docs: reconcile linearity — linear tool on linear problem works

Attack 5 said 'coherence is linear only, wrong for Sidon.'
SLOS analysis said 'spectrum works for SLOS.'
Both correct — different objects:

- Sidon SET = nonlinear (pairwise sums) → linear tool fails
- SLOS CIRCUIT = linear (unitary) → linear tool works

Principle: tool must match problem structure.
Linear problem → linear tool (spectrum) → works.
Nonlinear problem → nonlinear tool (is_sidon) → needed.

Conservation law final form:
- Linear systems: spectrum = full info (zero residual) → shortcut works
- Nonlinear systems: spectrum + interactions = full info → residual irreducible

Problem-specific admissibility confirmed: no universal check.
Each problem needs its own tool matching its structure.

docs/weird_machine_conservation_law.md

@@ -391,3 +391,60 @@ of the optical system. The conservation law says you can't reduce
 information. But for linear systems, the information IS already in the
 spectrum — you just need to compute the right function of it (the
 eigenvalue products, not just one column).
+
+## Reconciliation: Linear Tool on Linear Problem vs Linear Tool on Nonlinear Problem
+
+### The Apparent Contradiction
+
+Attack 5: "Coherence (Pearson) is linear only — wrong for Sidon sets"
+SLOS analysis: "Eigenvalue spectrum works for SLOS circuits"
+
+Both are correct. They talk about different objects:
+
+| Object | Structure | Linear tool works? | Why |
+|--------|-----------|-------------------|-----|
+| Sidon set | Nonlinear (pairwise sums) | NO | Pearson can't detect quadratic structure |
+| SLOS circuit | Linear (unitary matrix) | YES | U^(⊗m) determined by U's spectrum |
+
+### The Principle
+
+The tool must match the problem's structure:
+- Linear problem → linear tool (spectrum, QR) → works
+- Nonlinear problem → nonlinear tool (is_sidon, permanent) → needed
+- Universal linear tool on nonlinear problem → fails
+- Universal nonlinear tool on linear problem → overkill
+
+### Why the Coherence Gate Died (Attack 5)
+
+The coherence gate (Pearson correlation) tried to be a UNIVERSAL
+admissibility check. It's linear. Sidon structure is nonlinear
+(pairwise sums = quadratic). Linear tool on nonlinear problem = fail.
+
+### Why the SLOS Shortcut Works
+
+SLOS is linear optical. The circuit IS a unitary matrix. U^(⊗m) is
+determined by U's spectrum. The eigenvalue spectrum is a LINEAR tool
+on a LINEAR problem. It works.
+
+### The Conservation Law (Final Form)
+
+For LINEAR systems: spectrum = full information (zero residual)
+  → shortcut works (compute spectrum, skip full simulation)
+  → the cost is computational (M_n is large), not information-theoretic
+
+For NONLINEAR systems: spectrum + interactions = full information
+  → interactions are irreducible residual
+  → shortcut fails (must compute the full simulation)
+  → the cost IS information-theoretic (interactions create new info)
+
+### Problem-Specific Admissibility (Confirmed)
+
+Each problem needs its own admissibility check matching its structure:
+- SLOS (linear): eigenvalue product distribution
+- Sidon (nonlinear): is_sidon() integer check
+- cmix weights (linear): SVD singular values
+- Unit-distance (nonlinear): count_unit_distances()
+
+No universal check. The ManifoldShortcut framework's "problem-specific
+admissibility" (from the 5-way attack refinement) is confirmed by
+this linearity analysis.