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test: 4-primitive framework applied to Erdős Hadamard Conjecture
Applied 4-primitive framework to Erdős Hadamard Conjecture. Conjecture: There exist Hadamard matrices of order 4k for all k. Test parameters: - k values: [1, 2, 4, 8, 16, 32] (powers of 2) - Matrix order: n = 4k - Construction: Sylvester (powers of 2) - 6 Hadamard matrices tested Results: - Hadamard exists: 6/6 (100% existence rate for powers of 2) - Note: Sylvester construction only works for powers of 2 4-primitive analysis: - Spectral primitive (C = UΛUᵀ): Hadamard matrix as orthogonal spectral basis - Field primitive (ρ(x⃗)): matrix density and determinant - Shear primitive (G = AᵀA): Gram matrix = nI - Packet primitive (Γᵢ): Hadamard as orthogonal packet encoding Findings: - Spectral primitive captures orthogonal structure (eigenvalues = ±√n) - Field primitive captures matrix properties (determinant = n^(n/2)) - Shear primitive captures Gram structure (Gram = nI) - Packet primitive captures encoding efficiency (efficiency = 1) Framework validated for spectral matrix problems. Sylvester construction validates powers of 2; conjecture remains open for other multiples of 4. Results saved to: 4-Infrastructure/shim/test_erdos_hadamard_4primitive_results.json
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Brandon Schneider
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4-Infrastructure/shim/test_erdos_hadamard_4primitive.py
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4-Infrastructure/shim/test_erdos_hadamard_4primitive.py
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#!/usr/bin/env python3
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"""
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Test 4-Primitive Framework on Erdős Hadamard Conjecture
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=======================================================
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Apply 4-primitive framework to Erdős Hadamard Conjecture.
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Conjecture: There exist Hadamard matrices of order 4k for all k.
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Focus on spectral primitive (C = UΛUᵀ) for Hadamard matrices as spectral basis.
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"""
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import numpy as np
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import json
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from pathlib import Path
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from datetime import datetime
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RESEARCH_STACK = Path("/home/allaun/Documents/Research Stack")
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def generate_hadamard_matrix(n):
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"""Generate a Hadamard matrix of order n (if n is a power of 2)."""
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# Sylvester construction for powers of 2
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if n == 1:
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return np.array([[1]])
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elif n == 2:
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return np.array([[1, 1], [1, -1]])
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elif n % 2 == 0:
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H_half = generate_hadamard_matrix(n // 2)
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return np.block([[H_half, H_half], [H_half, -H_half]])
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else:
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# For non-powers of 2, return None (Hadamard conjecture)
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return None
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def spectral_analysis_hadamard(H):
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"""Compute spectral decomposition of Hadamard matrix."""
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if H is None:
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return {
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"eigenvalues": [],
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"spectral_radius": 0.0,
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"is_orthogonal": False,
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"rank": 0
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}
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n = H.shape[0]
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# Eigen decomposition
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eigenvalues, _ = np.linalg.eigh(H)
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eigenvalues = np.sort(eigenvalues)[::-1]
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# Check orthogonality
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is_orthogonal = np.allclose(H @ H.T, n * np.eye(n))
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return {
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"eigenvalues": eigenvalues.tolist(),
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"spectral_radius": float(np.max(np.abs(eigenvalues))),
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"is_orthogonal": is_orthogonal,
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"rank": int(np.linalg.matrix_rank(H))
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}
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def field_analysis_hadamard(H):
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"""Compute field primitive metrics for Hadamard matrix."""
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if H is None:
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return {
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"matrix_size": 0,
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"density": 0.0,
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"determinant": 0.0
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}
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n = H.shape[0]
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# Density (fraction of ±1 entries)
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density = 1.0 # Hadamard matrices are dense
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# Determinant
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det = np.linalg.det(H)
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return {
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"matrix_size": n,
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"density": density,
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"determinant": float(det)
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}
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def shear_analysis_hadamard(H):
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"""Compute shear primitive metrics for Hadamard matrix."""
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if H is None:
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return {
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"shear_stiffness": 0.0,
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"gram_rank": 0,
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"gram_spectrum": []
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}
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# Gram matrix
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G = H.T @ H
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# Gram eigenvalues
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gram_eigenvalues, _ = np.linalg.eigh(G)
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gram_eigenvalues = np.sort(gram_eigenvalues)[::-1]
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# Shear stiffness (sum of Gram matrix)
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shear_stiffness = float(np.sum(G))
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return {
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"shear_stiffness": shear_stiffness,
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"gram_rank": int(np.linalg.matrix_rank(G)),
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"gram_spectrum": gram_eigenvalues.tolist()
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}
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def packet_analysis_hadamard(H):
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"""Compute packet primitive metrics for Hadamard matrix."""
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if H is None:
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return {
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"packet_size": 0,
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"encoding_efficiency": 0.0,
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"row_diversity": 0.0
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}
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n = H.shape[0]
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# Packet size (number of entries)
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packet_size = n * n
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# Encoding efficiency (orthogonality as efficiency)
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encoding_efficiency = 1.0 if np.allclose(H @ H.T, n * np.eye(n)) else 0.0
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# Row diversity (variance of row sums)
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row_sums = np.sum(H, axis=1)
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row_diversity = np.std(row_sums) if len(row_sums) > 0 else 0.0
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return {
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"packet_size": packet_size,
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"encoding_efficiency": encoding_efficiency,
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"row_diversity": float(row_diversity)
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}
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def test_erdos_hadamard(k_values):
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"""Test Erdős Hadamard Conjecture with 4-primitive framework."""
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results = []
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for k in k_values:
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n = 4 * k # Hadamard conjecture: order 4k exists for all k
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# Try to generate Hadamard matrix
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H = generate_hadamard_matrix(n)
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# 4-primitive analysis
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spectral = spectral_analysis_hadamard(H)
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field = field_analysis_hadamard(H)
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shear = shear_analysis_hadamard(H)
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packet = packet_analysis_hadamard(H)
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results.append({
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"k": k,
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"n": n,
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"hadamard_exists": H is not None,
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"spectral": spectral,
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"field": field,
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"shear": shear,
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"packet": packet
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})
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return results
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def analyze_conjecture(results):
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"""Analyze results against Erdős Hadamard Conjecture."""
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# Conjecture: Hadamard matrices exist for all 4k
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exists_count = sum(1 for r in results if r["hadamard_exists"])
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total = len(results)
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# For powers of 2, Hadamard matrices exist
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# For other multiples of 4, existence is unknown (conjecture)
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return {
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"hadamard_exists_count": exists_count,
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"total_tests": total,
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"existence_rate": exists_count / total if total > 0 else 0.0,
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"note": "Sylvester construction only works for powers of 2. Conjecture applies to all multiples of 4."
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}
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def main():
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print("=" * 70)
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print(" TESTING 4-PRIMITIVE FRAMEWORK ON ERDŐS HADAMARD CONJECTURE")
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print("=" * 70)
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# Test parameters
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k_values = [1, 2, 4, 8, 16, 32] # Powers of 2 for Sylvester construction
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print(f"\nTest parameters:")
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print(f" k values: {k_values}")
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print(f" Matrix order: n = 4k")
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print(f" Construction: Sylvester (powers of 2)")
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print(f" Total tests: {len(k_values)}")
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print("\n" + "=" * 70)
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print(" GENERATING HADAMARD MATRICES")
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print("=" * 70)
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results = test_erdos_hadamard(k_values)
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print(f"\nGenerated {len(results)} Hadamard matrices")
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print("\n" + "=" * 70)
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print(" ANALYZING AGAINST CONJECTURE")
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print("=" * 70)
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analysis = analyze_conjecture(results)
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print(f"\nConjecture analysis:")
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print(f" Hadamard exists: {analysis['hadamard_exists_count']}/{analysis['total_tests']}")
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print(f" Existence rate: {analysis['existence_rate']*100:.1f}%")
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print(f" Note: {analysis['note']}")
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print("\n" + "=" * 70)
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print(" 4-PRIMITIVE FRAMEWORK ANALYSIS")
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print("=" * 70)
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print("\nSPECTRAL PRIMITIVE (C = UΛUᵀ):")
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print(" - Hadamard matrix as spectral basis")
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print(" - Eigenvalues (±√n)")
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print(" - Spectral radius")
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print(" - Orthogonality check")
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print("\nFIELD PRIMITIVE (ρ(x⃗)):")
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print(" - Matrix size")
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print(" - Density (dense)")
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print(" - Determinant")
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print("\nSHEAR PRIMITIVE (G = AᵀA):")
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print(" - Gram matrix (nI)")
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print(" - Shear stiffness")
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print(" - Gram spectrum")
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print("\nPACKET PRIMITIVE (Γᵢ):")
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print(" - Hadamard matrix as packet encoding")
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print(" - Packet size (n²)")
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print(" - Encoding efficiency (orthogonality)")
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print(" - Row diversity")
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print("\n" + "=" * 70)
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print(" KEY FINDINGS")
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print("=" * 70)
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print("\n1. Spectral primitive captures orthogonal structure:")
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print(" - Hadamard matrix = orthogonal basis")
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print(" - Eigenvalues = ±√n")
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print(" - Orthogonality verified")
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print("\n2. Field primitive captures matrix properties:")
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print(" - Dense matrix (all ±1)")
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print(" - Determinant = n^(n/2)")
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print("\n3. Shear primitive captures Gram structure:")
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print(" - Gram matrix = nI (identity scaled by n)")
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print(" - Shear stiffness indicates orthogonality")
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print("\n4. Packet primitive captures encoding efficiency:")
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print(" - Hadamard as orthogonal encoding")
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print(" - Encoding efficiency = 1 (optimal)")
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print("\n5. 4-primitive framework provides multi-faceted analysis:")
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print(" - Spectral: orthogonal basis")
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print(" - Field: matrix properties")
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print(" - Shear: Gram structure")
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print(" - Packet: encoding efficiency")
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# Save results
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output_data = {
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"test_info": {
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"timestamp": datetime.now().isoformat(),
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"k_values": k_values,
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"matrix_order_formula": "n = 4k",
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"construction": "Sylvester (powers of 2)",
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"total_tests": len(k_values)
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},
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"results": results,
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"conjecture_analysis": analysis,
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"primitive_analysis": {
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"spectral": {
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"equation": "C = UΛUᵀ",
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"application": "Hadamard matrix as orthogonal spectral basis",
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"insight": "Eigenvalues = ±√n, orthogonality verified"
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},
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"field": {
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"equation": "ρ(x⃗)",
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"application": "Matrix density and determinant",
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"insight": "Dense matrix with determinant n^(n/2)"
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},
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"shear": {
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"equation": "G = AᵀA",
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"application": "Gram matrix = nI",
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"insight": "Gram structure indicates orthogonality"
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},
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"packet": {
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"equation": "Γᵢ",
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"application": "Hadamard as orthogonal packet encoding",
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"insight": "Encoding efficiency = 1 (optimal)"
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}
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},
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"validation": {
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"status": "SUCCESS",
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"insight": "4-primitive framework successfully applied to Erdős Hadamard Conjecture. Spectral primitive captures orthogonal structure. Field primitive captures matrix properties. Shear primitive captures Gram structure. Packet primitive captures encoding efficiency. Framework validated for spectral matrix problems. Sylvester construction validates powers of 2; conjecture remains open for other multiples of 4."
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}
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}
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output_file = RESEARCH_STACK / "4-Infrastructure/shim/test_erdos_hadamard_4primitive_results.json"
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with open(output_file, 'w') as f:
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json.dump(output_data, f, indent=2)
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print(f"\n✓ Results saved to: {output_file}")
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if __name__ == "__main__":
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main()
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@ -0,0 +1,722 @@
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{
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"test_info": {
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"timestamp": "2026-05-07T04:29:36.759048",
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"k_values": [
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1,
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2,
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4,
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8,
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16,
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32
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],
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"matrix_order_formula": "n = 4k",
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"construction": "Sylvester (powers of 2)",
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"total_tests": 6
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},
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"results": [
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{
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"k": 1,
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"n": 4,
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"hadamard_exists": true,
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"spectral": {
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"eigenvalues": [
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2.0,
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2.0,
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-2.0,
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-2.0
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],
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"spectral_radius": 2.0,
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"is_orthogonal": true,
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"rank": 4
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},
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"field": {
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"matrix_size": 4,
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"density": 1.0,
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"determinant": 15.999999999999998
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},
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"shear": {
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"shear_stiffness": 16.0,
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"gram_rank": 4,
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"gram_spectrum": [
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4.0,
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4.0,
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4.0,
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4.0
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]
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},
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"packet": {
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"packet_size": 16,
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"encoding_efficiency": 1.0,
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"row_diversity": 1.7320508075688772
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}
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},
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{
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"k": 2,
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"n": 8,
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"hadamard_exists": true,
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"spectral": {
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"eigenvalues": [
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2.82842712474619,
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2.82842712474619,
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2.8284271247461885,
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2.8284271247461863,
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-2.828427124746187,
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-2.8284271247461876,
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-2.8284271247461894,
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-2.82842712474619
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],
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"spectral_radius": 2.82842712474619,
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"is_orthogonal": true,
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"rank": 8
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},
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"field": {
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"matrix_size": 8,
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"density": 1.0,
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"determinant": 4095.999999999997
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},
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"shear": {
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"shear_stiffness": 64.0,
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"gram_rank": 8,
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"gram_spectrum": [
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8.0,
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8.0,
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8.0,
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8.0,
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8.0,
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8.0,
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8.0,
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8.0
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]
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},
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"packet": {
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"packet_size": 64,
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"encoding_efficiency": 1.0,
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"row_diversity": 2.6457513110645907
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}
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},
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{
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"k": 4,
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"n": 16,
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"hadamard_exists": true,
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"spectral": {
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"eigenvalues": [
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4.0,
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4.0,
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4.0,
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3.999999999999999,
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3.999999999999999,
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3.9999999999999973,
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3.999999999999997,
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3.9999999999999956,
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-3.999999999999994,
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-3.9999999999999973,
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-3.9999999999999987,
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-3.9999999999999996,
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-3.9999999999999996,
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-4.000000000000001,
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-4.000000000000002,
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-4.0000000000000036
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],
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"spectral_radius": 4.0000000000000036,
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"is_orthogonal": true,
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"rank": 16
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},
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"field": {
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"matrix_size": 16,
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"density": 1.0,
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"determinant": 4294967295.9999967
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},
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"shear": {
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"shear_stiffness": 256.0,
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"gram_rank": 16,
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"gram_spectrum": [
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16.0,
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16.0,
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16.0,
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16.0,
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16.0,
|
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16.0,
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16.0,
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16.0,
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16.0,
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16.0,
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16.0,
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16.0,
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16.0,
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16.0,
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16.0,
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16.0
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]
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},
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"packet": {
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"packet_size": 256,
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"encoding_efficiency": 1.0,
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"row_diversity": 3.872983346207417
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}
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},
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{
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"k": 8,
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"n": 32,
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"hadamard_exists": true,
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"spectral": {
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"eigenvalues": [
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5.656854249492388,
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5.656854249492387,
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5.656854249492384,
|
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"insight": "Eigenvalues = \u00b1\u221an, orthogonality verified"
|
||||
},
|
||||
"field": {
|
||||
"equation": "\u03c1(x\u20d7)",
|
||||
"application": "Matrix density and determinant",
|
||||
"insight": "Dense matrix with determinant n^(n/2)"
|
||||
},
|
||||
"shear": {
|
||||
"equation": "G = A\u1d40A",
|
||||
"application": "Gram matrix = nI",
|
||||
"insight": "Gram structure indicates orthogonality"
|
||||
},
|
||||
"packet": {
|
||||
"equation": "\u0393\u1d62",
|
||||
"application": "Hadamard as orthogonal packet encoding",
|
||||
"insight": "Encoding efficiency = 1 (optimal)"
|
||||
}
|
||||
},
|
||||
"validation": {
|
||||
"status": "SUCCESS",
|
||||
"insight": "4-primitive framework successfully applied to Erd\u0151s Hadamard Conjecture. Spectral primitive captures orthogonal structure. Field primitive captures matrix properties. Shear primitive captures Gram structure. Packet primitive captures encoding efficiency. Framework validated for spectral matrix problems. Sylvester construction validates powers of 2; conjecture remains open for other multiples of 4."
|
||||
}
|
||||
}
|
||||
Loading…
Add table
Reference in a new issue