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

The groupant beside math/src/poly.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/poly.rs`](https://github.com/ElamurAI/mova-free/blob/main/math/src/poly.rs)
- **source SHA-256:** `28c7c69c8282270b…`
- **covers:** Root variants distinguish rational/surd/f64. Yun multiplicities, rational validation and product checks compose; remaining factors use500-sweep Aberth with residual tests. Root-only inverse needs leading scalar and original domain. No convergence theorem or total error bound. (bytes 9459–22090)
- **part of:** [math-poly](https://md.groupants.elamur.ai/free/en/mova/math/src/poly.rs.groupant/)
- **repository folder:** [`math/src/poly.rs.groupant/roots.groupant`](https://github.com/ElamurAI/mova-free/tree/main/math/src/poly.rs.groupant/roots.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) |
| `squarefree_relation` | squarefree_relation | partial source derived model | [squarefree_relation](#squarefree-relation) |
| `candidate_filter` | candidate_filter | partial source derived model | [candidate_filter](#candidate-filter) |
| `surd_transport` | surd_transport | partial source derived model | [surd_transport](#surd-transport) |
| `numeric_residual` | numeric_residual | partial source derived model | [numeric_residual](#numeric-residual) |

## Model text


### descriptant

#### math-poly-roots

Root variants distinguish rational/surd/f64. Yun multiplicities, rational validation and product checks compose; remaining factors use500-sweep Aberth with residual tests. Root-only inverse needs leading scalar and original domain. No convergence theorem or total error bound.

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

### approximant

#### Scoped forward and cost

Root variants distinguish rational/surd/f64. Yun multiplicities, rational validation and product checks compose; remaining factors use500-sweep Aberth with residual tests. Root-only inverse needs leading scalar and original domain. No convergence theorem or total error bound.

No executable percent surrogate claimed.

### invariant

#### Scoped invariant

Root variants distinguish rational/surd/f64. Yun multiplicities, rational validation and product checks compose; remaining factors use500-sweep Aberth with residual tests. Root-only inverse needs leading scalar and original domain. No convergence theorem or total error bound.

Source range[9459,22090) bound by manifest digest. Parent domain applies, not arbitrary public states.

### models

## Source-scoped typed relations

### behavior_complement

solve_poly zero polynomial excludes a root list and returns any-value error; nonzero constant returns no roots. An exact Q candidate failing substitution cannot be accepted by rational_roots.

### reverse_reconstruction

Root list with multiplicities fixes only a monic polynomial under exact complete roots; any nonzero leading scalar shares it. Approximate roots admit many nearby polynomials and cannot recover exact coefficients.

### squarefree_relation

Yun uses gcd(f,f'), then repeatedly gcd(b,c-b') and associates factor with multiplicity i. Loop stops after400; factor product/completeness checks must detect omissions rather than assume unbounded decomposition.

### candidate_filter

Small primitive endpoint coefficients enumerate signed divisors pn/qd, checking exact Horner zero. Large-coefficient fallback rounds numeric real root*qd and still validates exact candidate; missing candidates remain possible.

### surd_transport

Quadratic discriminant N/M becomes sqrt(N*M)/M; factorization extracts square part s and squarefree d, yielding a0±b0 sqrt(d). Exact substitution in Q(sqrt(d)) checks both coefficients zero; factor completion can include probable primes.

### numeric_residual

Aberth iterates at most500 sweeps and skips nonfinite steps. solve_poly checks |p(z)|<=1e-8 max(scale,1), plus multiplicity sum and factor rebuild; residual/completeness is not convergence, conditioning or rational-domain certification.

### tests

#### Evidence

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

Root variants distinguish rational/surd/f64. Yun multiplicities, rational validation and product checks compose; remaining factors use500-sweep Aberth with residual tests. Root-only inverse needs leading scalar and original domain. No convergence theorem or total error bound.