---
title: "combinatorial-structures — Mova code model"
description: "Mova code model of models/combinatorial-structures.groupant: components with their labels, obernants and the model text."
url: "https://groupants.elamur.ai/free/en/mova/models/combinatorial-structures.groupant/"
---
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# combinatorial-structures

The groupant beside models/combinatorial-structures.groupant in the Mova repository: what the code computes, what it never does, how to go back from its outputs, its invariants, cost and checks.

## Facts

- **kind:** groupant_group
- **status:** partial source bound group
- **bound sources:** 10
- **repository folder:** [`models/combinatorial-structures.groupant`](https://github.com/ElamurAI/mova-free/tree/main/models/combinatorial-structures.groupant)
- **repository commit:** `855da7bd4a7b`

> **Implementation material.** A Mova code model describes this code at one source revision: partial models, finite checks and their stated scope. It is not a reviewed general exposition, and its labels are shown as written, never raised.

## Components

| component | label | where |
|---|---|---|
| **approximant** | partial source model | [forward](#forward) |
| **ast** | not materialized |  |
| **composition** | partial source model | [composition](#composition) |
| **cost** | partial source model | [cost](#cost) |
| **descriptant** | partial source model | [epoch-memory](#epoch-memory) |
| **invariant** | partial source model | [invariant](#invariant) |
| **tests** | bounded executed | [wave26-evidence](#wave26-evidence) |

## Obernants

The typed relations around the forward model: the behaviour complement (what the code never does), the reverse reconstruction (every input compatible with an output) and specialised views.

| id | type | label | where |
|---|---|---|---|
| `combinatorial-structures-behavior-complement` | behavior_complement | partial relational specification | [behavior-complement](#behavior-complement) |
| `combinatorial-structures-reverse-reconstruction` | reverse_reconstruction | partial relational specification | [reverse-reconstruction](#reverse-reconstruction) |
| `combinatorial-structures-objective_projection` | objective_projection | partial relational specification | [objective_projection](#objective-projection) |
| `combinatorial-structures-feasibility_objective` | feasibility_objective | partial relational specification | [feasibility_objective](#feasibility-objective) |
| `combinatorial-structures-reduction_capacity` | reduction_capacity | partial relational specification | [reduction_capacity](#reduction-capacity) |
| `combinatorial-structures-component_ambiguity` | component_ambiguity | partial relational specification | [component_ambiguity](#component-ambiguity) |

## Model text


### model

## combinatorial-structures

### forward

Tagged exact combinatorial relations: general-graph maximum-cardinality matching, exact cover, functional-graph decomposition, maximum-weight closure and rooted minimum arborescence. Each retains its own feasible witness and objective. Matching maximizes |M| over disjoint edges; exact cover enforces sum_selected_rows 1[column in row]=1. Closure maximizes sum_{v in C}w_v under implications. Arborescence returns min_T sum_e w_e. Functional output satisfies f^distance(v)(v)=entry(v) on canonical cycle cycle_id(v).

### invariant

Witness indexes refer to the original graph/row universe. Matching disjointness, exact column coverage, functional successor components, closure implications and rooted directed reachability are separate invariants. Infeasible, invalid and successful-empty outcomes remain distinct.

### cost

Functional decomposition O(n) time/space. Exact cover may be exponential; matching and closure use concrete leaf bounds. Arborescence scans each nonroot parent once per contraction and labels disjoint cycles: O(V+E) per round, at most V rounds, O(VE+V²) total. The nested-contraction witness at V=257 reduced primary parent advances from 1065024 to 24640 with the same optimum; this is an operation count, not timing. Adapter construction and allocation add cost.

### composition

Closure uses exact reduction capacities and implication edges. Arborescence exposes Result<Option<i128>,Error> objective only, no edge witness. For heads define successor[0]=0 and successor[i]=head[i] after validation: token i reaches root iff cycle_id[i]=cycle_id[0]. Caller empty/root-label/first-error policy remains separate. Matching-to-cover needs explicit columns and objective handling.

### limitations

Successful allocations and each leaf's finite domain/error precedence. These are tagged relations, not automatic production adoption. Floating residuals are not solution-error guarantees. No generic 1/2/5/10% composition-wide profile; finite evidence remains separate from analytical claims.

### epoch-memory

Wave26, 2026-10-30. Tagged library tools; no automatic training adoption, source-volume quality claim or generic numerical profile.

### checks

["cargo test -p mova-algorithms blossom_matching::tests","cargo test -p mova-algorithms exact_cover::tests","cargo test -p mova-algorithms functional_graph_decomposition::tests","cargo test -p mova-algorithms maximum_weight_closure::tests","cargo test -p mova-algorithms minimum_arborescence::tests"]

### wave26-executed-checks

## combinatorial-structures: wave26 checks

13 source-local test declarations across five modules. Every module has a compiled failing production mutation followed by exact restoration. The final whole library passed 877 checks twice debug and twice release. See [integrated evidence](https://md.groupants.elamur.ai/free/en/mova/models/algorithm-library.groupant/#wave26-evidence) for scopes and limitations. These tagged APIs are not an invented serial pipeline.

### catalog

## Typed family relations

### behavior-complement

Type: behavior_complement.

Exclude overlapping matched edges, nonexact coverage, closure implication violations, wrong cycle/entry/distance, or nonminimum arborescence cost. None means infeasible, not cost zero; a returned arborescence edge witness is outside the API.

### reverse-reconstruction

Type: reverse_reconstruction.

Arborescence cost c admits all rooted graphs with optimum c; selected edges are unrecoverable. Matching/cover/closure witnesses admit different unused edges/rows/weights. Cycle/entry/distance summaries admit distinct successor forests between adjacent distance layers.

### objective_projection

Type: objective_projection.

Arborescence returns cost only; edge-witness lifting is not exposed. Matching, cover and closure retain their separate original-index witnesses.

### feasibility_objective

Type: feasibility_objective.

Maximum cardinality, exact coverage, closure weight and arborescence cost are different objectives with different feasibility domains.

### reduction_capacity

Type: reduction_capacity.

Closure reductions require checked capacity arithmetic and exact implication encoding; finite capacities cannot silently replace an intended hard implication.

### component_ambiguity

Type: component_ambiguity.

After validating heads and setting successor[0]=0, cycle_id[i]=cycle_id[0] iff i reaches root. This adapter preserves caller-specific empty, root-label and error precedence rules.

### wave26-evidence

#### combinatorial-structures: wave26 checks

13 source-local test declarations across five modules. Every module has a compiled failing production mutation followed by exact restoration. The final whole library passed 877 checks twice debug and twice release. See [integrated evidence](https://md.groupants.elamur.ai/free/en/mova/models/algorithm-library.groupant/#wave26-evidence) for scopes and limitations. These tagged APIs are not an invented serial pipeline.