fix(tools): apply #1175 casing fix to residual TOOLS/ call sites

[StarksLabs]

Summary

Follow-up to #1175 / #1259 / #1267 / #1273. Audit of Releases/v5.0.0/.claude/PAI/ for residual TitleCase directory names in path-construction calls found three more sites — all in TOOLS/. Same root cause, same fix shape, same risk profile as the prior PRs.

On macOS HFS+/APFS (case-insensitive default), TitleCase and UPPERCASE resolve to the same directory — bug is invisible. On Linux ext4/btrfs (case-sensitive), each join() resolves to a non-existent path and the affected feature silently fails.

Confirmed canonical casing in v5.0.0

ALGORITHM/  DOCUMENTATION/  MEMORY/  PULSE/  TOOLS/  USER/  TEMPLATES/  PAI-Install/  bin/

All top-level dirs except PAI-Install and bin are UPPERCASE.

Fix table

File	Line	Was	Now
TOOLS/DASchedule.ts	18	join(PAI_DIR, "Pulse", "state", "da")	"PULSE"
TOOLS/MemoryRetriever.ts	286	path.join(PAI_DIR, "Tools", "Inference.ts")	"TOOLS"
TOOLS/pai.ts	34	join(homedir(), ".claude", "PAI", "Tools", "Banner.ts")	"TOOLS"

Total: 3 string changes across 3 files.

User-visible Linux failures pre-patch:

• TOOLS/pai.ts:34 — BANNER_SCRIPT points at non-existent Tools/Banner.ts; banner load fails silently when the pai CLI runs.
• TOOLS/DASchedule.ts:18 — DA scheduled-task state directory not found; scheduling state cannot be read/written.
• TOOLS/MemoryRetriever.ts:286 — Inference.ts fallback path missing; compression takes the substring-truncation degraded path instead of the LLM compression path.

Reproduction (against the release tree)

$ ls -d Releases/v5.0.0/.claude/PAI/Pulse 2>&1
ls: cannot access 'Releases/v5.0.0/.claude/PAI/Pulse': No such file or directory

$ ls -d Releases/v5.0.0/.claude/PAI/Tools 2>&1
ls: cannot access 'Releases/v5.0.0/.claude/PAI/Tools': No such file or directory

$ ls -d Releases/v5.0.0/.claude/PAI/PULSE Releases/v5.0.0/.claude/PAI/TOOLS
Releases/v5.0.0/.claude/PAI/PULSE
Releases/v5.0.0/.claude/PAI/TOOLS

Each join(PAI_DIR, "Pulse"|"Tools", …) resolves to one of those ENOENT paths on Linux.

Audit method

cd Releases/v5.0.0/.claude/PAI
grep -rn --include='*.ts' -E '"(Pulse|Memory|Skills|Tools|Algorithm|Documentation|User)"' . \
  | grep -E 'join|path\.|resolve|readFile|existsSync|readdirSync'

Six hits on current main after the prior PRs landed. Three were false positives and are NOT touched here:

• PULSE/pulse-old.ts:49 — dead code (zero references anywhere in repo).
• PAI-Install/engine/actions.ts:1405 — per-skill Tools/ subdir convention (TitleCase), distinct from top-level TOOLS/.
• TOOLS/ArchitectureSummaryGenerator.ts:27 — DOCUMENTATION/Memory/ is the legitimate TitleCase subdir of DOCUMENTATION/.

The remaining three are this PR.

Risk

Very low — identical risk profile to #1175.

• Linux: affected call sites previously silently broken → now work (intended fix).
• macOS: behavior unchanged — case-insensitive HFS+/APFS already resolved TitleCase to the canonical UPPERCASE directory, so the same paths resolved fine before and after.
• No semantic changes — only path-string casing.
• No public API touched. No config schema changes.

Optional follow-up — CI lint

This is the fifth PR in the same bug class (after #1175 / #1259 / #1267 / #1273). A CI lint that fails the build when "Pulse"|"Memory"|"Algorithm"|"Tools" (TitleCase) appears inside join() / path.* / resolve() calls would prevent the next round. Out of scope here.