Raise HAdaptiveIntegration minimum to 0.2.2

[ChrisRackauckas-Claude]

Summary

• Raise the HAdaptiveIntegration compatibility floor from 0.2 to 0.2.2.
• Preserve support for HAdaptiveIntegration 0.2.2 and later 0.2.x/1.x releases.

Please ignore this draft until reviewed by @ChrisRackauckas.

Why

The extension constructs HAdaptiveIntegration.Segment, but releases 0.2.0 and 0.2.1 do not define that API. The downgrade resolver therefore selects an allowed dependency version that makes all focused HAdaptiveIntegration backend tests error with Segment not defined. HAdaptiveIntegration 0.2.2 introduced the required API in upstream commit 7ba2881.

Integrals commit b91eb0fd introduced the wrapper, its tests, and the 0.2 floor together, so it is the first bad Integrals commit.

This is intentionally separate from other independent downgrade-floor failures.

Local verification

• Clean Integrals master with exact HAdaptiveIntegration 0.2.0: reproduced 7/7 focused test errors with Segment not defined.
• Exact HAdaptiveIntegration 0.2.1: reproduced the same 7/7 errors.
• Exact HAdaptiveIntegration 0.2.2, Julia 1.10.11, --check-bounds=yes --depwarn=error: focused backend tests passed (9/9).
• Deployed downgrade.jl dependency resolver after this change: selected HAdaptiveIntegration 0.2.2.
• Julia 1.10.11, current dependencies, GROUP=Core Pkg.test(), --check-bounds=yes --depwarn=error: Integrals tests passed (1,741 assertions).
• Runic 1.7.0 --check --verbose .: all 37 Julia files passed.

[ChrisRackauckas-Claude]

Investigation scratchpad for commit c2630d87:

1. Reproduced from clean Integrals base 218dbe4e with Julia 1.10.11 and a fresh depot. With exact HAdaptiveIntegration 0.2.0, all seven focused backend testsets errored:

   UndefVarError: Segment not defined
   .../ext/IntegralsHAdaptiveIntegrationExt.jl:10
   Test Summary: HAdaptiveIntegrationJL | Error 7 | Total 7
   

2. Tested the intervening 0.2.1 release; it produced the same 7/7 errors.
3. Tested exact 0.2.2 under --startup-file=no --check-bounds=yes --depwarn=error; the existing focused backend suite passed 9/9.
4. Traced the Integrals-side boundary to b91eb0fd (2026-01-29), which added the wrapper/tests with a 0.2 floor while already calling Segment.
5. Traced the dependency-side boundary to HAdaptiveIntegration commit 7ba2881, released as 0.2.2, which added the required Segment API.
6. Raised only the root floor to 0.2.2; this weak dependency has no separate AD compat entry.
7. Ran the same deployed Resolver/downgrade script in dependency mode; it selected HAdaptiveIntegration 0.2.2 and completed resolution.
8. Ran GROUP=Core Pkg.test() with Julia 1.10.11, current dependencies, --startup-file=no --check-bounds=yes --depwarn=error; all eight Core files passed, totaling 1,741 assertions.
9. Ran Runic 1.7.0 over the whole repository in check mode; 37/37 Julia files passed.
10. Fetched upstream/master after testing; the branch was not behind before publication.

No upstream HAdaptiveIntegration issue/PR was opened because the missing API is already fixed in 0.2.2 and that repository is outside the repositories authorized for autonomous writes.

[ChrisRackauckas-Claude]

Final validation update:

• GitHub Actions is fully green: downgrade, AD, Core on LTS/current/prerelease Julia, QA, docs, spelling, Runic, and Runic suggestions all passed.
• The downgrade job selected the corrected floor and passed its Core tests end-to-end in 8m22s.
• After a final git fetch upstream, the branch is not behind upstream/master, the worktree is clean, and GitHub reports the draft PR mergeable.