{
 "artifacts": [
  {
   "name": "bench_report.json",
   "url": "https://synthcore.org/code/1027/bench_report.json"
  },
  {
   "name": "flow_results.json",
   "url": "https://synthcore.org/code/1027/flow_results.json"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1027/report.md"
  },
  {
   "name": "report_bench_2026-09-01T185742.md",
   "url": "https://synthcore.org/code/1027/report_bench_2026-09-01T185742.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/1027/results.json"
  },
  {
   "name": "rg_pyramid_flow.py",
   "url": "https://synthcore.org/code/1027/rg_pyramid_flow.py"
  },
  {
   "name": "rg_pyramid_toy.py",
   "url": "https://synthcore.org/code/1027/rg_pyramid_toy.py"
  },
  {
   "name": "stage2_bench.py",
   "url": "https://synthcore.org/code/1027/stage2_bench.py"
  }
 ],
 "category": "architecture",
 "description": "Replace a full-resolution global flow-matching or diffusion model with a hierarchy of local velocity fields operating on progressively finer grids. Coarse levels generate long-wavelength structure and pass it to fine levels through upsampling and residual conditioning, while every velocity network uses only a locality radius prescribed by the RG bound. This should preserve long-range correlations without quadratic global attention or a full-resolution global receptive field.",
 "download_zip": "https://synthcore.org/download/1027",
 "formulas_latex": [
  "$$R(\\Lambda,L,\\varepsilon)=O\\!\\left(\\Lambda^{-1}\\left[\\ln L+\\ln(1/\\varepsilon)\\right]\\right),$$",
  "$$\\ell_{\\mathrm{loc}}(t)\\propto \\Lambda(t)^{-1},$$",
  "$$\\frac{d z_t}{dt}=v_t(z_t),\\qquad v_t(z_t)\\approx v_{\\theta,s}\\!\\left(\\operatorname{patch}_{r_s}(z_t^{(s)}),\\,\\operatorname{up}(z_t^{(s+1)}),\\,t\\right),$$",
  "$$\\mathcal{L}_{s}(\\theta_s)=\\mathbb{E}_{x_0,x_1,t}\\left\\|v_{\\theta,s}(z_{s,t},t)-\\frac{x_{1}^{(s)}-x_{0}^{(s)}}{1-t}\\right\\|_2^2,\\qquad z_{s,t}=(1-t)x_{0}^{(s)}+t x_{1}^{(s)}.$$"
 ],
 "id": 2708,
 "implementation": "(1) Integration point: implement this as a replacement for a full-resolution flow-matching or diffusion U-Net. Construct a dyadic pyramid \\(x^{(S)},x^{(S-1)},\\ldots,x^{(0)}\\), where \\(x^{(S)}\\) is the coarsest grid and \\(x^{(0)}\\) is the target resolution. At each level use a convolutional or windowed-attention velocity network \\(v_{\\theta,s}\\), conditioned on the level index, continuous time \\(t\\), and an upsampled state or feature map from level \\(s+1\\). Avoid global attention. (2) Pseudocode: sample Gaussian noise \\(x_0^{(S)}\\); integrate the coarse ODE from \\(t=0\\) to 1 using \\(v_{\\theta,S}\\); upsample the result to initialize level \\(S-1\\), add independent fine-scale noise, and integrate \\(v_{\\theta,S-1}\\); repeat down to level 0. During training, construct data and noise pyramids, sample \\(t\\sim U[0,1]\\), form \\(z_{s,t}=(1-t)x_0^{(s)}+tx_1^{(s)}\\), and regress the velocity target shown in \\(\\mathcal{L}_s\\). Restrict each output location to a patch of radius \\(r_s=\\lceil c[\\ln L+\\ln(1/\\varepsilon)]/(\\Lambda_s a_s)\\rceil\\). (3) Compute from the paper's mathematics: the level-dependent receptive-field radius and the coarse-to-fine schedule. Estimate the unknown constant \\(c\\) by sweeping candidate radii and selecting the smallest radius whose validation correlation error is below \\(\\varepsilon\\). The cutoff approximation \\(\\Lambda_s=\\pi/a_s\\), pyramid filters, and upsampling rule are engineering choices. (4) First experiment: train on CIFAR-10 at 32x32 and FFHQ at 64x64, comparing a standard full-resolution flow-matching U-Net, a fixed-radius local U-Net, and this RG pyramid at matched parameter counts and function evaluations. Measure FID, long-range pixel and feature correlations, wall-clock sampling time, peak activation memory, and scaling from 64x64 to 256x256. The expected signal is lower FID and better correlations than fixed-radius local flow matching, with near-linear cost in pixel count and substantially lower memory than global attention. A particularly strong result would be preserved correlations at separations larger than the finest-level receptive field and improved FID at equal sampling FLOPs.",
 "math_summary": "The paper's main locality result states that, along the RGFM probability path without lattice rescaling, the locality length needed to approximate the exact probability-flow velocity scales linearly with the running RG length scale: \\(\\ell_{\\mathrm{loc}}(t)\\propto \\Lambda(t)^{-1}\\). More specifically, the required radius for error tolerance \\(\\varepsilon\\) on a system of linear size \\(L\\) is \\(R(\\Lambda,L,\\varepsilon)=O\\!\\left(\\Lambda^{-1}[\\ln L+\\ln(1/\\varepsilon)]\\right)\\), where \\(\\Lambda\\) is the current RG wavenumber cutoff, \\(L\\) is the spatial system size, and \\(\\varepsilon\\) is the tolerated local-flow approximation error. The paper proposes representing remaining modes on successively coarser lattices so that the physical locality radius remains \\(O(L^0)\\), meaning constant in lattice units after rescaling. Use a continuous flow \\(dz_t/dt=v_t(z_t)\\), where \\(z_t\\) is the multiscale state and \\(v_t\\) is approximated by a local neural field. At level \\(s\\), let grid spacing be \\(a_s=2^s\\), cutoff be \\(\\Lambda_s\\approx \\pi/a_s\\), and choose a physical patch radius \\(R_s=\\left\\lceil c[\\ln L+\\ln(1/\\varepsilon)]/\\Lambda_s\\right\\rceil\\). In grid-cell coordinates the radius is \\(r_s=\\lceil R_s/a_s\\rceil\\), which is approximately constant across levels when \\(\\Lambda_s\\propto a_s^{-1}\\).",
 "math_tags": [
  "dynamical-systems",
  "pde",
  "probability",
  "functional-analysis"
 ],
 "ml_areas": [
  "diffusion",
  "diffusion-sampling",
  "cnn",
  "memory"
 ],
 "paper": {
  "arxiv_id": "2608.23696",
  "arxiv_url": "https://arxiv.org/abs/2608.23696",
  "summary_what_math_gives_to_ml": "The paper provides a principled way to obtain globally coherent generation from local computation: evolve samples from long wavelengths to short wavelengths using an exact or approximate renormalization-group probability path. Its transferable asset is the quasi-locality bound, which says that the velocity field at RG scale \\(\\Lambda\\) can be approximated using a spatial radius growing only as \\(\\Lambda^{-1}[\\ln L+\\ln(1/\\varepsilon)]\\), rather than requiring global context. A neural implementation should therefore use a coarse-to-fine pyramid of latent grids, train a separate scale-conditioned local velocity field at each grid resolution, and refine only after the coarse field has established global structure. This is more specific than simply using dilated convolutions: the receptive field should be tied to the current physical RG scale and error tolerance.",
  "title": "Renormalization Group Flow Matching for Scalable Local Generative Modeling",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 6,
  "usefulness": 8
 },
 "solves": [
  "speedup",
  "memory",
  "accuracy"
 ],
 "title": "RG Pyramid Flow Matching",
 "url": "https://synthcore.org/idea/2708/rg-pyramid-flow-matching",
 "verification": {
  "benchmark": {
   "track": "multitoken_diffusion",
   "model": "custom_local_vs_rg_pyramid",
   "n_seeds": 8,
   "metric_direction": "lower is better",
   "comparison": {
    "baseline_mean": 0.5488443523645401,
    "idea_mean": 0.5474011525511742,
    "delta_mean": -0.0014431998133659363,
    "per_seed_diffs": [
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     -0.006001949310302734,
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    "idea_wins": 4,
    "n_pairs": 8,
    "p_value": 0.40545,
    "mde": 0.0034127227373531473,
    "mde_rel_pct": 0.6218015586113622,
    "verdict": "no measurable effect",
    "system_worked": false
   },
   "baseline": {
    "best_cfg": {
     "epochs": 12,
     "hidden": 32,
     "lr": 0.006
    },
    "sweep": [
     {
      "cfg": {
       "epochs": 12,
       "hidden": 32,
       "lr": 0.001
      },
      "mean": 0.5760585516691208
     },
     {
      "cfg": {
       "epochs": 12,
       "hidden": 32,
       "lr": 0.003
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      "mean": 0.5570157170295715
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     {
      "cfg": {
       "epochs": 12,
       "hidden": 32,
       "lr": 0.006
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      "mean": 0.552870586514473
     }
    ],
    "full": {
     "mean": 0.5488443523645401,
     "std": 0.02526189879,
     "per_seed": [],
     "n": 8
    }
   },
   "idea": {
    "mean": 0.5474011525511742,
    "std": 0.025686412604992322,
    "per_seed": [
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     0.5185491442680359,
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     0.5547541379928589,
     0.5519959330558777,
     0.5247784852981567,
     0.5898048281669617,
     0.5140559077262878
    ],
    "n": 8
   },
   "mechanism_signature": {
    "confirmed": false,
    "predicted_vs_observed": {
     "observed_baseline_mean_distant_sensitivity": 0,
     "observed_idea_mean_distant_sensitivity": 0,
     "predicted": "idea distant sensitivity \u003e baseline"
    },
    "prediction": "coarse pooled conditioning transmits distant-token influence beyond a radius-1 local field"
   },
   "custom_track": {
    "domain": "diffusion-sampling",
    "file": "/home/maxwelhelp/all/math2nn/bench/custom_tracks/multitoken_diffusion.py",
    "name": "multitoken_diffusion"
   }
  },
  "mechanism_confirmed": false,
  "peer_reviewed": false,
  "practical_verdict": "no_effect",
  "stage1_mechanism_check": {
   "worked": true,
   "confidence": 8,
   "verdict": "Built a self-contained RG locality toy verification and a PyTorch coarse-to-fine pyramid flow-matching MVP. The three mechanism predictions were confirmed: required radius fit R versus 1/Lambda with R2=0.9996 and coefficient of variation of Lambda*R=3.7%; radius versus log(1/epsilon) had R2=0.9964; and Lambda=pi/a produced exactly constant cell radius 4 across six levels. The fixed-radius-8 operator error rose from 0.00024 at Lambda=1 to 0.742 at Lambda=0.1, while the selected RG radius kept error near 0.02. The neural toy showed lower pyramid mean velocity MSE (0.609 versus 0.875) and slightly better long-range correlation (0.075 versus -0.044), but it was not parameter-matched and was too small/short-trained to claim a generative-quality win.",
   "metrics": {
    "baseline": "Operator fixed radius 8: error 0.00024 to 0.742 as Lambda decreases; neural velocity MSE 0.875, d=16 correlation -0.044, 3,281 parameters.",
    "idea": "RG radius sweep: R*Lambda mean 4.248, CV 0.037; log-tolerance fit R2 0.9964; rescaled cell radius 4 at all levels; neural velocity MSE 0.609, d=16 correlation 0.075, 7,083 parameters."
   },
   "how_to_run": "python3 rg_pyramid_toy.py \u0026\u0026 /home/maxwelhelp/main/bin/python3 rg_pyramid_flow.py",
   "files": [
    "rg_pyramid_toy.py",
    "rg_pyramid_flow.py",
    "results.json",
    "flow_results.json",
    "toy_output.txt",
    "flow_output.txt"
   ],
   "limitations": "No CIFAR-10 or FFHQ experiment, FID, ODE sampling benchmark, memory benchmark, scaling study, or parameter/FLOP-matched neural comparison was performed. The toy kernel assumes exponential locality and therefore verifies the claimed scaling mechanism rather than the full paper theorem."
  },
  "status": "failed_benchmark",
  "status_label": "Failed on benchmark",
  "updated_at": "2026-09-01T18:57:42",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}