[ZSH] Faster startup: instant prompt, single compinit, cached probes

Re-enable p10k instant prompt (sudo-key ssh-add moved above it). Drop
oh-my-zsh.sh (second compinit, update check); set up what OMZ plugins
need ourselves. Cache java_home, go env, fzf --zsh output. Disable the
kubecontext segment (ran kubectl per shell). Log starts over 1 s to
~/.cache/zsh/startup-slow.log to catch cold-start stalls.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

zsh/lib/startup.zsh

@@ -0,0 +1,53 @@
+# Startup helpers: output caching and slow-start logging.
+
+zmodload zsh/datetime
+zmodload -F zsh/stat b:zstat
+
+# Cache a command's output; regenerate when a dependency's resolved path or
+# mtime changes (brew upgrades swap Cellar paths but keep old mtimes).
+# Usage: _zsh_cached <name> [dep...] -- <cmd...>; sets REPLY to the cache file.
+_zsh_cached() {
+    local file=${ZDOTDIR:-$HOME}/.cache/zsh/$1 stamp old
+    local -a mt
+    shift
+    while [[ $# -gt 0 && $1 != -- ]]; do
+        mt=(); zstat -A mt +mtime -- $1 2>/dev/null
+        stamp+="${1:A}@${mt[1]:-none};"
+        shift
+    done
+    shift
+    [[ -r $file.stamp ]] && read -r old < $file.stamp
+    if [[ ! -s $file || $old != $stamp ]]; then
+        local tmp=$file.$$.tmp
+        if "$@" >| $tmp 2>/dev/null && [[ -s $tmp ]]; then
+            command mv -f $tmp $file
+            print -r -- $stamp >| $file.stamp
+        else
+            command rm -f $tmp
+            [[ -s $file ]] || return 1
+        fi
+    fi
+    REPLY=$file
+}
+
+# Slow-start log: per-step ms, written only if startup (until the first prompt)
+# exceeds ZSH_STARTUP_LOG_MS. Diagnostic for cold starts; remove once resolved.
+typeset -gF _zst_t0=$EPOCHREALTIME _zst_last=$EPOCHREALTIME
+typeset -ga _zst_steps
+_zst_mark() {
+    local ms
+    printf -v ms '%s=%.0f' $1 $(( (EPOCHREALTIME - _zst_last) * 1000 ))
+    _zst_steps+=($ms)
+    _zst_last=$EPOCHREALTIME
+}
+_zst_report() {
+    add-zsh-hook -d precmd _zst_report
+    _zst_mark precmd
+    local total=$(( (EPOCHREALTIME - _zst_t0) * 1000 ))
+    (( total >= ${ZSH_STARTUP_LOG_MS:-1000} )) || return 0
+    local ts login=0
+    strftime -s ts '%Y-%m-%dT%H:%M:%S' $EPOCHSECONDS
+    [[ -o login ]] && login=1
+    printf '%s total=%.0f tmux=%d login=%d | %s\n' $ts $total ${+TMUX} $login "$_zst_steps" \
+        >> ${ZDOTDIR:-$HOME}/.cache/zsh/startup-slow.log
+}