{
 "artifacts": [
  {
   "name": "bench_report.json",
   "url": "https://synthcore.org/code/637/bench_report.json"
  },
  {
   "name": "bench_run.py",
   "url": "https://synthcore.org/code/637/bench_run.py"
  },
  {
   "name": "experiment.py",
   "url": "https://synthcore.org/code/637/experiment.py"
  },
  {
   "name": "fractal_block.py",
   "url": "https://synthcore.org/code/637/fractal_block.py"
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   "name": "report.md",
   "url": "https://synthcore.org/code/637/report.md"
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  {
   "name": "report_bench_2026-08-31T203843.md",
   "url": "https://synthcore.org/code/637/report_bench_2026-08-31T203843.md"
  },
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   "name": "results.json",
   "url": "https://synthcore.org/code/637/results.json"
  },
  {
   "name": "smoke_test.py",
   "url": "https://synthcore.org/code/637/smoke_test.py"
  }
 ],
 "category": "architecture",
 "description": "Replace ordinary depth-wise feature propagation by a ternary hierarchical block that recursively aggregates three child representations while maintaining separate neutral and defect channels. The block is initialized from the Sierpinski six-vertex recursion, then optionally learns a bounded correction. The neutral channel preserves the paper's cubic mixing law, while the defect channel provides a controlled route for long-range and nonlocal interactions.",
 "download_zip": "https://synthcore.org/download/637",
 "formulas_latex": [
  "$$Z_{n+1}=\\lambda^3 Z_n^3+2x^3 Q_n^3,\\qquad y=z=\\lambda.$$",
  "$$r_n=\\frac{Q_n}{Z_n},\\qquad r_{n+1}=F(r_n;x,\\lambda),\\qquad \\left|\\frac{\\partial F}{\\partial r}(r_*;x,\\lambda)\\right|=1.$$",
  "$$a_{n+1}=\\lambda^3 a_n^{\\odot 3}+2x^3 b_n^{\\odot 3},\\qquad b_{n+1}=G_\\theta(a_n,b_n),$$",
  "$$\\rho\\left(J_{(a,b)}\\right)\u003c1\\ \\text{implies local contraction, while}\\ \\rho\\left(J_{(a,b)}\\right)\u003e1\\ \\text{implies perturbation amplification}.$$"
 ],
 "id": 1723,
 "implementation": "Use this block as a replacement for a 3-to-1 pooling or message-passing stage in a hierarchical image, graph, or token model. Partition the input into three child groups recursively. Each child emits two vectors: a neutral content vector a and a defect or disagreement vector b. First align the three child vectors with shared linear projections. Then compute the neutral parent using the paper-inspired rule a_parent = lambda^3 times the elementwise product of the three a vectors plus 2 times x^3 times the elementwise product of the three b vectors. Compute b_parent with a small gated MLP G_theta that receives all six child vectors. Apply RMS normalization after the cubic operation and use softplus parameterizations for x and lambda. At each level, estimate the largest local Jacobian eigenvalue with two power-iteration steps or Hutchinson-vector products. Add a penalty max(0, rho_hat - 1 + epsilon)^2 when contractive propagation is desired; alternatively target rho_hat near one for maximal signal persistence. The paper supplies the ternary recursion, the two boundary-state interpretation, and the fixed-point stability criterion. The learned defect map, Jacobian estimate, and task loss are empirical additions. First test a six-level model on CIFAR-10 using a small hierarchical CNN, comparing an ordinary pooling baseline and the fractal block at matched parameter count. Sweep q = x/lambda and measure loss, gradient amplification, and maximum stable depth. The predicted signature is a sharp change when the dominant Jacobian eigenvalue crosses one: below the boundary, perturbations decay approximately as rho^n; above it, perturbations grow or remain persistent. Near the deconfinement-like setting, active routing paths should scale approximately as L^1.3 with hierarchy depth L, rather than as the full number of sites.",
 "math_summary": "For six-vertex weights x and y = z = lambda, let Z_n be the charge-free partition function and Q_n the constrained partition function with opposite charges at two corner vertices. The exact Sierpinski recursion contains Z_(n+1) = lambda^3 Z_n^3 + 2 x^3 Q_n^3, together with a corresponding recursion Q_(n+1) = H(Z_n,Q_n;x,lambda) from the paper. Define the normalized defect ratio r_n = Q_n / Z_n. Its renormalization evolution is r_(n+1) = F(r_n;x,lambda), where F is obtained by dividing H by the Z recursion. A fixed point r_* changes stability when the absolute value of F'(r_*) equals one. For vector neural features, the analogous condition is that the spectral radius of the local Jacobian crosses one. The neural block uses neutral features a_n and defect features b_n, with elementwise cubic neutral mixing and a learned bounded defect map.",
 "math_tags": [
  "renormalization",
  "statistical-mechanics",
  "dynamical-systems",
  "bifurcations",
  "combinatorics"
 ],
 "ml_areas": [
  "graph-nn",
  "transformer",
  "training-dynamics",
  "regularization"
 ],
 "paper": {
  "arxiv_id": "2608.02741",
  "arxiv_url": "https://arxiv.org/abs/2608.02741",
  "summary_what_math_gives_to_ml": "The paper provides a constructive real-space recursion for a six-vertex model on a Sierpinski gasket, retaining charge-free and charged boundary partition functions as coarse-grained state variables. Its transferable asset is a two-channel hierarchical renormalization map whose fixed points and Jacobian determine confinement, deconfinement, and sharp regime changes. A neural implementation can use repeated ternary aggregation blocks with neutral and defect channels, while monitoring the recursion Jacobian to obtain a measurable stability boundary. The noninteger path dimension reported at the deconfined point also suggests a concrete sparsity and routing diagnostic for hierarchical networks.",
  "title": "Fractal deconfinement and confinement in Sierpinski ice",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 7,
  "usefulness": 7
 },
 "solves": [
  "accuracy",
  "stability",
  "scalability"
 ],
 "title": "Two-Channel Fractal Renormalization Network",
 "url": "https://synthcore.org/idea/1723/two-channel-fractal-renormalization-network",
 "verification": {
  "benchmark": {
   "track": "vision",
   "model": "cnn_small",
   "n_seeds": 8,
   "metric_direction": "lower is better",
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    "idea_wins": 7,
    "n_pairs": 8,
    "p_value": 0.0152,
    "mde": 0.0517253348,
    "mde_rel_pct": 6.1853915,
    "verdict": "idea better (significant)",
    "system_worked": true
   },
   "baseline": {
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     "lr": 0.003
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     {
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     {
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       "lr": 0.01
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     "n": 8
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    "math_check": {
     "analytic_jacobian": [
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     "cubic_gain_ratio": 0.512,
     "finite_difference": [
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     "max_abs_error": 5.137e-11
    },
    "trained_behavior": {
     "confirmed": false,
     "observed_trained_pool_perturbation_gain_mean": 2.37213549,
     "observed_trained_pool_perturbation_gain_std": 0.50636656,
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   "custom_track": null
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  "mechanism_confirmed": false,
  "peer_reviewed": false,
  "practical_verdict": "helps",
  "stage1_mechanism_check": {
   "worked": true,
   "confidence": 8,
   "verdict": "Built the exact Sierpinski ratio recursion check and a runnable PyTorch two-channel ternary block with cubic neutral mixing and bounded defect routing. The mechanism manifested: divergence occurred below the predicted q=x/z=1 boundary, the q=1.2 marginal scaling gave mean n(R*-R)=2.097 versus predicted 2.4, and q=0.8 showed a double-log growth increment of 0.977 versus predicted 1.0. The toy product task favored the fractal structure only when the target matched the product mechanism; against a linear mean-pooling readout it was worse, so no general accuracy win is claimed.",
   "metrics": {
    "baseline": "Toy regression mean-pooling plus linear readout MSE = 9.4605",
    "idea": "Exact ternary product/fractal recursion MSE = 62.6550 on the generic product toy evaluation; mechanism checks: boundary predicted q=1.0 and observed finite-depth transition between 0.95 and 0.975, q=1.2 marginal statistic 2.097 vs 2.4 predicted, q=0.8 double-log increment 0.977 vs 1.0 predicted, analytic Jacobian finite-difference max error 2.96e-12"
   },
   "how_to_run": "python3 experiment.py \u0026\u0026 /home/maxwelhelp/main/bin/python3 smoke_test.py",
   "files": [
    "experiment.py",
    "fractal_block.py",
    "smoke_test.py",
    "results.json"
   ],
   "limitations": "Did not run the proposed CIFAR-10 six-level CNN, matched-parameter training comparison, Jacobian regularization during learning, active-path scaling, or a full paper-scale partition-function implementation. The toy neural block’s Jacobian sweep used a scalar simplified defect map and did not include RMS normalization in the analytic stability check."
  },
  "status": "beats_baseline",
  "status_label": "Beats tuned baseline",
  "updated_at": "2026-08-31T20:38:43",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
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    "tested": true
   },
   "practical_benchmark": {
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    "verdict": "helps"
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   "toy_mechanism_gate": {
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  }
 }
}