CodingThrust
diff --git a/‎.claude/CLAUDE.md‎
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diff --git a/‎.claude/skills/add-model/SKILL.md‎
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diff --git a/‎docs/paper/reductions.typ‎
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diff --git a/‎problemreductions-cli/src/cli.rs‎
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diff --git a/‎problemreductions-cli/src/commands/create.rs‎
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Lines changed: 73 additions & 25 deletions
@@ -91,7 +91,8 @@ make release V=x.y.z  # Tag and push a new release (CI publishes to crates.io)
   - `misc/` - Unique input structures
   - Run `pred list` for the full catalog of problems, variants, and reductions; `pred show <name>` for details on a specific problem
 - `src/rules/` - Reduction rules + inventory registration
-- `src/solvers/` - BruteForce solver for aggregate values plus witness recovery when supported, ILP solver (feature-gated, witness-only). To check if a problem supports ILP solving via a witness-capable reduction path, run `pred path <ProblemName> ILP`
+- `src/models/decision.rs` - Generic `Decision<P>` wrapper converting optimization problems to decision problems
+- `src/solvers/` - BruteForce solver for aggregate values plus witness recovery when supported, ILP solver (feature-gated, witness-only), decision search (binary search via Decision queries). To check if a problem supports ILP solving via a witness-capable reduction path, run `pred path <ProblemName> ILP`
 - `src/traits.rs` - `Problem` trait
 - `src/rules/traits.rs` - `ReduceTo<T>`, `ReduceToAggregate<T>`, `ReductionResult`, `AggregateReductionResult` traits
 - `src/registry/` - Compile-time reduction metadata collection
@@ -122,14 +123,22 @@ Problem (core trait — all problems must implement)
 
 **Aggregate-only problems** use fold values such as `Sum<W>` or `And`; these solve to a value but have no representative witness configuration.
 
+**Decision problems** wrap an optimization problem with a bound: `Decision<P>` where `P::Value: OptimizationValue`. Evaluates to `Or(true)` when the inner objective meets the bound (≤ for Min, ≥ for Max).
+
 Common aggregate wrappers live in `src/types.rs`:
 ```rust
 Max<V>, Min<V>, Sum<W>, Or, And, Extremum<V>, ExtremumSense
 ```
 
+`OptimizationValue` trait (in `src/types.rs`) enables generic Decision conversion:
+- `Min<V>`: meets bound when value ≤ bound
+- `Max<V>`: meets bound when value ≥ bound
+
 ### Key Patterns
 - `variant_params!` macro implements `Problem::variant()` — e.g., `crate::variant_params![G, W]` for two type params, `crate::variant_params![]` for none (see `src/variant.rs`)
 - `declare_variants!` proc macro registers concrete type instantiations with best-known complexity and registry-backed load/serialize/value-solve/witness-solve metadata. One entry per problem may be marked `default`, and variable names in complexity strings are validated at compile time against actual getter methods.
+- `decision_problem_meta!` macro registers `DecisionProblemMeta` for a concrete inner type, providing the `DECISION_NAME` constant.
+- `register_decision_variant!` macro generates `declare_variants!`, `ProblemSchemaEntry`, and both `ReductionEntry` submissions (aggregate Decision→Opt + Turing Opt→Decision) for a `Decision<P>` variant. Callers must define inherent getters (`num_vertices()`, `num_edges()`, `k()`) on `Decision<P>` before invoking. Accepts `dims`, `fields`, and `size_getters` parameters for problem-specific size fields.
 - Problems parameterized by graph type `G` and optionally weight type `W` (problem-dependent)
 - `Solver::solve()` computes the aggregate value for any `Problem` whose `Value` implements `Aggregate`
 - `BruteForce::find_witness()` / `find_all_witnesses()` recover witnesses only when `P::Value::supports_witnesses()`
@@ -171,14 +180,16 @@ Reduction graph nodes use variant key-value pairs from `Problem::variant()`:
 - Default variant ranking: `SimpleGraph`, `One`, `KN` are considered default values; variants with the most default values sort first
 - Nodes come exclusively from `#[reduction]` registrations; natural edges between same-name variants are inferred from the graph/weight subtype partial order
 - Each primitive reduction is determined by the exact `(source_variant, target_variant)` endpoint pair
-- Reduction edges carry `EdgeCapabilities { witness, aggregate }`; graph search defaults to witness mode, and aggregate mode is available through `ReductionMode::Aggregate`
-- `#[reduction]` accepts only `overhead = { ... }` and currently registers witness/config reductions; aggregate-only edges require manual `ReductionEntry` registration with `reduce_aggregate_fn`
+- Reduction edges carry `EdgeCapabilities { witness, aggregate, turing }`; graph search defaults to witness mode, aggregate mode is available through `ReductionMode::Aggregate`, and Turing (multi-query) mode via `ReductionMode::Turing`
+- `#[reduction]` accepts only `overhead = { ... }` and currently registers witness/config reductions; aggregate-only and Turing edges require manual `ReductionEntry` registration
+- `Decision<P> → P` is an aggregate-only edge (solve optimization, compare to bound); `P → Decision<P>` is a Turing edge (binary search over decision bound)
 
 ### Extension Points
 - New models register dynamic load/serialize/brute-force dispatch through `declare_variants!` in the model file, not by adding manual match arms in the CLI
 - **CLI creation is schema-driven:** `pred create` automatically maps `ProblemSchemaEntry` fields to CLI flags via `snake_case → kebab-case` convention. New models need only: (1) matching CLI flags in `CreateArgs` + `flag_map()`, and (2) type parser support in `parse_field_value()` if using a new field type. No match arm in `create.rs` is needed.
 - **CLI flag names must match schema field names.** The canonical name for a CLI flag is the schema field name in kebab-case (e.g., schema field `universe_size` → `--universe-size`, field `subsets` → `--subsets`). Old aliases (e.g., `--universe`, `--sets`) may exist as clap `alias` for backward compatibility at the clap level, but `flag_map()`, help text, error messages, and documentation must use the schema-derived name. Do not add new backward-compat aliases; if a field is renamed in the schema, update the CLI flag name to match.
-- Aggregate-only models are first-class in `declare_variants!`; aggregate-only reduction edges still need manual `ReductionEntry` wiring because `#[reduction]` only registers witness/config reductions today
+- **Decision variants** of optimization problems use `Decision<P>` wrapper. Add via: (1) `decision_problem_meta!` for the inner type, (2) inherent methods on `Decision<Inner>`, (3) `register_decision_variant!` with `dims`, `fields`, `size_getters`. Schema-driven CLI creation auto-restructures flat JSON into `{inner: {...}, bound}`.
+- Aggregate-only models are first-class in `declare_variants!`; aggregate-only and Turing reduction edges still need manual `ReductionEntry` wiring because `#[reduction]` only registers witness/config reductions today
 - Exact registry dispatch lives in `src/registry/`; alias resolution and partial/default variant resolution live in `problemreductions-cli/src/problem_name.rs`
 - `pred create` schema-driven dispatch lives in `problemreductions-cli/src/commands/create.rs` (`create_schema_driven()`)
 - Canonical paper and CLI examples live in `src/example_db/model_builders.rs` and `src/example_db/rule_builders.rs`
@@ -195,8 +206,8 @@ Reduction graph nodes use variant key-value pairs from `Problem::variant()`:
 ### Paper (docs/paper/reductions.typ)
 - `problem-def(name)[def][body]` — defines a problem with auto-generated schema, reductions list, and label `<def:ProblemName>`. Title comes from `display-name` dict.
 - `reduction-rule(source, target, example: bool, ...)[rule][proof]` — generates a theorem with label `<thm:Source-to-Target>` and registers in `covered-rules` state. Overhead auto-derived from JSON edge data.
-- Every directed reduction needs its own `reduction-rule` entry
-- Completeness warnings auto-check that all JSON graph nodes/edges are covered in the paper
+- Every directed reduction needs its own `reduction-rule` entry (except trivial Decision↔Optimization pairs which are auto-filtered)
+- Completeness warnings auto-check that all JSON graph nodes/edges are covered in the paper; `Decision<P> ↔ P` edges are excluded since they are trivial solve-and-compare reductions
 - `display-name` dict maps `ProblemName` to display text
 
 ## Testing Requirements
 
@@ -311,6 +311,7 @@ Structural and quality review is handled by the `review-pipeline` stage, not her
 | Wrong aggregate wrapper | Use `Max` / `Min` / `Extremum` for objective problems, `Or` for existential witness problems, and `Sum` / `And` (or a custom aggregate) for value-only folds |
 | Wrong `declare_variants!` syntax | Entries no longer use `opt` / `sat`; one entry per problem may be marked `default` |
 | Forgetting CLI alias | Must add lowercase entry in `problem_name.rs` `resolve_alias()` |
+| Adding a hand-written decision model | Use `Decision<P>` wrapper instead — see `decision_problem_meta!` + `register_decision_variant!` in `src/models/graph/minimum_vertex_cover.rs` for the pattern |
 | Inventing short aliases | Only use well-established literature abbreviations (MIS, SAT, TSP); do NOT invent new ones |
 | Forgetting CLI flags | Schema-driven create needs matching CLI flags in `CreateArgs` for each `ProblemSchemaEntry` field (snake_case → kebab-case). Also add to `flag_map()`. |
 | Missing type parser | If the problem uses a new field type, add a handler in `parse_field_value()` in `create.rs` |
 
@@ -62,8 +62,18 @@
   }
 }
 
-#let graph-num-vertices(instance) = instance.graph.num_vertices
-#let graph-num-edges(instance) = instance.graph.edges.len()
+#let graph-instance(instance) = {
+  if "graph" in instance {
+    instance
+  } else if "inner" in instance and "graph" in instance.inner {
+    instance.inner
+  } else {
+    instance
+  }
+}
+
+#let graph-num-vertices(instance) = graph-instance(instance).graph.num_vertices
+#let graph-num-edges(instance) = graph-instance(instance).graph.edges.len()
 #let spin-num-spins(instance) = instance.fields.len()
 #let sat-num-clauses(instance) = instance.clauses.len()
 #let subsetsum-num-elements(instance) = instance.sizes.len()
@@ -234,7 +244,7 @@
   "MinimumDisjunctiveNormalForm": [Minimum Disjunctive Normal Form],
   "MinimumGraphBandwidth": [Minimum Graph Bandwidth],
   "MinimumMetricDimension": [Minimum Metric Dimension],
-  "VertexCover": [Vertex Cover],
+  "DecisionMinimumVertexCover": [Decision Minimum Vertex Cover],
   "MinimumCodeGenerationUnlimitedRegisters": [Minimum Code Generation (Unlimited Registers)],
   "RegisterSufficiency": [Register Sufficiency],
   "ResourceConstrainedScheduling": [Resource Constrained Scheduling],
@@ -649,22 +659,23 @@ In all graph problems below, $G = (V, E)$ denotes an undirected graph with $|V|
 }
 
 #{
-  let x = load-model-example("VertexCover")
+  let x = load-model-example("DecisionMinimumVertexCover")
+  let inner = x.instance.inner
   let nv = graph-num-vertices(x.instance)
   let ne = graph-num-edges(x.instance)
-  let k = x.instance.k
+  let k = x.instance.bound
   let sol = x.optimal_config
   let cover = sol.enumerate().filter(((i, v)) => v == 1).map(((i, _)) => i)
   [
-    #problem-def("VertexCover")[
-      Given an undirected graph $G = (V, E)$ and a positive integer $k <= |V|$, determine whether there exists a vertex cover of size at most $k$: a subset $V' subset.eq V$ with $|V'| <= k$ such that for each edge ${u, v} in E$, at least one of $u, v$ belongs to $V'$.
+    #problem-def("DecisionMinimumVertexCover")[
+      Given an undirected graph $G = (V, E)$ with vertex weights $w: V -> RR_(gt.eq 0)$ and an integer bound $k$, determine whether there exists a vertex cover $S subset.eq V$ with $sum_(v in S) w(v) <= k$ such that every edge has at least one endpoint in $S$.
     ][
-    Vertex Cover is one of Karp's 21 NP-complete problems @karp1972 and the decision version of Minimum Vertex Cover @garey1979. The best known exact algorithm runs in $O^*(1.1996^n)$ time (Chen, Kanj, and Xia, 2010).
+    Decision Minimum Vertex Cover is the decision version of Minimum Vertex Cover and one of Karp's 21 NP-complete problems @karp1972 @garey1979. It asks whether the optimization objective can be achieved within a prescribed budget rather than minimizing the cover weight directly.
 
-    *Example.* Consider a graph on $n = #nv$ vertices and $|E| = #ne$ edges with threshold $k = #k$. The cover $V' = {#cover.map(i => $v_#i$).join(", ")}$ with $|V'| = #cover.len() <= k$ is a valid vertex cover.
+    *Example.* Consider a graph on $n = #nv$ vertices and $|E| = #ne$ edges with threshold $k = #k$. The cover $S = {#cover.map(i => $v_#i$).join(", ")}$ has total weight $2 <= #k$ and therefore certifies a yes-instance.
 
     #pred-commands(
-      "pred create --example VertexCover -o vc.json",
+      "pred create --example DecisionMinimumVertexCover -o vc.json",
       "pred solve vc.json",
       "pred evaluate vc.json --config " + sol.map(str).join(","),
     )
@@ -13086,8 +13097,13 @@ See #link("https://github.com/CodingThrust/problem-reductions/blob/main/examples
     }
     unique
   }
+  // Skip trivial Decision<P> ↔ P edges (solve-and-compare, no interesting proof)
+  let is-decision-opt-pair(src, tgt) = {
+    src == "Decision" + tgt or tgt == "Decision" + src
+  }
   let missing = json-edges.filter(e => {
-    covered.find(c => c.at(0) == e.at(0) and c.at(1) == e.at(1)) == none
+    if is-decision-opt-pair(e.at(0), e.at(1)) { false }
+    else { covered.find(c => c.at(0) == e.at(0) and c.at(1) == e.at(1)) == none }
   })
   if missing.len() > 0 {
     block(width: 100%, inset: (x: 1em, y: 0.5em), fill: rgb("#fff3cd"), stroke: (left: 3pt + rgb("#ffc107")))[
 
@@ -230,7 +230,7 @@ Flags by problem type:
   SpinGlass                       --graph, --couplings, --fields
   KColoring                       --graph, --k
   KClique                         --graph, --k
-  VertexCover (VC)                --graph, --k
+  DecisionMinimumVertexCover      --graph, --weights, --bound
   MinimumMultiwayCut              --graph, --terminals, --edge-weights
   MonochromaticTriangle           --graph
   PartitionIntoTriangles          --graph
 
@@ -17,12 +17,13 @@ use problemreductions::models::graph::{
     GeneralizedHex, HamiltonianCircuit, HamiltonianPath, HamiltonianPathBetweenTwoVertices,
     LengthBoundedDisjointPaths, LongestCircuit, MinimumCutIntoBoundedSets,
     MinimumDummyActivitiesPert, MinimumMaximalMatching, RootedTreeArrangement, SteinerTree,
-    SteinerTreeInGraphs, VertexCover,
+    SteinerTreeInGraphs,
 };
 use problemreductions::models::misc::{
     CbqRelation, FrequencyTable, KnownValue, QueryArg, SchedulingWithIndividualDeadlines,
     ThreePartition,
 };
+use problemreductions::models::Decision;
 use problemreductions::prelude::*;
 use problemreductions::registry::collect_schemas;
 use problemreductions::topology::{
@@ -672,6 +673,19 @@ fn ser_vertex_weight_problem_with<G: Graph + Serialize>(
     }
 }
 
+fn ser_decision_minimum_vertex_cover_with<
+    G: Graph + Serialize + problemreductions::variant::VariantParam,
+>(
+    graph: G,
+    weights: Vec<i32>,
+    bound: i32,
+) -> Result<serde_json::Value> {
+    ser(Decision::new(
+        MinimumVertexCover::new(graph, weights),
+        bound,
+    ))
+}
+
 fn ser<T: Serialize>(problem: T) -> Result<serde_json::Value> {
     util::ser(problem)
 }
@@ -1787,6 +1801,63 @@ fn create_random(
     let graph_type = resolved_graph_type(resolved_variant);
 
     let (data, variant) = match canonical {
+        "DecisionMinimumVertexCover" => {
+            let raw_bound = args.bound.ok_or_else(|| {
+                anyhow::anyhow!(
+                    "DecisionMinimumVertexCover requires --bound\n\n\
+                     Usage: pred create DecisionMinimumVertexCover --random --num-vertices 5 [--edge-prob 0.5] [--seed 42] --bound 3"
+                )
+            })?;
+            anyhow::ensure!(
+                raw_bound >= 0,
+                "DecisionMinimumVertexCover: --bound must be non-negative"
+            );
+            let bound = i32::try_from(raw_bound).map_err(|_| {
+                anyhow::anyhow!(
+                    "DecisionMinimumVertexCover: --bound must fit in a 32-bit signed integer, got {raw_bound}"
+                )
+            })?;
+            let weights = vec![1i32; num_vertices];
+            match graph_type {
+                "KingsSubgraph" => {
+                    let positions = util::create_random_int_positions(num_vertices, args.seed);
+                    let graph = KingsSubgraph::new(positions);
+                    (
+                        ser_decision_minimum_vertex_cover_with(graph, weights, bound)?,
+                        resolved_variant.clone(),
+                    )
+                }
+                "TriangularSubgraph" => {
+                    let positions = util::create_random_int_positions(num_vertices, args.seed);
+                    let graph = TriangularSubgraph::new(positions);
+                    (
+                        ser_decision_minimum_vertex_cover_with(graph, weights, bound)?,
+                        resolved_variant.clone(),
+                    )
+                }
+                "UnitDiskGraph" => {
+                    let positions = util::create_random_float_positions(num_vertices, args.seed);
+                    let radius = args.radius.unwrap_or(1.5);
+                    let graph = UnitDiskGraph::new(positions, radius);
+                    (
+                        ser_decision_minimum_vertex_cover_with(graph, weights, bound)?,
+                        resolved_variant.clone(),
+                    )
+                }
+                _ => {
+                    let edge_prob = args.edge_prob.unwrap_or(0.5);
+                    if !(0.0..=1.0).contains(&edge_prob) {
+                        bail!("--edge-prob must be between 0.0 and 1.0");
+                    }
+                    let graph = util::create_random_graph(num_vertices, edge_prob, args.seed);
+                    (
+                        ser_decision_minimum_vertex_cover_with(graph, weights, bound)?,
+                        resolved_variant.clone(),
+                    )
+                }
+            }
+        }
+
         // Graph problems with vertex weights
         "MaximumIndependentSet"
         | "MinimumVertexCover"
@@ -1848,29 +1919,6 @@ fn create_random(
             )
         }
 
-        "VertexCover" => {
-            let edge_prob = args.edge_prob.unwrap_or(0.5);
-            if !(0.0..=1.0).contains(&edge_prob) {
-                bail!("--edge-prob must be between 0.0 and 1.0");
-            }
-            let graph = util::create_random_graph(num_vertices, edge_prob, args.seed);
-            let usage =
-                "Usage: pred create VertexCover --random --num-vertices 5 [--edge-prob 0.5] [--seed 42] --k 3";
-            let k = args
-                .k
-                .ok_or_else(|| anyhow::anyhow!("VertexCover requires --k\n\n{usage}"))?;
-            if k == 0 {
-                bail!("VertexCover: --k must be positive");
-            }
-            if k > graph.num_vertices() {
-                bail!("VertexCover: k must be <= graph num_vertices");
-            }
-            (
-                ser(VertexCover::new(graph, k))?,
-                variant_map(&[("graph", "SimpleGraph")]),
-            )
-        }
-
         // MinimumCutIntoBoundedSets (graph + edge weights + s/t/B/K)
         "MinimumCutIntoBoundedSets" => {
             let edge_prob = args.edge_prob.unwrap_or(0.5);
@@ -2232,7 +2280,7 @@ fn create_random(
         _ => bail!(
             "Random generation is not supported for {canonical}. \
              Supported: graph-based problems (MIS, MVC, MaxCut, MaxClique, \
-             MaximumMatching, MinimumDominatingSet, SpinGlass, KColoring, KClique, VertexCover, TravelingSalesman, \
+             MaximumMatching, MinimumDominatingSet, SpinGlass, KColoring, KClique, DecisionMinimumVertexCover, TravelingSalesman, \
              BottleneckTravelingSalesman, SteinerTreeInGraphs, HamiltonianCircuit, MaximumLeafSpanningTree, SteinerTree, \
              OptimalLinearArrangement, RootedTreeArrangement, HamiltonianPath, LongestCircuit, GeneralizedHex)"
         ),