[Montana]

Part of #961: lifts the item "A $-fixed species in a model with no conservation law and no rule species is refused as 'dY_ss/dp does not exist'". It also lifts the same refusal beside a conservation law, which the item didn't list.

What was wrong

A $-fixed species has no equation, so its row of J is 0. compute_ss_sensitivity solved the columns −J⁻¹·∂f/∂p over the unknowns, and kept a fixed species among them in two cases:

• No law, no mask: the full-species solve, and the fixed species' zero row made J singular.
• Beside a conservation law (unmasked): ss_unknown_species returns the law's independents verbatim, with the fixed species among them. Singular the same way.

On main, both raise SimulationError: Steady-state dY_ss/dp does not exist for this model: the Jacobian at the steady state is singular … for columns that exist:

model	exact	main	here
$S makes X at k1, X lost at k2; S set by S0	dS/d[k1,k2,S0] = [0,0,1], dX = [S/k2, −k1·S/k2², k1/k2] = [6, −24, 4]	refused	exact
A ⇌ B (kf, kr), and $S turns A into B at k1; A + B = 1	B* = a/(a+kr), a = kf + k1·S	refused	exact

The change

The columns' unknowns no longer include a fixed species. The solve for the state is unchanged: ss_unknown_species keeps its contract for its other two callers, the Newton solve and the eigenvalues.

The diff (src/steady_state.cpp)

@@ compute_ss_sensitivity
-    if (!all_laws.empty() || sub.masked) {
+    // A `$`-fixed species is no unknown either (issue #961): solved for as one,
+    // its zero row made J singular in a model with no law and no mask, and the
+    // columns were refused as "dY_ss/dp does not exist".
+    const bool any_fixed = std::any_of(model.species().begin(), model.species().end(),
+                                       [](const Species &s) { return s.fixed; });
+    if (!all_laws.empty() || sub.masked || any_fixed) {
         // Reduced-space sensitivity solve.
@@
-        ss_kept_species_system(model, all_laws, sub, ss_unknown_species(model, all_laws, sub), cl,
-                               solve_idx);
+        // A fixed species is no unknown of the columns (issue #961): its row of
+        // J is 0, and kept, it made the solve singular, beside a law as well as
+        // with none. Its own row of dY_ss/dp is what a parameter sets it to.
+        std::vector<int> unknowns = ss_unknown_species(model, all_laws, sub);
+        unknowns.erase(std::remove_if(unknowns.begin(), unknowns.end(),
+                                      [&](int i) { return model.species()[i].fixed; }),
+                       unknowns.end());
+        ss_kept_species_system(model, all_laws, sub, unknowns, cl, solve_idx);

Tests

python/tests/test_steady_state_sensitivity_fixed_species.py (5 tests):

• the two models above, held to their closed forms at rtol 1e-6, including the fixed species' own row (dS/dS0 = 1). Both fail on main;
• a model with no fixed species (control);
• the state itself, under integration and Newton, unchanged (control).

Mutation tests (the core rebuilt for each):

mutant	killed by
no any_fixed gate	the no-law test
no filter at the call site	the beside-a-law test

Corpus

steady_state(sensitivity_params=<first 4 parameters>) on 1,500 models (1,323 BioModels, 177 .net files in the repo), each in its own process, on main and here:

outcome	models
solved on both, bit-identical (state and columns)	566
refused on main, solved here	44
refused on both, the same reason	881
crashed or timed out on both	9
solved on main, refused here	0
solved on both, changed	0

The 44 were checked against central differences of the steady state itself, at relative steps 1e-5 and 1e-3 per parameter:

• 34 agree to 1e-4 of their column's scale or better.
• 3 have columns that are zero to rounding on both sides: BIOMD0000000289 and 290 (1e-22 to 1e-62 against differences of 1e-14 to 1e-27 that change sign with the step), and sink_compart.net (0 against 1e-10 to 1e-9 of noise).
• 5 agree within the differences' own noise, settling at one of the two steps. For example, BIOMD0000000325's BR is 2.51032 analytic, against 5.46 at step 1e-5 and 2.51011 at 1e-3. The others are MODEL6655501972, BIOMD0000000221, Motivating_example.net and BIOMD0000000015.
• 2 agree to 6e-4 at the single step 1e-5 and were not inspected further: BIOMD0000000190 (5.9e-4) and 222 (2.5e-4).

Checks (macOS arm64)

• python/tests: 17,209 passed, 0 failed.
• parity_checks/tests: 1,015 passed. The 2 failures (test_bngsim_provenance and the reference-archive replay) also fail on main here: their subprocess can't import bngsim without my local import shim.

pre-commit run --all-files, sphinx -W, and ci/changelog.py check/required (changelog.d/961.fixed.3.md) are clean.