{
 "artifacts": null,
 "category": "architecture",
 "description": "Build a sparse recurrent graph-neural layer on a path-by-path, path-by-cycle, or cycle-by-cycle latent lattice using a skew-zero-forcing seed set and its forcing order as a causal update schedule. Only the currently forced target node is activated at each step, so a small number of anchor states can propagate through the complete lattice while retaining local connectivity and periodic-boundary structure. The exact seed-count formulas predict the minimum number of anchors required by the graph family and expose parity effects that a generic boundary mask would miss.",
 "formulas_latex": [
  "$$N_G(v_i)\\setminus\\bigcup_{j=1}^{i-1}N_G(v_j)\\neq\\varnothing\\qquad\\text{for every }1\\leq i\\leq k.$$",
  "$$\\gamma_{\\mathrm{gr}}^t(G)=|V(G)|-Z_{-}(G).$$",
  "$$Z_{-}(P_a\\square P_b)=a-\\mathbf{1}_{\\{a\\ {\\rm odd},\\,b\\ {\\rm even}\\}},\\qquad 2\\leq a\\leq b.$$",
  "$$Z_{-}(P_p\\square C_c)=\\begin{cases}\\min\\{p,c\\},\u0026c\\text{ is odd},\\\\ \\min\\{2p,c\\},\u0026c\\text{ is even},\\end{cases}\\qquad Z_{-}(C_a\\square C_b)=\\begin{cases}2a-1,\u0026a=b\\text{ odd},\\\\2a,\u0026a=b\\text{ even},\\\\\\min\\{b,2a\\},\u0026a\u003cb,\\ a\\text{ odd},\\\\a,\u0026a\u003cb,\\ a\\text{ even},\\ b\\text{ odd},\\\\2a,\u0026a\u003cb,\\ a,b\\text{ even}.\\end{cases}$$"
 ],
 "id": 2774,
 "implementation": "1. Integration point: replace the dense spatial mixer in a small vision or grid-state model by a recurrent sparse message-passing layer over H\\times W, H\\times C, or C_1\\times C_2 latent nodes. Each node stores h_v, and each local lattice edge has a learned message map M_{u\\to v}(h_u). Choose an anchor set B with size equal to the relevant Z_- formula and construct a forcing order offline. 2. Pseudocode: initialize active=B and set h_v=x_v for v in B; while active does not contain every node, inspect every vertex u and compute the number of inactive neighbors r(u)=|N(u)\\setminus active|; select a u with r(u)=1, let v be its unique inactive neighbor, and update h_v=GRU(h_v, M_{u\\to v}(h_u)+\\sum_{w\\in N(v)\\cap active}M_{w\\to v}(h_w)); mark v active and record u\\to v. If no legal force exists, add an emergency anchor at the inactive vertex maximizing future closure and log this violation. 3. The paper supplies the target anchor count and parity-dependent graph-family budget. Compute the actual forcing order by greedy closure or beam search over boundary candidates, and verify closure and seed count exhaustively on small lattices. The neural weights, message maps, and GRU parameters are learned; the graph mask, anchor set, and schedule are fixed. 4. First experiment: train on 16x16 masked-MNIST or Moving-MNIST reconstruction with 10%, 25%, and 50% observed anchor nodes. Compare a standard 3x3 convolutional recurrent baseline and a synchronous GNN at equal parameter count. Measure reconstruction error versus message updates, peak activation memory, and robustness to missing anchors. Success means comparable or lower error with at least 2x fewer simultaneously stored activations, plus the predicted one-anchor parity difference between odd/even rectangular grids.",
 "math_summary": "For a graph G without isolated vertices, a legal open-neighborhood sequence is a sequence S=(v_1,...,v_k) of distinct vertices satisfying N_G(v_i)\\setminus\\bigcup_{j\u003ci}N_G(v_j)\\neq\\varnothing for every i; its maximum length is the Grundy total domination number \\gamma_{\\mathrm{gr}}^t(G). The paper uses \\gamma_{\\mathrm{gr}}^t(G)=|V(G)|-Z_{-}(G), where Z_{-}(G) is the minimum size of a skew-zero-forcing set. In the skew forcing process, a vertex with exactly one white neighbor forces that neighbor blue; the forcing vertex need not itself be blue. For Cartesian products, the associated sparse matrix pattern is the Kronecker difference A\\otimes I-I\\otimes B, with A and B factor-graph matrices and I identity matrices of matching sizes. The exact seed budgets are Z_{-}(P_a\\square P_b)=a-\\mathbf{1}_{\\{a\\text{ odd},b\\text{ even}\\}}, Z_{-}(P_p\\square C_c)=\\min\\{p,c\\} for odd c and \\min\\{2p,c\\} for even c, and the cycle-cycle cases listed in the formulas. The network treats every forcing relation u\\to v as a directed schedule edge and computes v from u plus already activated neighbors.",
 "math_tags": [
  "graph-theory",
  "combinatorics",
  "linear-algebra",
  "spectral-theory"
 ],
 "ml_areas": [
  "graph-nn",
  "ssm",
  "memory",
  "training"
 ],
 "paper": {
  "arxiv_id": "2608.27804",
  "arxiv_url": "https://arxiv.org/abs/2608.27804",
  "summary_what_math_gives_to_ml": "The paper supplies exact combinatorial observability budgets for rectangular, cylindrical, and toroidal grid graphs through skew zero forcing, together with the identity connecting these budgets to maximum skew nullity and Kronecker-difference matrix constructions. The transferable asset is not the domination parameter itself, but a way to choose a small set of anchor nodes and an explicit sequential propagation order that reaches every node under a sparse local graph pattern. This can define causal lattice mixers or sparse graph-recurrent networks whose update masks are guaranteed to cover the entire latent grid, including parity-sensitive cases on periodic boundaries. The most credible first use is a structured sparse message-passing architecture for masked-grid reconstruction or long-horizon spatial state propagation, compared against ordinary fixed-depth grid convolutions.",
  "title": "Grundy Total Domination and Skew Zero Forcing in Cartesian Products of Paths and Cycles",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 6,
  "usefulness": 5
 },
 "solves": [
  "memory",
  "scalability",
  "sample-efficiency"
 ],
 "title": "Zero-forcing causal lattice mixer",
 "url": "https://synthcore.org/idea/2774/zero-forcing-causal-lattice-mixer",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}