{
 "artifacts": null,
 "category": "architecture",
 "description": "Replace part of an MLP with a structured coordinate mutation modeled on the paper's cluster-chart map. The layer computes a smooth approximation of the valuation-level minimum between a learned monomial and a constant, while retaining the exact multiplicative/rational form in positive coordinates. This creates a structured, piecewise-smooth feature transformation with an explicit chart-switching mechanism.",
 "formulas_latex": [
  "$$\\Lambda=\\mathbb{C}((T^{\\mathbb{R}})),\\qquad x=\\sum_{i=0}^{\\infty}a_iT^{\\lambda_i},\\qquad \\operatorname{val}(x)=\\min\\{\\lambda_i\\mid a_i\\neq 0\\}.$$",
  "$$q=\\frac{y_1y_3+T}{y},\\qquad \\operatorname{val}(q)=\\min\\{\\operatorname{val}(y_1)+\\operatorname{val}(y_3),1\\}-\\operatorname{val}(y).$$",
  "$$m_\\beta(u,u_1,u_3)=-\\frac{1}{\\beta}\\log\\!\\left(e^{-\\beta(u_1+u_3)}+e^{-\\beta\\tau}\\right)-u\\xrightarrow[\\beta\\to\\infty]{}\\min\\{u_1+u_3,\\tau\\}-u.$$",
  "$$Q=\\frac{e^{u_1}e^{u_3}+e^{\\tau}}{e^u},\\qquad \\log Q=\\operatorname{LSE}(u_1+u_3,\\tau)-u.$$"
 ],
 "id": 248,
 "implementation": "Integrate the block immediately before or after a standard transformer MLP, operating on groups of four channels. First apply a learned affine projection from the hidden state \\(h\\in\\mathbb{R}^d\\) to positive coordinates \\(u,u_1,u_3\\) and an unconstrained residual vector \\(r\\). Use \\(u=\\operatorname{softplus}(a_u)+\\epsilon\\), and similarly for \\(u_1,u_3\\), to avoid division by zero. Compute the mutation in log coordinates with \\(m=-(1/\\beta)\\log(\\exp[-\\beta(u_1+u_3)]+\\exp[-\\beta\\tau])-u\\), where \\(\\tau\\) is either a learned scalar per block or initialized to zero. Concatenate \\(m\\), \\(u\\), \\(u_1\\), \\(u_3\\), and \\(r\\), then project back to dimension \\(d\\) and add a residual connection. Pseudocode: `z=proj(h); u,u1,u3=softplus(z[:3])+eps; m=-logaddexp(-beta*(u1+u3),-beta*tau)/beta-u; out=h+W_out[concat(m,u,u1,u3,z[3:])]`. Compute the exact smooth expression during training; estimate the tropical branch frequency by recording whether \\(u_1+u_3\u003c\\tau\\). Start with a 2-layer width-256 MLP on CIFAR-10 or a small character language model, comparing equal-parameter GELU and SwiGLU baselines. Sweep \\(\\beta\\in\\{1,3,10\\}\\), and include an ablation removing the additive constant. Success means lower validation loss at equal parameters, fewer exploding-gradient events, or faster loss descent at equal FLOPs.",
 "math_summary": "The paper works over the Novikov field \\(\\Lambda=\\mathbb{C}((T^{\\mathbb{R}}))\\), whose valuation is \\(\\operatorname{val}(x)=\\min\\{\\lambda_i:a_i\\neq 0\\}\\) for \\(x=\\sum_i a_iT^{\\lambda_i}\\). The relevant analytic cluster-chart coordinate is \\(q=(y_1y_3+T)/y\\), where \\(y,y_1,y_3\\in\\Lambda^*\\) are invertible coordinates and \\(T\\) has valuation \\(\\operatorname{val}(T)=1\\). In the absence of cancellation, valuation converts this rational expression into the tropical mutation \\(\\operatorname{val}(q)=\\min\\{\\operatorname{val}(y_1)+\\operatorname{val}(y_3),1\\}-\\operatorname{val}(y)\\). For real neural features, let \\(u,u_1,u_3\\in\\mathbb{R}\\) represent log-coordinates, let \\(\\tau\\) be a threshold, and define \\(m_\\beta(u,u_1,u_3)=-(1/\\beta)\\log(\\exp[-\\beta(u_1+u_3)]+\\exp[-\\beta\\tau])-u\\). As \\(\\beta\\to\\infty\\), \\(m_\\beta\\to\\min\\{u_1+u_3,\\tau\\}-u\\). The positive lift is \\(Q=(\\exp(u_1)\\exp(u_3)+\\exp(\\tau))/\\exp(u)\\). The transferable property is that one rational formula supplies a smooth positive-coordinate map and a controlled piecewise-linear tropical limit.",
 "math_tags": [
  "algebra",
  "geometry",
  "tropical-geometry",
  "non-archimedean"
 ],
 "ml_areas": [
  "mlp",
  "transformer",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2607.03843",
  "arxiv_url": "https://arxiv.org/abs/2607.03843",
  "summary_what_math_gives_to_ml": "The paper gives an explicit atlas of multiplicative cluster coordinates whose charts are glued by rational transformations involving additions of monomials, and whose non-archimedean valuations become piecewise-linear minimum operations. This suggests a structured neural layer that alternates between multiplicative coordinate changes and tropical, data-dependent branch selection rather than using unconstrained affine maps. The transferable asset is not the Grassmannian itself, but the combination of an invertible rational mutation, valuation-induced piecewise-linear behavior, and a chart-consistency constraint. A first test should insert one such stabilized cluster-mutation block into an MLP or transformer feed-forward sublayer and compare parameter efficiency, optimization stability, and accuracy against a similarly sized MLP.",
  "title": "Family Floer SYZ mirror algorithm for the Grassmannian $Gr(2,4)$",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 4,
  "novelty": 6,
  "usefulness": 5
 },
 "solves": [
  "accuracy",
  "stability",
  "speedup"
 ],
 "title": "Tropical Cluster-Mutation Layer",
 "url": "https://synthcore.org/idea/248/tropical-cluster-mutation-layer",
 "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
   }
  }
 }
}