diff --git a/foresight/src/main/scala/foresight/eqsat/commands/CommandSchedule.scala b/foresight/src/main/scala/foresight/eqsat/commands/CommandSchedule.scala
index 1658f62e..1311cea9 100644
--- a/foresight/src/main/scala/foresight/eqsat/commands/CommandSchedule.scala
+++ b/foresight/src/main/scala/foresight/eqsat/commands/CommandSchedule.scala
@@ -115,15 +115,11 @@ final case class CommandSchedule[NodeT](batchZero: (ArraySeq[EClassSymbol.Virtua
    * Executes the command schedule against an immutable e-graph.
    *
    * @param egraph
-   * Destination e-graph. Implementations may either return it unchanged or produce a new immutable
-   * e-graph snapshot.
-   * @param reification
-   * Mapping from virtual symbols to concrete calls available before this command runs. This is used
-   * to ground virtual references present in [[uses]].
+   *   Destination e-graph. A new immutable e-graph snapshot is produced if any change occurs.
    * @param parallelize
-   * Parallelization strategy to label and distribute any internal work.
+   *   Parallelization strategy to label and distribute any internal work.
    * @return
-   * `Some(newGraph)` if any change occurred, or `None` for a no-op.
+   *   `Some(newGraph)` if any change occurred, or `None` for a no-op.
    */
   def applyImmutable[
     Repr <: immutable.EGraphLike[NodeT, Repr] with immutable.EGraph[NodeT]
diff --git a/foresight/src/main/scala/foresight/eqsat/immutable/EGraph.scala b/foresight/src/main/scala/foresight/eqsat/immutable/EGraph.scala
index 013a9600..5f54348d 100644
--- a/foresight/src/main/scala/foresight/eqsat/immutable/EGraph.scala
+++ b/foresight/src/main/scala/foresight/eqsat/immutable/EGraph.scala
@@ -5,8 +5,8 @@ import foresight.eqsat.{EClassCall, MixedTree}
 
 /**
  * An e-graph compactly represents many equivalent expressions at once by grouping structurally
- * * compatible nodes into e-classes. This enables fast equality saturation and rewrite exploration
- * * without duplicating common substructure.
+ * compatible nodes into e-classes. This enables fast equality saturation and rewrite exploration
+ * without duplicating common substructure.
  *
  * `EGraph` is a convenience alias of [[EGraphLike]] where the self type equals the trait itself:
  * `EGraph[NodeT]` extends `EGraphLike[NodeT, EGraph[NodeT]]`. Use this when you don’t need a custom
diff --git a/foresight/src/main/scala/foresight/eqsat/package.scala b/foresight/src/main/scala/foresight/eqsat/package.scala
index 2f617eff..dd7dc7c2 100644
--- a/foresight/src/main/scala/foresight/eqsat/package.scala
+++ b/foresight/src/main/scala/foresight/eqsat/package.scala
@@ -10,9 +10,12 @@ package foresight
  * This representation supports efficient, parallel, and reversible rewrite-driven
  * optimization, with explicit tracking of symbolic bindings (*slots*) and rich metadata.
  *
- * E-graphs in Foresight are purely functional: every operation returns a new value
- * rather than mutating in place. This immutability underpins reproducibility,
- * concurrent analysis, and caching across snapshots.
+ * E-graphs in Foresight come in two flavors: an **immutable** variant
+ * ([[eqsat.immutable.EGraph]]) where every operation returns a new e-graph value
+ * without modifying the original (underpinning reproducibility, concurrent analysis, and
+ * caching across snapshots), and a **mutable** variant ([[eqsat.mutable.EGraph]]) that
+ * supports in-place additions and unions for workloads where allocation overhead matters.
+ * Both variants implement the same read-only query interface ([[eqsat.readonly.EGraph]]).
  *
  * ## Core Concepts
  *
@@ -87,13 +90,13 @@ package foresight
  * // Create an empty e-graph for a custom node type
  * val g0: EGraph[MyNode] = EGraph.empty
  *
- * // Build a command queue
- * val builder = new CommandQueueBuilder[MyNode]
- * val root    = builder.add(myENode)
- * val queue   = builder.queue.optimized
+ * // Build a command schedule using a builder
+ * val builder = CommandScheduleBuilder.newConcurrentBuilder[MyNode]
+ * builder.add(EClassSymbol.virtual(), myENodeSymbol, batch = 0)
+ * val schedule = builder.result()
  *
- * // Apply commands to produce a new graph
- * val (g1, reif) = queue.applyImmutable(g0, Map.empty, ParallelMap.sequential)
+ * // Apply the schedule to produce a new immutable graph
+ * val g1 = schedule.applyImmutable(g0, ParallelMap.sequential).getOrElse(g0)
  *
  * // Run a saturation strategy
  * val rules: Seq[Rule[MyNode, MyMatch, EGraph[MyNode]]] = Seq(r1, r2)
diff --git a/foresight/src/main/scala/foresight/eqsat/rewriting/Applier.scala b/foresight/src/main/scala/foresight/eqsat/rewriting/Applier.scala
index d02c78cb..c7d75c9f 100644
--- a/foresight/src/main/scala/foresight/eqsat/rewriting/Applier.scala
+++ b/foresight/src/main/scala/foresight/eqsat/rewriting/Applier.scala
@@ -123,7 +123,7 @@ object Applier {
    * Pre-flatMap the input match type: expand one match into **many** then apply and batch them.
    *
    * Useful when a single outer match implies multiple inner edits (e.g., expand an equivalence to
-   * a set of updates). The resulting [[CommandQueue]] preserves the order of the produced
+   * a set of updates). The resulting [[foresight.eqsat.commands.CommandSchedule]] preserves the order of the produced
    * inner commands (though the engine may later optimize/deduplicate).
    *
    * @param applier Downstream applier that consumes `MatchT2`.
diff --git a/foresight/src/main/scala/foresight/eqsat/rewriting/PortableMatch.scala b/foresight/src/main/scala/foresight/eqsat/rewriting/PortableMatch.scala
index bdb79d47..dcc0d467 100644
--- a/foresight/src/main/scala/foresight/eqsat/rewriting/PortableMatch.scala
+++ b/foresight/src/main/scala/foresight/eqsat/rewriting/PortableMatch.scala
@@ -22,7 +22,7 @@ import foresight.eqsat.readonly.EGraph
  *   2. **Structural-only matches** — the match stores only stable, structural data (e.g., constants,
  *   shapes). In this case, `port` can simply return `this`. Even so, implementing `PortableMatch`
  *   is valuable because the caching and recording infrastructure requires all matches to be portable:
- *     - [[foresight.eqsat.saturation.EGraphWithRecordedApplications]] records applied matches per rule
+ *     - [[foresight.eqsat.immutable.EGraphWithRecordedApplications]] records applied matches per rule
  *      and re-ports them after unions.
  *     - [[foresight.eqsat.saturation.SearchAndApply$.immutableWithCaching]] filters out already-applied matches; portability
  *       keeps equality and hashing consistent as the graph evolves.
@@ -54,7 +54,7 @@ import foresight.eqsat.readonly.EGraph
  *   def port(egraph: EGraph[NodeT]): ConstShapeMatch[NodeT] = this
  * }
  * }}}
- * @see [[foresight.eqsat.saturation.EGraphWithRecordedApplications]]
+ * @see [[foresight.eqsat.immutable.EGraphWithRecordedApplications]]
  *      for how applied matches are recorded and re-ported after unions; and
  *      [[foresight.eqsat.saturation.SearchAndApply$.immutableWithCaching]]
  *      for how cached applications are filtered using `PortableMatch`.
diff --git a/foresight/src/main/scala/foresight/eqsat/rewriting/Rewrite.scala b/foresight/src/main/scala/foresight/eqsat/rewriting/Rewrite.scala
index 4f9f4e14..4473e877 100644
--- a/foresight/src/main/scala/foresight/eqsat/rewriting/Rewrite.scala
+++ b/foresight/src/main/scala/foresight/eqsat/rewriting/Rewrite.scala
@@ -20,8 +20,9 @@ import foresight.eqsat.readonly.EGraph
  *  - [[apply]]/[[applyImmutable]]/[[tryApply]]: Search and apply all matches immediately, returning
  *    either the updated e-graph or a flag indicating whether any change occurred.
  *
- * The staged command produced by [[delayed]] can be enqueued into a [[CommandQueue]] along with
- * other rules, allowing callers to batch multiple rewrites into a single saturation step.
+ * The staged command produced by [[delayed]] can be batched with commands from other rules by
+ * sharing a [[foresight.eqsat.commands.CommandScheduleBuilder]], allowing callers to
+ * combine multiple rewrites into a single saturation step.
  *
  * @tparam NodeT   Node type for expressions represented by the e-graph.
  * @tparam MatchT  Type of matches produced by this rule's search phase.
@@ -64,7 +65,7 @@ trait Rewrite[NodeT, MatchT, -EGraphT <: EGraph[NodeT]] {
    * @param matches     Matches to apply.
    * @param egraph      Target e-graph from which matches were derived.
    * @param parallelize Parallel strategy used when building per-match commands.
-   * @return An optimized [[CommandQueue]] encapsulated as a [[CommandSchedule]].
+   * @return A single, optimized [[foresight.eqsat.commands.CommandSchedule]] that applies all current matches of this rule.
    * @throws Rule.ApplicationException
    * if constructing the per-match commands fails.
    */
@@ -152,14 +153,11 @@ trait Rewrite[NodeT, MatchT, -EGraphT <: EGraph[NodeT]] {
   }
 
   /**
-   * Search and apply all matches immediately.
+   * Search and apply all matches immediately, mutating the e-graph in place.
    *
-   * If the rule makes any changes, `true` is returned without mutating the e-graph.
-   * If no changes occur, `false` is returned.
-   *
-   * @param egraph      Target e-graph.
+   * @param egraph      Target e-graph. Mutated in place if any changes occur.
    * @param parallelize Parallel strategy used for both search and apply.
-   * @return The updated e-graph if changes occurred; otherwise the original `egraph`.
+   * @return `true` if any change occurred; `false` otherwise.
    */
   def apply[
     MutEGraphT <: EGraphT with mutable.EGraph[NodeT]
diff --git a/foresight/src/main/scala/foresight/eqsat/rewriting/Rule.scala b/foresight/src/main/scala/foresight/eqsat/rewriting/Rule.scala
index be2f0072..3e9e7d9e 100644
--- a/foresight/src/main/scala/foresight/eqsat/rewriting/Rule.scala
+++ b/foresight/src/main/scala/foresight/eqsat/rewriting/Rule.scala
@@ -15,7 +15,7 @@ import foresight.eqsat.readonly.EGraph
  *
  * After search and apply, a rule performs a **composition step**:
  *  - Command aggregation – all per-match commands are combined into a single, optimized
- *    [[CommandQueue]] so they can be executed as one logical unit.
+ *    [[foresight.eqsat.commands.CommandSchedule]] so they can be executed as one logical unit.
  *  - This aggregation deduplicates operations, removes no-ops, and preserves an execution order
  *    that is valid but not necessarily identical to the match order.
  *  - The result can be executed immediately (via [[apply]] / [[tryApply]]) or returned for later
@@ -57,15 +57,17 @@ import foresight.eqsat.readonly.EGraph
  * }}}
  * @example Staging a rule and applying later (batching with other rules)
  * {{{
- * val r1Cmd = r1.delayed(egraph) // stage first rule
- * val r2Cmd = r2.delayed(egraph) // stage second rule
+ * import foresight.eqsat.commands.CommandScheduleBuilder
+ * import foresight.eqsat.parallel.ParallelMap
  *
- * // Merge both into one optimized command
- * val batched = CommandQueue(Seq(r1Cmd, r2Cmd)).optimized
+ * // Batch two rules into a single command schedule using a shared builder
+ * val builder = CommandScheduleBuilder.newConcurrentBuilder[MyNode]
+ * r1.delayed(egraph, ParallelMap.default, builder) // stage first rule
+ * r2.delayed(egraph, ParallelMap.default, builder) // stage second rule
+ * val batched = builder.result()
  *
  * // Somewhere later in the saturation loop:
- * val (maybeNewEGraph, _) = batched(egraph, Map.empty, ParallelMap.default)
- * val next = maybeNewEGraph.getOrElse(egraph) // fallback to original if no changes
+ * val next = batched.applyImmutable(egraph, ParallelMap.default).getOrElse(egraph)
  * }}}
  */
 final case class Rule[NodeT, MatchT, EGraphT <: EGraph[NodeT]](override val name: String,
@@ -88,7 +90,7 @@ final case class Rule[NodeT, MatchT, EGraphT <: EGraph[NodeT]](override val name
    * Build a staged command that, when executed, applies this rule's matches to `egraph`.
    *
    * This does not mutate the e-graph now; instead it returns a [[Command]] that you can:
-   *  - enqueue into a [[CommandQueue]] with other rules; and
+   *  - enqueue alongside other rules via a [[foresight.eqsat.commands.CommandScheduleBuilder]]; and
    *  - execute later as part of a larger saturation step.
    *
    * @param egraph      The e-graph to search for matches. The staged command is intended to be run
diff --git a/foresight/src/main/scala/foresight/eqsat/rewriting/package.scala b/foresight/src/main/scala/foresight/eqsat/rewriting/package.scala
index aa083d01..d2d49d89 100644
--- a/foresight/src/main/scala/foresight/eqsat/rewriting/package.scala
+++ b/foresight/src/main/scala/foresight/eqsat/rewriting/package.scala
@@ -27,7 +27,7 @@ package foresight.eqsat
  * Matches are also **portable**. By implementing [[PortableMatch]], a match can
  * remain meaningful across multiple e-graph snapshots, enabling caching and
  * reapplication through
- * [[foresight.eqsat.saturation.EGraphWithRecordedApplications]] and
+ * [[foresight.eqsat.immutable.EGraphWithRecordedApplications]] and
  * [[foresight.eqsat.saturation.SearchAndApply$.immutableWithCaching]].
  *
  * ## Typical workflow
@@ -74,7 +74,7 @@ package foresight.eqsat
  * a new snapshot, preserving structural meaning. Purely structural matches can
  * treat `port` as a no-op, while ID-bearing matches perform translations such as
  * canonicalization. Recording and replaying matches is supported by
- * [[foresight.eqsat.saturation.EGraphWithRecordedApplications]], and caching is
+ * [[foresight.eqsat.immutable.EGraphWithRecordedApplications]], and caching is
  * provided by [[foresight.eqsat.saturation.SearchAndApply$.immutableWithCaching]].
  *
  * ## Design contracts
@@ -91,7 +91,7 @@ package foresight.eqsat
  * [[ReversibleSearcher]]; for application:
  * [[Applier]] and [[ReversibleApplier]]; for composition: [[Rule]]; for matches
  * across snapshots: [[PortableMatch]]; and for caching and recording:
- * [[foresight.eqsat.saturation.EGraphWithRecordedApplications]] and
+ * [[foresight.eqsat.immutable.EGraphWithRecordedApplications]] and
  * [[foresight.eqsat.saturation.SearchAndApply$.immutableWithCaching]]. Strategies for
  * high-level orchestration include
  * [[foresight.eqsat.saturation.Strategy]],
@@ -103,7 +103,7 @@ package foresight.eqsat
  * {{{
  * import foresight.eqsat.rewriting._
  * import foresight.eqsat.parallel.ParallelMap
- * import foresight.eqsat.commands.CommandQueue
+ * import foresight.eqsat.commands.CommandScheduleBuilder
  *
  * // 1) Search: find matches in the e-graph
  * val s: Searcher[MyNode, MyMatch, MyEGraph] = ...
@@ -114,9 +114,13 @@ package foresight.eqsat
  * // 3) Rule: run now or stage/batch
  * val r = Rule("my-rule", s, a)
  * val updated = r(egraph, ParallelMap.default)                     // immediate
- * val staged  = r.delayed(egraph)                                  // staged
- * val batched = CommandQueue(Seq(staged, r2.delayed(egraph))).optimized
- * val (next, _) = batched(egraph, Map.empty, ParallelMap.default)  // execute batch
+ *
+ * // Batch two rules via a shared builder
+ * val builder = CommandScheduleBuilder.newConcurrentBuilder[MyNode]
+ * r.delayed(egraph, ParallelMap.default, builder)
+ * r2.delayed(egraph, ParallelMap.default, builder)
+ * val batched = builder.result()
+ * val next = batched.applyImmutable(egraph, ParallelMap.default).getOrElse(egraph)
  * }}}
  * @example Run with a saturation strategy
  * {{{
@@ -141,9 +145,10 @@ package foresight.eqsat
  * @example Portability and caching
  * {{{
  * import foresight.eqsat.saturation._
+ * import foresight.eqsat.immutable.EGraphWithRecordedApplications
  *
  * type M = MyMatch with PortableMatch[MyNode, M]
- * val sa = SearchAndApply.withCaching[MyNode, MyEGraph, M]
+ * val sa = SearchAndApply.immutableWithCaching[MyNode, MyEGraph, M]
  *
  * val found: Map[String, Seq[M]] =
  *   sa.search(Seq(r1, r2), EGraphWithRecordedApplications(egraph), ParallelMap.default)
diff --git a/foresight/src/main/scala/foresight/eqsat/saturation/StochasticRuleApplication.scala b/foresight/src/main/scala/foresight/eqsat/saturation/StochasticRuleApplication.scala
index 61dbb6d2..24f2774e 100644
--- a/foresight/src/main/scala/foresight/eqsat/saturation/StochasticRuleApplication.scala
+++ b/foresight/src/main/scala/foresight/eqsat/saturation/StochasticRuleApplication.scala
@@ -23,7 +23,7 @@ import foresight.util.random.{Random, Sample}
  * {{{
  * val strategy = StochasticRuleApplication(
  *   rules = myRules,
- *   searchAndApply = SearchAndApply.withoutCaching,
+ *   searchAndApply = SearchAndApply.immutable,
  *   priorities = MyCustomPriorities,
  *   random = Random(42)
  * )
@@ -151,7 +151,7 @@ object StochasticRuleApplication {
      priorities: MatchPriorities[NodeT, Rewrite[NodeT, MatchT, EGraphT], EGraphT, MatchT],
      random: Random
   ): StochasticRuleApplication[NodeT, Rewrite[NodeT, MatchT, EGraphT], EGraphT, MatchT] = {
-    apply(rules, SearchAndApply.immutable[NodeT, EGraphT, MatchT], priorities)
+    new StochasticRuleApplication(rules, SearchAndApply.immutable[NodeT, EGraphT, MatchT], priorities, random)
   }
 
   /**