# synTagma

Spatial coordinate space computing system built on Tagma

Author

Affiliation

SSCCS Foundation [](mailto:contact@ssccs.org)

[SSCCS Foundation](https://ssccs.org)

Published

August 3, 2026

Whitepapers

[Tagma](https://doi.org/10.5281/zenodo.21302508)

[Tagma-KV](https://doi.org/10.5281/zenodo.21499437)

References

[Primitive](../../projects/syntagma/tagma/)

[Devlog](https://github.com/ssccsorg/syntagma/tree/main/sw/rust/devlog)

[with neXus](../../projects/nexus/development/syntagma.llms.md)

Code

[GitHub](https://github.com/ssccsorg/syntagma)

Other Formats

[LLMs](https://docs.ssccs.org/projects/syntagma/index.llms.md)

Identity is coordinate, and address is space.

## Overview

synTagma is a spatial coordinate space computing system. Its core primitive, [Tagma](../../projects/syntagma/tagma/), is a 16-bit coordinate embedded in a fixed Unicode block (U+AC00–U+D7AF) that replaces hash-based addressing with direct structural addressing. Every valid 16-bit value is simultaneously a 1-D address (Unicode code point), a 3-D coordinate (Axis 0, Axis 1, Axis 2), and a displayable Unicode character.[^1]

The system spans two layers:

**Tagma (core primitive).** The atomic coordinate: a 16-bit value with closed-form composition, three independent axes, zero collision probability, and single-cycle combinational decoding (~300 gates). A variable-length Coord sequence extends the address space from \\1.12 \times 10^4\\ to \\2^{256}\\ while keeping lookup cost linear in Coord count.

**synTagma coordination system.** The distributed extension of Tagma arithmetic across physical topologies. Each axis of a multi-Coord coordinate can reside on a different node. No distributed consensus is required because each axis is independently stored and accessed. The full recursive formulation is defined in the [Tagma](../../projects/syntagma/tagma/wp).

## Tagma core primitive

The core Tagma coordinate is defined by the composition formula (ISO/IEC 10646) for block U+AC00–U+D7AF:

\\C(i,m,f) = \text{U+AC00} + 588i + 28m + f, \quad 0 \leq i \< 19,\\ 0 \leq m \< 21,\\ 0 \leq f \< 28\\

Of 65,536 representable 16-bit states, 11,172 satisfy this formula. The remaining 54,364 are structurally invalid and hardware-detectable. Each valid value carries three interpretations: a Unicode code point for flat addressing, a triple-axis coordinate for structural queries, and a Unicode character for display.

CoordPath composition extends the address space linearly with Coord count:

| Coords | Axes | Identifier space        | Equivalent to   |
|--------|------|-------------------------|-----------------|
| 1      | 3    | \\1.12 \times 10^4\\    | Sensor tags     |
| 6      | 18   | \\1.94 \times 10^{24}\\ | UUID scale      |
| 10     | 30   | \\2.69 \times 10^{40}\\ | Exceeds 128-bit |
| 19     | 57   | \\1.94 \times 10^{77}\\ | SHA-256 scale   |

The decoder extracts three axis fields from a 16-bit input in one combinational cycle: range check, field extraction (division by 588 and 28), and axis validation. Total gate count is approximately 300 gates in 28nm – smaller than a single 32-bit multiplier.

Core data structures include:

- **CoordSpace (N=1)** – Dense inline array, 22 KB stack allocation, single-load O(1) access. No heap, no hashing, no collisions.
- **CoordSpace2 (N=2)** – Dense heap allocation, 119 MB, same single-load O(1) guarantee.
- **CoordSpaceN\<N\>** – Sparse tree for N-Coord paths, memory proportional to stored entries.
- **CoordSet** – Fixed-size bit array over the coordinate space, bitwise set operations over 175 machine words.

## Coordination layer

The coordination layer extends Tagma’s arithmetic to physical topologies without modifying the core:

**Recursive coordinate space.** The composition formula admits unbounded \\k\\ levels of recursion. A 19-Coord sequence at \\k=0\\ occupies the SHA-256 scale; at \\k=1\\ each of its three axes is itself a full 19-Coord sequence.

**Self-routing.** A Coord sequence carries its own address. Given a sequence and current recursion depth, any node determines whether each sub-axis is local or remote by applying the topology function \\\phi_k\\. No routing table lookup is required.

**Axis fungibility.** The three axes carry no intrinsic semantics. Axis 0 may represent “region”, “shard”, or “timestamp” depending on deployment. The coordinate arithmetic – composition, decomposition, linearisation – is invariant.

## Performance

Based on a software reference implementation on ARMv8.4-A Firestorm:

| Metric | SHA-256 | Tagma dense (N=1) | Tagma tree (N=19) |
|----|----|----|----|
| Latency | 227 ns | 0.39 ns (582x) | 58.6 ns (3.9x) |
| Collision | probabilistic | zero | zero |
| Nonexistent prefix (10M) | 23.05 ms (HashMap scan) | 1.65 ns (14.0Mx) | – |

Spatial query: CoordSet bitwise AND resolves compound axis filters at 329 Melem/s – 137x faster than HashMap scan. All operations are scale-invariant: lookup cost depends on Coord Depth, not data volume.

Cross-validation from hardware verification: the [Ibex RV32IMCB Exhaustive Verification Report](../../projects/ev/ibex.llms.md) using ev (ExaVerif) confirms the same structural advantage on real RISC-V instruction encoding spaces. Tagma-based structural enumeration verifies 524,288 combinations in 49.5 ms versus 3.63 s for the standard pipeline (73x).

## Application domains

**Embedded systems.** The 11,172-identifier space fits in a 22 KB no-allocator array: one load, no hashing, no collisions.

**LLM inference cache.** KV caches indexed by token prefixes use CoordPath-based direct access: production cache sizes (\\10^4\\–\\10^7\\) are covered by 2–4 Coords, with zero hash computation.

**Graph and multi-dimensional query.** Each node maps to a Coord, each edge type to a CoordSet. Adjacency reduces to a bitwise AND over 175 machine words – no index intersection.

**General-purpose addressing.** Replaces UUIDs, hash keys, and sequence numbers with shorter, faster, deterministic identifiers.

## Boundaries

Tagma replaces hash-based identity generation and addressing. SHA-256 remains for signatures, Merkle proofs, and integrity verification. Encryption, authentication, and key derivation are outside the primitive’s scope. The two strategies compose: SHA-256 output encoded as 19 Coords is more readable than 64 hex characters while preserving \\2^{-256}\\ collision probability.

## Documents

| Document | Description |
|----|----|
| [Tagma](../../projects/syntagma/tagma/wp) | Complete Tagma specification: coordinate space, decoder, hardware, benchmarks |
| [Tagma](../../projects/syntagma/tagma/wp#sec-appendix-benches) benchmarks | 51 microbenchmarks across 12 criterion groups: identity generation, spatial query, edge cases at 10M entries, deep trees, mixed-operation stress tests |
| [Tagma Core Primitive](../../projects/syntagma/tagma/) | Core types, data structures, API reference |
| [Tagma-ID](../../projects/syntagma/tagma/id) | Content-addressable identity without hash functions |
| [Tagma-KV](../../projects/syntagma/tagma/kv) | Hashless Key-Value Storage |
| [Ibex RV32IMCB Exhaustive Verification Report](../../projects/ev/ibex.llms.md) | Exhaustive RISC-V verification using structural enumeration: 73x speedup |
| [CVA6 CV-X-IF Verification](../../projects/ev/cva6) | At 2M scale, structural enumeration achieves 846x speedup |

## Contact

For inquiries or research collaboration: <syntagma@ssccs.org>.

## Footnotes

[^1]: The ranges 19, 21, and 28 derive from the Unicode block U+AC00–U+D7AF, which encodes the compositional writing system. The composition formula is defined in ISO/IEC 10646.