# g_math
Pure-Rust, zero-float, deterministic multi-domain fixed-point arithmetic.
Built by **Niels Erik Toren** · published as `g_math` on [crates.io](https://crates.io/crates/g_math)
---
> **Upgrading from 0.4.x?** 0.5.0 is more accurate, and as a side effect some
> results end in a different final digit. It matters only if you store, hash, or
> compare results across versions. The four-line summary is at the top of
> [CHANGELOG.md](CHANGELOG.md).
>
> **0.6.0** adds certified arithmetic (`Interval`, `DecimalInterval`), a proven
> positive-definiteness verdict (`pd_verdict`) and exact geometric predicates,
> all additive. **0.5.1** carries only the three defect fixes that work found
> in 0.5.0 (two on the scientific profile, one in the wide decimal integers);
> every `^0.5` user receives them without adopting any new API.
>
> **0.6.4 to 0.6.7**: the operators and every narrowing fail loudly instead of
> wrapping, and each multi-step computation rounds once (0.6.4). The decimal
> type has an explicit tie rule and a one-rounding `mul_div` (0.6.5). The wide
> tier gains gate functions, a correctly rounded square root, a full-width
> integer `mul_div` and exact decimal ratios, and the batch kernels can write
> into caller-provided buffers (0.6.7). 0.6.5 to 0.6.7 change no in-range
> result. Details per release: [CHANGELOG.md](CHANGELOG.md).
>
> **Known issue**: on the compact profile `exp` is one unit from the nearest
> value on about 1 input in 100 when the result is large. Small arguments and
> the realtime profile are not affected. See "Open defect" in
> [ROADMAP.md](ROADMAP.md).
---
## What it is
`g_math` computes with scaled integers only. No `f32`/`f64` appears anywhere in
the arithmetic, validation, or comparison paths (float conversions exist solely as
caller-convenience `from_f64`/`to_f64` wrappers). Because every operation is
integer arithmetic, results are bit-identical on every architecture - which makes
the crate suitable for blockchain consensus, financial auditing, and reproducible
scientific computation.
It is a library, not a service: pure computation, no I/O, no ambient state. Values
route across four numeric domains (binary, decimal, balanced ternary, symbolic
rational), and 18 transcendental functions plus linear algebra, geometry, and
ternary-quantized inference are built on top. It is one component of the Geodineum
ecosystem and is consumed by its geometric and inference projects.
## Public build surface
The build-with surface is five layers; everything beneath them is internal.
- **Canonical / router** - `g_math::canonical`: `gmath("…")` builds a lazy
expression, `evaluate(…)` runs it with cross-domain routing and chain
persistence.
- **Imperative** - `g_math::fixed_point::{FixedPoint, FixedVector, FixedMatrix}`
and `DecimalFixed<DECIMALS>`: direct `Copy` types, one call is one engine
invocation.
- **Fused** - `g_math::fixed_point::imperative::fused`: whole accumulation patterns
at the wide compute tier (norms, softmax and its mixes, `rms_norm`,
`sigmoid_mul`, `entropy`, the correctly rounded quadratic form), several with
forms that write into caller-provided slices.
- **Geometric** - `imperative::{decompose, derived, matrix_functions, manifold,
lie_group, curvature, projective, fiber_bundle, ode, tensor, tensor_decompose,
serialization}`.
- **Wide** - `g_math::wide`: Q64.64 `exp`/`ln`/`sin`/`cos`, `sigmoid`/`softplus`/
`silu`, a correctly rounded square root, π constants, an exact
decimal-literal parser, exact decimal ratios and a full-width integer
`mul_div`, for quantities outside the storage tiers. Each function's
measured error is in the module documentation.
- **Compute tier** - `g_math::compute_tier` (feature `inference`): the tier-N+1
engines over raw compute-tier integers.
- **TQ1.9** - `g_math::tq19` (feature `inference`): standalone ternary inference,
with the five-trit row-scaled form, the weight-bit decoders and quantisers,
and batch matvecs that write into one caller-provided buffer.
- **Internal** - the `universal`/`fasc` evaluator, the domain implementations
beneath `DecimalFixed` and the ternary types, the wide-integer types, and the
shadow/router internals are not part of the surface and may change between
releases.
The complete per-symbol index - every public item and a one-line summary - is
**generated** from source into **[`PUBLIC_API.md`](PUBLIC_API.md)**
(`rustc -O scripts/gen-public-api.rs -o /tmp/gen-public-api && /tmp/gen-public-api`).
Live signatures are on [docs.rs](https://docs.rs/g_math). This README never
re-hosts either.
## Guides
Each layer and cross-cutting concept has a focused guide under `docs/`.
| [Routing (FASC)](docs/README_ROUTING.md) | The `gmath()`/`evaluate()` layer: lazy trees, cross-domain routing, chain persistence (tier N+1), UGOD graceful overflow, the shadow system, mode routing |
| [Imperative](docs/README_IMPERATIVE.md) | `FixedPoint`/`FixedVector`/`FixedMatrix` + `DecimalFixed`: direct, no-routing, `Copy` types for hot loops |
| [Transcendentals](docs/README_TRANSCENDENTALS.md) | The 18 functions: dedicated engines, composed forms, fused `sincos`/`sinhcosh`, `try_*` variants |
| [Fused operations](docs/README_FUSED.md) | Compute-tier `sqrt_sum_sq`, `euclidean_distance`, `softmax`, `softmax_mix`, `rms_norm_factor`, `silu` |
| [Linear algebra](docs/README_LINALG.md) | LU/QR/Cholesky/SVD/eigen/Schur, solvers, matrix functions |
| [Geometry](docs/README_GEOMETRY.md) | Manifolds, Lie groups, curvature, projective geometry, fiber bundles, ODE, tensors |
| [Ternary & TQ1.9](docs/README_TERNARY.md) | Balanced-ternary arithmetic and the trit-plane inference weight formats |
| [Precision & profiles](docs/README_PRECISION.md) | The five profiles, `GMATH_FRAC_BITS`, rounding, validation, feature flags, build |
## Capabilities
- **Four numeric domains** - each value is held in the representation that is exact
for it (binary, decimal, balanced ternary, or symbolic rational), routed
automatically.
- **18 transcendentals** - computed one tier wider than storage and rounded once,
reachable lazily (`gmath`), directly (`FixedPoint`), or natively in decimal
(`DecimalFixed`).
- **Two computation paths** - a routed canonical layer for mixed or chained work
and a direct imperative layer for known-domain hot loops. Results are
path-independent: one rounding rule per domain on every path (unified
0.5.0, permanently test-gated: see [CONTRACT.md](CONTRACT.md) §3).
- **Certified arithmetic and exact predicates** (0.6.0) - `Interval` and
`DecimalInterval` return a bracket that provably contains the exact result
for `+ - * /`, square root, dot products and quadratic forms (0.6.1: the
quadratic form's bracket is at most 1 ulp wide, and `fused::quadratic_form`
is its correctly rounded scalar); `pd_verdict`
proves a matrix positive definite (or not, or says it cannot decide and hands
back the straddling bracket); `orient2d`, `orient3d`, `incircle`, `insphere`
return an exact three-way sign. Transcendentals are deliberately excluded
from the certified set until their bounds are proven. Design and measured
widths: `docs/design/CERTIFIED_INTERVALS.md`.
- **Numerics on top** - fused ML ops, dense linear algebra, differential geometry
and Lie groups, and standalone TQ1.9 ternary inference.
- **Five precision profiles** - Q16.16 through Q256.256, chosen at compile time;
results are bit-identical across platforms within a profile.
- **Zero float internally, zero runtime dependencies by default.**
## Contract
The precise contract - the supported surface and its stability boundary, the
determinism guarantee, the per-domain rounding rules, cross-profile semantics, and
what the crate requires - is in **[`CONTRACT.md`](CONTRACT.md)**. Agents should
prime from **[`CONTRACT.scn.md`](CONTRACT.scn.md)**.
## Quick start
```bash
cargo add g_math
```
```rust
use g_math::canonical::{gmath, evaluate};
// "100" and "50" route to binary, "3" keeps the result exact via symbolic routing.
let value = evaluate(&((gmath("100") + gmath("50")) / gmath("3"))).unwrap();
println!("{value}");
```
From here, follow the [guides](#guides): the [routing guide](docs/README_ROUTING.md)
for mixed-domain and chained work, or the [imperative guide](docs/README_IMPERATIVE.md)
for direct hot-loop arithmetic.
## Limits worth knowing
- **The binary domain can't hold every literal exactly.** `0.3` and `1/3` repeat
in binary, so the direct `FixedPoint` path approximates them. The canonical
`gmath(...)` router avoids this for you: it classifies each literal and routes it
to a domain where it is exact (`0.3`→decimal, `1/3`→symbolic), carrying the exact
value in a `CompactShadow` - the limit only applies when you pin the binary
domain.
- **Ill-conditioned systems amplify input error.** Solver error scales with the
matrix condition number; this is a property of finite precision, not of the
implementation.
- **Precision differs by profile.** A Q16.16 result cannot hold Q64.64 digits;
choose the profile for your accuracy and range needs.
- **Certified means the arithmetic, not your model.** A certified bracket
encloses the exact result of the computation on the values you passed in; it
cannot see rounding that happened while those values were produced. On the
realtime profile the brackets of a 50-dimensional interval Cholesky are a few
percent of the pivot; from the compact profile upward they are negligible.
- **Switching profiles needs a clean incremental cache** (`rm -rf
target/*/incremental/`), or stale `cfg` artifacts can crash the build.
## Collaborate
Contributions are welcome. Open issues and pick up work on the ecosystem board
at [geodineum.com](https://geodineum.com); issues tagged `good-first-issue` are
a good place to start.
- Fork, branch, and open a pull request against `main`.
- Any change to a wire contract must update **both** `CONTRACT.md` and
`CONTRACT.scn.md` in the same commit.
- A change to a signed extension must be re-signed in the same commit.
## Author & support
Built by **Niels Erik Toren**.
If you want to support the work:
| Bitcoin (BTC) | `bc1qwf78fjgapt2gcts4mwf3gnfkclvqgtlg4gpu4d` |
| Ethereum (ETH) | `0xf38b517Dd2005d93E0BDc1e9807665074c5eC731` / `nierto.eth` |
| Monero (XMR) | `8BPaSoq1pEJH4LgbGNQ92kFJA3oi2frE4igHvdP9Lz2giwhFo2VnNvGT8XABYasjtoVY2Qb3LVHv6CP3qwcJ8UnyRtjWRZ5` |
## Disclaimer
This software is provided **"as is"**, without warranty of any kind, express or
implied. Use of this software is entirely at your own risk. In no event shall the
author or contributors be held liable for any damages arising from the use or
inability to use this software.
**"Certified" is a technical term, not a warranty.** Where this documentation
says *certified*, *proven* or *enclosure*, it means exactly what
[CONTRACT.md](CONTRACT.md) §3 and `docs/design/CERTIFIED_INTERVALS.md` define:
a statement about the exact integer computation performed on the values passed
in, established by construction and gated by tests. It is not a warranty of
fitness for any purpose, and the paragraph above applies to it in full.
## License
Licensed under either of
* [Apache License, Version 2.0](LICENSE-APACHE)
* [MIT License](LICENSE-MIT)
at your option.