rhombic

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): two papers are released — published from this repository, internally audited, not at a formal venue: Asset-1, The Gauge Is the Obstacle (July 29, 2026) and XR-001, Typed State Beats Prose (July 14, 2026). Paper 1, The Shape of the Cell, is complete but neither published nor 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 Asset-1 evidence base is a 480-run adapter bank (two model families × six tasks × forty seeds), now complete. Every analysis of it (D1/D2/D3/D-aux) was pre-registered before the bank finished, 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.

Released Research

Asset-1: The Gauge Is the Obstacle (July 2026)

Task Identity, Cross-Family Transfer, and Merge-Conflict Prediction from Adapter Weights Alone — Timothy Bielec and Minta Carlson. Paper · page · features dataset · audit ledger

A pre-registered bank of 480 LoRA adapters (2 model families × 6 tasks × 40 seeds) trained on one local GPU, then read out by weights alone. Raw adapter weights fail to identify the training task (leave-one-out 0.0792 / 0.1375 against chance 0.1667) while GL(r)-gauge-canonical features reach 1.0000 — the obstacle is the parameterization, not the information. The pre-registered no-transfer prediction was refuted by its own triviality control: raw transfer-at-chance was a family-scale artifact, and standardized transfer runs 0.7375–0.7833. Swapping trained bridges across tasks costs less than 0.001 nats, where destroying the identity backbone costs +2.8 / +3.8. Weight-only features predict midpoint-merge conflict at group-aware AUC 0.995 / 0.962. A pilot correlation of r = 0.888 shrank honestly to r = 0.300 at bank scale. Two of five outcomes went against the pre-registration, and they are the credibility of the other three. Audited by a 117-finding adversarial ladder, with every headline number independently re-derived by an isolated Director instance.

XR-001: Typed State Beats Prose (July 2026)

A Pre-Registered Measurement of Numeric Corruption Under Agent Context Compaction — Timothy Bielec and Minta Carlson. Read it · paper · pre-registration

At matched token budgets, prose re-encoding corrupts 36.4% of numeric facts where typed state blocks corrupt 9.4% (paired exact McNemar p = 3.524 × 10⁻²¹, direction unanimous across three models). A write-time decomposition puts most of the prose deficit at the moment of compaction: typed blocks retain 99.8% of entity values across cascaded checkpoints, prose 65.2%.

The Numbers

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.

Scope: what the comparison does and does not show (EE-001)

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).

Under Structured Weights (Paper 2)

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.

• Raw data and tables
• What the numbers mean

The Question

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.

Read the full thesis →

Quick Start

pip install rhombic            # minimal (numpy + networkx)
pip install "rhombic[viz]"     # add matplotlib for plots
pip install "rhombic[all]"     # everything including dev tools

Reproduce all results:

python -m rhombic.benchmark

Use 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 nodes

Results

Rung 1: Graph Theory (complete)

Four 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.

• Raw data and tables
• What the numbers mean

Rung 2: Spatial Operations (complete)

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.

• Raw data and tables
• What the numbers mean

Rung 3: Signal Processing (complete)

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.

• Raw data and tables
• What the numbers mean

Rung 4: Context Architecture (complete)

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.

• Raw data and tables
• What the numbers mean

Full experimental ladder →

FCC Embedding Index (complete)

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)

• Raw data and tables
• What the numbers mean

Paper 2: Weighted Extensions (complete)

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.

• Raw data and tables
• What the numbers mean

Paper 3: The Learnable Bridge (13 experiments) — working draft

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

Paper 4: The Topology Programmer — working draft

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_graph structural-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.

• Outline and results · Draft

Synthesis

The complete argument across all four rungs — cultural genealogy, empirical evidence, cybernetic interpretation, and practical recommendations.

• The Shape of the Cell: Cubic vs FCC Lattice Topology Across Four Domains

Philosophy

• 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.

Interactive Agent

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.

Hackathon Demo →

Ecosystem

• 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 — Neural Adapter Geometry

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

Papers

Status: Asset-1 and XR-001 are released (published from this repository, not at a formal venue). Paper 1 is complete (not yet published). Papers 2–4 are working drafts — internally audited, under revision, not submitted.

• Asset-1: The Gauge Is the Obstacle — task identity, cross-family transfer, and merge-conflict prediction from adapter weights alone (480-adapter pre-registered bank). Released July 29, 2026. Page · dataset
• XR-001: Typed State Beats Prose — numeric corruption under agent context compaction, format as the sole variable. Released July 14, 2026. Page
• 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.

Contributing

See CONTRIBUTING.md. We're looking for new topologies, new metrics, and new rungs on the experimental ladder.

License

MPL-2.0 — Use freely. Modifications to library files shared back to the commons.

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