Summary

Fixes the mtkcompile/structural_simplify failure ArgumentError: The equations have at least one cycle. (from topsort_equations) when chaining two or more rotation-matrix (Euler) orientation states — SciML/ModelingToolkit.jl#4608.

Stacks on #106, which adds the opt-in _assert_b_free_of_scc_vars invariant. This PR makes that invariant hold by construction (and thus actually fixes #4608), rather than only checking it under MTKTEARING_CHECK_REDUCTION.

Root cause

get_linear_scc_linsol builds the inline A \ b block by extracting each SCC variable's coefficient out of every (total_sub-substituted) residual, gated on the structural incidence graph via has_edge(graph, eq, var). That graph is built before total_sub substitution and is mutated during reassembly, so it can desync from the residual: total_sub can reintroduce an SCC variable into a residual whose structural edge was already torn. The has_edge gate then skips expanding that variable into the coefficient matrix A, leaving it buried in b. __reduce_linear_system! — which only inspects coefficient rows, never b — subsequently smuggles it onto the RHS, producing a self-referential A \ b (the solution vector appears inside its own RHS). The resulting observed equations form a cycle that topsort_equations rejects.

Confirmed on the #4608 MWE: the buried variables are exactly the second joint's Euler rates j2.phid[1:3], all with has_edge == false. The world-rooted first joint stays in sync (constant R, zero angular velocity), which is why a single Euler joint compiles but two chained do not.

Fix

Gate extraction on the actual residual incidence (get_variables on the substituted residual) instead of the stale structural graph. Every SCC variable genuinely present in a residual is expanded into A, so b is free of SCC variables by construction. Blocks whose structural graph is already in sync get identical extraction and identical output — no behavior change on healthy models.

Verification (registry MTK 11.26.8)

A/B over models that expose the cycle and controls that do not:

All controls keep identical unknown counts → no regression. With MTKTEARING_CHECK_REDUCTION=1 the #106 self-check confirms the previously self-referential j2.phid block is now consistent (relres ≈ 4e-17).

Scope

Draft. This fixes the structural cycle only. Runtime solving of these models hits a separate, non-rank-tolerant inline-\ (LU) issue at degenerate configurations — independent of the cycle and out of scope here.

🤖 Generated with Claude Code