docs: SLOS direction analysis — pipeline needs SLOS only for Omega

The pipeline runs SLOS (K=2) to compute ONE number (Omega) from the
full M_n-state distribution. The other 4 queries (Sidon check, GCCL
gate, QR rank, collision count) don't need SLOS at all — they use
integer arithmetic or eigenvalue decomposition.

The shortcut: skip SLOS for 4/5 queries. 5x speedup from not running
expensive quantum simulations for queries that only need O(n²) or O(n³)
classical computation.

The Omega computation itself still needs full SLOS (K=2). The K=2
interference pattern IS the irreducible residual — the part the K=1
spectrum can't predict. Conservation law: spectrum (model) + K=2
interference (residual) = full distribution. Can't predict Omega from
spectrum alone.

This is the honest quantum advantage: SLOS computes something the
spectrum can't recover. Not quantum speedup — information content.
The K=2 correlations are fundamentally denser than the K=1 spectrum.

docs/weird_machine_conservation_law.md

@@ -222,3 +222,91 @@ doesn't matter whether the tape is stored, computed, or given by
 physics. The sparse structure you target must be genuinely sparse
 (low-rank, k-sparse, RIP-compliant), not just "looked up from a
 big table." Look-up = base conversion = no gain.
+
+## SLOS Direction: What the Pipeline Actually Needs
+
+### The Two Computation Paths
+
+| Path | What it computes | Cost | Gives |
+|------|-----------------|------|-------|
+| SLOS (Perceval, K=2) | Full output distribution, all M_n states | O(n × M_n) | Exact Omega |
+| QR/spectral (numpy, K=1) | Eigenvalue spectrum → first column of U | O(n³) | Approximate Omega |
+
+K=2 (SLOS) gives the correct Sidon vs non-Sidon signal (4/4).
+K=1 (spectral) gave an INVERTED signal initially.
+
+### What Each Pipeline Query Actually Needs
+
+| Query | Needs SLOS? | Why |
+|-------|-------------|-----|
+| Omega (exhaust mode probabilities) | YES (K=2) | K=1 approximation was wrong |
+| Sidon property check | NO | is_sidon() = pure integer arithmetic |
+| GCCL coherence gate | NO | IS the eigenvalue spectrum |
+| QR rank/spectrum | NO | O(n³) eigendecomposition |
+| Collision count | NO | Integer pairwise-sum check |
+
+The pipeline runs SLOS to compute ONE number (Omega) from the
+full M_n-state distribution. Everything else is computed without SLOS.
+
+### The Conservation Law Applied to SLOS
+
+The K=2 interference pattern IS the "residual" — the part the K=1
+spectrum can't predict. This is the same conservation law:
+
+  K=1 spectrum (program/model) + K=2 interference (residual) = full distribution
+
+The spectrum captures the sparse structure (block-diagonal modes,
+dominant eigenvalues). The K=2 interference captures the dense
+correlations (photon-photon interactions, Hong-Ouandel interference).
+
+You can't predict Omega from the spectrum alone because Omega depends
+on the K=2 interference — which IS the incompressible residual.
+
+### The Honest Application
+
+The shortcut for the SLOS direction is NOT "skip SLOS using the spectrum."
+The shortcut is: **don't run SLOS for queries that don't need it.**
+
+- Sidon check: run is_sidon() (O(n²) integer arithmetic, no SLOS)
+- GCCL gate: compute eigenvalue spectrum (O(n³), no SLOS)
+- QR rank: compute eigenvalue spectrum (O(n³), no SLOS)
+- Collision count: integer pairwise-sum check (O(n²), no SLOS)
+- Omega: MUST run SLOS (K=2 interference is irreducible)
+
+The pipeline was running SLOS for ALL queries when only Omega needed it.
+The shortcut: skip SLOS for 4/5 queries, run it only for Omega.
+
+### What This Saves
+
+For an n=8, m=8 system:
+- Full SLOS: O(8 × 6435) = O(51,480) operations
+- Skip SLOS for 4/5 queries: save 4 × 51,480 = 205,920 operations
+- Only run SLOS for Omega: 51,480 operations (1/5 of before)
+
+This is a real 5x speedup — not from compression, but from not
+running expensive quantum simulations for queries that only need
+integer arithmetic or eigenvalue decomposition.
+
+### What This Does NOT Save
+
+The Omega computation itself still needs full SLOS (K=2). The K=2
+interference pattern is the irreducible residual. The conservation
+law says: you can't predict it from the K=1 spectrum. The spectrum
+is the model; the K=2 interference is the tape. Both must be computed.
+
+### The Quandela SLOS as "Weird Machine"
+
+The SLOS IS a weird machine: a photonic quantum simulator repurposed
+as a Sidon set classifier. The "weirdness" is that a quantum optics
+experiment (designed for photon counting) is being used to detect
+combinatorial properties (Sidon sets) of integer sets.
+
+But the conservation law applies: the quantum computation (K=2 SLOS)
+gives information that the classical computation (K=1 spectrum) can't
+recover. This IS the quantum advantage — not in computation speed,
+but in information content. The K=2 interference pattern contains
+genuinely new information (the residual) that the K=1 spectrum lacks.
+
+This is the honest quantum advantage: SLOS computes something the
+spectrum can't predict. Not because of quantum speedup, but because
+the K=2 correlations are fundamentally denser than the K=1 spectrum.