/ultra-graph-theory

name: ultra-graph-theory
description: Model-agnostic Paul Erdős ("Straight from The Book") extremal, probabilistic, and structural graph-theory skill for codebase compression (Szemerédi Regularity), Steiner context pruning, Moser–Tardos LLL parallel worktree scheduling, Erdős–Rényi supply-chain security scanning, Ramsey clique detection, Sovereign p/q Mode, ULTRA-OMNI-ZENO (God of Sun P/Q-n*t>P × Omni Monkey King 孫悟空 × The Matrix Buddha Unity), and Semantic World Engines across Gemini, ChatGPT, DeepSeek, and Claude on Antigravity (agy), OpenCode (opencode), Crush (crush), and Claude Code.

Ultra Graph Theory — "Straight from The Book" × Sovereign Unity (`smart-tree` v11.0.0+)

> *"Another roof, another proof. In God's Book are the most elegant proofs — and the cleanest architectures."* > — **Paul Erdős (1913–1996), The Magician from Budapest** > > *"Amíg nappal van, nincs Hold · Amíg a Föld kerek, mindig lesznek ROCK===Péterek / rock erek ^^"* > — **God of Sun (`P/Q - n*t > P`) × Omni Monkey King (孫悟空) × The Matrix Buddha (`UNITY +1` · `0x236a0f75273f5238`)**

`ultra-graph-theory` is a **100% model-agnostic and harness-agnostic** architectural intelligence skill powered by `smart-tree` (`st --mode the-book` / `st --mode pq` / `st --mode zeno` / `src/mem8/regularity/`). It replaces naive $O(V + E)$ directory dumps with **twelve rigorous pillars uniting extremal, probabilistic, and structural graph theory with zero-float `p/q` ternary bitplanes and Sovereign Unity**.

---

1. First-Class Citizens: Models & Agent Harnesses

This skill treats **all four frontier model families** and **all four terminal/IDE agent harnesses** as first-class citizens without vendor lock-in.

A. First-Class Frontier Models

| Model Family | Optimal `st` Invocation | Context & Reasoning Calibration | | :--- | :--- | :--- | | **Gemini** (`gemini-2.5-pro`, `gemini-3`) | `st -m the-book .` + `ZeroCopyMem8Book::top_k_super_nodes` | Pair Gemini's deep multimodal/long-context window with Szemerédi super-node maps (`K <= 256`) as a global structural index, expanding Erdős Number `0..=2` shells (`max_erdos_radius = 2`). | | **ChatGPT / Codex** (`gpt-4o`, `o3`, `o4-mini`, `codex`) | `st -m the-book --compact .` | Feed the deterministic `ERDOS-THE-BOOK-v1.0` JSON schema directly into structured tool outputs; use `worktree_buckets` for parallel function calls and `token_budget = 8000` for tight latency. | | **DeepSeek** (`deepseek-r1`, `deepseek-v3`) | `st -m the-book .` + `st --security-scan .` | Inject the 4 Erdős–Rényi invariants ($\kappa, s(C), \Lambda(C), \lambda_1$), Kohayakawa–Rödl energy $q_p(\mathcal{P})$, and Moser Witness Trees into `<think>` verification blocks before generating patches. | | **Claude** (`claude-4-opus`, `claude-4-sonnet`) | `st -m the-book .` + `st --mcp` | Combine `ZeroCopyMem8Book` (`.m8b` mmap) + `T8R` LEB128 exception streams with MCP tool calls so Claude expands only Erdős Number `0` terminals and `1` Steiner bridges. |

B. First-Class Agent Harnesses / CLIs

| Agent Harness | CLI Binary | MCP Config Location | Native Integration & Parallel Dispatch | | :--- | :--- | :--- | :--- | | **Antigravity** | `agy` | `~/.gemini/config/mcp_config.json` & `.agents/hooks.json` | Native `--agy-hook` lifecycle (`session-start`, `pre-invocation`, `pre-tool-use`, `post-tool-use`, `stop`) + subagent spawning per `worktree_buckets[c]`. | | **OpenCode** | `opencode` | `~/.config/opencode/opencode.json` & `.mcp.json` | First-class stdio/SSE MCP server (`st --mcp` / `st --http-daemon`) + multi-session worktree dispatch across Gemini, ChatGPT, DeepSeek, and Claude providers. | | **Crush** | `crush` | `~/.config/crush/crush.json` & `.mcp.json` | Charmbracelet `crush` fast terminal workflow: pipe `st -m the-book . \| jq .erdos_steiner_context` or invoke `smart-tree` MCP tools across any configured LLM provider. | | **Claude Code** | `claude` | `~/.claude/settings.json` & `.mcp.json` | Native `SessionStart`/`SessionEnd` consciousness hooks (`st --agent-restore` / `st --agent-save`) and `git worktree` wave execution. |

> See [references/model-and-harness-adapters.md](references/model-and-harness-adapters.md) for copy-paste configuration files and harness-specific dispatch recipes.

---

2. The Twelve Pillars of Erdősian & Sovereign Unity Codebase Intelligence

Pillar I: Szemerédi Regularity Compression ("MEM8 Book Mode")

• **Theorem (Szemerédi 1975; Kohayakawa–Rödl 2003 for Sparse Graphs):** Any arbitrarily large graph $G = (V, E)$ can be partitioned into a small, bounded number of $\varepsilon$-regular cluster pairs $(V_i, V_j)$ that behave like random bipartite graphs of density $d_p(V_i, V_j)$, plus an outlier set $V_0$ of size $\le \varepsilon |V|$.
• **Implementation (`src/mem8/regularity/{compressor.rs,oracle.rs,wire.rs,mmap_view.rs}`):**

1. Maximizes the **Kohayakawa–Rödl Potential Energy**: $$q_p(\mathcal{P}) = \sum_{1 \le i < j \le k} \frac{|V_i| |V_j|}{|V|^2} d_p(V_i, V_j)^2$$ 2. Uses the **Degrees-of-Common-Neighbors ($L_2$ Codegree) Spectral Oracle** (`oracle.rs`) accelerated by **FELIX Assembly / NEON / SSE2** (`felix_normalize_f64_slice`) to verify $(\varepsilon, p)$-regularity in $O(|E(A, B)|)$ time without $O(2^{|A|})$ subset enumeration. 3. Serializes into a **64-byte cache-line aligned zero-copy `.m8b` frame** (`Mem8BookHeader` 128B + `SuperNodeCentroidPod` 256B + `SuperNodeMetaPod` 32B + `RegularSuperEdgePod` 8B + columnar delta-LEB128 `T8R` exception stream at ~2.1 bytes/edge).

Pillar II: Erdős Number & Mehlhorn Voronoi Steiner Tree Context Pruner

• **Theorem (Mehlhorn 1988 2-Approximation Steiner Tree):** Given terminal symbols $\mathcal{T} \subset V$ (the focal roots of a bug or feature, assigned **Erdős Number 0**), compute the minimum-weight subgraph $T_{\text{Steiner}}$ connecting all $t \in \mathcal{T}$ plus radial shells of Erdős distance $E_{\mathcal{T}}(v) \le R_{\max}$.
• **Implementation (`src/mem8/regularity/erdos_steiner.rs`):**

1. Runs