ToricGT

The ToricGT row is not claiming accepted leaderboard status. It is a pending experimental/non-record candidate until the Parameter Golf PR, tokenizer accounting, reproducibility checks, and official review are complete.

• graphcg_loss0.011413

  Full-rank GraphCG pressure: disentangles latent concept axes so graph and byte objectives can expose stable directions rather than collapsing into one entangled basis.

• analogy_loss0.064827

  Analogical-retrieval lattice loss: encourages maps between reasoning memories through simplex-tree and vectorized persistent-homology similarity gates.

• tokengt_graph_loss1.162810

  First-class TokenGT graph objective: trains causal node, edge, endpoint, distance, torus, and tokenization-DAG features used by the OAI FineWeb adapter.

• memory_loss1.140689

  Trajectory-memory retrieval loss: selects reusable reasoning traces and stabilizes retrieval-conditioned auxiliary routing.

• toric_geometry_loss5.884988

  Toric embedding and fan audit loss: keeps tropical ring attention active faces embedded into toric charts with interpretable cone/fan structure.

• toric_vector_bundle_1d_cone_ce_loss0.704291

  Toric vector-bundle one-dimensional-cone/sheaf CE: regularizes per-cone filtrations and sheaf-style compatibility across fan neighborhoods.

• toric_bgg_loss0.164997

  Toric BGG category-O supervision: keeps finite category-O certificates, standard filtrations, Gale-dual labels, and differential consistency visible.

• koszul_persistence_loss0.045398

  Koszul/persistence loss: checks filtered complexes, multigraded modules, and chain-complex consistency for topological reasoning traces.

• toric_cca_topology_loss1.045991

  Combinatorial toric commutative algebra/topology loss: tracks monomial staircases, syzygies, and CAS-backed toric module diagnostics.

• derived_signature_loss3.243159

  Derived-signature distillation: compresses resolution, chain-map, and derived-category evidence into low-rank signatures for training-time audits.

• oai_gflownet_loss1.715701

  Embedding-space GFlowNet loss: samples graph-of-thought continuations with reward tied to byte-likelihood, retrieval utility, and structural diversity.

• oai_fot_loss5.027384

  Embedding-space Forest-of-Thought loss: searches branching reasoning forests and uses BPB-delta reward to prioritize branches that improve compression.

• oai_mtp_loss11.487040

  Multi-token prediction loss: adds short-horizon targets that improve early byte likelihood while staying compatible with graph output flattening.

• graph_lm_loss0.368675

  Graph-LM primary loss: trains graph-structured records directly while FineWeb output flattening remains optional and BPB-focused.

Continuous Structure Tokenization

ConvexTok suggests a practical recipe: build a candidate graph, solve or relax a global path/cover objective, then round into a compact token inventory. The same idea can be lifted from byte strings to continuous 3D structures by replacing byte-boundary edges with geometric motifs: protein backbone fragments, residue-neighborhood contact patches, ligand pharmacophore neighborhoods, mesh cells, point-cloud patches, and local dynamical states. The tokenization objective should remain tropical-toric: dynamic programming supplies active paths, LP or optimal-transport relaxations supply lower bounds and regret metrics, and toric charts organize the active motif complex into cones, divisors, and sheaf-compatible local neighborhoods.

ToricBLM Continuation

The ToricBLM direction builds on ToricGT by continuing training toward a universal-modality biomedical reasoning model: sequences, atom graphs, protein contact graphs, RNA/DNA structures, complexes, assay facts, trajectories, and scientific claims all become typed graphs with optional continuous coordinates. The mathematical foundation is the bounded-domain universal equivariant graph-to-graph approximation theorem: within fixed graph budgets, a TokenGT-style model can approximate continuous equivariant graph maps, while tropical heads add dynamic-programming and Boolean-circuit-like active support, and ConvexTok/TokenGT positional encodings expose graph grammar structure that ordinary token-only transformers must infer indirectly.

Reasoning, Memory, and Control

Embedding-space Forest-of-Thought and GFlowNet training make reasoning a controlled search process, not a single chain. GraphCG full-rank concept axes make the search directions inspectable. Persistent homology and Toric BGG certificates decide when a retrieved memory is an analogy rather than a superficial nearest neighbor. The proposed thought-alcove control layer then treats reasoning trajectories as dynamical systems inside tropical-toric cells: fan walls are decision boundaries, Lyapunov-style energies discourage unstable exits, ergodic averages diagnose repeated itinerary behavior, and optimal-control objectives choose interventions that keep the model inside productive alcoves while preserving BPB and task reward.

Pinched-Embedding Disambiguation

Jakubowski, Gašić, and Zibrowius argue that static word vectors are better modeled as a pinched manifold: a quotient of a meaning manifold in which several meaning sheets are identified at one word vector. Their diagnostic is operationally useful for ToricGT because our hidden states also identify many roles at one surface token, graph node, ConvexTok edge, or memory key. The proposed extension is to puncture the local neighborhood of an ambiguous token/edge/state, normalize the outgoing directions, compute degree-zero persistent homology with GUDHI, and use the resulting component structure as a sense, role, or analogy gate.

This gives a rigorous path from topological word-sense induction to ToricGT: ConvexTok supplies tokenization DAG neighborhoods; TokenGT supplies equivariant vertex/edge neighborhoods; tropical attention supplies active dynamic-programming sheets; toric charts turn those sheets into cones, walls, divisors, and one-dimensional-cone audits; GraphCG supplies interpretable coordinates; analogical memory requires simplex-map and PH agreement inside the same component; and Toric BGG certificates prevent standard-object leakage across incompatible local sheets. The expected benefit is fewer sense-collisions during retrieval and graph flattening, cleaner memory selection, and lower BPB when ambiguous token choices were causing early likelihood spikes.

Thought Fluid Dynamics

The Tropical Quivers of Archs program frames learned graph-to-graph functions as composable operators with tropical/polyhedral local charts. For ToricGT, that suggests a fluid view of reasoning: hidden trajectories have divergence, circulation, vorticity, boundary flux, and energy. Navier-Stokes-inspired regularizers should be used conservatively, as audits and small penalties, but they give a precise vocabulary for branch merging, turbulence in unstable reasoning zones, and dissipative correction when FoT branches drift. The long-run objective is optimal control over a learned, graph-valued, tropical-toric dynamical system.

Topology of Word Embeddings · ToricBLM blog · Tropical Quivers of Archs · GeoBPE reference · protein structure tokenization benchmark