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

The groupant beside models/convex-projections.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/convex-projections.groupant`](https://github.com/ElamurAI/mova-free/tree/main/models/convex-projections.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 |
|---|---|---|---|
| `convex-projections-behavior-complement` | behavior_complement | partial source model | [behavior-complement](#behavior-complement) |
| `convex-projections-reverse-reconstruction` | reverse_reconstruction | partial source model | [reverse-reconstruction](#reverse-reconstruction) |
| `convex-projections-signed-simplex` | projection_transport | partial source model | [signed-simplex](#signed-simplex) |
| `convex-projections-threshold-kkt` | proximal_kkt_relation | partial source model | [threshold-kkt](#threshold-kkt) |
| `convex-projections-isotonic-residual` | order_cone_certificate | partial source model | [isotonic-residual](#isotonic-residual) |

## Model text


### model

## convex-projections

### forward

Project finite vectors onto monotone, simplex and L1-ball constraints and apply the L1 proximal soft-threshold map.

### invariant

Real maps have unique minimizers and KKT conditions; native f64 routines preserve explicit validation and acceptance tolerances rather than universal exact feasibility.

### cost

Unit-weight PAVA and soft threshold O(n) work/storage; sorted simplex and L1 projection O(n log n) work/O(n) storage. Length <=10^6 with fallible allocation.

### composition

Outside the L1 ball, its projection equals signed simplex projection of magnitudes and soft threshold at the common active magnitude difference. Isotonic regression preserves total sum in real arithmetic; finite checks separately test residual prefixes. Residual-prefix evidence is in the leaf partition oracle; the parent harness checks totals, order, idempotence and selected normal-cone inequalities.

### 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

["projections_join_threshold_reconstruction_and_mean"]

### limitations

Real normal-cone and KKT laws do not by themselves prove floating-point outputs. Simplex relative mass residual acceptance is <=64*n*EPSILON, not objective error. L1 may exceed radius within tolerance. Overflow/cancellation can reject mathematically solvable inputs.

### catalog

## Typed family relations

Each relation keeps its own domain and assurance.

### behavior-complement

Type: behavior_complement.

At the real-model level exclude KKT-incompatible outputs for the tagged convex set/proximal problem. At the native level exclude different rounded transitions and invalid accepted results; arithmetic/allocation errors are retained, and tolerance-sized feasibility deviations are not silently discarded.

### reverse-reconstruction

Type: reverse_reconstruction.

For fixed convex set C and output y in C, compatible real inputs are y+N_C(y). Soft-threshold zero coordinate has input fiber [-lambda,lambda]; nonzero coordinate reconstructs x_i=y_i+lambda*sign(y_i). This many-to-one model does not uniquely invert rounded implementations.

### signed-simplex

Type: projection_transport.

When ||x||1>r, projection onto the L1 ball is sign(x)*P_simplex(abs(x),r) in real arithmetic. On the interior the identity branch applies instead. Floating mass residual is bounded only by the leaf acceptance condition.

### threshold-kkt

Type: proximal_kkt_relation.

A shared nonnegative threshold lambda satisfies y_i=sign(x_i)*max(abs(x_i)-lambda,0), with sum(abs(y))=r on an active L1 boundary. A common threshold witness links simplex and soft-threshold fixtures, with explicit additive numerical tolerance.

### isotonic-residual

Type: order_cone_certificate.

For real unit-weight isotonic regression, residual prefix sums sum(x-y) are nonnegative, total residual is zero, output is nondecreasing, and active block sums agree. These KKT observations are checked independently on bounded fixtures.

### 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 ["projections_join_threshold_reconstruction_and_mean"] 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.