docs: formal literature — octagon question answered at O(n^2)

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.

docs/research/OCTAGON_PRINCIPLE.md

@@ -336,3 +336,88 @@ either:
 
 This is honest research: no claim of solving P vs NP, just a
 systematic measurement program with clear yes/no outcomes per problem.
+
+## Formal Literature: The Question Has Been Asked and Partially Answered
+
+### Paper: "On NP-hard graph properties characterized by the spectrum"
+arXiv: 1912.07061 (Etesami & Haemers, 2019)
+
+**Question 1 (their formalization):**
+"Does there exist a graph property that is computationally hard to check
+but that can be characterized by the spectrum?"
+
+**Their answer: YES (affirmative).**
+
+They prove: you can encode n bits of information inside the spectrum of
+a graph with O(n²) vertices, and the n bits can be recovered from the
+spectrum. This means ANY property of n bits (including NP-hard properties
+like 3-colorability) CAN be translated into a graph property that IS
+characterizable by the spectrum.
+
+**BUT:**
+- The embedding uses O(n²) vertices to encode n bits (polynomial, not linear)
+- Eigendecomposition of O(n²)×O(n²) matrix = O(n⁶) — polynomial but expensive
+- The graph is ENGINEERED (not a standard adjacency matrix)
+
+**They also prove the NEGATIVE for standard matrices:**
+- For every k ≥ 6, they construct k-regular cospectral graphs where one
+  is Hamiltonian and the other isn't
+- This proves: the adjacency spectrum CANNOT determine Hamiltonicity
+- Similar cospectral counterexamples exist for chromatic number, clique number
+
+### The Three-Way Split (Confirmed by Literature)
+
+| Matrix type | Octagon works? | Dimension | Cost |
+|-------------|----------------|-----------|------|
+| Standard (adjacency, Laplacian) | NO | n×n | O(n³) — but cospectral counterexamples |
+| Custom (engineered embedding) | YES | O(n²)×O(n²) | O(n⁶) — polynomial but expensive |
+| Optimal custom (does it exist?) | OPEN | O(n)×O(n)? | O(n³) — the user's question |
+
+### What GPT Got Right
+
+GPT's analysis matches the paper exactly:
+- "For all standard spectral constructions → decisively negative" ✓ (paper proves cospectral Hamiltonian pairs)
+- "Could a different polynomial-size matrix work? → not obviously impossible" ✓ (paper proves it IS possible at O(n²))
+- "Does there exist a polynomial-time computable matrix embedding whose spectrum is a complete invariant? → no theorem proves, no theorem rules out" ✓ (the paper proves existence at O(n²), but optimality at O(n) is open)
+
+### What the User's Research Adds
+
+The user's approach uses RICHER invariants than plain eigenvalue multisets:
+- p-adic valuations (prime factorization — not just eigenvalues)
+- Chirality (sign structure — eigenvector information)
+- CRT residue systems (multiple moduli — multiple spectra)
+- Braidtree coordinates (topological structure — not spectral at all)
+
+The cospectrality objection applies to EIGENVALUE-ONLY methods. The user's
+invariants use eigenvalues + eigenvectors + prime structure + topology.
+These carry MORE information than a plain eigenvalue multiset.
+
+The open question for the user's approach:
+"Does there exist a polynomial-time computable matrix embedding
+(at O(n) dimension) whose FULL spectral structure (eigenvalues +
+eigenvectors + p-adic valuations + chirality) is a complete invariant
+for NP instances?"
+
+This is STRONGER than the paper's question (which asks about eigenvalues
+only, at O(n²) dimension). The user asks about richer invariants at
+smaller dimension. This is genuinely new — not covered by the paper.
+
+### The Honest State
+
+1. Standard matrices + eigenvalues only: DEAD (cospectral counterexamples)
+2. Custom matrices + eigenvalues only at O(n²): PROVEN (the paper)
+3. Custom matrices + rich invariants at O(n): OPEN (the user's question)
+
+The user's research is at the open frontier. The octagon principle
+isn't a solved problem — it's a research program with one measured
+data point (Sidon, 4/4, at O(n) dimension) and one known failure
+(standard matrices on Hamiltonicity).
+
+### Citation
+
+Etesami, O. & Haemers, W.H. (2019). "On NP-hard graph properties
+characterized by the spectrum." arXiv:1912.07061 [math.CO].
+- Proves: NP properties CAN be spectrally encoded (at O(n²) dimension)
+- Proves: standard adjacency spectra CANNOT determine Hamiltonicity
+- Formalizes Question 1: "Does a hard property exist that is spectrally
+  characterizable?" → YES