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

The groupant beside models/certified-approximation.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/certified-approximation.groupant`](https://github.com/ElamurAI/mova-free/tree/main/models/certified-approximation.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 |
|---|---|---|---|
| `certified-approximation-behavior-complement` | behavior_complement | partial source model | [behavior-complement](#behavior-complement) |
| `certified-approximation-reverse-reconstruction` | reverse_reconstruction | partial source model | [reverse-reconstruction](#reverse-reconstruction) |
| `certified-approximation-subset-embedding` | objective_embedding | partial source model | [subset-embedding](#subset-embedding) |
| `certified-approximation-absolute-transport` | lipschitz_error_transport | partial source model | [absolute-transport](#absolute-transport) |
| `certified-approximation-cancellation` | normalization_countermodel | partial source model | [cancellation](#cancellation) |

## Model text


### model

## certified-approximation

### forward

Compute bounded-loss feasible objectives, exact rational interpolation and quantized integer dot observations with distinct certificates.

### invariant

Feasibility, objective loss, conditional interpolation error and dot absolute/L1 error remain separately typed. Unsupported percentages and resource/arithmetic errors are categorical.

### cost

Knapsack value scaling O(n^3/epsilon); trimmed subset list O(n*L*log L), L=O(n*log(target)/epsilon) in the mathematical model. Interpolation validates O(n) then searches O(log n); dot O(n). Work and allocation limits apply.

### composition

Embedding subset values as equal knapsack weights/profits preserves the exact optimum. The two approximate outputs may differ while satisfying their own loss bounds. Conditional absolute interpolation error transports through a known Lipschitz map; dot L1 normalization does not imply relative scalar error under cancellation.

### 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

["fptas_objectives_join_exhaustive_subset_fibers","rational_certificates_join_polynomials_and_compensated_products"]

### limitations

Curvature bound is a caller premise, not inferred from knots. Overflow, work limits and allocation failures do not count as small approximation errors. Integer certificates do not provide a shared parent percentage or universal speed gain.

### catalog

## Typed family relations

Each relation keeps its own domain and assurance.

### behavior-complement

Type: behavior_complement.

Exclude infeasible or over-optimal FPTAS outputs, violated declared loss bounds, extrapolated interpolation, and true dot errors outside the absolute certificate. Exclude promotion of caller curvature to checked fact and promotion of L1-normalized dot error to cancellation-relative error.

### reverse-reconstruction

Type: reverse_reconstruction.

Objective certificates admit many item lists and optimal subsets. One interpolated ordinate admits many functions through the supplied knots; dot values admit sign/permutation and cancellation fibers. Reconstruct only constraints retained by the certificate, not unique original inputs.

### subset-embedding

Type: objective_embedding.

On representable inputs, each subset value v becomes a knapsack item of weight v and profit v with capacity target. Exact optima coincide. Each accepted epsilon result lies between (1-epsilon)*OPT and OPT; results need not coincide.

### absolute-transport

Type: lipschitz_error_transport.

If interpolation error <=e and downstream real map is L-Lipschitz on the reached interval, downstream discrepancy <=L*e. Any independent downstream error adds separately; floating conversion and missing curvature premises require extra treatment.

### cancellation

Type: normalization_countermodel.

For signed products near cancellation, sum(abs(products)) may be large while abs(sum(products)) is zero or tiny. A <=10% L1-normalized dot certificate supplies an absolute bound, not <=10% relative error of the scalar dot.

### 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 ["fptas_objectives_join_exhaustive_subset_fibers","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.