Fix MIRK extrema boundary conditions under AD

[ChrisRackauckas-Claude]

Ignore this PR until reviewed by @ChrisRackauckas.

Summary

• Make MIRK dense interpolation evaluate against dual-compatible stage caches during boundary-condition AD.
• Use the dense AD mode for MIRK boundary residual construction.
• Strip nested dual values before solving the auxiliary extrema root problems used by maxsol/minsol, via BoundaryValueDiffEqCore.nodual_value rather than direct MIRK access to ForwardDiff internals.
• Add a regression test for MIRK4 with maxsol/minsol boundary conditions.
• Keep direct interpolation reads on primal DiffCache stages so sparse constraint tracing does not read uninitialized AD temporaries.

Local Verification

• Exact pendulum maxsol/minsol MIRK4 reproducer under Julia 1.10 -> SciMLBase.successful_retcode(extrema_sol) = true
• Exact pendulum maxsol/minsol MIRK4 reproducer under Julia 1.12 -> SciMLBase.successful_retcode(extrema_sol) = true
• Targeted Julia 1.12 optimizer reproducer with local Core/MIRK and OptimizationIpopt: bounded constraint path -> SciMLBase.successful_retcode(sol1) = true; no-user-bounds optimization path -> SciMLBase.successful_retcode(sol2) = true
• BOUNDARYVALUEDIFFEQ_TEST_GROUP=QA timeout 3600 julia +1.10 --project=lib/BoundaryValueDiffEqMIRK -e 'using Pkg; Pkg.test()' -> Quality Assurance 15/15 passed
• git diff --check
• Runic formatter check with Runic.main(["-c", "-d", "."]) passed under Julia 1.12

CI Status on 6614a935

• MIRK Core passed on Julia 1, lts, and pre.
• MIRK AD passed on Julia 1 and lts.
• MIRK QA passed.
• Repo Misc/QA/Wrappers, docs, benchmarks, Runic, runic-suggestions, and spell check passed.
• MIRK AD on Julia pre fails with the master-side Mooncake uppercase(::Char) issue tracked in MIRK AD Julia pre fails with Mooncake uppercase(::Char) during matvec differentiation #543.
• FIRK/MIRK downgrade failures are the SciMLBase compat-floor issue split into Raise FIRK/MIRK downgrade compat floors #540.
• Shooting downgrade failure is the dual ODE cache issue split into Fix MultipleShooting caches for tagged AD states #541.
• ModelingToolkit downstream failed because the self-hosted runner lost communication with GitHub; no downstream test failure log was produced, and GitHub refused a rerun for that workflow run.

[github-actions]

Benchmark Results (Julia v1.11)

Time benchmarks

	master	4ef52d1...	master / 4ef52d1...
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK2()	0.584 ± 0.012 s	0.573 ± 0.009 s	1.02 ± 0.026
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK3()	12.3 ± 0.21 ms	12.1 ± 0.24 ms	1.02 ± 0.027
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK4()	2.7 ± 0.1 ms	2.65 ± 0.091 ms	1.02 ± 0.052
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK5()	3.29 ± 0.11 ms	3.18 ± 0.13 ms	1.03 ± 0.056
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK6()	1.58 ± 0.063 ms	1.54 ± 0.051 ms	1.02 ± 0.053
Simple Pendulum/IIP/MultipleShooting(10, Tsit5; grid_coarsening = false)	1.55 ± 0.098 ms	1.52 ± 0.072 ms	1.02 ± 0.081
Simple Pendulum/IIP/MultipleShooting(10, Tsit5; grid_coarsening = true)	3.29 ± 0.19 ms	3.17 ± 0.15 ms	1.04 ± 0.077
Simple Pendulum/IIP/MultipleShooting(100, Tsit5; grid_coarsening = false)	0.0406 ± 0.0038 s	0.0397 ± 0.0033 s	1.02 ± 0.13
Simple Pendulum/IIP/MultipleShooting(100, Tsit5; grid_coarsening = true)	0.0641 ± 0.0022 s	0.0629 ± 0.0027 s	1.02 ± 0.057
Simple Pendulum/IIP/Shooting(Tsit5())	0.177 ± 0.076 ms	0.174 ± 0.073 ms	1.02 ± 0.61
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK2()	0.74 ± 0.0075 s	0.72 ± 0.0057 s	1.03 ± 0.013
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK3()	15.9 ± 5.5 ms	15.3 ± 4.6 ms	1.04 ± 0.48
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK4()	3.23 ± 0.16 ms	3.16 ± 0.15 ms	1.02 ± 0.07
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK5()	3.97 ± 0.21 ms	3.84 ± 0.26 ms	1.03 ± 0.09
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK6()	1.86 ± 0.11 ms	1.85 ± 0.12 ms	1 ± 0.086
Simple Pendulum/OOP/MultipleShooting(10, Tsit5; grid_coarsening = false)	3.4 ± 0.68 ms	3.37 ± 0.85 ms	1.01 ± 0.32
Simple Pendulum/OOP/MultipleShooting(10, Tsit5; grid_coarsening = true)	6.56 ± 6 ms	6.5 ± 5.5 ms	1.01 ± 1.3
Simple Pendulum/OOP/MultipleShooting(100, Tsit5; grid_coarsening = false)	0.0856 ± 0.003 s	0.0833 ± 0.0022 s	1.03 ± 0.045
Simple Pendulum/OOP/MultipleShooting(100, Tsit5; grid_coarsening = true)	0.135 ± 0.0032 s	0.128 ± 0.0032 s	1.05 ± 0.036
Simple Pendulum/OOP/Shooting(Tsit5())	0.607 ± 0.047 ms	0.595 ± 0.048 ms	1.02 ± 0.11
time_to_load	7.21 ± 0.018 s	7.06 ± 0.019 s	1.02 ± 0.0038

Memory benchmarks

	master	4ef52d1...	master / 4ef52d1...
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK2()	0.389 M allocs: 0.0443 GB	0.389 M allocs: 0.0443 GB	1
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK3()	0.0436 M allocs: 4.88 MB	0.0436 M allocs: 4.88 MB	1
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK4()	15.7 k allocs: 1.65 MB	15.7 k allocs: 1.65 MB	1
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK5()	22.1 k allocs: 2.03 MB	22.1 k allocs: 2.03 MB	1
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK6()	12.8 k allocs: 1.05 MB	12.8 k allocs: 1.05 MB	1
Simple Pendulum/IIP/MultipleShooting(10, Tsit5; grid_coarsening = false)	25.5 k allocs: 1.82 MB	25.5 k allocs: 1.82 MB	1
Simple Pendulum/IIP/MultipleShooting(10, Tsit5; grid_coarsening = true)	0.0491 M allocs: 3.39 MB	0.0491 M allocs: 3.39 MB	1
Simple Pendulum/IIP/MultipleShooting(100, Tsit5; grid_coarsening = false)	0.553 M allocs: 0.0535 GB	0.553 M allocs: 0.0535 GB	1
Simple Pendulum/IIP/MultipleShooting(100, Tsit5; grid_coarsening = true)	0.833 M allocs: 0.0778 GB	0.833 M allocs: 0.0778 GB	1
Simple Pendulum/IIP/Shooting(Tsit5())	4.63 k allocs: 0.224 MB	4.63 k allocs: 0.224 MB	1
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK2()	0.89 M allocs: 0.984 GB	0.89 M allocs: 0.984 GB	1
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK3()	0.0932 M allocs: 24.8 MB	0.0932 M allocs: 24.8 MB	1
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK4()	0.0324 M allocs: 3.96 MB	0.0324 M allocs: 3.96 MB	1
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK5()	0.045 M allocs: 4.99 MB	0.045 M allocs: 4.99 MB	1
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK6()	25.3 k allocs: 2.17 MB	25.3 k allocs: 2.17 MB	1
Simple Pendulum/OOP/MultipleShooting(10, Tsit5; grid_coarsening = false)	0.142 M allocs: 10.2 MB	0.142 M allocs: 10.2 MB	1
Simple Pendulum/OOP/MultipleShooting(10, Tsit5; grid_coarsening = true)	0.266 M allocs: 18.7 MB	0.266 M allocs: 18.7 MB	1
Simple Pendulum/OOP/MultipleShooting(100, Tsit5; grid_coarsening = false)	2.52 M allocs: 0.279 GB	2.52 M allocs: 0.279 GB	1
Simple Pendulum/OOP/MultipleShooting(100, Tsit5; grid_coarsening = true)	3.82 M allocs: 0.404 GB	3.82 M allocs: 0.404 GB	1
Simple Pendulum/OOP/Shooting(Tsit5())	0.0373 M allocs: 1.69 MB	0.0373 M allocs: 1.69 MB	1
time_to_load	0.159 k allocs: 11.2 kB	0.159 k allocs: 11.2 kB	1

[ChrisRackauckas-Claude]

Update pushed in c306f64.

Additional local verification for the Mooncake Julia-pre failure:

timeout 3600 env JULIA_DEPOT_PATH=/home/crackauc/sandbox/tmp_20260708_121627_10236/julia_depot_mirk_mooncake_matvec /home/crackauc/.juliaup/bin/julia +1.13 --startup-file=no --project=lib/BoundaryValueDiffEqMIRK/test/AD -e 'import Pkg; Pkg.instantiate(); include("lib/BoundaryValueDiffEqMIRK/test/AD/ad_tests.jl")'\n```\n\nObserved result:\n\n```\nTest Summary:              | Pass  Total      Time\nDifferent AD compatibility |    9      9  15m01.1s\n```\n\nAlso ran:\n\n```\ntimeout 3600 julia +1 --startup-file=no --project=.runic_env -m Runic --check lib/BoundaryValueDiffEqMIRK/src/interpolation.jl lib/BoundaryValueDiffEqMIRK/src/collocation.jl\n```\n\nRunic exited successfully.

[ChrisRackauckas-Claude]

Final-state verification after the variable rename commit 4ef52d14:

timeout 3600 env JULIA_DEPOT_PATH=/home/crackauc/sandbox/tmp_20260708_121627_10236/julia_depot_mirk_mooncake_matvec /home/crackauc/.juliaup/bin/julia +1.13 --startup-file=no --project=lib/BoundaryValueDiffEqMIRK/test/AD -e 'import Pkg; Pkg.instantiate(); include("lib/BoundaryValueDiffEqMIRK/test/AD/ad_tests.jl")'\n```\n\nObserved result:\n\n```\nTest Summary:              | Pass  Total      Time\nDifferent AD compatibility |    9      9  14m57.5s\n```\n\nLocal checks after `4ef52d14`:\n\n```\ntimeout 3600 julia +1 --startup-file=no --project=.runic_env -m Runic --check lib/BoundaryValueDiffEqMIRK/src/interpolation.jl lib/BoundaryValueDiffEqMIRK/src/collocation.jl\n/home/crackauc/.local/bin/typos lib/BoundaryValueDiffEqMIRK/src/interpolation.jl lib/BoundaryValueDiffEqMIRK/src/collocation.jl\ngit diff --check\n```\n\nAll exited successfully.

[ChrisRackauckas-Claude]

Downstream audit replay for head 4ef52d1443062cf0e6877fa3338752c35af5939f:

I ran the exact current ModelingToolkitBase BVPOnly target under Julia 1.12 with this PR's local BoundaryValueDiffEq source projects. The test group genuinely executed; it did not silently select zero tests.

Observed terminal summary:

Optimal Control + Constraints Tests | 20 pass, 2 fail, 1 error, 5 broken

The two failures are the cost-function compilation checks evaluating NaN ≈ NaN; the error is the parameter-estimation optimization path. The testset reported 51m36s before exiting 1. I therefore cannot claim the complete MTKBase BVP target passes against this head.

This result is relevant to the caller-workflow correction in #563: that PR points CI at the right moved test group, and the remaining red result is genuine source/integration behavior rather than a workflow startup failure or no-op.

[ChrisRackauckas-Claude]

Rebased this branch onto current master (5e9eb4e) and reran the affected lanes locally.

PR-specific validation:

• Julia 1.12 MIRK AD group: 9/9 passed (MIRK AD Tests, 18m03.1s).
• Julia 1.12 MIRK Core reached the changed interpolation block: 80/80 passed.
• repository-wide Runic check passed; git diff --check passed.

The broader current-dependency Core run ended at 195 passed, 2 failed, and 1 errored. All three outcomes are independently reproduced on untouched current master and are unrelated to this interpolation patch:

• The error is the OptimizationBase 5.2.1 / OptimizationIpopt 1.3.0 generic-cache regression introduced by Optimization PR #1257; Optimization PR #1270 passes the exact BVP reproduction.
• The two failures are successful Ipopt solves rejected only by existing @test_nowarn assertions after OptimizationBase began emitting missing_second_order_ad. Both focused testsets reproduce identically on clean BVP master with OptimizationIpopt 1.3.0 and pass with 1.2.3. Optimization PR #1269 covers the explicit sparse-second-order handling.

No test was skipped, disabled, silenced, or loosened. The branch changes themselves remain the original five focused commits.

[ChrisRackauckas-Claude]

CI classification for rebased head 3ed990db:

• MIRK AD passed on Julia lts and pre. The Julia 1 AD job lost communication with its self-hosted runner while the test step was active, so it produced no test failure.
• MIRK Core on Julia lts/1/pre and the MIRK downgrade job reach the same current-dependency result reproduced on clean master: 195 passed, 2 failed, 1 errored. The changed interpolation testset is 80/80 and the maxsol/minsol regression is 3/3. The residuals are the OptimizationBase/OptimizationIpopt issues covered by Optimization.jl #1269 and #1270.
• FIRK downgrade also lost communication with its self-hosted runner and produced no test failure.
• Shooting downgrade reproduces the dual ODE-cache error already split into Fix MultipleShooting caches for tagged AD states #541.
• ModelingToolkit downstream genuinely ran to 7242 passed, 2 failed, 1 errored, 5 broken. Its BVP residuals match the same Optimization integration failures reproduced locally on clean BVP master; this is not an interpolation regression.
• The benchmark check failed before checkout because the trusted pull_request_target workflow refused to execute fork PR code. No benchmark code ran in that check.
• Repository Misc/QA/Wrappers, docs, Runic, spell check, MIRK QA, MIRK AD lts/pre, and the remaining downgrade sublibraries are green.

No additional PR-specific failure is exposed by this run.