SimpleNonlinearSolve: single branchless loops that trace under Reactant

[ChrisRackauckas-Claude]

Draft. Ignore this PR until reviewed by @ChrisRackauckas. Stacked on #1197 (NonlinearSolveBase/first-order Reactant support); review the last commit only.

What changed and why

Each SimpleNonlinearSolve iteration (SimpleNewtonRaphson, SimpleTrustRegion, SimpleBroyden, SimpleKlement, SimpleHalley) is now written once as a branchless loop that runs unchanged on the host and under Reactant.@compile:

• for _ in 1:maxiters with solved && return becomes ReactantCore.@trace track_numbers = false while (!solved) & (iterations < maxiters); the "initial guess is already a root" early return is subsumed (the loop does not execute and simple_solution reports Success). @trace is a compile-time no-op outside Reactant.
• Decisions that used to return or branch (SimpleHalley's factorization failure, SimpleTrustRegion's shrink-threshold exit and step acceptance, SimpleKlement's Jacobian reset) are loop conditions or ifelse selections. dogleg_method!! keeps its two host early-outs behind !within_compile() with the selection as the fallthrough.
• Utils.init_loop_state promotes the return code and counter with NonlinearSolveBase.maybe_traced and makes the loop-carried arrays distinct; Utils.fresh re-applies that after check_termination (whose best-iterate result aliases the termination cache) and after @bb copyto! (which rebinds rather than copies for traced arrays). Both are no-ops on the host.
• Utils.simple_solution builds the NonlinearSolution from solved with ifelse/select; the previous _build_simple_solution, _copy_for_reactant, reactant.jl and the nonlinear_solution_new_alg shadow-solution hack are removed, as is the Base.@noinline on the safe-best check_termination (its tests pass on 1.10 without it).

SimpleTrustRegion's host loop now forms the candidate Jacobian on every iteration and selects it (previously only accepted steps recomputed it). The result-type assertions in simple_solution exist because SimpleTrustRegion with autodiff = nothing has a run-time-typed Jacobian (prepare_jacobian returns a non-concrete DIExtras), which is pre-existing but previously never reached the return type; with them @inferred solve(...) holds on 1.10 and 1.12.

Diff on top of the core PR: 9 files, +237/−102.

Verification

Same environment as the core PR (Reactant#3232, SciMLBase#1563, DI 571fc178), Julia 1.12.4 unless noted.

Reactant group with the full matrix (Simple solvers included):

Test Summary:        | Pass  Total     Time
Reactant Integration |  147    147  8m42.9s

SimpleNonlinearSolve's own groups with this branch's NonlinearSolveBase developed in:

GROUP=Core  Julia 1.10.11   Testing SimpleNonlinearSolve tests passed
GROUP=Core  Julia 1.12.4    Testing SimpleNonlinearSolve tests passed
GROUP=QA    Julia 1.12.4    Testing SimpleNonlinearSolve tests passed

Before the select/simple_solution result-type assertions, the 1.10 Core group failed @inferred solve(prob, SimpleTrustRegion()) in the root-finding snippet (return type NonlinearSolution{_A, _B, _C, _D, …}); with them it passes on both versions.

Formatted with Runic 1.10.0; typos clean.

Not verified

• SimpleDFSane and SimpleLimitedMemoryBroyden are unchanged here (SimpleDFSane's line search branches on traced values and is still unsupported under Reactant; limited-memory Broyden already traced).
• GPU groups; GROUP=Everything.

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🤖 Generated with Claude Code (model: claude-fable-5) on behalf of Chris Rackauckas. Session: https://claude.ai/code/session_016LsC6pp9z6s5EABX9DnVjE

[ChrisRackauckas-Claude]

Re-stacked on the rebased #1197 (after #1072); NonlinearSolveBase compat is now 2.50. Full Reactant matrix 147/147 and Simple Core 1.10/1.12 + QA pass locally on this stack.

🤖 Posted by an AI agent (Claude Code, model claude-fable-5) on behalf of Chris Rackauckas. Session: https://claude.ai/code/session_016LsC6pp9z6s5EABX9DnVjE

[ChrisRackauckas-Claude]

CI triage: the red OrdinaryDiffEq.jl/* lanes (Downstream workflow) fail during environment resolution — Unsatisfiable requirements detected for package AlgebraicMultigrid against NonlinearSolveBase's TimerOutputs = "1" compat — and they fail identically on master's own push run 33303882132 (and on #1072's last run), after having passed earlier the same day, so it is a registry-side change unrelated to this PR. It is being investigated separately.

🤖 Posted by an AI agent (Claude Code, model claude-fable-5) on behalf of Chris Rackauckas. Session: https://claude.ai/code/session_016LsC6pp9z6s5EABX9DnVjE