---
title: "math-linsys-elimination — Mova code model"
description: "Mova code model of math/src/linsys.rs: components with their labels, obernants and the model text."
url: "https://groupants.elamur.ai/free/en/mova/math/src/linsys.rs.groupant/elimination.groupant/"
---
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# math-linsys-elimination

The groupant beside math/src/linsys.rs 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_fragment
- **status:** partial source bound model
- **source:** [`math/src/linsys.rs`](https://github.com/ElamurAI/mova-free/blob/main/math/src/linsys.rs)
- **source SHA-256:** `dd9f5f6f6c55e4f9…`
- **covers:** gauss classifies rank with first nonzero pivots and free-zero particular solutions on rectangular matched data. check_solution rejects count then width mismatches before exact substitution; empty equations accept arbitrary x. Ax=b and rank constrain inverse fibers. Shape scan O(m), elimination rational cubic-scale work. (bytes 2922–5197)
- **part of:** [math-linsys](https://md.groupants.elamur.ai/free/en/mova/math/src/linsys.rs.groupant/)
- **repository folder:** [`math/src/linsys.rs.groupant/elimination.groupant`](https://github.com/ElamurAI/mova-free/tree/main/math/src/linsys.rs.groupant/elimination.groupant)
- **repository commit:** `48f2e6f74146`

> **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 | [approximant](#approximant) |
| **ast** | partial | `../syntax.json` |
| **descriptant** | present | [descriptant](#descriptant) |
| **invariant** | partial | [invariant](#invariant) |
| **tests** | executed bounded checks | [tests](#tests) |

## 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 |
|---|---|---|---|
| `behavior_complement` | behavior_complement | partial source derived model | [behavior_complement](#behavior-complement) |
| `reverse_reconstruction` | reverse_reconstruction | partial source derived model | [reverse_reconstruction](#reverse-reconstruction) |
| `pivot_selection` | pivot_selection | partial source derived model | [pivot_selection](#pivot-selection) |
| `rank_classification` | rank_classification | partial source derived model | [rank_classification](#rank-classification) |
| `shape_precedence` | shape_precedence | partial source derived model | [shape_precedence](#shape-precedence) |
| `arithmetic_cost` | arithmetic_cost | partial source derived model | [arithmetic_cost](#arithmetic-cost) |

## Model text


### descriptant

#### math-linsys-elimination

gauss classifies rank with first nonzero pivots and free-zero particular solutions on rectangular matched data. check_solution rejects count then width mismatches before exact substitution; empty equations accept arbitrary x. Ax=b and rank constrain inverse fibers. Shape scan O(m), elimination rational cubic-scale work.

Epoch 2026-10-31: complete source read, parent AST reused; partial finite evidence only.

### approximant

#### Scoped forward and cost

gauss classifies rank with first nonzero pivots and free-zero particular solutions on rectangular matched data. check_solution rejects count then width mismatches before exact substitution; empty equations accept arbitrary x. Ax=b and rank constrain inverse fibers. Shape scan O(m), elimination rational cubic-scale work.

No executable percent surrogate claimed.

### invariant

#### Scoped invariant

gauss classifies rank with first nonzero pivots and free-zero particular solutions on rectangular matched data. check_solution rejects count then width mismatches before exact substitution; empty equations accept arbitrary x. Ax=b and rank constrain inverse fibers. Shape scan O(m), elimination rational cubic-scale work.

Source range[2922,5197) bound by manifest digest. Parent domain applies, not arbitrary public states.

### models

## Source-scoped typed relations

### behavior_complement

check_solution rejects A.len!=b.len, then any row.len!=x.len, before arithmetic; a malformed shape cannot pass by truncated zip. Matching shapes with any dot product !=b_i also reject.

### reverse_reconstruction

Accepted x constrains b=A*x for every shape-valid A, so many systems share it. Unique additionally requires full column rank. Infinite particular/free does not encode a nullspace basis or recover A.

### pivot_selection

gauss chooses first nonzero row at/after current row for each column, normalizes entire augmented row and eliminates pivot from all other rows. Exact arithmetic needs no epsilon tie policy.

### rank_classification

Any remaining reduced row0=c!=0 gives Inconsistent; otherwise pivot coordinates take RHS and all free coordinates remain0. Pivots count n means Unique, fewer gives Infinite with ascending free indices.

### shape_precedence

Checker count mismatch is O(1) and precedes O(m) width scan. Empty A,b accepts arbitrary x. gauss retains its own unchecked rectangular/matched precondition and is not made total by checker repair.

### arithmetic_cost

After shape validation substitution evaluates m length-n rational dot products O(mn) plus mismatch text. Gauss is O(m*n*min(m,n)) rational operations/O(mn) storage; bit growth is additional.

### tests

#### Evidence

See ../tests.md for actual checks and source mutant; this fragment adds no runtime test.

gauss classifies rank with first nonzero pivots and free-zero particular solutions on rectangular matched data. check_solution rejects count then width mismatches before exact substitution; empty equations accept arbitrary x. Ax=b and rank constrain inverse fibers. Shape scan O(m), elimination rational cubic-scale work.