Research
Four interconnected lines of inquiry — all emerging from the structure of the Fano plane and its 168 automorphisms — converging on a trained neural architecture that uses this algebra as inductive bias.
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
Where This Led: Fano-7
These four research threads converged into a single architecture — Fano-7, a transformer variant where 7 attention heads are arranged on the projective plane and cross-head communication is gated by octonion associator norms.
This model now powers our learning platform. Read how it works →
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)
Open Research
90+ discovery experiments completed. Code and model weights being prepared for open-source release. Join our Discord to follow along, or join the Learn waitlist to be notified when the platform launches.