---
title: "math-linsys-linear — 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/linear.groupant/"
---
Elamur: [Mova](https://md.mova.elamur.ai/free/en/) · [Dictionary](https://md.dictionary.elamur.ai/free/en/) · [Research](https://md.research.elamur.ai/free/en/) · **Groupants**

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# math-linsys-linear

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:** linear_form maps supported syntax to affine Q coefficients after selected-variable absence test; constant calc eval may mutate Env. Products need a constant operand, division constant nonzero denominator. Coefficients do not recover syntax. Traversal/vector costs compose with constant evaluation. (bytes 392–2640)
- **part of:** [math-linsys](https://md.groupants.elamur.ai/free/en/mova/math/src/linsys.rs.groupant/)
- **repository folder:** [`math/src/linsys.rs.groupant/linear.groupant`](https://github.com/ElamurAI/mova-free/tree/main/math/src/linsys.rs.groupant/linear.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) |
| `absence_priority` | absence_priority | partial source derived model | [absence_priority](#absence-priority) |
| `affine_join` | affine_join | partial source derived model | [affine_join](#affine-join) |
| `variable_coordinate` | variable_coordinate | partial source derived model | [variable_coordinate](#variable-coordinate) |
| `failure_effect` | failure_effect | partial source derived model | [failure_effect](#failure-effect) |

## Model text


### descriptant

#### math-linsys-linear

linear_form maps supported syntax to affine Q coefficients after selected-variable absence test; constant calc eval may mutate Env. Products need a constant operand, division constant nonzero denominator. Coefficients do not recover syntax. Traversal/vector costs compose with constant evaluation.

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

### approximant

#### Scoped forward and cost

linear_form maps supported syntax to affine Q coefficients after selected-variable absence test; constant calc eval may mutate Env. Products need a constant operand, division constant nonzero denominator. Coefficients do not recover syntax. Traversal/vector costs compose with constant evaluation.

No executable percent surrogate claimed.

### invariant

#### Scoped invariant

linear_form maps supported syntax to affine Q coefficients after selected-variable absence test; constant calc eval may mutate Env. Products need a constant operand, division constant nonzero denominator. Coefficients do not recover syntax. Traversal/vector costs compose with constant evaluation.

Source range[392,2640) bound by manifest digest. Parent domain applies, not arbitrary public states.

### models

## Source-scoped typed relations

### behavior_complement

Selected-variable product with both nonzero coefficient vectors excludes an affine result; division by a variable denominator excludes success before the denominator-zero constant check.

### reverse_reconstruction

Coefficient vector c and constant k admit all supported syntax representing sum c_i*x_i+k, e.g x+x and2*x. Variable-free expressions can share coefficients while leaving different calc Env effects.

### absence_priority

Collect free variable spellings first; no selected name means eval entire subtree through calc, requiring V::Q. Unsupported syntax can still be admitted when constant relative to selected vars.

### affine_join

Add/sub combine coefficients and constants componentwise. For multiplication one side must have all-zero coefficients, scaling the other's vector and constant. Division scales by reciprocal constant denominator.

### variable_coordinate

E::Var selects the first matching vars index and places1 there; duplicates in vars therefore do not represent distinct coordinates. Caller should supply distinct names.

### failure_effect

Recursive left branch may evaluate constants/change Env before right error. Conversion returns String without rollback; callers cannot infer unchanged Env from Err.

### tests

#### Evidence

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

linear_form maps supported syntax to affine Q coefficients after selected-variable absence test; constant calc eval may mutate Env. Products need a constant operand, division constant nonzero denominator. Coefficients do not recover syntax. Traversal/vector costs compose with constant evaluation.