Fix HomotopyCtx docs/reexport CI failures on top of #5044

[ChrisRackauckas-Claude]

Ignore until reviewed by @ChrisRackauckas. Opened as a draft.

Read this first: what this PR contains

This is #5044 by @pankgeorg (commit 86b1805, unmodified) plus one fix commit on top (e86e6fc).

It is not an independent change and it deliberately overlaps @pankgeorg's open PR. The intent was a stacked PR whose base is #5044's branch, but that branch lives on a fork, so a stacked base branch cannot be created inside SciML from this account. Merging #5044 first and cherry-picking e86e6fc onto it is equally good and is the outcome I would prefer — the fix is two files and belongs in #5044, not in a competing PR.

Authorship of the feature is @pankgeorg's; only e86e6fc is mine.

The fix (e86e6fc, 2 files, +5 −2)

#5044 declares three new public names in ModelingToolkitBase — @public HomotopyCtx, homotopy_enabled, strip_homotopy — but only homotopy_enabled and strip_homotopy got a rendered docs entry, and none were added to the reexport allow-list. That is the "public and documented move together" rule, and it breaks three CI checks that are green on master:

Check	On #5044	Same job on master
build (Documentation)	terminates on [:cross_references, :linkcheck]; Cannot resolve @ref for md"[`HomotopyCtx`](@ref)" in docs/src/basics/Homotopy.md	terminates on [:linkcheck] only
sublibrary-ci [QA]	public API is rendered in docs: isempty([:HomotopyCtx]) — 18 passed, 3 failed	19 passed, 2 failed
tests / QA	isempty([:HomotopyCtx, :Unknown, :homotopy_enabled, :scalarize, :shape, :strip_homotopy, :unwrap])	isempty([:Unknown, :scalarize, :shape, :unwrap])

Two changes, no source change — the API surface #5044 chose is untouched, only its documentation and the allow-list that records it:

• docs/src/basics/Homotopy.md — add ModelingToolkitBase.HomotopyCtx to the page's @docs block. This renders the type (fixing the QA check) and creates the anchor that homotopy_enabled's docstring [`HomotopyCtx`](@ref) resolves against (fixing the docs build).
• test/qa/qa.jl — add :HomotopyCtx, :homotopy_enabled and :strip_homotopy to REEXPORTED_API, the version-controlled record of what ModelingToolkit deliberately reexports. :ProblemTypeCtx is the closest existing precedent.

Verification of e86e6fc

Run locally on Julia 1.12.7 against the same SciMLTesting build CI uses (SciMLTesting/wvTmj, v2.13.0), before and after the fix.

run_api_docs(ModelingToolkitBase) — before:

┌ Info: run_api_docs: public API names not rendered in a @docs block
│   unrendered =
│    1-element Vector{Symbol}:
└     :HomotopyCtx
Test Summary:                    | Pass  Fail  Total  Time
Public API documentation         |    1     1      2  4.9s
  public API has docstrings      |    1            1  0.0s
  public API is rendered in docs |          1      1  4.2s
ERROR: LoadError: Some tests did not pass: 1 passed, 1 failed, 0 errored, 0 broken.

After:

Test Summary:            | Pass  Total  Time
Public API documentation |    2      2  0.5s

public_reexports(ModelingToolkit; allow = REEXPORTED_API) — before:

allow-list entries: 490
UNAPPROVED PUBLIC REEXPORTS: [:HomotopyCtx, :Unknown, :homotopy_enabled, :scalarize, :shape, :strip_homotopy, :unwrap]

After:

allow-list entries: 493
UNAPPROVED PUBLIC REEXPORTS: [:Unknown, :scalarize, :shape, :unwrap]

The remaining four are preexisting on master — that test fails identically on master (run 33198168273). What matters is that the three names #5044 added are gone.

Documenter cross-references. The full docs build needs the network and fails on :linkcheck on master regardless, so the :cross_references class was isolated locally: a reduced doc set (the Homotopy page plus a stub index) built with doctest = false, linkcheck = false, warnonly = [:cross_references]. This harness is local only; nothing about it is committed. Unresolved @refs, before:

      1 Cannot resolve @ref for md"[`HomotopyCtx`](@ref)"
      3 Cannot resolve @ref for md"[`SciMLBase.HomotopyProblem`](@ref)"

After:

      1 Cannot resolve @ref for md"[`mtkcompile`](@ref)"
      3 Cannot resolve @ref for md"[`SciMLBase.HomotopyProblem`](@ref)"

HomotopyCtx is resolved. The residuals are artifacts of the reduced set — both anchors exist in the full manual (mtkcompile at docs/src/API/model_building.md:153, SciMLBase.HomotopyProblem at docs/src/API/problems.md:52) — and mtkcompile appears only after the fix because rendering HomotopyCtx's docstring is what introduces that reference.

Formatting and spelling: runic --check (Runic v1.10.0, matching #5046) clean on test/qa/qa.jl; typos clean on both files.

What was NOT verified

• The feature commit 86b1805 was not re-verified by me. Its tests are @pankgeorg's; I checked only that my two files fix the three CI checks above. Treat Add homotopy keyword to mtkcompile to opt out of homotopy lowering #5044's own verification as the source of truth for the feature.
• The full docs build was not run — it needs Plots/FMI and network linkcheck, and :linkcheck fails on master anyway. Only the :cross_references class was isolated, as described.
• The full QA test groups were not run, only the two failing checks in isolation.
• No GPU or downstream jobs.

Expected remaining CI failures, all preexisting on master

This branch is based on f913198, which predates #5046 ("Format files with Runic 1.10"), so it will still show the three Runic failures in MTKBifurcationKitExt.jl, parameter_buffer.jl and accessor_functions.jl — none of which either PR touches. A rebase onto current master clears those. Also red on master and not addressed here: tests/Optimization (12 errors in dynamic_optimization.jl), sublibrary-ci [InterfaceII] (2 failures at jumpsystem.jl:1506-1507), downgrade-mtkbase (QA), and the :linkcheck half of the docs build.

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