This gallery is the conceptual map for the ToricGT training stack. It connects tropical dynamic programming, toric embeddings, tableau combinatorics, tensor operations, multiparameter persistence, intersection-style certificates, and control of reasoning trajectories. Every diagram is a lightweight inline SVG with accompanying text so the page stays readable on older hardware.
Multiparameter persistence PDF · Ergodic theory reference · Tropical Quivers of Archs
Tropical attention is piecewise-linear. The active candidate set is a face of a lifted Newton polytope, and the corresponding normal cone is the toric chart used by the audits.
Training use: fan-margin, active-face stability, toric embedding metrics, and BPB-safe routing diagnostics.
Tableaux encode structured bases for polynomial and exterior constructions. They are useful when graph-token features behave like tensor products but need symmetry constraints.
Training use: candidate future losses for symmetrizer/shuffle consistency and combinatorial certificate compression.
Graph-token reasoning creates many pairwise and higher-order feature products. Algebraic symmetry gives a principled way to constrain which products should be identified or separated.
Training use: low-rank probes for wedge/symmetric consistency and future Schur-functor-inspired regularizers.
The full reasoning trajectory can be filtered by reasoning level and embedding radius. Algebraically, this is a two-parameter persistence module over F₂[x_level,y_radius].
Training use: GUDHI landscapes/images/entropy, simplex-map gates for analogical memory, and Macaulay2 resolution checks.
Reasoning paths move through fan cells. A stable path should cross walls for reasons tied to the task, not because the latent state is noisy.
Training use: fan-margin scheduling, branch pruning, FoT reward shaping, and future control-theoretic audits.
Toric divisors, one-dimensional cones, and intersection numbers provide compact summaries of how a local piecewise-linear decision bends across walls.
Training use: toric vector-bundle one-dimensional-cone CE, sheaf compatibility, and divisor/fan audit metrics.