The bottleneck is not the processor. It is the shape of the cell.
A benchmarking library that compares cubic (6-connected) and FCC/rhombic dodecahedral (12-connected) lattice topologies across graph theory, spatial operations, and signal processing.
Research status (July 2026): Paper 1, The Shape of the Cell, is the program's one completed paper; it has not yet been published or submitted. Papers 2–4 are working drafts under active revision. Corrections and retractions are tracked openly — see the Retracted table in results/EXPERIMENT_TRACKER.md and the equal-edge control in results/EE-001-equal-edge-control/.
Pre-registration discipline (July 2026): the program's next evidence base is a 480-run adapter bank (two model families × six tasks × forty seeds), currently training. Every analysis of it (D1/D2/D3/D-aux) was pre-registered before the bank completes, and the analysis tools refuse to run on a partial bank — a completeness interlock in code, not a convention. Null-model calibration runs before trained arms (BM-000, BM-000b hub-motif nulls), and the BM-004 runner refuses to launch any arm until its positive-control gate passes. Designs are reviewed by an independent referee whose rulings are recorded verbatim (docs/DIRECTOR_DECISIONS_2026-07-06.md, docs/DIRECTOR_RULING_PREREG_A3A5_2026-07-07.md); deviations are dated amendments, never silent revisions.
| Metric | FCC vs Cubic | Scale |
|---|---|---|
| Average shortest path | 30% shorter | 125 – 8,000 nodes |
| Graph diameter | 40% smaller | 125 – 8,000 nodes |
| Algebraic connectivity | 2.3–2.5× higher | 125 – 8,000 nodes |
| Flood fill reach | 55% more nodes | 125 – 8,000 nodes |
| NN query speed | 17% faster | 125 – 8,000 nodes |
| Signal reconstruction | 4-10× lower MSE | 216 – 1,000 samples |
| Reconstruction isotropy | 5-20× more uniform | 216 – 1,000 samples |
| Embedding neighbor recall | +15-26pp at 1-hop | 125 – 1,000 nodes |
| Information diffusion | 1.4-2× faster | 125 – 1,000 nodes |
| Edge cost | ~2× more edges | (the price) |
These ratios are stable across all tested scales. They hold at every size tested, consistent with derivation from Voronoi cell geometry rather than sample size.
The table above compares two spatially-embeddable lattices. It does not make FCC a near-optimal graph in the abstract: the equal-edge-count control (EE-001, pre-registered) shows that a degree-preserving random rewire of the FCC's own edges beats it 4–17× on algebraic connectivity and halves path lengths. Random expanders win at a fixed edge budget — but they require mean wire lengths 3.5–7.2× the FCC's nearest-neighbor edges, growing without bound (~N^⅓), and so cannot be physically embedded with local wiring. The honest claim: among spatially-embeddable lattice topologies at matched node count, FCC dominates cubic — and spatial embeddability is exactly the constraint that physical domains (routing fabrics, meshes, neighbor lists, spatial data structures) impose. Where a task has no spatial metric, this library's own null results say the geometry does not help (LEARNINGS.md: L-001, Exp 2.5).
| Metric | Corpus vs Uniform | Scale |
|---|---|---|
| Fiedler ratio (direction-weighted) | 2.3x → 6.1x | 125 – 8,000 nodes |
| Path advantage | 30% → 60% shorter | 125 – 1,000 nodes |
| Consensus speedup | 1.0x → 6.7x | 125 nodes |
| Prime-vertex coherence | p = 0.000025 | Single cell (40,320 permutations) |
Heterogeneous edge weights amplify the FCC advantage. Direction-based weighting — mapping structured values to the 6 direction pairs of the FCC lattice — nearly triples the Fiedler ratio. The mechanism is bottleneck resilience: FCC routes around suppressed edges that strangle cubic lattices.
Computation is built on the cube. Memory is linear. Pixels are square. Voxels are cubic. Nobody chose this — it accumulated. Descartes gave us orthogonal coordinates. Von Neumann gave us linear memory. The cubic lattice is the spatial expression of Cartesian geometry.
Is the cube optimal? This library measures the alternative: the face-centered cubic lattice, whose Voronoi cells are rhombic dodecahedra. 12 faces instead of 6. The densest sphere packing in three dimensions (Kepler, proved by Hales 2005, formally verified 2017). The lattice that nature uses for copper, aluminum, and gold.
pip install rhombic # minimal (numpy + networkx)
pip install "rhombic[viz]" # add matplotlib for plots
pip install "rhombic[all]" # everything including dev toolsReproduce all results:
python -m rhombic.benchmarkUse in code:
from rhombic.lattice import CubicLattice, FCCLattice
cubic = CubicLattice(n=10) # 1000 nodes, 6-connected
fcc = FCCLattice(n=6) # ~864 nodes, 12-connected
# Convert to networkx for any graph analysis
G_cubic = cubic.to_networkx()
G_fcc = fcc.to_networkx()Compare topologies:
from rhombic.lattice import CubicLattice, FCCLattice
cubic, fcc = CubicLattice(5), FCCLattice(5)
print(f"Cubic: {cubic.stats().connectivity}-connected, {cubic.stats().node_count} nodes")
print(f"FCC: {fcc.stats().connectivity}-connected, {fcc.stats().node_count} nodes")
# Cubic: 6-connected, 125 nodes
# FCC: 12-connected, 500 nodesFour metrics, three scales, consistent ratios. The FCC lattice outperforms the cubic lattice on every measure of routing efficiency and structural robustness. The cost is bounded: ~2× edges for ~30% shorter paths and ~2.4× robustness.
The routing advantage translates. FCC flood fill reaches 55% more nodes per hop. Nearest-neighbor queries are 17% faster. Range queries return 24% more nodes per volume (denser packing). The cost: range query time scales with density — 3-5× slower for sphere/box queries at 8,000 nodes.
Direct empirical measurements confirm the FCC advantage. FCC spatial sampling produces 4-10× lower MSE and 5-20× more isotropic reconstruction than cubic sampling at matched sample counts. The advantage peaks in the mid-frequency range (10-60% of Nyquist) and grows with scale — from +6 dB at 216 samples to +10 dB at 1,000. Above Nyquist, both lattices alias and cubic's axis alignment accidentally helps.
Does the FCC advantage survive when the lattice organizes high-dimensional embedding data? FCC captures 15-26 more percentage points of an embedding's true nearest neighbors at 1-hop. Information diffuses 1.4-2× faster. Consensus converges 1.58× faster at moderate scale (500 nodes), though per-neighbor weight dilution reduces the advantage at 1,000 nodes.
A proof-of-concept ANN index that organizes high-dimensional embeddings on lattice topology. At matched node counts, the FCC index captures +7 to +20 percentage points more true nearest neighbors at 1-hop than the cubic index. The only variable is the connectivity pattern.
from rhombic.index import FCCIndex, CubicIndex, brute_force_knn
fcc = FCCIndex.from_target_nodes(dim=384, target_nodes=500).build(embeddings)
results = fcc.query(query_vector, k=10, hops=1)
recall = fcc.recall_at_k(queries, ground_truth, k=10, hops=1)What happens when edges carry heterogeneous weights? Seven experiments across two scales (lattice and single-cell). The FCC advantage amplifies under structured weights — direction-based corpus weighting pushes the Fiedler ratio from 2.3x to 6.1x. Prime-vertex coherence is significant at the optimal mapping (p = 0.000025 vs 40,320 alternatives). Spectral bottleneck creation is universal across 24-edge polytopes, not RD-specific.
Thirteen experiments across four model families (1.1B–14B parameters), demonstrating that a cybernetic feedback mechanism (the Steersman) programs a specified coupling topology into multi-channel LoRA bridge matrices. The rhombic dodecahedron is the canonical target, not a discovery: the geometry does not emerge from data (Exp 2.5 null: co/cross = 1.002, p = 0.474), and the same controller imprints octahedral, tesseract, and 24-cell targets just as cleanly (see Paper 4). The topology is chosen; the mechanism is what's demonstrated.
Key finding: When the Steersman (contrastive + spectral feedback) is active at channel count n=6 with RD face-pair supervision, 100% of bridge matrices develop block-diagonal structure aligned to the three coordinate planes of the rhombic dodecahedron. Without the Steersman: 0%. The co-planar/cross-planar coupling ratio peaks at 82,854:1. Structure locks in by step 200, survives adversarial initialization, and costs 0.17% validation loss.
What the bridge is for: interpretable diagnostics, at benchmark parity. Bridge fingerprints classify task type at 72.3% leave-one-out accuracy (linear SVM over 336 individual 36-parameter bridge matrices; chance 33.3%), bridge deviation tracks the generalization gap at r = 0.888, and a benchmark head-to-head (BM-001) shows TeLoRA matching standard LoRA on MMLU/ARC-C/HellaSwag/WinoGrande (aggregate Δ +0.0012). The structure comes at zero benchmark cost. (An earlier 84.5% fingerprinting figure was retracted 2026-04-06 — never backed by reproducible computation; see the Retracted table in results/EXPERIMENT_TRACKER.md.)
| Finding | Value |
|---|---|
| Block-diagonal rate (cybernetic n=6) | 100% (42,500+ matrices) |
| Block-diagonal rate (non-cybernetic) | 0% (570 matrices) |
| Peak co-planar/cross-planar ratio | 82,854:1 |
| Lock-in speed | ~200 steps (half-life 123 steps) |
| Adversarial initialization suppression | 99.5% in 900 steps |
| Bridge Fiedler bifurcation (n=6 vs n≠6) | 1,020× |
| Val loss cost of topology | 0.17% max |
| Scale invariance | 1.1B, 7B, 14B (Fiedler converges ~0.10) |
7-round adversarial audit. 232 findings across 7 rounds, 87 fixed. The audit covers Papers 1–3 and predates the April 2026 fingerprinting correction; Papers 2 and 3 remain working drafts — internally audited, not published, not submitted.
- Audit trail — full findings, hub validations, rewrite log
- Cross-phase synthesis
The Steersman is a general-purpose topology programmer, not an RD-specific mechanism. Four polytopes confirmed: octahedron (473,622:1 co/cross, per-bridge mean), rhombic dodecahedron (70,404:1), tesseract (41,564:1; three same-seed runs mutually consistent at co/cross r ≥ 0.995 — see the provenance record, which documents that the original run's results file was destroyed by a same-seed relaunch and recovered from its frozen log), 24-cell (35,808:1).
⚠ Provisional (frozen 2026-07-03): these ratios were measured in runs governed by the adaptive controller, whose stability detector has a known defect (declares STABLE while its metric is still moving — see BM_BATTERY_PLAN.md, Known Issues). They are held out of any publication until the detector is fixed and the runs re-measured; the
rd_graphstructural-mask results (BM-003, no controller) are unaffected. Fixed-weight runs (FC-001, FO-001) are less implicated. (Update 2026-07-07: the Paper 4 audit is complete — all six blockers resolved; per-number controller-exposure classification in PAPER4_EXPOSURE_CLASSIFICATION.md. Re-measurement under the fixed detector remains queued for the post-bank GPU window.)
Four training regimes mapped: Block-Diagonal, Spectral Attractor (universal band, Fiedler ≈ 0.09), Hierarchical Coherence, Collapse. Negative controls sharpen the claim: wrong-label pairs and prime-derived pairs both collapse — geometric coherence in the pair specification is required. Removing the controller dissolves the topology within ~100 steps: the Steersman is homeostatic maintenance, not one-time crystallization.
The complete argument across all four rungs — cultural genealogy, empirical evidence, cybernetic interpretation, and practical recommendations.
- Reproducible by default. Every result has code that generates it.
- The geometry is the argument. The numbers are the evidence.
- Cost is always reported alongside benefit.
- Sparse results are data, not failure.
rhombic-agent — a Hermes Agent
that thinks in 12 dimensions. 9 custom tools + 3 conversational skills. Ask it
to run experiments, generate visualizations, and explain the geometry.
- Interactive demo — try it in your browser
- The essay — the thesis for humans
- PyPI —
pip install rhombic - Full synthesis — the complete argument across all four rungs
- Weighted extensions — what happens under heterogeneous weights
TeLoRA adds a learnable n×n coupling matrix — the bridge — between the A and B projections in LoRA, adding n² parameters per layer. When the bridge is the identity matrix, the architecture reduces exactly to standard LoRA.
The bridge does not improve fine-tuning loss, and it does not hurt it
(BM-001: benchmark parity with standard LoRA across four lm-eval tasks).
It provides something LoRA cannot: a compact, interpretable diagnostic of
adapter behavior — an n²-parameter summary of what training did, readable
without inference or evaluation. A cybernetic feedback mechanism (the
Steersman) can program a chosen topology into the bridge; the rhombic
dodecahedron is the canonical instance, and the mechanism is
topology-agnostic (Paper 4). A structural variant (rd_graph mode) builds
the topology in by construction — a fixed adjacency mask with learnable
edge weights, no controller — and is the current experimental frontier
(BM-003).
- Architecture:
rhombic.nn—RhombiLoRALinear, topology, bridge init, exact absorption to standard LoRA - Training:
scripts/train_cybernetic.py— full Steersman pipeline - 28 experimental learnings, nulls and retractions included: LEARNINGS.md
Status: Paper 1 is complete (not yet published). Papers 2–4 are working drafts — internally audited, under revision, not submitted.
- Paper 1: The Shape of the Cell — four-domain topology comparison. Complete.
- Paper 2: Structured Edge Weights Amplify FCC Lattice Topology — bottleneck resilience under heterogeneous weights. Draft.
- Paper 3: The Learnable Bridge — cybernetic feedback programs coupling topology in multi-channel LoRA; interpretable adapter diagnostics (13 experiments, 4 model families). Draft.
- Paper 4: The Topology Programmer — the Steersman as a general topology programmer across four polytopes; four-regime taxonomy. Draft.
See CONTRIBUTING.md. We're looking for new topologies, new metrics, and new rungs on the experimental ladder.
MPL-2.0 — Use freely. Modifications to library files shared back to the commons.