Quaternion-Monoid Algebra

A compositional algebra over fixed-width quaternionic-symbolic state packets. Defines a closed associative binary operation with two-sided identity on a packet space whose elements carry a unit-quaternion rotation plus symbolic metadata plus a scaling factor. Implements the construction on CPU and GPU with bit-exact correspondence between the two implementations.

The result is that a packet space which would otherwise be a passive data container becomes an active algebraic element — composable, chainable, and admitting a single-cycle hardware implementation.

What this is

Given a fixed-width packet structure of the form:

packet = (quaternion, symbolic_fields..., scaling_factor)

this library defines a binary operation packet_product (denoted ⊗) such that for any two packets p1 and p2:

• p1 ⊗ p2 is itself a valid packet of the same width (closure)
• there exists a packet I such that I ⊗ p = p ⊗ I = p for all p (two-sided identity)
• (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c) for all valid triples (associativity)
• iterating the operation state[t+1] = state[t] ⊗ stim[t] preserves the H₁ persistent-homology signature of the input stream within a bounded ratio (topology preservation)

Together, closure + associativity + identity make the packet space a monoid. On topology preservation, paper/topology_notes.md separates what is proved from what is empirical: composing a whole configuration with a common packet preserves persistence diagrams exactly (translations by unit quaternions are isometries — proved, machine-checked), iterated composition is an isometric development whose step lengths exactly equal the stimulus offsets (proved), and the bounded-H₁-ratio band for iterated composition is an empirical property of the tested stimulus classes, documented by the validation suite.

Why this might be useful

Three application classes the construction supports natively:

1. Composable agent state evolution. state[t+1] = state[t] ⊗ stimulus[t] is a single bounded-width operation per time step. On a CPU/GPU pipeline it runs at tens to hundreds of millions of operations per second. On a single-cycle FPGA reference design it runs at the clock rate of the fabric.

2. Algebraic chain verification. For an N-packet chain produced sequentially, the verifier can compute I ⊗ p₁ ⊗ p₂ ⊗ ... ⊗ pₙ and compare to a signed chain-head packet. This is an alternative to Merkle-tree-style O(log n) verification with O(n) operations at constant per-operation cost, with an additional algebraic-consistency check beyond cryptographic verification.

3. Multi-source state composition. Combining N independently-evolving state packets via left-fold composition yields a single composed-state packet that algebraically encodes the multi-source state.

Status

Installable Python package (src/quaternion_monoid_algebra/) with a scalar API, a vectorized batch API, a library of alternative sub-field constructions, and a CuPy GPU port. Three layers of validation in tests/: a pytest + Hypothesis property suite (including machine checks of the proved topology statements), the original 8-test property runner, and 7 stress tests (including real-data behavior on public TUM RGB-D and EuRoC MAV sequences). White paper and formal topology notes in paper/. Use cases discussed in examples/.

Install

pip install git+https://github.com/consigcody94/quaternion-monoid-algebra

or, for development:

git clone https://github.com/consigcody94/quaternion-monoid-algebra
cd quaternion-monoid-algebra
pip install -e .[test]
pip install gudhi            # optional: enables the topology-preservation test
pip install cupy-cuda12x     # optional: enables the GPU tests

Validation summary

[Identity laws]           Left and right identity hold across 100 packets
[Associativity]           512 random triples, 0 violations
[Closure]                 500 random pairs, 0 invalid products
[Stability]               1000-step self-product chain stays unit-norm
[GPU bit-exact]           max GPU vs CPU diff on Hamilton product = 0.00e+00
[Topology preservation]   H₁ persistence ratio in target band [0.3, 5.0]

TOTAL: 8 of 8 property tests pass, plus 7 of 7 stress tests
       (including real TUM RGB-D and EuRoC MAV data) and a
       77-test pytest + Hypothesis suite

Reproduce with:

pytest                          # property-based suite (Hypothesis-driven)
python tests/run_all.py         # the 8-test property runner above
python tests/stress_tests.py    # stress tests (downloads TUM data, SHA-256 verified)
python benchmarks/bench.py      # throughput benchmarks

A note on floating point: the symbolic sub-fields are integer-exact under any association. The quaternion sub-operation is associative exactly over the reals and up to rounding (~1e-16) in IEEE 754 arithmetic; packet equality compares the quaternion at absolute 1e-9 per component and the scale at relative 1e-9.

Quick start

from quaternion_monoid_algebra import packet_product, identity_packet, Packet
import numpy as np

p1 = Packet.random()
p2 = Packet.random()
I = identity_packet()

p3 = packet_product(p1, p2)          # ⊗
assert packet_product(I, p3) == p3   # left identity
assert packet_product(p3, I) == p3   # right identity

# Associativity
a, b, c = Packet.random(), Packet.random(), Packet.random()
assert packet_product(packet_product(a, b), c) == packet_product(a, packet_product(b, c))

# Agent state evolution
state = identity_packet()
for stim in stimulus_stream:
    state = packet_product(state, stim)

Batch API

For throughput, hold N packets as a struct-of-arrays and compose them with vectorized NumPy kernels. Chain reduction runs as an O(log N)-depth pairwise tree, which is legal because ⊗ is associative:

from quaternion_monoid_algebra import PacketArray, packet_product_batch, reduce_packets

pa = PacketArray.random(1_000_000)
pb = PacketArray.random(1_000_000)

pc = packet_product_batch(pa, pb)   # 1M elementwise compositions
head = reduce_packets(pa)           # chain head p₀ ⊗ p₁ ⊗ ... ⊗ pₙ₋₁, tree-reduced

Measured on a Ryzen 5700G (see benchmarks/bench.py to reproduce on your machine):

Choosing sub-field operations

Custom sub-field operations can be swapped in via lookup tables. quaternion_monoid_algebra.tables ships a library of verified monoid constructions with different trade-offs:

Groups mix and preserve information (best avalanche/distinguishability); bands are idempotent one-way registers by design. Check any hand-built table once with validate_monoid_table(table, identity=...) (it verifies closure, the identity laws, and full associativity, with a counterexample on failure) before passing it to packet_product. Gotcha: for tables whose identity element is not 0 (AND, min, mod-mul), build the matching monoid identity with identity_packet(field_a=15) etc. — the default identity_packet() assumes identity 0.

Paper

A standalone write-up of the construction and its properties is in paper/compositional_algebra.md. It covers the field-by-field construction, the proof sketches for associativity of each sub-field operation, the topology-preservation claim, and the three application classes above. Formal statements and proofs of the topology results (exact preservation under common composition; the Lipschitz development bound; why iterated preservation is empirical, not a theorem) are in paper/topology_notes.md, each machine-checked in tests/test_topology.py.

Use case examples

Generic examples (no domain-specific framing) in examples/:

• 01_agent_state.py — composable state evolution under a stream of stimuli
• 02_swarm_composition.py — multi-agent state combination
• 03_chain_verification.py — algebraic audit-chain alternative to Merkle verification

Author and provenance

This work is by Cody Churchwell (@consigcody94). The construction was developed with AI as an engineering assistant for implementation, validation, GPU porting, and documentation. The conceptual direction, evaluative choices, and the specific algebraic structure imposed on the packet space were the author's. Released under MIT license to encourage open use and extension. If you build on it, an attribution is appreciated but not required.

Contributing

PRs welcome. Particularly interested in:

• Alternative sub-field constructions beyond the families shipped in quaternion_monoid_algebra.tables (XOR, mod-add, mod-mul, max, min, AND, OR) that preserve associativity
• Real-world dataset validation results beyond the TUM RGB-D and EuRoC MAV streams in the stress suite
• Hardware implementations (Verilog/SystemVerilog references, ASIC synthesis numbers)
• Theoretical results sharpening paper/topology_notes.md — in particular, sufficient conditions on stimulus streams for a two-sided H₁-ratio bound under iterated composition
• Connections to existing algebraic-structure literature this construction may be related to

Citing

If you use this library in academic work, cite the Zenodo archive (the concept DOI below always resolves to the latest release; version DOIs are on the releases page):

@software{churchwell_quaternion_monoid_algebra,
  author    = {Churchwell, Cody},
  title     = {Quaternion-Monoid Algebra: A Compositional Algebra over
               Fixed-Width Quaternionic-Symbolic State Packets},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.20301069},
  url       = {https://github.com/consigcody94/quaternion-monoid-algebra}
}

GitHub's "Cite this repository" button (from CITATION.cff) produces the same reference in APA/BibTeX.

License

MIT — see LICENSE.