synTagma
Spatial coordinate space computing system built on Tagma
Identity is coordinate, and address is space.
Overview
synTagma is a spatial coordinate space computing system. Its core primitive, 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.
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 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 | Complete Tagma specification: coordinate space, decoder, hardware, benchmarks |
| Tagma 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 | Core types, data structures, API reference |
| Tagma-ID | Content-addressable identity without hash functions |
| Tagma-KV | Hashless Key-Value Storage |
| Ibex RV32IMCB Exhaustive Verification Report | Exhaustive RISC-V verification using structural enumeration: 73x speedup |
| CVA6 CV-X-IF Verification | At 2M scale, structural enumeration achieves 846x speedup |
Contact
For inquiries or research collaboration: syntagma@ssccs.org.
Footnotes
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.↩︎