A research program interrogating computation's unexamined geometric defaults — pre-registered, adversarially verified, artifacts public. Cubic networks leave up to 6.1× connectivity on the table.
Every computation — neural networks, simulations, databases — defaults to cubic lattice topology: 6-connected grids. Each node talks to 6 neighbors along 3 axis-aligned directions. It's the geometry of spreadsheets, voxel grids, and attention heads. Nobody chose it. It was just… there.
The FCC lattice gives each node 12 neighbors along 6 face-diagonal directions. Its Voronoi cell is the rhombic dodecahedron — a 14-vertex polyhedron whose 8 cube corners and 6 octahedral bridges tessellate all of 3D space. Kepler proved it packs spheres optimally in 1611. Hales confirmed it in 2005.
Heterogeneous weights create near-disconnection cuts — bottlenecks where information barely flows. The Fiedler value (algebraic connectivity) measures the worst such bottleneck.
In a cubic lattice, each node has 5 alternative routes around a blocked connection. In the FCC lattice, it has 11. The more heterogeneous the weights, the more the FCC advantage grows.
The program runs on a fixed discipline: hypotheses are pre-registered before data exists, implementations are adversarially verified by fresh-context reviewers before anything runs, every number is re-derived from committed artifacts, and null results are published with the same care as wins. What follows is the current state of the record.
RELEASED
The Gauge Is the Obstacle (Asset-1, July 2026) — a pre-registered bank of 480 LoRA adapters (two model families × six tasks × forty seeds), read out by weights alone. Raw weights fail to identify the training task (0.0792 / 0.1375 against chance 0.1667); GL(r)-gauge-canonical features reach 1.0000. Weight-only merge-conflict prediction hits group-aware AUC 0.995 / 0.962. Two of five pre-registered outcomes went against us, including a no-transfer prediction refuted by our own triviality control, and both are reported in full.
RELEASED
Typed State Beats Prose (July 2026) — a pre-registered, matched-budget measurement of numeric corruption under agent context compaction. Prose summaries corrupt 36.4% of numeric facts; typed state blocks 9.4%; paired McNemar p = 3.5×10−21, unanimous across models. Protocol, raw transcripts, and scoring code are public.
MEASURED
The obedient builder (E-5, July 2026) — the Steersman's mechanism executes any count-preserving coupling specification, coherent or deliberately frustrated, with per-pair all-or-none crystallization: across 36,960 trained couplings, every one lands fully on or fully off, and alignment with the true geometry follows an exact combinatorial ladder. Coherence decides what gets built, never whether.
BOUNDED
Where geometry pays (EE-001) — at matched edge budget with free rewiring, random expander-like graphs beat the FCC lattice on connectivity. The lattice advantage is real and scoped: it holds for spatially-embeddable substrates, where wiring has cost — chips, networks, files, agent memory. We found the boundary of our own claim and published it.
AUDITED
The adversarial ladder — the Asset-1 paper was put through seven rounds and nineteen lenses of agent-driven audit: 117 confirmed findings, 92 fixed, every headline number independently re-derived from per-item data by an isolated reviewer. The ledger ships with the paper, dispositions and all.
The cube isn't the enemy — it's the skeleton. The rhombic dodecahedron contains the cube: its 8 trivalent vertices ARE the cube's corners. The 6 tetravalent vertices are the bridges that convert cubic topology into space-filling FCC.
TeLoRA adds a learnable n×n bridge matrix between LoRA's A and B projections. A cybernetic feedback mechanism (the Steersman) programs a chosen coupling topology into the bridge — the rhombic dodecahedron is the canonical target, not a discovery: 100% block-diagonal bridges across 42,500+ matrices, four model families (1.1B–14B), coupling ratios of order 10⁴–10⁵:1 (provisional pending controller re-verification), at benchmark parity with standard LoRA. The mechanism is topology-agnostic (octahedron, tesseract, and 24-cell program just as cleanly), and the geometry does not emerge from data on its own — our own null results say so, and we publish them. The July 2026 bifurcation sweep sharpened the picture: the mechanism builds whatever specification it is handed, fully and bistably — so the specification, not the optimizer, is where coherence must come from. Papers 1–4 are working drafts, internally audited across 7 adversarial rounds; the program's released papers are XR-001 and Asset-1.
"Data symmetry analytically determines representation geometry." — Karkada et al., 2026
If data symmetry determines embedding geometry, then the adapter's topology should match the data's symmetry structure. TeLoRA is that match.
pip install rhombic
@techreport{bielec2026gauge,
author = {Timothy Bielec and Minta Carlson},
title = {The Gauge Is the Obstacle: Task Identity, Cross-Family
Transfer, and Merge-Conflict Prediction from Adapter
Weights Alone},
institution = {TASUMER MAF},
year = {2026},
month = {July},
url = {https://tasumermaf.com/rhombic/gauge/}
}
@techreport{bielec2026typedstate,
author = {Timothy Bielec and Minta Carlson},
title = {Typed State Beats Prose: A Pre-Registered Measurement
of Numeric Corruption Under Agent Context Compaction},
institution = {TASUMER MAF},
year = {2026},
month = {July},
url = {https://tasumermaf.com/rhombic/typed-state/}
}
@unpublished{bielec2026shape,
author = {Timothy Paul Bielec},
title = {The Shape of the Cell: Empirical Comparison of
Cubic and {FCC} Lattice Topologies},
note = {Promptcrafted LLC},
year = {2026}
}
@unpublished{bielec2026weighted,
author = {Timothy Paul Bielec},
title = {Structured Edge Weights Amplify {FCC} Lattice
Topology Advantages via Bottleneck Resilience},
note = {Promptcrafted LLC},
year = {2026}
}
@unpublished{bielec2026bridge,
author = {Timothy Paul Bielec},
title = {The Learnable Bridge: Task Fingerprinting and Adapter
Composition via Structured Coupling in Low-Rank Adaptation},
note = {Promptcrafted LLC. Working draft},
year = {2026}
}