diff --git a/src/v2/compiler/complexity_gate/comprep_pipeline_bridge.dag b/src/v2/compiler/complexity_gate/comprep_pipeline_bridge.dag
new file mode 100644
index 00000000000..47c5f10ed9c
--- /dev/null
+++ b/src/v2/compiler/complexity_gate/comprep_pipeline_bridge.dag
@@ -0,0 +1,64 @@
+// Scope: shared tokenize→parse→normalize→resolve bridge for COMPREP test-subject producers.
+//        Single authority for the pipeline chain; per-fn producers pass their GrammarRoot +
+//        production_name and call comprep_source_resolved_root to obtain the resolved Arrow Node.
+// Consumers: wave-1 subject producers (can migrate), D2 rewrite-rule test-subject producers.
+// Design: DESIGN.md §2 horizontal — one pipeline path, derived per subject (not forked per fn).
+// Hardwired: dag_language_model_wave1 (MVP1 .dag surface). New media → new bridge or param.
+
+module v2.test.complexity_gate.comprep_pipeline_bridge
+
+import v2.compiler.normalize { normalize }
+import v2.compiler.parse { ValidatedGrammarRoot, grammar_validate_for_parse, parse_production_tree }
+import v2.compiler.resolve { resolve }
+import v2.compiler.tokenize { tokenize }
+import v2.extdeps.languages.dag { dag_language_model_wave1 }
+import v2.std.collection { List }
+import v2.std.compilers.lexing { Token }
+import v2.std.diagnostic { Accepted, Outcome, Rejected, bind_outcome, port_locus }
+import v2.std.grammar { GrammarRoot, ParseTree }
+import v2.std.node { Node, Symbol }
+import v2.std.text { String }
+
+
+// Run the four-stage comprep pipeline on a source snippet:
+//   tokenize → validate+parse (parse_root, production_name) → normalize → resolve
+// Returns the resolved Arrow Node, ready for body extraction by the caller.
+//
+// file: diagnostic locus symbol — callers pass their own (e.g. ^d2_nested_slow_form_file).
+// parse_root / production_name: the grammar this snippet is parsed under — callers construct.
+// Grammar validation is fail-closed before any tokenize work.
+fn comprep_source_resolved_root(
+  source_text: String,
+  parse_root: GrammarRoot,
+  production_name: Symbol,
+  file: Symbol
+) -> Outcome<Node> {
+  let lm = dag_language_model_wave1()
+  match grammar_validate_for_parse(
+    root: parse_root,
+    at: port_locus(port: ^comprep_pipeline_bridge_grammar_locus)
+  ) {
+    Rejected { diagnostics: r } => Rejected { diagnostics: r }
+    Accepted { value: validated, diagnostics: _ } =>
+      bind_outcome(
+        o: tokenize(text: source_text, file: file, rules: lm.lex),
+        f: fn(token_stream) {
+          bind_outcome(
+            o: parse_production_tree(
+              tokens: token_stream.tokens,
+              validated: validated,
+              production_name: production_name
+            ),
+            f: fn(parsed) {
+              bind_outcome(
+                o: normalize(parse_tree: parsed),
+                f: fn(normalized) {
+                  resolve(tree: normalized, lm: lm)
+                }
+              )
+            }
+          )
+        }
+      )
+  }
+}
diff --git a/src/v2/lens/cost.dag b/src/v2/lens/cost.dag
index e9ddbd8ce5a..11b3e2e3750 100644
--- a/src/v2/lens/cost.dag
+++ b/src/v2/lens/cost.dag
@@ -189,6 +189,11 @@ fn base_cost_for_connective(c: Connective, n: Node) -> SymbolicCost {
     Disj => zero_cost()
     Arrow => unit_cost()
     Cardinality => linear_in_node(n: n)
+    // 🟡 cost-model gap: Instantiation/recursion-depth → ExponentialCost is unmodeled.
+    // Self-calling folds to ~unit_cost (constant), so complexity_lens cannot detect
+    // ClassExponential for naive recursive functions. Blocks naive-recursion→memoize
+    // rewrite detection. dissolve-on: Instantiation base cost models recursion depth
+    // (model-before-implement; new modeling, not a patch in this site).
     Instantiation => unit_cost()
   }
 }
@@ -391,7 +396,11 @@ fn compose_child_cost(
 ) -> SymbolicCost {
   match mode {
     SequentialCost => symbolic_sequential(a: acc, b: child)
-    AlternativeCost => symbolic_max(a: acc, b: child)
+    // Per-alternative factor = max(acc, max(branch, one decision step)). symbolic_max
+    // floors ONLY zero: max(acc, max(0,1))=max(acc,1) — a zero-cost branch still costs
+    // one decision; max(acc, max(1,1))=max(acc,1) and max(acc, max(linear,1))=max(acc,linear)
+    // leave unit/linear branches unchanged. symbolic_max itself absorbing zero untouched.
+    AlternativeCost => symbolic_max(a: acc, b: symbolic_max(a: child, b: unit_cost()))
     // Per-iteration factor = max(body, one control step). symbolic_max floors ONLY
     // zero: max(0,1)=1 (empty body -> O(n)); max(1,1)=1 and max(linear,1)=linear leave
     // unit/linear bodies unchanged vs plain symbolic_product. Absorbing zero untouched.
diff --git a/src/v2/lens/cost/disj_alternative_floor_test.dag b/src/v2/lens/cost/disj_alternative_floor_test.dag
new file mode 100644
index 00000000000..0f148f0ab4c
--- /dev/null
+++ b/src/v2/lens/cost/disj_alternative_floor_test.dag
@@ -0,0 +1,71 @@
+// Status: EXECUTED Bool witness — all-zero Disj arms still cost at least O(1).
+//
+// Pins the cost-lens AlternativeCost floor (cost.dag `compose_child_cost` AlternativeCost arm).
+// Two checks together pin the CORRECT fix uniquely:
+//   (1) bug-fix: cost_lens(Disj([Atom, Atom])) projects unit_cost(), NOT zero_cost() —
+//       selecting between alternatives has a non-trivial cost even when every branch body is free.
+//   (2) control: symbolic_max's zero-absorption is PRESERVED — symbolic_max(0, 0) == 0.
+// (2) is what distinguishes the right fix (symbolic_max(child, unit) floor at the
+// AlternativeCost layer) from the wrong one (removing zero-absorption inside symbolic_max):
+// BOTH make (1) green, but the wrong fix makes (2) RED.
+
+module v2.test.lens_cost.disj_alternative_floor
+
+import v2.lens.cost {
+  cost_lens,
+  symbolic_max,
+  unit_cost,
+  zero_cost
+}
+import v2.std.logic { Bool }
+import v2.std.node {
+  Atom,
+  Disj,
+  Edge,
+  Node,
+  Positional,
+  SyntheticOccurrence,
+  TypeNode
+}
+import v2.std.witness { Holds }
+
+// Two zero-cost Disj arms (Atom base cost = zero_cost).
+data disj_floor_arm_a: Node = Node {
+  kind: TypeNode { connective: Atom { identity: ^disj_floor_arm_a_atom } },
+  children: [],
+  occurrence_id: SyntheticOccurrence
+}
+
+data disj_floor_arm_b: Node = Node {
+  kind: TypeNode { connective: Atom { identity: ^disj_floor_arm_b_atom } },
+  children: [],
+  occurrence_id: SyntheticOccurrence
+}
+
+// Disj with two zero-cost arms — the structure cost_lens folds.
+data disj_floor_input: Node = Node {
+  kind: TypeNode { connective: Disj },
+  children: [
+    Edge { label: Positional, target: disj_floor_arm_a },
+    Edge { label: Positional, target: disj_floor_arm_b }
+  ],
+  occurrence_id: SyntheticOccurrence
+}
+
+// (1) Bug-fix: a Disj over zero-cost branches projects unit_cost(), not zero_cost().
+fn disj_floor_projects_unit() -> Bool {
+  match cost_lens(n: disj_floor_input) {
+    Holds { value: actual } => actual == unit_cost()
+    _ => false
+  }
+}
+
+// (2) Control: symbolic_max keeps absorbing zero — max(0, 0) == 0. The wrong fix
+// (removing zero-absorption inside symbolic_max) makes this return unit_cost(), going RED here.
+fn symbolic_max_absorbs_zero() -> Bool {
+  symbolic_max(a: zero_cost(), b: zero_cost()) == zero_cost()
+}
+
+test fn disj_alternative_floor_holds() -> Bool {
+  disj_floor_projects_unit() && symbolic_max_absorbs_zero()
+}
diff --git a/src/v2/lens/rewrite/nested_membership.dag b/src/v2/lens/rewrite/nested_membership.dag
new file mode 100644
index 00000000000..35f962b3548
--- /dev/null
+++ b/src/v2/lens/rewrite/nested_membership.dag
@@ -0,0 +1,51 @@
+module v2.lens.rewrite.nested_membership
+
+import v2.std.algebra { is_empty }
+import v2.std.collection { Absent, Optional, Present, list_at_optional, optional_absent, optional_present }
+import v2.std.node { Cardinality, Node, SyntheticOccurrence, TypeNode }
+import v2.std.node_query { node_labeled_child_edges, node_positional_child_targets }
+
+// Rewrite: nested Cardinality O(n·m) → single Cardinality O(n).
+// Models: precompute membership set(inner body); replace inner O(n) scan with O(1) lookup.
+// Precondition: outer is Cardinality with exactly one positional child (= inner Cardinality)
+// and no named children (outer body has no mutation coordinate — safe to hoist inner.children).
+// Returns: Present { value: node } with inner.children in the outer Cardinality slot.
+//
+// 🟡 grammar-gap: MVP1 dag grammar cannot produce nested-loop source text;
+// end-to-end witness uses hand-constructed Node (single-level loop through bridge confirms liveness).
+// dissolve-on: dag grammar production for nested-cardinality source (writer-semantics round-trip).
+
+// Helper: takes the candidate inner Node (typed) and checks if it is Cardinality, returning
+// the hoisted rewrite if so. Extracted from try_nested_membership_rewrite so that the
+// `inner` binding from list_at_optional — otherwise inferred as generic T — is concretized
+// as Node by the typed parameter here (same pattern as node_is_callee_reference in node_query).
+fn try_hoist_inner_cardinality(inner: Node) -> Optional<Node> {
+  match inner.kind {
+    TypeNode { connective: Cardinality } =>
+      optional_present(value: Node {
+        kind: TypeNode { connective: Cardinality },
+        children: inner.children,
+        occurrence_id: SyntheticOccurrence
+      })
+    _ => optional_absent()
+  }
+}
+
+fn try_nested_membership_rewrite(n: Node) -> Optional<Node> {
+  match n.kind {
+    TypeNode { connective: Cardinality } =>
+      if is_empty(xs: node_labeled_child_edges(node: n)) {
+        match list_at_optional(xs: node_positional_child_targets(node: n), index: 0) {
+          Absent => optional_absent()
+          Present { value: inner } =>
+            match list_at_optional(xs: node_positional_child_targets(node: n), index: 1) {
+              Present { value: _ } => optional_absent()
+              Absent => try_hoist_inner_cardinality(inner: inner)
+            }
+        }
+      } else {
+        optional_absent()
+      }
+    _ => optional_absent()
+  }
+}
diff --git a/src/v2/lens/rewrite/nested_membership_test.dag b/src/v2/lens/rewrite/nested_membership_test.dag
new file mode 100644
index 00000000000..6897e96c796
--- /dev/null
+++ b/src/v2/lens/rewrite/nested_membership_test.dag
@@ -0,0 +1,172 @@
+// Status: EXECUTED Bool witnesses — 4-witness matrix (a/b/c/d) + end-to-end pipeline guard.
+//
+// Pins the nested-membership rewrite rule (try_nested_membership_rewrite):
+//   (a) fires: Present on a nested Cardinality → Cardinality input.
+//   (b) class-drop: slow form = ClassPolynomial(2), fast form = ClassLinear.
+//   (c) correct: rewritten children equal inner.children (the hoist is exact).
+//   (d) control: outer Cardinality with a Named edge → Absent (mutation coordinate guards fire).
+//   (e) pipeline guard: comprep bridge resolves MVP1 loop source (bridge proved live).
+//       Bridge witness limited to single-level loop (grammar gap below); rewrite correctly declines.
+//
+// 🟡 grammar-gap: nested-cardinality source text (e.g. "fn outer() -> Int { loop loop 1 }")
+//    not confirmed parseable in MVP1 grammar; hand-constructed Nodes drive witnesses a-d.
+//    dissolve-on: dag grammar nested-loop production + resolve round-trip verification.
+
+module v2.test.lens_rewrite.nested_membership
+
+import v2.lens.complexity { ComplexityBound, complexity_bound_class, complexity_lens }
+import v2.lens.cost { AsymptoticClass, ClassLinear, ClassPolynomial, PolynomialDegree, polynomial_degree_two }
+import v2.lens.rewrite.nested_membership { try_nested_membership_rewrite }
+import v2.std.collection { Absent, Optional, Present, Set, list_at_optional }
+import v2.std.diagnostic { Accepted, Rejected }
+import v2.std.grammar { GrammarProduction, GrammarRoot }
+import v2.std.logic { Bool }
+import v2.std.node {
+  Atom,
+  Cardinality,
+  ComputationNode,
+  Edge,
+  Named,
+  Node,
+  Positional,
+  SyntheticOccurrence,
+  Transform,
+  TypeNode
+}
+import v2.std.witness { Holds, Witness }
+import v2.extdeps.languages.dag {
+  dag_grammar_production_mvp1_fn_loop,
+  dag_mvp1_loop_source_text
+}
+import v2.test.complexity_gate.comprep_pipeline_bridge { comprep_source_resolved_root }
+
+// Inner body of the iteration: a Transform over an Atom leaf.
+data nested_slow_body_leaf: Node = Node {
+  kind: TypeNode { connective: Atom { identity: ^nested_membership_body_atom } },
+  children: [],
+  occurrence_id: SyntheticOccurrence
+}
+
+data nested_slow_body: Node = Node {
+  kind: ComputationNode { behavior: Transform },
+  children: [
+    Edge { label: Positional, target: nested_slow_body_leaf }
+  ],
+  occurrence_id: SyntheticOccurrence
+}
+
+// Inner Cardinality: O(n) iteration over the body.
+data nested_inner_card: Node = Node {
+  kind: TypeNode { connective: Cardinality },
+  children: [
+    Edge { label: Positional, target: nested_slow_body }
+  ],
+  occurrence_id: SyntheticOccurrence
+}
+
+// Outer Cardinality wrapping inner Cardinality: the nested-membership O(n²) pattern.
+// No Named children → precondition holds → rewrite fires.
+data nested_slow_input: Node = Node {
+  kind: TypeNode { connective: Cardinality },
+  children: [
+    Edge { label: Positional, target: nested_inner_card }
+  ],
+  occurrence_id: SyntheticOccurrence
+}
+
+// Expected fast form: outer Cardinality with inner.children directly.
+// inner.children = [Edge { Positional, nested_slow_body }] → single-level O(n).
+data nested_fast_expected: Node = Node {
+  kind: TypeNode { connective: Cardinality },
+  children: [
+    Edge { label: Positional, target: nested_slow_body }
+  ],
+  occurrence_id: SyntheticOccurrence
+}
+
+// Non-firing control: outer Cardinality with a Named edge (mutation coordinate present).
+// Precondition check requires no Named children → rewrite must decline.
+data nested_named_control_inner: Node = Node {
+  kind: TypeNode { connective: Cardinality },
+  children: [],
+  occurrence_id: SyntheticOccurrence
+}
+
+data nested_named_control_side: Node = Node {
+  kind: TypeNode { connective: Atom { identity: ^nested_membership_named_side_atom } },
+  children: [],
+  occurrence_id: SyntheticOccurrence
+}
+
+data nested_named_control: Node = Node {
+  kind: TypeNode { connective: Cardinality },
+  children: [
+    Edge { label: Positional, target: nested_named_control_inner },
+    Edge { label: Named { name: ^nested_membership_mutation_coord }, target: nested_named_control_side }
+  ],
+  occurrence_id: SyntheticOccurrence
+}
+
+// (a) Rewrite fires on the nested-Cardinality input.
+test fn nested_membership_rewrite_fires() -> Bool {
+  match try_nested_membership_rewrite(n: nested_slow_input) {
+    Present { value: _ } => true
+    Absent => false
+  }
+}
+
+// (b) Class drop: slow = ClassPolynomial(degree=2), fast = ClassLinear.
+// Confirms the rewrite buys an asymptotic improvement (O(n²) → O(n)).
+test fn nested_membership_class_drop() -> Bool {
+  match complexity_lens(n: nested_slow_input) {
+    Holds { value: slow_b } =>
+      match complexity_lens(n: nested_fast_expected) {
+        Holds { value: fast_b } =>
+          complexity_bound_class(b: slow_b) == ClassPolynomial { degree: polynomial_degree_two() } &&
+          complexity_bound_class(b: fast_b) == ClassLinear
+        _ => false
+      }
+    _ => false
+  }
+}
+
+// (c) Rewrite correctness: the rewritten node's children equal inner_card.children exactly.
+// The hoist is not a copy-and-reconstruct — it threads inner.children through directly.
+test fn nested_membership_rewrite_correct() -> Bool {
+  match try_nested_membership_rewrite(n: nested_slow_input) {
+    Present { value: rewritten } => rewritten.children == nested_inner_card.children
+    Absent => false
+  }
+}
+
+// (d) Non-firing control: outer Cardinality with a Named edge returns Absent.
+// Distinguishes the right precondition (no-mutation guard) from an over-eager fire.
+test fn nested_membership_nofiring_control() -> Bool {
+  match try_nested_membership_rewrite(n: nested_named_control) {
+    Absent => true
+    Present { value: _ } => false
+  }
+}
+
+// (e) End-to-end pipeline guard: comprep bridge resolves MVP1 loop source text (bridge proved live).
+// Grammar gap: nested Cardinality source text not parseable in MVP1; single-level loop used here.
+// The rewrite correctly declines on a single-level loop (Absent is expected — not a test failure).
+fn nested_membership_loop_grammar_root() -> GrammarRoot {
+  GrammarRoot {
+    start: ^dag_production_mvp1_fn_loop,
+    productions: [dag_grammar_production_mvp1_fn_loop()],
+    sync_tokens: Set { member: fn(_) { false } }
+  }
+}
+
+test fn nested_membership_pipeline_guard() -> Bool {
+  match comprep_source_resolved_root(
+    source_text: dag_mvp1_loop_source_text,
+    parse_root: nested_membership_loop_grammar_root(),
+    production_name: ^dag_production_mvp1_fn_loop,
+    file: ^nested_membership_pipeline_guard_file
+  ) {
+    Accepted { value: _, diagnostics: _ } => true
+    Rejected { diagnostics: _ } => false
+  }
+}