A complete cognitive architecture on one manifold: S³, the unit 3-sphere. Every brain state is a unit quaternion. Every update is a Hamilton product. One arithmetic operation. Five layers. Turing-complete.
Paper: A Geometric Computer · Zenodo Zeroth Law: Sequence Integrity · Zenodo
The brain runs a single loop: ingest a carrier, compare the current state against what the genome predicts, measure the geodesic distance σ. When σ crosses π/4 — the BKT phase boundary on S³, the Hopf equator — the brain closes: writes to the genome, updates the hierarchy, corrects its prediction. Learning is repeated closure. Convergence is a fixed-point theorem.
Five layers build upward from S³:
substrate → memory → execution → brain → learning
Each layer uses only the layer beneath it. The same Hamilton product that encodes a unit quaternion rotation also runs a Minsky machine, reads the genome, and integrates the perceptual field. There is no other arithmetic.
Run in the browser → Conway's GoL on S³ and the Gray Game (three interference modes) running live in JS — no install required.
Terminal visualizers (require true-color support):
cargo run --example gol_live --release # Conway's GoL on S³, pattern selector
cargo run --example gray_game_live --release # Gray Game: spectrum / resonance / edge| # | experiment | subject |
|---|---|---|
| 1 | exp_arithmetic |
Exact modular arithmetic on S³ for Z/nZ orbits |
| 2 | exp_bkt_phase_transition |
BKT phase boundary at σ = π/4 |
| 3 | exp_riemann_zeros |
Riemann zeros as local minima in the carrier field |
| 4 | exp_associative_memory |
Associative recall by σ-radius |
| 5 | exp_turing |
Turing completeness via 2-counter Minsky machine |
| 6 | exp_fractran |
FRACTRAN: Turing-complete prime-native computation |
| 7 | exp_prime_resonance |
Riemann zeros as prime eigenstates on S³ |
| 8 | exp_collatz |
Collatz sequence: orbit convergence on the carrier lattice |
| 9 | exp_fiber_memory |
Fiber memory: axis and angle as independent address channels |
| — | exp_su2_gates |
SU(2) gate dictionary and single-qubit completeness |
| — | exp_neuromodulated_learning |
Arousal and coherence modulation of the learning regime |
SUBSTRATE
├── sphere Hamilton product · inverse · sigma · slerp · IDENTITY
├── embed SHA-256 → carrier on S³ · Vocabulary · MusicEncoder
├── verify A?=A · σ · VerificationEvent · Hopf decomposition
└── hopf S³ → S² × S¹ · factorized addressing · AddressMode
MEMORY
├── buffer Transient input window (EMBED writes, ZREAD reads)
├── genome DNA · Epigenetic · Category · nearest-neighbor reads · BKT control
└── field zread · resonate · coalition accumulation · cos(σ) falloff
EXECUTION
├── carrier VerificationCell · EulerPlane · TwistSheet · CouplingState
├── execution MinskyMachine · FractranMachine · OrbitRuntime
└── zeta Zeta functions over the carrier lattice
BRAIN
├── hierarchy Recursive closure detection · genome emission
├── localization O(log n) minimal closure interval search
├── consolidation Merge · prune · reorganize epigenetic · DNA is permanent
├── neuromodulation arousal_tone · coherence_tone (observational, session-ephemeral)
└── three_cell ThreeCell::ingest → buffer → ZREAD → RESONATE → VERIFY
→ Cell A composition → closure → Cell C integration → genome
LEARNING
└── teach (input, target) pairs · σ-gap error · geometric convergence
cargo run --example exp_arithmetic --release
cargo run --example exp_fiber_memory --release
cargo run --example gol_live --release
cargo run --example gray_game_live --release
cargo test --release # 55 tests, all passingIdentity maintenance A?=A at any scale. One comparison.
σ = 0 at perfect coherence.
σ = π/4 at the Hopf equator — the closure threshold.
Two incident types Missing record (W axis breaks) or reorder (RGB axes break).
Algebraic inverses. There is no third type.
Carrier channels R = salience (X) — the [Total, Unknown] commutator.
G = total (Y) — the full field; the prior.
B = unknown (Z) — what has not been integrated.
Known = G − B. Yellow = learned. Blue = novel.
Factorized addressing Axis queries find semantic type (S²).
Angle queries find cyclic position (S¹).
Full queries find the exact carrier (S³).
Field resonance ZREAD reads the genome by proximity, not exact match.
Query strength falls as cos(σ), cuts off at σ = π/3.
Genome persistence DNA layer bootstraps from orbit seeds. Permanent.
Epigenetic layer learns from ingest. Consolidates.
Turing completeness 2-counter Minsky machine and FRACTRAN on the carrier substrate.
BrainState serialization Full state serializes to JSON. Exact round-trip.
An agent with two drives: hunger σ(state, food) and cold σ(state, shelter). Both are geodesic distances on S³. When both are zero the agent is at IDENTITY — the algebraic identity element is homeostasis. One comparison per tick. No reward function. No learned policy. The geometry produces the gradient.
The same Hamilton product applied to data integrity. Any ordered byte sequence composes on S³; if the sequence is intact the product closes; if not, the gap tells you exactly what broke and where. Built on the same Rust core.
pip install closure-sdkFull SDK documentation · CLI documentation
| Path | What it is |
|---|---|
closure_ea/ |
The Geometric Computer — Rust crate. Paper |
docs/ |
Site + live experiments — experiments.html |
closure_sdk/ |
pip install closure-sdk — data integrity SDK |
closure_dna/ |
pip install closure-dna — geometric database |
closure_cli/ |
CLI surface |
rust/ |
Shared Rust core |
cargo test --release --manifest-path closure_ea/Cargo.toml # 55 tests
pytest closure_sdk/tests -q
pytest closure_dna/tests -qIndependent research, done on personal time, released for free.
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| PIX (Brazil) | walter.h057@gmail.com |
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