---
title: "math-comb — Mova code model"
description: "Mova code model of math/src/comb.rs: components with their labels, obernants and the model text."
url: "https://groupants.elamur.ai/free/en/mova/math/src/comb.rs.groupant/"
---
Elamur: [Mova](https://md.mova.elamur.ai/free/en/) · [Dictionary](https://md.dictionary.elamur.ai/free/en/) · [Research](https://md.research.elamur.ai/free/en/) · **Groupants**

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# math-comb

The groupant beside math/src/comb.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 bound model
- **source:** [`math/src/comb.rs`](https://github.com/ElamurAI/mova-free/blob/main/math/src/comb.rs)
- **source SHA-256:** `c5937a3576697c05…`
- **repository folder:** [`math/src/comb.rs.groupant`](https://github.com/ElamurAI/mova-free/tree/main/math/src/comb.rs.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` |
| **composition** | partial | [composition](#composition) |
| **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) |
| `product_identity` | product_identity | partial source derived model | [product_identity](#product-identity) |
| `recurrence_direction` | recurrence_direction | partial source derived model | [recurrence_direction](#recurrence-direction) |
| `limit_precedence` | limit_precedence | partial source derived model | [limit_precedence](#limit-precedence) |
| `certificate_availability` | certificate_availability | partial source derived model | [certificate_availability](#certificate-availability) |

## Parts

- [math-comb-products](https://md.groupants.elamur.ai/free/en/mova/math/src/comb.rs.groupant/products.groupant/)
- [math-comb-recurrences](https://md.groupants.elamur.ai/free/en/mova/math/src/comb.rs.groupant/recurrences.groupant/)

## Model text


### descriptant

#### comb

factorial uses balanced range products with empty product1; binomial applies negative-k zero and C(n,k)=(-1)^k C(k-n-1,k) for negative n, then product ratios. Legendre checks use nt sieve. Stirling II descends j with j*S(i-1,j)+S(i-1,j-1), I uses (i-1)*c(i-1,j)+c(i-1,j-1). Bell triangle, signed pentagonal partitions, coin DP, descending distinct/ascending odd-part counts, derangements seeds(1,0), and Fibonacci doubling (a(2b-a),a²+b²) produce Z values with alternate checks.

Epoch 2026-10-31: complete source read before edits, bounded actual-code tests and source controls. Linsys shape defect was witnessed on old code and repaired; other semantics retained. External receipt name octopus-wave28-mathcore-2026-10-31 includes prospective.md. Registration is partial, not verification.

### approximant

#### Forward and cost

factorial uses balanced range products with empty product1; binomial applies negative-k zero and C(n,k)=(-1)^k C(k-n-1,k) for negative n, then product ratios. Legendre checks use nt sieve. Stirling II descends j with j*S(i-1,j)+S(i-1,j-1), I uses (i-1)*c(i-1,j)+c(i-1,j-1). Bell triangle, signed pentagonal partitions, coin DP, descending distinct/ascending odd-part counts, derangements seeds(1,0), and Fibonacci doubling (a(2b-a),a²+b²) produce Z values with alternate checks.

Product trees O(n) big multiplies/O(log n) recursion; binomial O(min(k,n-k)) nonnegative case after guards. StirlingII O(nk)/O(k) storage, I/Bell O(n²)/O(n). Pentagonal O(n sqrt n)/O(n); coin/distinct O(n²)/O(n), partitions_k O(nk) time/space. Fibonacci O(log n) growing-width steps, derangements O(n). Legendre sieve O(n log log n) plus exponentiation, bit costs additional.

No new executable Rust surrogate: exact values, branching, domains and effects require an observation-specific metric before supported 1/2/5/10% profiles. Existing floating diagnostics are not such a profile; no speed claim.

### invariant

#### Domain and invariant

Factorial and selected product work cap1e6; Stirling n<=3000, Bell<=2000/check<=400, partitions<=200000/DP<=3000, partitions_k/distinct<=5000, derangements<=100000/sum<=3000. binomial converts k before k>n/symmetry; Stirling n gate precedes k>n. Catalan unchecked2*n,n+1 and direct helper dimensions lack universal machine-overflow protection. Big values still require allocation.

factorial uses balanced range products with empty product1; binomial applies negative-k zero and C(n,k)=(-1)^k C(k-n-1,k) for negative n, then product ratios. Legendre checks use nt sieve. Stirling II descends j with j*S(i-1,j)+S(i-1,j-1), I uses (i-1)*c(i-1,j)+c(i-1,j-1). Bell triangle, signed pentagonal partitions, coin DP, descending distinct/ascending odd-part counts, derangements seeds(1,0), and Fibonacci doubling (a(2b-a),a²+b²) produce Z values with alternate checks.

A Z count has many compatible function/index antecedents: 0!=1!=1, D0=D2=1. Known function and admitted domain constrain recurrence fibers but generally do not invert uniquely. Cassini alone admits more pairs than intended Fibonacci values; unavailable checks impose no value constraint.

### composition

#### Composition

factorial uses balanced range products with empty product1; binomial applies negative-k zero and C(n,k)=(-1)^k C(k-n-1,k) for negative n, then product ratios. Legendre checks use nt sieve. Stirling II descends j with j*S(i-1,j)+S(i-1,j-1), I uses (i-1)*c(i-1,j)+c(i-1,j-1). Bell triangle, signed pentagonal partitions, coin DP, descending distinct/ascending odd-part counts, derangements seeds(1,0), and Fibonacci doubling (a(2b-a),a²+b²) produce Z values with alternate checks.

calc dispatches to comb/nt. poly/linsys consume calc E and stateful Env for constants; deriv consumes E and calc::approx. fcheck consumes separate legacy expr::Expr/units, with no implicit lossless E adapter. Q/Z joins use big. Parent symbolic-computation models these edges; local checks do not certify application wiring. Nested scopes reuse parent syntax, and remaining helpers/types/tests are parent-modeled.

### models

## Typed models

### behavior_complement

For fixed admitted input/state, exclude outputs and effect traces contradicting this actual relation: factorial uses balanced range products with empty product1; binomial applies negative-k zero and C(n,k)=(-1)^k C(k-n-1,k) for negative n, then product ratios. Legendre checks use nt sieve. Stirling II descends j with j*S(i-1,j)+S(i-1,j-1), I uses (i-1)*c(i-1,j)+c(i-1,j-1). Bell triangle, signed pentagonal partitions, coin DP, descending distinct/ascending odd-part counts, derangements seeds(1,0), and Fibonacci doubling (a(2b-a),a²+b²) produce Z values with alternate checks. These exclusions are conditional on the stated domain. Unavailable checks, passing residuals and resource limits do not establish global correctness.

### reverse_reconstruction

A Z count has many compatible function/index antecedents: 0!=1!=1, D0=D2=1. Known function and admitted domain constrain recurrence fibers but generally do not invert uniquely. Cassini alone admits more pairs than intended Fibonacci values; unavailable checks impose no value constraint.

### product_identity

lo>hi gives1; negative k gives0 before conversion. Generalized negative n uses parity sign.

### recurrence_direction

Descending Stirling indices preserve previous row. Distinct sum descends to forbid reuse; odd-parts sum ascends to allow reuse.

### limit_precedence

stirling2 rejects n>3000 before k>n. Oversized k conversion may fail before useful symmetry. factorial_legendre has no factorial cap.

### certificate_availability

check_bell beyond400 and binomial outside independent domain return ok=true with unavailability text, not independent proof.

### tests

#### Actual-code evidence 2026-10-31

Each selected named test passed twice debug/twice release, sorted outcomes identical:

- `test comb::tests::classic_values ... ok`
- `test comb::wave28::zero_seeds_and_limit_precedence ... ok`

Compiled source control: control-comb.patch.txt; compiler exit0, test nonzero, exact source restoration and focused positive exit0. Full logs and numeric exits retained externally in octopus-wave28-mathcore-2026-10-31. Linsys also retains shape-old compile 0 / test 101 witness before shape repair. Seven module controls plus the additional linsys width-guard control; fragment declarations are not extra tests.

Finite synthetic fixtures only; no teacher/API/corpus/GPU/heldout. No allocation/stack faults, huge sizes, deadline proof, or universal numerical bound.

Final validation: eight compiled mutants total (seven module boundaries plus linsys width guard), all compile 0 / test 101 and exact restoration / positive 0. Old-code count mismatch witness compile 0 / test 101 separately. Full-registry structural run reports only ancestor integration bindings; isolated source snapshots validate each owned sidecar. Strict Clippy: dependency invocation blocked in existing global build script; --no-deps reaches math and reports 113 existing library findings, retained externally, no clean lint claim. Added test blocks formatted independently; production layout preserved.