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

The groupant beside models/numerical-evaluation.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:** 8
- **repository folder:** [`models/numerical-evaluation.groupant`](https://github.com/ElamurAI/mova-free/tree/main/models/numerical-evaluation.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 | [checks](#checks) |

## 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 |
|---|---|---|---|
| `numerical-evaluation-behavior-complement` | behavior_complement | partial source model | [behavior-complement](#behavior-complement) |
| `numerical-evaluation-reverse-reconstruction` | reverse_reconstruction | partial source model | [reverse-reconstruction](#reverse-reconstruction) |
| `numerical-evaluation-basis-conversion` | polynomial_basis_transport | partial source model | [basis-conversion](#basis-conversion) |
| `numerical-evaluation-analytic-integral` | antiderivative_certificate | partial source model | [analytic-integral](#analytic-integral) |
| `numerical-evaluation-sample-blindness` | finite_observation_complement | partial source model | [sample-blindness](#sample-blindness) |

## Model text


### model

## numerical-evaluation

### forward

Evaluate power and Chebyshev polynomials, compensated finite sums and budgeted adaptive quadrature with a separately labeled heuristic error estimate.

### invariant

Keep coefficient conventions, finite results, callback evaluations, budgets and numerical assurance distinct. A successful heuristic tolerance status is not an analytical integration certificate.

### cost

Horner, Clenshaw and Neumaier O(n) arithmetic/O(1) scratch; Simpson O(evaluations) with bounded depth<=64 and fixed panel stack. Callback costs are additional.

### composition

Fixed-degree Chebyshev-to-power conversion and analytic polynomial integrals provide independent identities. The harness uses the cubic [2,3,-2,1] in Chebyshev form and [4,0,-4,4] in power form on 33 dyadic points. The separate quadratic formula is an analytical relation. Native comparisons use explicit additive tolerances.

### epoch-memory

Wave19, 2026-10-30. Models retain real-arithmetic assumptions separately from native finite checks. Kernels are library tools, not automatically adopted into training or inference.

### checks

["polynomial_basis_join_derivative_and_quadrature","rational_certificates_join_polynomials_and_compensated_products"]

### limitations

Finite intermediate overflow and cancellation are explicit. Simpson's local estimate can miss unsampled structure; callbacks may violate smoothness. Neither residual samples nor finite successful tests certify all functions or a shared <=10% error.

### catalog

## Typed family relations

Each relation keeps its own domain and assurance.

### behavior-complement

Type: behavior_complement.

Exclude outputs outside each tagged native transition, successful nonfinite values, excess callback budgets and Simpson estimates promoted to universal guarantees. Distinct error/status observations are not numerical approximation loss.

### reverse-reconstruction

Type: reverse_reconstruction.

At x=0 Horner retains c0,c1 and loses higher coefficients; Clenshaw retains an alternating even-coefficient sum and loses odd terms. Sum fibers include [1,2] and [3]. Quadrature samples permit arbitrary off-sample perturbations, so reports do not identify a unique integrand.

### basis-conversion

Type: polynomial_basis_transport.

Chebyshev coefficients [a,b,c] represent power coefficients [a-c,b,2c]. Evaluations agree in real arithmetic. Floating tests require scale-aware additive tolerance; derivative comparison must use transformed coefficients.

### analytic-integral

Type: antiderivative_certificate.

A power polynomial has exact real integral sum a_k*(b^(k+1)-a^(k+1))/(k+1). This supplies an independent finite quadrature oracle; general callback error remains heuristic.

### sample-blindness

Type: finite_observation_complement.

A function vanishing at all sampled abscissae can have a nonzero integral between them. Equal sampled values and error estimates therefore admit different exact integrals; an explicit hidden-bump test witnesses this gap. The witness is in the leaf validation_and_unsampled_fiber test.

### wave19-evidence

#### Family evidence — 2026-10-30

This family binds eight implementation/test sources through its manifest; views are partial source-derived models. Selected named joins are ["polynomial_basis_join_derivative_and_quadrature","rational_certificates_join_polynomials_and_compensated_products"] in algorithms/tests/composition_wave19.rs. Those checks pass four times as part of the 616-check suite. Leaf oracles and compiled production controls remain separately bound in adjacent evidence.

[Integrated evidence](https://md.groupants.elamur.ai/free/en/mova/models/algorithm-library.groupant/#wave19-evidence) records receipts; conditional laws outside these fixture scopes are not promoted to executed checks. No joint universal percentage bound follows from family membership.