Tagma is the core coordinate primitive of the synTagma: a 16-bit coordinate space embedded in a fixed Unicode block (U+AC00–U+D7AF) that replaces hash-based addressing with direct structural addressing. The Tagma composition formula maps three independent axes (Axis 0, Axis 1, Axis 2) to a unique 16-bit value in a single cycle. The result is collision-free by construction, hash-less by design, and decodable at the gate level.

The coordinate space that enables this is an open international standard (Unicode) and a public good. For the full system including recursive coordination and distributed topology, see synTagma.

Property	Value
Coordinate space	19 (Axis 0)¹ x 21 (Axis 1) x 28 (Axis 2) = 11,172 valid 16-bit values
Invalidity margin	54,364 of 65,536 states are structurally invalid (hardware-detectable)
1-D interpretation	Unicode code point U+AC00–U+D7AF (character encoding)
3-D interpretation	(Axis 0, Axis 1, Axis 2) coordinate (structural positioning)
Display interpretation	Unicode character (debugging aid, zero exposure in the process)

This combination of properties — a contiguous 16-bit block with a closed-form composition formula, complete three-axis decomposition, and a built-in invalidity margin — exists in no other Unicode character range. The block U+AC00–U+D7AF is the only one that satisfies all conditions simultaneously.

What problem Tagma solves

Hash-based identity generation costs approximately 10,000 gates and 64-75 cycles per operation. The Tagma decoder replaces this with approximately 300 gates and one cycle — a 30x reduction in hardware cost and a 64-75x reduction in latency.

For larger identifier spaces, Coords compose linearly. Each Coord adds a factor of 11,172 to the addressable space (see benchmark table for addressable space per variant).

Benchmark

Based on a software reference implementation, measured lookup latency: 0.38-0.40 ns for dense CoordSpace variants vs 227 ns for SHA-256 (567-597x) at single-Coord; 54.9 ns vs 227 ns (4.1x) at SHA-256 scale (19 Coords via CoordSpaceN fallback). Structural prefix queries show the decisive advantage: Tagma answers nonexistent prefix in 1.65 ns while a general-purpose hash table scans 10M entries in 23.05 ms (14.0Mx). See benchmarks section in Tagma Whitepaper: All measured on ARMv8.4-A Firestorm with cargo bench code.

Metric	SHA-256	CoordSpace (N=1)	CoordSpace2 (N=2)	CoordSpaceM3 (N=3)	CoordSpaceN (any N)
Latency per lookup (ARMv8.4-A Firestorm)	227 ns	0.39 ns	0.39 ns	0.40 ns	0.94-40.8 ns
Allocation	—	Inline 22 KB	Heap 119 MB	Mmap 1.27 TB	Sparse tree
Identity size	32 bytes	2 bytes	4 bytes	6 bytes	\(2N\) bytes
Addressable space	\(2^{256}\)	\(1.12 \times 10^4\)	\(1.25 \times 10^8\)	\(1.39 \times 10^{12}\)	variable
Collision	probabilistic (\(2^{-128}\))	deterministic zero	deterministic zero	deterministic zero	deterministic zero
Tagma	—	Complete	Complete	Complete	Fallback

Comparison

Method	Identifier size	Generation cost	Collision	Lookup
Pointer	32-64 bits	zero	none	direct
Hash (SHA-256)	256 bits	~10K gates, 64-75 cycles	probabilistic	hash table + resolution
UUID	128 bits	entropy-dependent	probabilistic	hash table
CAM	per-bit comparison	9-16 transistors/bit	none	associative
Tagma	16 bits	1 cycle (combinational)	none (formulaic)	direct (coordinate = address)

What Tagma Replaces and What It Does Not

Domain	Hash Role	Tagma Replaces?
ID generation	hash(data) for unique identifier	Yes
Hash map key	hash(key) for bucket index	Yes
Content addressing	data hash as storage address	Partial
Integrity verification	data hash for tamper detection	No (retain SHA-256)
Digital signatures	hash-then-sign for non-repudiation	No (retain Ed25519)
Key derivation	HKDF for key expansion	No (retain HKDF)

The entries marked No are not limitations; they reflect a deliberate composition strategy. An identifier can be generated by SHA-256 and then encoded as Tagma coordinates at write time with near-zero overhead — a single modulo operation per hash block. From that point forward, every read operation uses Tagma’s native O(1) direct access regardless of how the identifier was originally produced. The cryptographic cost is paid exactly once at write time; all subsequent accesses benefit from the structural address space.

This composition pattern extends beyond cryptography to every domain in the table. Each domain can adopt Tagma at its own pace — replacing the hash step where possible, wrapping it where necessary — while always reading at Tagma speed. The result is a universal addressing layer that different applications (cryptographic content addressing, distributed node identification, sensor networks, real-time object tracking) share without compromising their domain-specific requirements.

Data Structures

The Tagma coordinate space is implemented as a Rust library providing:

Coord — A 16-bit value guaranteed structurally valid. Constructable from raw index, Unicode code point, character, or the three axes.
CoordSet — A 1.4 KB bit array over the coordinate space. Insert, remove, membership: single-bit operations. Union, intersection, difference: bitwise over 175 machine words.

The CoordSpace family follows a three-tier design that scales from embedded to datacenter:

Variant	Allocation	Access	Latency	Tagma
CoordSpace (N=1)	Inline 22 KB	Single load	0.39 ns	Complete
CoordSpace2 (N=2)	Heap 119 MB	Single load	0.39 ns	Complete
CoordSpaceM3 (N=3)	Mmap 1.27 TB	Single load	0.40 ns	Complete
CoordSpaceN<N> (any N)	Sparse tree	N dereferences	0.94-40.8 ns	Fallback

CoordSpace (N=1) — Dense inline array, 22 KB stack allocation for the full single-Coord space. No heap, no allocator. Single-load O(1) access, zero collisions, zero hashing.
CoordSpace2 (N=2) — Dense heap allocation, 119 MB covering the full two-Coord space. Same single-load O(1) guarantee as N=1.
CoordSpaceM3 (N=3) — Dense mmap allocation, 1.27 TB with MAP_NORESERVE for the full three-Coord space. Physical pages fault on demand. Same single-load O(1) latency.
CoordSpaceN<N, V> (any N) — Multi-Coord direct-address tree indexed by CoordPath. Fixed compile-time depth N (2, 6, 12, 19). N dereferences per lookup. Variable-depth DynCoordSpace for &[Coord] runtime paths. Used as fallback when the dense variants exceed system memory.

Papers

The full technical specification and domain-specific analyses are available as standalone papers:

Paper	Description
synTagma (System Overview)	Spatial coordinate space computing system built on Tagma
Tagma	Complete specification: coordinate space, decoder, coprocessor, compliance, benchmarks, SEU analysis
Tagma-ID	Content-addressable identity without hash functions
Tagma-Memory (forthcoming)	Three-dimensional SRAM decode using structural coordinates
Tagma-Hardware (forthcoming)	A 300-gate combinational decoder for processor pipeline attachment
Tagma-Encoding (forthcoming)	Base11172: human-readable, self-validating serialization

Contact

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

Whitepaper: PDF / HTML DOI: 10.5281/zenodo.18759106 via CERN/Zenodo, indexed by OpenAIRE. Licensed under CC BY-NC-ND 4.0.
Official repository: GitHub. Authenticated via GPG: BCCB196BADF50C99. Licensed under Apache 2.0.
Governed by the Foundational Charter and Statute of the SSCCS Foundation (in formation).
Provenance: Human-in-Command, AI-assisted. Aligns with ISO/IEC JTC 1/SC 42 and C2PA-certified. Full intellectual responsibility with author(s).

Footnotes

The constants 19, 21, and 28 derive from the compositional writing system for which the Unicode block U+AC00–U+D7AF was originally allocated.↩︎