb{R}^{4} $ = shadow dimensions:\n",
        "    - $ S_1 = $ Emotional Volatility\n",
        "    - $ S_2 = $ Harsh Critique Intensity\n",
        "    - $ S_3 = $ Ego Dominance\n",
        "    - $ S_4 = $ Unpredictability\n",
        "\n",
        "## 1. Leadership Attractor\n",
        "\n",
        "Define the leadership attractor strength as:\n",
        "$$\n",
        "A_L = M \\cdot \\frac{1}{21} \\sum_{i=1}^{21} L_i^2\n",
        "$$\n",
        "This encodes:\n",
        "\n",
        "- Higher coherence $ M $ strengthens the attractor.\n",
        "- Higher, well-aligned leadership coordinates $ L_i $ increase overall field strength.\n",
        "\n",
        "## 2. Shadow Attractor\n",
        "\n",
        "Model the shadow as a separate quadratic potential:\n",
        "$$\n",
        "A_S = \\sum_{j=1}^{4} w_j S_j^2\n",
        "$$\n",
        "with weights representing their destructive leverage:\n",
        "\n",
        "- $ w_1 = 0.9 $   (Emotional Volatility)\n",
        "- $ w_2 = 0.7 $   (Harsh Critique)\n",
        "- $ w_3 = 1.1 $   (Ego Dominance)\n",
        "- $ w_4 = 0.8 $   (Unpredictability)\n",
        "\n",
        "## 3. Coupling Tensor\n",
        "\n",
        "Let $ C_{ij} $ encode how each shadow dimension perturbs each leadership dimension:\n",
        "$$\n",
        "L'_j = L_j + \\sum_{i=1}^{4} C_{ij} S_i\n",
        "$$\n",
        "where:\n",
        "\n",
        "- Positive $ C_{ij} $ = amplification (e.g., Harsh Critique → Bar Setting).\n",
        "- Negative $ C_{ij} $ = attenuation (e.g., Emotional Volatility → Iteration Cadence).\n",
        "\n",
        "The effective attractor becomes:\n",
        "$$\n",
        "A_L' = M \\cdot \\frac{1}{21} \\sum_{j=1}^{21} (L'_j)^2\n",
        "$$\n",
        "## 4. Net System Attractor\n",
        "\n",
        "Introduce a shadow penalty factor $ \\lambda > 0 $:\n",
        "$$\n",
        "A_{\\text{net}} = A_L' - \\lambda A_S\n",
        "$$\n",
        "- If $ A_L' \\gg \\lambda A_S $: the leadership engine dominates (coherent attractor).\n",
        "- If $ \\lambda A_S \\approx A_L' $: the system becomes marginal, brittle, and high-variance.\n",
        "- If $ \\lambda A_S \\gg A_L' $: the shadow dominates; collapse of coherence (e.g., firing, implosion).\n",
        "\n",
        "Empirically, for Jobs you can interpret:\n",
        "\n",
        "- Early Jobs (pre-1985): high $ A_L $, very high $ A_S $, large $ \\lambda $ from weak dampers → breakdown.\n",
        "- Later Jobs (2000s): high $ A_L $, moderated $ A_S $, stronger dampers (Cook, Ive, culture) → stable edge regime.\n",
        "\n",
        "## 5. 22+4 Dimensional Bifurcation View\n",
        "\n",
        "Consider a control parameter $ \\mu \\in \\mathbb{R} $ that scales the effective shadow:\n",
        "$$\n",
        "\\mathbf{S}_{\\text{eff}} = \\mu \\mathbf{S}\n",
        "$$\n",
        "and let the damper matrix $ D \\in \\mathbb{R}^{4 \\times 4} $ (diagonal, $ 0 \\le d_{ii} \\le 1 $) represent\n",
        "ethical dampers that reduce each shadow component:\n",
        "$$\n",
        "\\tilde{\\mathbf{S}} = (I - D)\\, \\mathbf{S}_{\\text{eff}} = (I - D)\\, \\mu \\mathbf{S}\n",
        "$$\n",
        "The shadow attractor becomes:\n",
        "$$\n",
        "A_S(\\mu, D) = \\sum_{j=1}^{4} w_j \\tilde{S}_j^2 .\n",
        "$$\n",
        "Define the **bifurcation condition** as the locus where net coherence is zero:\n",
        "$$\n",
        "A_{\\text{net}}(\\mu, D) = 0 \\quad \\Rightarrow \\quad\n",
        "M \\cdot \\frac{1}{21} \\sum_{j=1}^{21} (L'_j)^2 = \\lambda A_S(\\mu, D) .\n",
        "$$\n",
        "Qualitatively:\n",
        "\n",
        "- For $ \\mu $ small (shadow under-expressed) or $ D $ strong (dampers effective), the system sits in a **single stable attractor**: high-performance, high-pressure, but coherent.\n",
        "- As $ \\mu $ increases or $ D $ weakens:\n",
        "    - First, you get **edge-of-chaos behavior**: large swings in morale and output, but still orbiting a recognizable attractor.\n",
        "    - Beyond a critical $ \\mu_c(D) $, the system bifurcates into:\n",
        "        - a high-output / high-damage regime, and\n",
        "        - a collapse regime (burnout, attrition, political revolt).\n",
        "\n",
        "This is a **22+4 dimensional bifurcation surface** in the $(\\mathbf{L}, M, \\mathbf{S}, D)$ space. Practically, you can treat:\n",
        "\n",
        "- $ \\mu $ as \"how much of the dark side is expressed\", and\n",
        "- $ D $ as \"how well the organization has wrapped the leader with buffers\".\n",
        "\n",
        "## 6. Practical Use\n",
        "\n",
        "- For coaching: estimate rough scores for $ \\mathbf{L}, M, \\mathbf{S} $, and dampers $ D $.\n",
        "- Track whether interventions (coaching, governance, process) move the system away from the bifurcation surface.\n",
        "- The goal is not to drive $ \\mathbf{S} $ to zero, but to contain it so $ A_L' $ consistently dominates.\n"
      ]
    }
  ],
  "metadata": {
    "language_info": {
      "name": "python"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 2
}