---
title: "math-nt-arithmetic — 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/arithmetic.groupant/"
---
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# math-nt-arithmetic

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:** phi/tau/sigma/mobius require positive complete-factor inputs; divisors accepts signs, rejects zero and checks count before sorted materialization. Prime count/rank sieve is capped, next_prime unbounded. digits strips sign; from_digits accepts arbitrary Horner coefficients. Canonical restrictions are needed for inversion. (bytes 18019–22371)
- **part of:** [math-nt](https://md.groupants.elamur.ai/free/en/mova/math/src/nt.rs.groupant/)
- **repository folder:** [`math/src/nt.rs.groupant/arithmetic.groupant`](https://github.com/ElamurAI/mova-free/tree/main/math/src/nt.rs.groupant/arithmetic.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) |
| `multiplicative_formula` | multiplicative_formula | partial source derived model | [multiplicative_formula](#multiplicative-formula) |
| `divisor_materialization` | divisor_materialization | partial source derived model | [divisor_materialization](#divisor-materialization) |
| `prime_search` | prime_search | partial source derived model | [prime_search](#prime-search) |
| `zero_digits` | zero_digits | partial source derived model | [zero_digits](#zero-digits) |
| `wave39-canonical_sign_fiber` | canonical_digit_sign_preimage | partial source derived model | `../model.md` |
| `wave39-positional_carry_kernel` | fixed_length_integer_kernel | partial source derived model | `../model.md` |
| `wave39-base_admission_partition` | typed_base_admission_partition | partial source derived model | `../model.md` |
| `wave39-magnitude_normalizer` | idempotent_magnitude_retraction | partial source derived model | `../model.md` |

## Model text


### descriptant

#### math-nt-arithmetic

phi/tau/sigma/mobius require positive complete-factor inputs; divisors accepts signs, rejects zero and checks count before sorted materialization. Prime count/rank sieve is capped, next_prime unbounded. digits strips sign; from_digits accepts arbitrary Horner coefficients. Canonical restrictions are needed for inversion.

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

Wave39 reviews integer digit/Horner transport and its own typed model sections. Finite checks, dependent formula derivations and algebraic claims retain separate scopes; no production speed or universal source proof is inferred.

### approximant

#### Scoped forward and cost

phi/tau/sigma/mobius require positive complete-factor inputs; divisors accepts signs, rejects zero and checks count before sorted materialization. Prime count/rank sieve is capped, next_prime unbounded. digits strips sign; from_digits accepts arbitrary Horner coefficients. Canonical restrictions are needed for inversion.

No executable percent surrogate claimed.

### invariant

#### Scoped invariant

phi/tau/sigma/mobius require positive complete-factor inputs; divisors accepts signs, rejects zero and checks count before sorted materialization. Prime count/rank sieve is capped, next_prime unbounded. digits strips sign; from_digits accepts arbitrary Horner coefficients. Canonical restrictions are needed for inversion.

Source range[17881,22233) bound by manifest digest. Parent domain applies, not arbitrary public states.

### models

## Source-scoped typed relations

### behavior_complement

phi/tau/sigma/mobius reject nonpositive n; divisors first takes magnitude and rejects zero but accepts negative values. digits base<2 rejects, from_digits imposes no such gate.

### reverse_reconstruction

digits(n,b) loses sign: n and -n share output. Canonical digits and known base recover abs(n); from_digits with arbitrary coefficients admits carrying transformations and leading zeros.

### multiplicative_formula

Given complete factors, phi=product p^(e-1)(p-1), tau=product(e+1), sigma_k=product sum(j=0..e)p^(kj), mobius=0 if repeated factor else(-1)^count.

### divisor_materialization

Count product(e+1) in u128 is checked against max before allocation, then products are enumerated and sorted. Count arithmetic itself is unchecked; complete probable factors remain accepted.

### prime_search

prime_pi caps n at2e8; nth_prime k in1..1e7 chooses analytic sieve bound and indexes k-1. next_prime/prev_prime step one integer at a time and accept prime-like values, with no iteration cap.

### zero_digits

digits0=[0], including nondecimal base; from_digits([],b)=0 for any b. Decimal path converts magnitude text, others repeatedly divide and reverse remainders.

### tests

#### Evidence

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

phi/tau/sigma/mobius require positive complete-factor inputs; divisors accepts signs, rejects zero and checks count before sorted materialization. Prime count/rank sieve is capped, next_prime unbounded. digits strips sign; from_digits accepts arbitrary Horner coefficients. Canonical restrictions are needed for inversion.