---
title: "algorithm-integer-sqrt — Mova code model"
description: "Mova code model of algorithms/src/integer_sqrt/mod.rs: components with their labels, obernants and the model text."
url: "https://groupants.elamur.ai/free/en/mova/algorithms/src/integer_sqrt/mod.rs.groupant/"
---
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# algorithm-integer-sqrt

The groupant beside algorithms/src/integer_sqrt/mod.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
- **status:** partial source derived instance
- **source:** [`algorithms/src/integer_sqrt/mod.rs`](https://github.com/ElamurAI/mova-free/blob/main/algorithms/src/integer_sqrt/mod.rs)
- **source SHA-256:** `791a2a301deae750…`
- **repository folder:** [`algorithms/src/integer_sqrt/mod.rs.groupant`](https://github.com/ElamurAI/mova-free/tree/main/algorithms/src/integer_sqrt/mod.rs.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** | bounded checked | [Forward](#forward) |
| **ast** | partial | `syntax.json` |
| **cost** | symbolic unmeasured | [Cost](#cost) |
| **descriptant** | present | [descriptant](#descriptant) |
| **invariant** | source derived | [Invariant](#invariant) |
| **tests** | bounded executed | [evidence](#evidence) |

## 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 |
|---|---|---|---|
| `nonbehavior` | behavior_complement | partial source derived | [nonbehavior](#nonbehavior) |
| `square_fibers` | reverse_reconstruction | partial source derived | [square_fibers](#square-fibers) |
| `square_boundary` | boundary_specialization | partial source derived | [square_boundary](#square-boundary) |
| `adjacent_inputs` | metamorphic_relation | partial source derived | [adjacent_inputs](#adjacent-inputs) |
| `search_budget` | resource_observation | partial source derived | [search_budget](#search-budget) |

## Rust approximants

| id | metric | error | assurance |
|---|---|---|---|
| `integer_sqrt-1pct` | abs(result-floor(sqrt(n)))/floor(sqrt(n)); zero input has zero error | 1 % cap, 0.78125 % stated | analytical |
| `integer_sqrt-2pct` | abs(result-floor(sqrt(n)))/floor(sqrt(n)); zero input has zero error | 2 % cap, 1.5625 % stated | analytical |
| `integer_sqrt-5pct` | abs(result-floor(sqrt(n)))/floor(sqrt(n)); zero input has zero error | 5 % cap, 3.125 % stated | analytical |
| `integer_sqrt-10pct` | abs(result-floor(sqrt(n)))/floor(sqrt(n)); zero input has zero error | 10 % cap, 6.25 % stated | analytical |

## Model text


### descriptant

#### integer_sqrt mod epoch

2026-10-29. Repository search found math/src/calc.rs delegating arbitrary-precision roots to a dependency. This fixed-width binary search is newly authored; standard isqrt is test-only. No external solutions or LeetCode imports.

Prospective assumptions are retained in prospective.md. The mathematical combiner reviews source, partial syntax, forward/typed models, invariants and finite evidence. It preserves predictions and failed controls when revising claims; it is not an autonomous proof generator.

Replacing <= by < in the square predicate compiled and exactly tests::square_neighbors_and_maximum failed (exit 101; one failed, two filtered).

Implementation and tests form separate leaf groupants. Parent registry wiring remains integration work. Partial syntax has opaque macros and no semantic/type resolution. No universal proof or application behavior is claimed.

### model

## integer_sqrt mod model

### Forward

For every u64 n, return r satisfying r^2<=n<(r+1)^2. Powers in this formula are mathematical integers.

### Invariant

0<=lo<=hi<=2^32; all k<lo satisfy k^2<=n, all k>=hi up to 2^32 satisfy k^2>n. The interval strictly shrinks. Zero is always accepted, so final lo>=1. Midpoint squares fit u128.

### Cost

At most 33 iterations, constant space; no allocations or floating point.

### nonbehavior

Type: behavior_complement. U=u64 x u64; R(n,r) iff r^2<=n<(r+1)^2. U minus R contains exactly candidates violating either inequality. Evaluate candidate squares mathematically, including invalid large r; exact target with bounded evidence.

### square_fibers

Type: reverse_reconstruction. Observed r<=u32::MAX admits all n in [r^2,min((r+1)^2-1,u64::MAX)]; other r admit none. Many inputs share a root, except zero.

### square_boundary

Type: boundary_specialization. For r<=u32::MAX, root(r^2)=r; if r>0, root(r^2-1)=r-1. Explicit square-neighbor tests check inclusive equality.

### adjacent_inputs

Type: metamorphic_relation. For n<u64::MAX, root(n)<=root(n+1)<=root(n)+1; increments occur only when n+1 is a square. Fixed-seed checks exercise the inequality.

### search_budget

Type: resource_observation. Initial width 2^32 shrinks through binary splitting; at most 33 iterations with u128 comparisons and O(1) state. This is a source-derived bound, not measured throughput.

### evidence

#### Evidence 2026-10-29

Three focused tests pass twice with identical normalized statuses. 65,537 inputs 0..=65536, nine square-root values with square/neighbor cases, u64::MAX and 4,096 fixed-seed full-width inputs with adjacent monotonicity.

Standard u64::isqrt is independent of this source; u128 square bounds provide a second constraint.

Replacing <= by < in the square predicate compiled and exactly tests::square_neighbors_and_maximum failed (exit 101; one failed, two filtered).

Commands: cargo test -p mova-algorithms --offline integer_sqrt::tests:: -- --test-threads=1; cargo clippy -p mova-algorithms --offline --all-targets -- -D warnings. Owned-source rustfmt checks pass. Temporary library exports permit local crate checks but are not committed; parent owns integration. External copies compile with rustc --edition 2024 --test and run with --exact. Production sources stay unchanged.

Source-bound syntax is generated twice after formatting and compared. Structural evidence does not establish semantic correctness, school/inference wiring or throughput.

### prospective

#### Prospective epoch 2026-10-29

Before implementation: Use a half-open binary search over possible root values through 2^32, with u128 square comparisons; expect r^2 <= n < (r+1)^2 for every u64. Compare against standard u64::isqrt and explicit square neighbors.

Predict at most 33 search steps or 64 exponent-bit steps respectively, constant auxiliary space. Fixed-seed and boundary fixtures, two repeated focused tests, strict Clippy and a compiled source mutant rejected by one specific test will supply bounded evidence. No general proof, timing gain, imported solution or downstream integration claim.