T15 — GPU LDLᵀ/LDLᴴ in regimes A/B/C, diagonal solve, statistics #15

Task T15 of TASKS.md — milestone M4 — Symmetric indefinite LDLᵀ/LDLᴴ (MadNLP-ready).

Reads: PLAN §2.4, §3.3, §2.6 (sytrf), §1.4.

Depends on: #14 (T14) must be merged first; tasks are strictly sequential.

Scope and deliverables: the section T15 of TASKS.md on main is the single source of truth; it is not copied here so it cannot drift. Read it together with the PLAN.md sections it names and the Report blocks of the previous tasks.

Definition of done (tests: test/test_numeric_ldlt.jl, ported tests, CPU + CUDA):

• Every test listed in the task passes on the KA CPU backend (GitHub runner) and on the CUDA/AMDGPU self-hosted runners (Run tests and Aqua checks green on the PR).
• The task's status marker in TASKS.md is set and its Report block is filled in honestly (CUDA counts come from CI).
• A PR titled T15: … from branch task/T15-<slug> with Closes #15 in its body, reviewed by the Claude reviewer with VERDICT: APPROVE.
• Out-of-scope bugs or follow-ups found on the way are opened as separate issues (label found-by-agent), not fixed silently.

How it starts: add the label claude:implement (the pipeline does this automatically after the previous task's PR merges; label on-hold to prevent that).

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