Research
Four interconnected lines of inquiry — all emerging from the structure of the Fano plane and its 168 automorphisms. Each experiment is a numbered, self-contained Python script with a hypothesis and measurable results.
The Connecting Thread
Everything here starts with the Fano plane PG(2,2) — the smallest finite projective plane. 7 points, 7 lines, each point on 3 lines, each line through 3 points. Its automorphism group PSL(2,7) has exactly 168 elements — the same group that governs octonion multiplication.
We didn't set out to apply one structure to four domains. The connections emerged from the experiments. The same algebra that makes octonion navigation composable also decomposes proteins, generates music, and corrects errors. The structure is deep enough that it keeps working in contexts we didn't design it for.
"We follow the math where it leads. When the same structure solves problems in four different fields, it's telling you something about the structure, not the fields."
Composable Non-Associative Algebra
D01 – D20
We discovered that non-associativity is not a limitation — it's a recording medium. The associator encodes which Fano plane boundary you crossed. ComponentTransforms form a fully associative group on top of non-associative octonion multiplication.
Key result: 100% navigation success vs 14% with raw multiplication
Spectral Protein Structure Analysis
D77 – D111
IBP-ENM: a single spectral decomposition of the protein contact network simultaneously yields domain boundaries, hinge locations, and per-residue structural roles — without any training data. Validated against CATH, DynDom, and literature ground truth.
Key result: 78% domain k-accuracy, 4.49× allosteric enrichment, 100% archetype classification (12/12)
Fano-Structured Music & Acoustic Protocols
D25 – D47, D63 – D72
The Fano plane maps perfectly onto Western music: 7 points become 7 scale degrees, 7 lines become 7 triads, and 168 automorphisms become harmonic transformations. Non-associativity means path history is encoded in the sound — the same chord after different paths sounds different.
Key result: Math-music isomorphism with error correction as jazz, acoustic data transfer at ~2.8 bits/pair
Algebraic Video Repair
Applied research
If corruption can be modeled as one of the 168 automorphisms of PSL(2,7) applied incorrectly, then repair is simply finding that automorphism and applying its inverse. This reframes error correction from a statistical problem into a group-theoretic search.
Key result: Group-theoretic error correction over PSL(2,7) with spatial algebra and belief systems
How We Work
Hypothesis
Each discovery starts with a specific, testable question. The docstring at the top of every experiment file states exactly what we're testing and why.
Experiment
Self-contained Python scripts that run the hypothesis. No hidden dependencies, no black boxes. When we benchmark, we use ground truth data from established databases (CATH, DynDom, PDB).
Results
Numbers, not narratives. When something works, we report accuracy, p-values, and enrichment ratios. When it doesn't, we say so and explain what we learned.
Recent Experiments
7 highlightedMulti-Mode Hinge Detection
confirmedT4 lysozyme is a "hinge enzyme" — mode 1 looks allosteric, but modes 2–5 concentrate at the catalytic cleft. Hinge occupation ratio (hinge_R₂₋₅ > 1.0) cleanly distinguishes enzymes from allosteric proteins. Completes the 12/12 benchmark.
100% archetype classification (12/12), 0 false barrel, 0 regressions
Enzyme Lens — Asymmetric Entropy Detection
confirmedEnzymes have localized active-site dynamics (high IPR in low modes), while allosteric proteins show delocalized propagation. An asymmetric entropy detector fixes DHFR classification.
92% archetype classification (11/12 proteins)
Thermodynamic Band — 7 Fano-Mapped Instruments
confirmedAdding full thermodynamic observables (vibrational entropy, heat capacity, Helmholtz free energy, IPR mode localization) and expanding to 7 Fano-mapped instruments. The breakthrough experiment for archetype classification.
83% accuracy (10/12), up from 17%
Quantum Carver — 7D Hilbert Space Alignment
confirmedUnifies spectral surgery with quantum-style alignment over a 7D Hilbert space mapped to Fano points. Includes shadow inspection (preview cuts before committing) and a diagnostic handbook.
58% blind archetype accuracy
Allosteric Validation — Ground Truth
confirmedGround-truth validation of spectral surgery lock contacts against UniProt/PDB functional sites using Fisher's exact test. The algebra finds real biology.
4.49× allosteric site enrichment (p < 0.05)
Papers
View all →Spectral Decomposition of Protein Contact Networks for Unsupervised Domain, Hinge, and Structural Role Analysis
Joshua Byrom
Thermodynamic Band Classification of Protein Structural Archetypes
Joshua Byrom
The (2,2,2) Origin of the QPS Algebra: Deriving Particle Characterization from Combinatorial Selection
Joshua Byrom
Notebooks & Visualizations
Alongside the discovery scripts, we maintain 25+ Jupyter notebooks with interactive visualizations spanning Fano plane geometry, protein spectral decomposition, 3D torus simulations, parallel universe computation, and automorphism orbit plots.
Highlights include a Three.js 3D Heawood Torus with real-time cross-diffusion simulation, protein B-factor correlation plots benchmarked against 200 PDB structures, and 3D gear visualizations of parallel universe computation where timeless universes drive their neighbors.
These will be published alongside the code when we open-source the research.
Current Status
91 discovery experiments completed. Two preprints available on our papers page. Code is not yet public — we're preparing the codebase for open-source release. Join our Discord to follow along, or leave your email to be notified when we publish.