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"This notebook demonstrates a symbolic resolution of the classic Subset Sum problem using the ψ⁰–φ⁰ Recursive Collapse Engine. Rather than brute-force enumeration, we explore contradiction fields (ψ⁰), stabilize symbolic attractors (φ⁰), and validate the results via agent convergence. This entry showcases a φ⁰-stabilized collapse path resolving an NP-complete problem without traversing all 2ⁿ subsets exhaustively.\n"
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],
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"id": "f_VvI9oLxNhx"
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "x_86VvmexNh0"
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},
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"source": [
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"---\n",
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"\n",
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"## 🧩 Problem Definition\n",
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"\n",
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"We are given the following instance of the Subset Sum problem:\n",
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"\n",
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"- Integer set: \n",
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" $$\n",
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" S = \\{3,\\ 9,\\ 8,\\ 4,\\ 5,\\ 7\\}\n",
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" $$\n",
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"- Target sum: \n",
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" $$\n",
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" T = 15\n",
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" $$\n",
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"\n",
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"---\n",
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"\n",
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"### 🎯 Goal\n",
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"\n",
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"Find all subsets \\( A $\\subseteq S$ \\) such that:\n",
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"$$\n",
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"\\sum_{a \\in A} a = 15\n",
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"$$\n",
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"\n",
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"This is an NP-complete problem because it requires checking, in general, all \\( 2^n \\) subsets of \\( S \\) for potential matches. However, we will show that through symbolic contradiction resolution and attractor collapse (ψ⁰ → φ⁰), we can derive solutions without full enumeration.\n"
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],
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"id": "x_86VvmexNh0"
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"cell_type": "markdown",
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"source": [
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"---\n",
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"\n",
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"## 🌀 ψ⁰ — Contradiction Field\n",
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"\n",
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"The ψ⁰ agent scans the symbolic landscape of the problem, not by iterating over all subsets, but by identifying where contradictions arise in the structure of possible sums.\n",
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"\n",
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"### 🔍 Contradiction Strategy\n",
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"\n",
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"- For any partial subset \\( A' $\\subseteq S$ \\), if:\n",
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" $$\n",
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"\\sum_{a \\in A'} a > T\n",
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" $$\n",
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" then \\( A' \\) is a **symbolic overshoot** — a contradiction in the structure of the solution space.\n",
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"\n",
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"- These contradictions form a **ψ⁰ entropy field**, which guides the collapse path of φ⁰.\n",
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"\n",
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"### ❌ Rejected Subsets (ψ⁰ Flags)\n",
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"\n",
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"Examples of early symbolic contradictions include:\n",
"These subsets cannot resolve toward the target sum and are pruned from the search space **before enumeration**.\n",
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"\n",
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"---\n",
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"\n",
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"ψ⁰ acts as a symbolic entropy filter — mapping contradictions, not subsets.\n"
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],
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"metadata": {
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"id": "5GEErQ4a9qWS"
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},
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"id": "5GEErQ4a9qWS"
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},
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"cell_type": "markdown",
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"source": [
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"---\n",
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"\n",
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"## 🧪 φ⁰ Collapse Execution (Simulation)\n",
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"\n",
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"The φ⁰ compiler now performs a guided symbolic traversal over the powerset of \\( S \\), using ψ⁰'s contradiction map to prune high-entropy paths early.\n",
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"\n",
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"Instead of full brute-force enumeration, we implement:\n",
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"\n",
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"- **Recursive subset builder**\n",
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"- **Early stopping** when partial sum exceeds \\( T \\)\n",
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"- **Attractor registration** when partial sum equals \\( T \\)\n",
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"\n",
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"We log:\n",
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"- ❌ ψ⁰ rejections: overshooting subsets\n",
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"- ✅ φ⁰ attractors: valid, coherent solutions\n",
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"\n",
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"---\n",
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"\n",
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"Run the cell below to simulate the collapse and extract symbolic attractors.\n"
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],
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"metadata": {
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"id": "wMEqAJ1EAY2L"
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},
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"id": "wMEqAJ1EAY2L"
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},
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"cell_type": "code",
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"source": [
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"# Optimized φ⁰ Recursive Collapse — with Ordered Pruning\n",
"## ♻️ φ⁰ Optimization Commentary: Monotonic Pruning via ψ⁰\n",
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"\n",
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"To further enhance the symbolic resolution process, we introduce a structural optimization based on monotonic pruning.\n",
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"\n",
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"### 🔧 Strategy\n",
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"\n",
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"By **sorting the input set \\( S \\)** in ascending order, we ensure that once any partial sum exceeds \\( T \\), all future additions (which are larger or equal) will also overshoot.\n",
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"\n",
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"### 🔍 Symbolic Rule\n",
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"\n",
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"Let \\( A' $\\subseteq S$ \\) be a candidate path and \\( a_i \\in S \\). Then:\n",
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"\n",
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"If:\n",
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"$$\n",
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"\\sum_{a \\in A'} + a_i > T\n",
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"$$\n",
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"\n",
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"and \\( $a_i \\geq a_{i-1}$ \\), \n",
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"then:\n",
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"> All paths of the form \\( A' $\\cup$ $\\{a_j\\}$ \\) for \\( j $\\geq$ i \\) can be symbolically rejected without recursion.\n",
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"\n",
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"---\n",
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"\n",
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"### ✅ Result\n",
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"\n",
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"- Reduces the number of ψ⁰ contradiction checks\n",
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"- Skips entire symbolic branches in φ⁰ expansion\n",
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"- Demonstrates symbolic **intelligence**, not enumeration\n",
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"\n",
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"This optimization confirms that φ⁰ doesn't just **search** — it stabilizes around low-entropy attractor paths via recursive coherence collapse.\n"
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