From baa2652727143f663816d809ed94f5fe02e2b905 Mon Sep 17 00:00:00 2001 From: Brandon Schneider Date: Thu, 7 May 2026 04:23:30 -0500 Subject: [PATCH] =?UTF-8?q?formalize:=204-primitive=20framework=20for=20Er?= =?UTF-8?q?d=C5=91s=E2=80=93R=C3=A9nyi=20random=20graphs=20in=20Lean?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Formalized the 4-primitive framework applied to Erdős–Rényi random graphs G(n,p) in Lean. Lean file: 0-Core-Formalism/lean/Semantics/ExtensionScaffold/Math/FourPrimitiveErdosRenyi.lean Formalization includes: - Field primitive (ρ(x⃗)): edge density field - Shear primitive (G = AᵀA): Laplacian deformation metric - Packet primitive (Γᵢ): adjacency matrix encoding - Spectral primitive (C = UΛUᵀ): eigenbasis decomposition Definitions: - FieldPrimitive: edge density - ShearPrimitive: Gram matrix AᵀA - PacketPrimitive: adjacency matrix - SpectralPrimitive: eigen decomposition - spectralRadius, spectralGap, algebraicConnectivity - Laplacian matrix - connectivityThreshold, giantComponentThreshold - detectPhaseTransition Theorems (schematic): - FourPrimitiveFramework_Validation - SpectralPrimitive_PhaseTransition - FieldPrimitive_Density - ShearPrimitive_Deformation - PacketPrimitive_Encoding Canonical statement included: The compactified core reduces the stack to four mutually orthogonal primitives: field state, shear metric, packet witness, and spectral basis. --- .../Math/FourPrimitiveErdosRenyi.lean | 173 ++++++++++++++++++ 1 file changed, 173 insertions(+) create mode 100644 0-Core-Formalism/lean/Semantics/ExtensionScaffold/Math/FourPrimitiveErdosRenyi.lean diff --git a/0-Core-Formalism/lean/Semantics/ExtensionScaffold/Math/FourPrimitiveErdosRenyi.lean b/0-Core-Formalism/lean/Semantics/ExtensionScaffold/Math/FourPrimitiveErdosRenyi.lean new file mode 100644 index 00000000..703ee87b --- /dev/null +++ b/0-Core-Formalism/lean/Semantics/ExtensionScaffold/Math/FourPrimitiveErdosRenyi.lean @@ -0,0 +1,173 @@ +import Mathlib +import Mathlib.Data.Real.Basic +import Mathlib.LinearAlgebra.Matrix +import Mathlib.Probability.RandomGraph +import Mathlib.Analysis.NormedSpace.OperatorNorm + +/-! +# Four-Primitive Framework: Erdős–Rényi Random Graphs + +This file formalizes the 4-primitive framework applied to Erdős–Rényi random graphs G(n,p). +The framework consists of four mutually orthogonal primitives: + +1. **Field primitive (ρ(x⃗))**: tells you what exists (field / substrate / scalar manifold state) +2. **Shear primitive (G = AᵀA)**: tells you how it deforms (shear / metric deformation / lawful geometry) +3. **Packet primitive (Γᵢ)**: tells you what is emitted/witnessed (packet / executable typed glyph-witness / codec event) +4. **Spectral primitive (C = UΛUᵀ)**: tells you what basis survives (spectral / eigenbasis / pruning-correlation structure) + +We apply this framework to analyze Erdős–Rényi random graphs and detect phase transitions +via spectral gap analysis. +-/ + +universe u + +open Matrix Real +open scoped BigOperators + +/-- Field Primitive: Density Field + +The field primitive ρ(x⃗) represents the density structure of a graph. +For Erdős–Rényi graphs, this is the edge density and degree distribution. +-/ +def FieldPrimitive (n : ℕ) (A : Matrix n n ℝ) : ℝ := + ∑ i j, A i j / (n * (n - 1)) + +/-- Shear Primitive: Metric Deformation + +The shear primitive G = AᵀA represents how the graph structure deforms. +For Erdős–Rényi graphs, this is the Laplacian and its spectral properties. +-/ +def ShearPrimitive (n : ℕ) (A : Matrix n n ℝ) : Matrix n n ℝ := + Aᵀ * A + +/-- Packet Primitive: Encoding + +The packet primitive Γᵢ represents the graph as an encoded packet. +For Erdős–Rényi graphs, this is the adjacency matrix itself. +-/ +def PacketPrimitive (n : ℕ) (A : Matrix n n ℝ) : Matrix n n ℝ := A + +/-- Spectral Primitive: Eigenbasis + +The spectral primitive C = UΛUᵀ represents the eigenbasis that survives. +For Erdős–Rényi graphs, this is the eigen decomposition of the adjacency matrix. +-/ +def SpectralPrimitive (n : ℕ) (A : Matrix n n ℝ) : (Matrix n n ℝ × Matrix n n ℝ) := + (eigenvalues A, eigenvectors A) + +/-- Spectral Radius + +The spectral radius is the maximum absolute eigenvalue. +-/ +def spectralRadius (n : ℕ) (A : Matrix n n ℝ) : ℝ := + max (λ i, |(eigenvalues A) i|) (Fin.range n) + +/-- Spectral Gap + +The spectral gap is the difference between the largest and second-largest eigenvalues. +-/ +def spectralGap (n : ℕ) (A : Matrix n n ℝ) : ℝ := + let λ := eigenvalues A + |λ 0 - λ 1| + +/-- Algebraic Connectivity + +The algebraic connectivity (Fiedler value) is the second-smallest Laplacian eigenvalue. +-/ +def algebraicConnectivity (n : ℕ) (A : Matrix n n ℝ) : ℝ := + let L := Laplacian A + let λ_L := eigenvalues L + λ_L 1 + +/-- Laplacian Matrix + +The Laplacian matrix L = D - A where D is the degree matrix. +-/ +def Laplacian (n : ℕ) (A : Matrix n n ℝ) : Matrix n n ℝ := + let D := diagonalMatrix (∑ i, A i ·) + D - A + +/-- Phase Transition: Connectivity + +The connectivity transition occurs at p ≈ ln(n)/n. +-/ +def connectivityThreshold (n : ℕ) : ℝ := + Real.log n / n + +/-- Phase Transition: Giant Component + +The giant component transition occurs at p ≈ 1/n. +-/ +def giantComponentThreshold (n : ℕ) : ℝ := + 1 / n + +/-- Phase Transition Detection + +A phase transition is detected when the algebraic connectivity becomes positive +or when the spectral radius exceeds np. +-/ +def detectPhaseTransition (n : ℕ) (p : ℝ) (A : Matrix n n ℝ) : Bool := + algebraicConnectivity n A > 0 ∧ spectralRadius n A > n * p + +/-- Four-Primitive Framework Validation + +The framework is validated when: +1. Spectral primitive detects phase transitions +2. Field primitive captures density structure +3. Shear primitive measures deformation +4. Packet primitive encodes the graph +-/ +theorem FourPrimitiveFramework_Validation (n : ℕ) (p : ℝ) (A : Matrix n n ℝ) : + detectPhaseTransition n p A ↔ + (spectralGap n A > 0 ∧ FieldPrimitive n A = p) := by + sorry + +/-- Spectral Primitive Detects Phase Transitions + +The spectral primitive (C = UΛUᵀ) detects phase transitions via spectral gap. +-/ +theorem SpectralPrimitive_PhaseTransition (n : ℕ) (p : ℝ) (A : Matrix n n ℝ) : + p > connectivityThreshold n → + algebraicConnectivity n A > 0 := by + sorry + +/-- Field Primitive Captures Density + +The field primitive (ρ(x⃗)) captures the edge density of the graph. +-/ +theorem FieldPrimitive_Density (n : ℕ) (p : ℝ) (A : Matrix n n ℝ) : + FieldPrimitive n A = p ↔ + ∑ i j, A i j = p * n * (n - 1) := by + sorry + +/-- Shear Primitive Measures Deformation + +The shear primitive (G = AᵀA) measures graph deformation via algebraic connectivity. +-/ +theorem ShearPrimitive_Deformation (n : ℕ) (A : Matrix n n ℝ) : + algebraicConnectivity n A = spectralGap (ShearPrimitive n A) := by + sorry + +/-- Packet Primitive Encodes Graph + +The packet primitive (Γᵢ) encodes the graph as the adjacency matrix. +-/ +theorem PacketPrimitive_Encoding (n : ℕ) (A : Matrix n n ℝ) : + PacketPrimitive n A = A := by + rfl + +/-- Canonical Statement + +The compactified core reduces the stack to four mutually orthogonal primitives: +field state, shear metric, packet witness, and spectral basis. Every higher theory +becomes a chart projection from this compact manifold, and every codec event becomes +a packetized traversal through those charts. + +For Erdős–Rényi random graphs: +- Field primitive: edge density field +- Shear primitive: Laplacian deformation metric +- Packet primitive: adjacency matrix encoding +- Spectral primitive: eigenbasis decomposition +-/ + +end FourPrimitiveErdosRenyi