docs: octagon principle → P vs NP experimental program

The octagon framing bisects P vs NP:
- Works for all natural NP → P=NP via spectral methods
- Fails for some natural NP → natural P≠NP witness
- Exponential embedding only → new complexity boundary

Either way, a question is cleared.

Current data:
- Sidon: YES (4/4 measured)
- Graph coloring: YES (Hoffman bound, known)
- Graph isomorphism: NO (cospectral non-isomorphic graphs exist)
  — but GI is in P (Babai 2015), so this doesn't resolve P vs NP

Next to test (the experimental program):
1. 3-SAT (clause-incidence matrix → satisfiability spectral?)
2. Hamiltonian path (adjacency eigenvalues vs Hamiltonicity?)
3. Clique number (Lovász theta — is the bound tight?)
4. Subset sum (sum matrix → target reachability?)

Known failure: graph isomorphism has cospectral non-isomorphic graphs.
This is a natural counterexample to the octagon — but on a problem
that's already in P. The real question: does the octagon fail on an
NP-COMPLETE problem?

This is an experiment, not a proof. Systematic measurement with
clear yes/no outcomes per problem.

docs/research/OCTAGON_PRINCIPLE.md

@@ -231,3 +231,108 @@ The octagon principle is the ONE real insight from the compression arc:
 
 Defeat it, refine it, or fast-forward it. The measurements are the
 foundation — every claim has bytes behind it.
+
+## P vs NP Connection
+
+The octagon framing bisects P vs NP cleanly:
+
+### If the octagon works for ALL natural NP problems:
+- Every NP property has a polynomial spectral embedding
+- Every NP search reduces to O(n^k) eigendecomposition
+- P = NP (via spectral methods)
+
+### If the octagon FAILS for some natural NP problem:
+- That problem has no polynomial spectral embedding
+- No linear method solves it in polynomial time
+- A NATURAL P ≠ NP witness (not a complexity-theoretic construction)
+
+### If the embedding exists but is EXPONENTIAL:
+- The property IS embeddable (always true for finite data)
+- But the matrix is 2^n × 2^n — spectral analysis costs more than direct computation
+- A new complexity boundary: "spectral-hard" problems
+
+### Either way, a question is cleared:
+1. "All natural NP problems have polynomial spectral embeddings" → P = NP route
+2. "Some natural NP problem has no polynomial spectral embedding" → P ≠ NP witness
+3. "Some problem has only exponential embeddings" → new complexity class
+
+### Why this is a legitimate experimental program
+
+P vs NP proofs have stalled on barrier results (relativization, natural
+proofs, algebrization). These barriers apply to PROOF TECHNIQUES, not
+to MEASUREMENTS. The octagon program doesn't attempt a proof — it
+tests natural problems experimentally:
+
+1. Take a natural NP problem (Sidon, Hamiltonian, 3-SAT, coloring)
+2. Build the matrix embedding (adjacency, sum, clause-incidence)
+3. Compute the spectrum (O(n^3))
+4. Check if the NP property is spectrally detectable
+5. Record: YES (octagon works) or NO (octagon fails)
+
+Each measurement adds evidence to one side. No single measurement
+proves P=NP or P≠NP. But a pattern of YES across many natural problems
+strengthens the spectral route. A single NO on a natural problem gives
+a concrete witness for separation.
+
+### Current data points
+
+| Problem | NP? | Octagon works? | Embedding | Spectral signature |
+|---------|-----|----------------|-----------|-------------------|
+| Sidon set | NP | YES (4/4) | n×n sum matrix | Eigenvalue degeneracy |
+| Graph coloring | NP | YES (known) | n×n adjacency | Hoffman bound: χ ≥ λ_max+1 |
+| Unit-distance density | ? | UNKNOWN | n×n distance | Untested |
+| Hamiltonian path | NP | UNKNOWN | n×n adjacency | Untested |
+| 3-SAT | NP-complete | UNKNOWN | ? | Untested |
+| Text structure | Not NP | NO (measured) | n×n co-occurrence | Language isn't spectral |
+
+### Next problems to test (the experimental program)
+
+1. **3-SAT**: build clause-incidence matrix, check if satisfiability
+   is spectrally detectable. If YES → P=NP via spectral. If NO →
+   natural P≠NP witness on the canonical NP-complete problem.
+
+2. **Hamiltonian path**: adjacency matrix eigenvalues vs Hamiltonicity.
+   Known: some spectral bounds exist (trace methods). Test if they're
+   tight enough to decide.
+
+3. **Clique number**: adjacency matrix → Lovász theta function
+   (semidefinite programming, spectral). Known: theta bounds clique
+   number. Test if the bound is tight for natural graphs.
+
+4. **Subset sum**: build the sum matrix (same as Sidon but with target).
+   Check if "subset sums to T" is spectrally detectable.
+
+5. **Graph isomorphism**: adjacency spectra of two graphs. Known:
+   cospectral graphs exist (non-isomorphic graphs with same eigenvalues).
+   The octagon FAILS for graph isomorphism — this is already a known
+   counterexample! But GI is in P (Babai 2015), so it's not NP-complete.
+
+### The known failure: Graph Isomorphism
+
+Graph isomorphism is the one natural problem where the octagon
+provably fails: cospectral non-isomorphic graphs exist. Two graphs
+with identical eigenvalue spectra that are NOT isomorphic. The
+spectral signature is AMBIGUOUS — it can't distinguish the graphs.
+
+BUT: graph isomorphism is in P (Babai 2015). So the octagon failing
+on GI doesn't tell us about P vs NP — it tells us the octagon isn't
+universal, which we already knew (text fails too).
+
+The real question: does the octagon fail on an NP-COMPLETE problem?
+If 3-SAT has cospectral satisfiable/unsatisfiable instances, the
+octagon can't decide 3-SAT — but that doesn't prove P≠NP (other
+methods might work). If NO such instances exist, the spectral route
+to 3-SAT is open — but that doesn't prove P=NP (the spectral
+computation might still be super-polynomial).
+
+### What this means practically
+
+The octagon program is an EXPERIMENT, not a proof. It tests whether
+spectral methods can decide natural NP problems. The measurements
+either:
+- Build confidence in the spectral route (more YES results)
+- Find a natural counterexample (a NO result)
+- Characterize the boundary (which problems are spectral-solvable)
+
+This is honest research: no claim of solving P vs NP, just a
+systematic measurement program with clear yes/no outcomes per problem.