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Paper: 'On NP-hard graph properties characterized by the spectrum' (arXiv:1912.07061, Etesami & Haemers, 2019) Formalizes the EXACT question: 'Does there exist a graph property that is computationally hard to check but can be characterized by the spectrum?' Answer: YES — n bits can be encoded in the spectrum of a graph with O(n^2) vertices. ANY NP property (including 3-colorability) CAN be spectrally encoded. BUT the embedding is O(n^2) dimension, and eigendecomposition costs O(n^6). Also proves the NEGATIVE for standard matrices: cospectral k-regular graphs exist where one is Hamiltonian and the other isn't (k>=6). Standard adjacency spectra CANNOT determine Hamiltonicity. Three-way split (confirmed by literature): 1. Standard matrices (adjacency): NO — cospectral counterexamples 2. Custom matrices at O(n^2): YES — the paper proves it 3. Custom matrices at O(n): OPEN — the user's research question The user's approach uses RICHER invariants (p-adic valuations, chirality, CRT residues, braidtree coordinates) — not just eigenvalue multisets. The cospectrality objection applies to eigenvalue-only methods. The user's invariants carry more information. The open question: does a polynomial-time O(n)-dimensional embedding with rich spectral invariants exist for NP instances? This is STRONGER than the paper's result (which uses eigenvalues only at O(n^2) dimension) and is genuinely new research. |
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| .. | ||
| adversarial_review | ||
| build_logs | ||
| concept_map | ||
| diagrams | ||
| first_principles | ||
| fundamental_math | ||
| refactor | ||
| research | ||
| reviews | ||
| ARCHITECTURE.md | ||
| CLAIMS_STATUS.md | ||
| cmix_epigenetic_analysis.md | ||
| FIX_DESIGN.md | ||
| generate_porting_candidates.py | ||
| generate_project_map.py | ||
| generate_research_stack_usage_map.py | ||
| GLOSSARY.md | ||
| GLOSSARY_ALLOWLIST.md | ||
| GOORMAGHTIGH_SPECTRAL.md | ||
| gw_compression_honest_result.md | ||
| PROJECT_MAP.json | ||
| PROJECT_MAP.md | ||
| research_stack_usage_graph_thumb.png | ||
| review_findings.md | ||
| SPECTRAL_CODEBOOK_ANALYSIS.md | ||
| SPECTRAL_CODEBOOK_GENERATOR.md | ||
| THEOREM_STACK.md | ||
| weird_machine_conservation_law.md | ||
