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

The groupant beside math/src/nt.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/nt.rs`](https://github.com/ElamurAI/mova-free/blob/main/math/src/nt.rs)
- **source SHA-256:** `27f4bf91578ab245…`
- **covers:** Euclid certificates and normalized modular operations compose through CRT gcd compatibility. Empty/mismatched CRT rejects but check_crt zip-truncates. Antecedents are congruence fibers; bare modulus helpers can panic on zero. Costs depend on bit widths and pairs. (bytes 1492–5534)
- **part of:** [math-nt](https://md.groupants.elamur.ai/free/en/mova/math/src/nt.rs.groupant/)
- **repository folder:** [`math/src/nt.rs.groupant/modular.groupant`](https://github.com/ElamurAI/mova-free/tree/main/math/src/nt.rs.groupant/modular.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) |
| `bezout_certificate` | bezout_certificate | partial source derived model | [bezout_certificate](#bezout-certificate) |
| `inverse_domain` | inverse_domain | partial source derived model | [inverse_domain](#inverse-domain) |
| `crt_merge` | crt_merge | partial source derived model | [crt_merge](#crt-merge) |
| `checker_shape` | checker_shape | partial source derived model | [checker_shape](#checker-shape) |

## Model text


### descriptant

#### math-nt-modular

Euclid certificates and normalized modular operations compose through CRT gcd compatibility. Empty/mismatched CRT rejects but check_crt zip-truncates. Antecedents are congruence fibers; bare modulus helpers can panic on zero. Costs depend on bit widths and pairs.

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

### approximant

#### Scoped forward and cost

Euclid certificates and normalized modular operations compose through CRT gcd compatibility. Empty/mismatched CRT rejects but check_crt zip-truncates. Antecedents are congruence fibers; bare modulus helpers can panic on zero. Costs depend on bit widths and pairs.

No executable percent surrogate claimed.

Epoch36 boundary: public gcd(0,0)=0 via explicit guard; all other signed operands use the existing dashu gcd. This agrees with ext_gcd/check_gcd. The original wrapper's zero-zero panic was observed before repair; bare modular helpers still retain their separate nonzero-modulus preconditions.

### invariant

#### Scoped invariant

Euclid certificates and normalized modular operations compose through CRT gcd compatibility. Empty/mismatched CRT rejects but check_crt zip-truncates. Antecedents are congruence fibers; bare modulus helpers can panic on zero. Costs depend on bit widths and pairs.

Source range[1492,5396) bound by manifest digest. Parent domain applies, not arbitrary public states.

### models

## Source-scoped typed relations

### behavior_complement

modpow modulus0 excludes Some; modulus1 returnsSome0 even for negative exponent. CRT empty/mismatched vectors or zero modulus excludes a successful pair.

### reverse_reconstruction

CRT(x,M) admits all residue/modulus lists with lcm|mi|=M and ri congruent x modulo|mi|. Reordering, redundant congruences and modulus signs are lost; x alone also loses M.

### bezout_certificate

ext_gcd returns g>=0 and s*a+t*b=g. check_gcd also tests divisibility and matching independently computed g; at g=0 both inputs must0.

### inverse_domain

modinv requires m!=0 and gcd(a,|m|)=1; modpow negative e delegates inverse except m=1 shortcut. mod_floor alone does not guard m=0.

### crt_merge

For accumulated(x,m), g=gcd(m,mi); reject if g does not divide r-x. Solve t modulo mi/g, then x'=x+m*t normalized modulo m*(mi/g).

### checker_shape

check_crt uses zip and does not test counts; empty zip accepts and zero reached modulus panics. Its true result is not proof that producer crt admitted the arrays.

### tests

#### Evidence

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

Euclid certificates and normalized modular operations compose through CRT gcd compatibility. Empty/mismatched CRT rejects but check_crt zip-truncates. Antecedents are congruence fibers; bare modulus helpers can panic on zero. Costs depend on bit widths and pairs.

Epoch36 parent evidence: zero_pair_agrees_with_independent_certificate exposed the original public gcd zero-zero panic, then passed with the guard. signed_grid_and_large_operands_preserve_euclid checks1088 signed grid pairs and selected big operands. Both pass debug/release jobs2; parent tests.md records source bindings and external logs. Modular-zero and CRT shape preconditions remain distinct.