refactor(background-task): delete dead code in modules/ directory

[iyoda]

Summary

Delete the entire src/tools/background-task/modules/ directory (6 files, ~827 lines) which is confirmed dead code — zero imports from anywhere in the codebase.

Changes

• Deleted: src/tools/background-task/modules/background-cancel.ts (116 lines)
• Deleted: src/tools/background-task/modules/background-output.ts (137 lines)
• Deleted: src/tools/background-task/modules/background-task.ts (105 lines)
• Deleted: src/tools/background-task/modules/formatters.ts (315 lines)
• Deleted: src/tools/background-task/modules/message-processing.ts (75 lines)
• Deleted: src/tools/background-task/modules/utils.ts (65 lines)

Verification

• bun run typecheck ✅
• bun test ✅ (2693 pass, 0 fail)
• bun run build ✅

PR Stack

This is PR 0 of 9 in the code deduplication refactor series. Merge this first.

#	PR	Description
0	This PR	Delete dead code in modules/
2	(next)	Consolidate formatDuration (4→1)
3		Consolidate getMessageDir (7→1)
4		Unify fuzzyMatchModel
5		Add json-cache utility
6		Apply json-cache to consumers
7		Add ensure-directory utility
8		Apply ensure-directory to production
9		Consolidate ToolContextWithMetadata

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Summary by cubic

Removed dead background-task modules. Trims ~827 LOC with no behavior changes.

• Refactors
  • Deleted src/tools/background-task/modules/ (6 unused files).

^{Written for commit f2be9b4. Summary will update on new commits.}

[cubic-dev-ai]

No issues found across 10 files

Confidence score: 5/5

• Automated review surfaced no issues in the provided summaries.
• No files require special attention.

_{Requires human review: PR removes background-task modules. Despite no AI issues, this significantly changes functionality and may cause regressions. Not 100% sure no regressions.}