{
 "artifacts": null,
 "category": "architecture",
 "description": "Replace dense token-to-token mixing on a 2D grid by a learned or fixed periodic perfect-matching route. Each source coordinate is connected to exactly one destination coordinate, and the same matching pattern is translated across periods, producing sparse, parameter-shared aggregation with predictable equivariance.",
 "formulas_latex": [
  "$$W(i,j)=W(i+r,j-r)$$",
  "$$(s_{1},s_{2},i)\\xmapsto{\\,v^{q}\\,}(s_{1}+qr,s_{2}-qr,i)$$",
  "$$y_{s_{1},s_{2}}=\\phi\\!\\left(Bx_{s_{1},s_{2}}+\\sum_{(i,j):\\,W(i,j)\\neq 0}W(i,j)\\,U_{i,j}x_{\\pi_{W}(s_{1},s_{2};i,j)}\\right)$$"
 ],
 "id": 304,
 "implementation": "Integrate this at the token-mixing operation of a small ViT, grid transformer, or 2D SSM. Let the input be X with shape [batch, height, width, d]. Choose a period r and a matching table containing one destination offset for each source residue class modulo r; enforce the matching constraint by sampling a permutation or by using Sinkhorn during initialization and then projecting to the nearest permutation. For every token (h,w), compute its routed partner using the periodic rule W(i,j)=W(i+r,j-r), gather that token, apply a shared linear map U for its orbit, and add the result to a local residual projection Bx. Pseudocode: `res = B(X); for each residue edge e=(i,j): src = periodic_index(h,w,e); res += alpha[e] * U[e](gather(X,src)); Y = phi(res)`. Use one or a few matching layers before a dense MLP; optionally stack K independently shifted matchings so the receptive field grows while each layer remains O(HW d) rather than O((HW)^2 d). Compute the mathematical route exactly from the stored permutation; estimate no spectral quantities. First experiment: CIFAR-10 or ImageNet-100 with a 4-layer ViT at equal parameter count, comparing dense attention, local-window attention, and one to four matching layers. Measure validation accuracy, wall-clock throughput, and memory. Success is at least 2x lower attention-memory cost with no more than 1 percentage-point accuracy loss, or better accuracy than local windows at equal FLOPs.",
 "math_summary": "The paper uses a perfect matching W: \\mathbb{Z}^{2}\\to\\mathbb{Z} as a sparse weight pattern and assumes periodicity W(i,j)=W(i+r,j-r). Here W(i,j) is interpreted as the edge weight between coordinate i on one axis and coordinate j on the other, with the perfect-matching constraint meaning that every active source and destination participates in exactly one edge. The periodic identity induces a shift operator v on matching elements: v:(s_{1},s_{2},i)\\mapsto(s_{1}+r,s_{2}-r,i), and v^{q} shifts by q periods. We transfer these two properties into a sparse neural mixing matrix A_W: each token has O(1) routed partners, and translating the input by one period translates the route rather than changing its parameters. If W is binary, A_W is a permutation/incidence matrix; if W has learned scalar weights, use one scalar per matching orbit and share it across all q.",
 "math_tags": [
  "combinatorics",
  "graph-theory",
  "linear-algebra"
 ],
 "ml_areas": [
  "attention",
  "graph-nn",
  "moe-routing",
  "inference-speedup"
 ],
 "paper": {
  "arxiv_id": "2607.00238",
  "arxiv_url": "https://arxiv.org/abs/2607.00238",
  "summary_what_math_gives_to_ml": "The paper turns a Riemann function with perfect-matching weight into a finite diagram of vector spaces whose differential, kernel, cokernel, and Euler characteristic exactly encode the function and its dual. The transferable asset is not the Riemann-function application itself, but a compact incidence/sheaf calculus: local linear maps can be assembled into global cohomology dimensions, while periodicity produces weight-sharing actions on infinite index sets. This suggests neural modules with explicit sparse matching routes and measurable kernel/cokernel channels, rather than ordinary unconstrained message passing. The strongest practical tests are a periodic sparse routing layer for grid or sequence tokens and an Euler-characteristic consistency regularizer.",
  "title": "Duality and a Canonical Sheaf in Periodic Riemann Functions",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 4,
  "novelty": 6,
  "usefulness": 7
 },
 "solves": [
  "scalability",
  "speedup",
  "accuracy"
 ],
 "title": "Periodic Perfect-Matching Router",
 "url": "https://synthcore.org/idea/304/periodic-perfect-matching-router",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}