Use PureGebal for the Base-exp balancing path

[ChrisRackauckas-Claude]

Please ignore until reviewed by @ChrisRackauckas. Draft.

⚠️ Blocked on registration

PureGebal is not in the General registry yet — JuliaRegistries/General#163748 is open, and new packages sit for three days. CI on this branch will fail at Pkg.instantiate until it merges. Local verification below used a dev'd PureGebal. Nothing to fix here; it just cannot go green yet.

Summary

Replace the hand-written LAPACK GEBAL ccall with PureGebal.

That wrapper existed for one reason: LinearAlgebra.LAPACK.gebal! allocates its scale vector on every call, which the Krylov hot path cannot afford, so this package duplicated ~20 lines of stdlib ccall to get a caller-supplied buffer. PureGebal provides that buffer as a first-class workspace, so the duplication goes away.

• the cache's bare scale vector becomes a PureGebal.GebalWorkspace
• the hand-rolled undo — the diagonal rescale plus the two permutation replays that decode LAPACK's packed scale layout — becomes PureGebal.unbalance!

+15 / −81 across four files.

This removes all dependence on non-public LAPACK internals

Six ExplicitImports exceptions existed solely for that wrapper and are deleted: @blasfunc, BlasInt, chkfinite, chklapackerror, chkstride1, libblastrampoline — along with the entire all_qualified_accesses_via_owners block. What remains (promote_op, arithmetic_closure, parameterless_type) are other packages' non-public names, unrelated to balancing.

Behaviour

Unchanged. PureGebal is bit-identical to LAPACK.gebal! on the inputs this path sees (asserted in its own suite across dense, wide-dynamic-range, sparse, triangular and block-isolated families), and the balanced Padé evaluation still runs at zero allocations — checked directly, with cache[7]::PureGebal.GebalWorkspace{Float64}.

Local verification

• Julia 1.12.6 Core: 742 passed, 1 pre-existing broken
• Julia 1.12.6 QA: 22 passed, 2 pre-existing broken
• A master baseline run of the same suite gives 742 passed, 1 broken — identical, so no test silently went missing
• Runic 1.5.1 and git diff --check: clean
• Julia 1.10 Core and QA: in flight; I will post the numbers here rather than assume them

Deliberately not in this PR

src/exp_noalloc.jl still balances via GenericSchur.balance! for ExpMethodHigham2005, which allocates (~672 B at n=30). PureGebal could replace that too and remove the allocation, plus the separate prow/pcol undo path — but it changes the generic (non-BLAS) path's behaviour, so it belongs in its own PR.

This is independent of #267, which is documentation only.

[ChrisRackauckas-Claude]

Local verification complete

Both Julia versions, with a master baseline run of the same suite on each so the numbers are attributable rather than assumed.

Run	This branch	master baseline
Core, Julia 1.12.6	742 passed, 1 broken	742 passed, 1 broken
Core, Julia 1.10.11	741 passed, 1 broken	741 passed, 1 broken
QA, Julia 1.12.6	22 passed, 2 broken	—
QA, Julia 1.10.11	20 passed, 2 broken	—

Every broken is pre-existing (the AllocCheck static-analysis markers in the QA group, and the one in Core).

Two count differences that are not attributable to this PR:

• Core is 741 on 1.10 vs 742 on 1.12 — master shows exactly the same split, so it is a version-conditional assertion in the suite.
• QA is 20 on 1.10 vs 22 on 1.12 — the same split appeared on an unrelated branch of this repo earlier today, so it is the Julia version, not the exception list this PR shortens.

Also: Runic 1.5.1 and git diff --check clean, and the balanced Padé path re-checked at zero allocations with cache[7]::PureGebal.GebalWorkspace{Float64}.

CI here will stay red until JuliaRegistries/General#163748 merges — the local runs above used a dev'd PureGebal.

For what it is worth, PureGebal's own CI is green on Julia 1 and lts across Ubuntu, macOS and Windows, so the balancing code this now depends on has cross-platform evidence behind it.