fix(formulas): replace hardcoded ~/gt/ paths with $GT_ROOT (#758)

* fix(formulas): replace hardcoded ~/gt/ paths with $GT_ROOT

Formula files contained hardcoded ~/gt/ paths that break when running
Gas Town from a non-default location (e.g., ~/gt-private/). This causes:

- Dogs stuck in working state (can't write to wrong path)
- Cross-town contamination when ~/gt/ exists as separate town
- Boot triage, deacon patrol, and log archival failures

Replaces all ~/gt/ and $HOME/gt/ references with $GT_ROOT which is
set at runtime to the actual town root directory.

Fixes #757

* chore: regenerate embedded formulas

Run go generate to sync embedded formulas with .beads/formulas/ source.

.beads/formulas/gastown-release.formula.toml

@@ -47,7 +47,7 @@ Check all crew workspaces and the mayor rig:
 
 ```bash
 # Check each workspace
-for dir in ~/gt/gastown/crew/* ~/gt/gastown/mayor; do
+for dir in $GT_ROOT/gastown/crew/* $GT_ROOT/gastown/mayor; do
   if [ -d "$dir/.git" ] || [ -d "$dir" ]; then
     echo "=== Checking $dir ==="
     cd "$dir" 2>/dev/null || continue

.beads/formulas/mol-boot-triage.formula.toml

@@ -47,7 +47,7 @@ bd show hq-deacon 2>/dev/null
 gt feed --since 10m --plain | head -20
 
 # Recent wisps (operational state)
-ls -lt ~/gt/.beads-wisp/*.wisp.json 2>/dev/null | head -5
+ls -lt $GT_ROOT/.beads-wisp/*.wisp.json 2>/dev/null | head -5
 ```
 
 **Step 4: Check Deacon mail**
@@ -221,7 +221,7 @@ Then exit. The next daemon tick will spawn a fresh Boot.
 **Update status file**
 ```bash
 # The gt boot command handles this automatically
-# Status is written to ~/gt/deacon/dogs/boot/.boot-status.json
+# Status is written to $GT_ROOT/deacon/dogs/boot/.boot-status.json
 ```
 
 Boot is ephemeral by design. Each instance runs fresh.

.beads/formulas/mol-deacon-patrol.formula.toml

@@ -480,7 +480,7 @@ needs = ["zombie-scan"]
 description = """
 Execute registered plugins.
 
-Scan ~/gt/plugins/ for plugin directories. Each plugin has a plugin.md with TOML frontmatter defining its gate (when to run) and instructions (what to do).
+Scan $GT_ROOT/plugins/ for plugin directories. Each plugin has a plugin.md with TOML frontmatter defining its gate (when to run) and instructions (what to do).
 
 See docs/deacon-plugins.md for full documentation.
 
@@ -497,7 +497,7 @@ For each plugin:
 
 Plugins marked parallel: true can run concurrently using Task tool subagents. Sequential plugins run one at a time in directory order.
 
-Skip this step if ~/gt/plugins/ does not exist or is empty."""
+Skip this step if $GT_ROOT/plugins/ does not exist or is empty."""
 
 [[steps]]
 id = "dog-pool-maintenance"
@@ -736,13 +736,13 @@ Maintain daemon logs and state files.
 **Step 1: Check daemon.log size**
 ```bash
 # Get log file size
-ls -la ~/.beads/daemon*.log 2>/dev/null || ls -la ~/gt/.beads/daemon*.log 2>/dev/null
+ls -la ~/.beads/daemon*.log 2>/dev/null || ls -la $GT_ROOT/.beads/daemon*.log 2>/dev/null
 ```
 
 If daemon.log exceeds 10MB:
 ```bash
 # Rotate with date suffix and gzip
-LOGFILE="$HOME/gt/.beads/daemon.log"
+LOGFILE="$GT_ROOT/.beads/daemon.log"
 if [ -f "$LOGFILE" ] && [ $(stat -f%z "$LOGFILE" 2>/dev/null || stat -c%s "$LOGFILE") -gt 10485760 ]; then
     DATE=$(date +%Y-%m-%dT%H-%M-%S)
     mv "$LOGFILE" "${LOGFILE%.log}-${DATE}.log"
@@ -754,7 +754,7 @@ fi
 
 Clean up daemon logs older than 7 days:
 ```bash
-find ~/gt/.beads/ -name "daemon-*.log.gz" -mtime +7 -delete
+find $GT_ROOT/.beads/ -name "daemon-*.log.gz" -mtime +7 -delete
 ```
 
 **Step 3: Prune state.json of dead sessions**

.beads/formulas/mol-shutdown-dance.formula.toml

@@ -8,7 +8,7 @@ goroutine (NOT a Claude session) that runs the interrogation state machine.
 
 Dogs are lightweight workers in Boot's pool (see dog-pool-architecture.md):
 - Fixed pool of 5 goroutines (configurable via GT_DOG_POOL_SIZE)
-- State persisted to ~/gt/deacon/dogs/active/<id>.json
+- State persisted to $GT_ROOT/deacon/dogs/active/<id>.json
 - Recovery on Boot restart via orphan state files
 
 ## State Machine
@@ -151,7 +151,7 @@ If target doesn't exist:
 - Skip to EPITAPH with outcome=already_dead
 
 **3. Initialize state file:**
-Write initial state to ~/gt/deacon/dogs/active/{dog-id}.json
+Write initial state to $GT_ROOT/deacon/dogs/active/{dog-id}.json
 
 **4. Set initial attempt counter:**
 attempt = 1
@@ -477,11 +477,11 @@ bd close {warrant_id} --reason "{epitaph_summary}"
 
 **3. Move state file to completed:**
 ```bash
-mv ~/gt/deacon/dogs/active/{dog-id}.json ~/gt/deacon/dogs/completed/
+mv $GT_ROOT/deacon/dogs/active/{dog-id}.json $GT_ROOT/deacon/dogs/completed/
 ```
 
 **4. Report to Boot:**
-Write completion file: ~/gt/deacon/dogs/active/{dog-id}.done
+Write completion file: $GT_ROOT/deacon/dogs/active/{dog-id}.done
 ```json
 {
   "dog_id": "{dog-id}",

.beads/formulas/mol-town-shutdown.formula.toml

@@ -132,7 +132,7 @@ gt daemon rotate-logs
 gt doctor --fix
 ```
 
-Old logs are moved to `~/gt/logs/archive/` with timestamps.
+Old logs are moved to `$GT_ROOT/logs/archive/` with timestamps.
 """
 
 [[steps]]