L1 ⚛ L1-060 ⊙ Testnet

MR Fingerprinting (MRF)

Medical Imaging · Pseudo-random acquisition with dictionary matching for T1/T2/rho

⬇ Download bundle 🚀 Submit to mainnet — earn 100 PWM + 5% royalty

Markdown — human-readable source of truth

⚙ auto-generated

⬇ L1-060.md

# ⚛ L1 Principle — MR Fingerprinting (MRF) **ID:** `L1-060` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Medical Imaging — *Pseudo-random acquisition with dictionary matching for T1/T2/rho* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 4D parameter maps > **📡 Carrier:** radio_wave · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41553386 --- ## 🧠 1. Introduction **MR Fingerprinting (MRF)** is a **nonlinear inverse problem** whose unknown lives in **4D parameter maps** space, within the **Pseudo-random acquisition with dictionary matching for T1/T2/rho** sub-domain of **Medical Imaging**. Measurements consist of radio-frequency electromagnetic waves via a **mri fingerprinting** sensing mechanism. The forward operator applies, in order: L · rf excitation pseudorandom operator; L · kspace undersample operator; L · dict match operator; detector accumulates flux over the exposure window. Observations are corrupted by additive Gaussian noise. Existence of the recovered 4D parameter maps is guaranteed within the declared Omega bounds. Uniqueness is local rather than global (non-convex landscape); convergence depends on initialisation and priors. Stability is moderately conditioned (kappa_eff ~= 25); B0_inhomogeneity dominates the stability cliff; undersampling_artifacts and the remaining mismatch parameters contribute higher-order bias terms. Additive gaussian thermal/electronic noise sets the irreducible data-fidelity floor, while TV / wavelet-sparsity / deep priors stabilise recovery at the ill-conditioned end of Omega. ## ⚙ 2. Forward Model Physical chain: **x** → L · rf excitation pseudorandom → L · kspace undersample → L · dict match → Temporal integration → **y** (detector). ``` y = ∫_t dt `L.dict_match` `L.kspace_undersample` `L.rf_excitation_pseudorandom` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.rf_excitation_pseudorandom` | L · rf excitation pseudorandom operator | | `L.kspace_undersample` | L · kspace undersample operator | | `L.dict_match` | L · dict match operator | | `int.temporal` | Detector accumulates flux over the exposure window | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Medical Imaging | | Sub domain | Pseudo-random acquisition with dictionary matching for T1/T2/rho | | Carrier | radio_wave | | Problem class | nonlinear_inverse | | Solution space | 4D_parameter_maps | | Noise model | gaussian | | Integration axis | temporal | | Difficulty delta | 10 | | L dag | 4.5 | ## 📡 4. Measurement Model Existence of the recovered 4D parameter maps is guaranteed within the declared Omega bounds. Uniqueness is local rather than global (non-convex landscape); convergence depends on initialisation and priors. Stability is moderately conditioned (kappa_eff ~= 25); B0_inhomogeneity dominates the stability cliff; undersampling_artifacts and the remaining mismatch parameters contribute higher-order bias terms. Additive gaussian thermal/electronic noise sets the irreducible data-fidelity floor, while TV / wavelet-sparsity / deep priors stabilise recovery at the ill-conditioned end of Omega. | Metric | Value | |---|---| | Metric | PSNR_dB | | Secondary | SSIM | ## 📏 5. Operating Range (Ω) **Center problem class:** `mrf` · **Forward operator:** `mrf_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 256 | | W | px | 256 | | Z | — | 32 | | Snr db | dB | 18 | | Motion | — | 0 | | N frames | — | 1000 | | Tr ms variable | — | True | | Partial volume | — | 0 | | B0 inhomogeneity | — | 0 | | Undersampling artifacts | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 64 | | W | px | 64 | | Z | — | 4 – 64 | | Snr db | dB | 0.0 – 30.0 | | Motion | — | 0.0 – 0.3 | | N frames | — | 100 – 5000 | | Partial volume | — | 0.0 – 0.3 | | B0 inhomogeneity | — | 0.0 – 0.3 | | Undersampling artifacts | — | 0.0 – 0.5 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 18.0 | Metric | Range | |---|---| | Psnr db | 5.0 – 45.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `500` and δ = `10`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x805d539964b777fbc574e4af2e24fa558651f1481aed5d845c5509e00fccd209` - **Chain tx hash:** `0x6f0ed8273e59ffe3fe9d4628a7e765a2e0dd618c7a34227f3228382923983379` - **Chain block:** `41553386` --- ## File Mapping This bundle consists of: `L1-060.md`, `L1-060.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-060.md` | Source of truth — edit this | Human or LLM | | `L1-060.json` | Structured metadata for the registry | LLM regenerates from the sections above | **Prompt for your LLM after editing this Markdown:** > Read the attached Markdown. Regenerate the sibling `.json` so every field matches. > Preserve the schema documented in the rows above. > Output each file in its own fenced code block tagged with the filename. > Output only the JSON object. _This Markdown was auto-synthesized from the catalog row for `L1-060`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

📝 Polish this artifact in your LLM

Download the files on the right.
Paste the Markdown into ChatGPT / Claude / Gemini and edit freely.
Send your edited Markdown back with the prompt below.
Save the regenerated JSON, then submit via /submit.

Prompt to give your LLM

Read the attached Markdown (L1-060.md).
Regenerate L1-060.json so every field matches the Markdown.
Preserve the schema documented in the File Mapping footer at the bottom of the MD.
Output each file in its own fenced code block tagged with the filename.

⌥ Hierarchy

Upstream lineage and downstream children

↓ L2 Spec 1 child ▾

L2-060-001 MR Fingerprinting (MRF) Mismatch-Only Spec

⬇ Download bundle

2 files · click to download

📋 JSON metadata

{
  "artifact_id": "L1-060",
  "chain_block": 41553386,
  "chain_hash": "0x805d539964b777fbc574e4af2e24fa558651f1481aed5d845c5509e00fccd209",
  "chain_tx_hash": "0x6f0ed8273e59ffe3fe9d4628a7e765a2e0dd618c7a34227f3228382923983379",
  "domain": "Medical Imaging",
  "hardness_fn": {
    "delta": 10,
    "kappa": 500,
    "metric": "PSNR_dB",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
    {
      "ipfs_cid": null,
      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "PSNR_dB",
    "regime": "Existence of the recovered 4D parameter maps is guaranteed within the declared Omega bounds. Uniqueness is local rather than global (non-convex landscape); convergence depends on initialisation and priors. Stability is moderately conditioned (kappa_eff ~= 25); B0_inhomogeneity dominates the stability cliff; undersampling_artifacts and the remaining mismatch parameters contribute higher-order bias terms. Additive gaussian thermal/electronic noise sets the irreducible data-fidelity floor, while TV / wavelet-sparsity / deep priors stabilise recovery at the ill-conditioned end of Omega.",
    "secondary": "SSIM"
  },
  "physics_fingerprint": {
    "L_DAG": 4.5,
    "carrier": "radio_wave",
    "difficulty_delta": 10,
    "domain": "Medical Imaging",
    "integration_axis": "temporal",
    "noise_model": "gaussian",
    "primitives": [
      "L.rf_excitation_pseudorandom",
      "L.kspace_undersample",
      "L.dict_match",
      "int.temporal"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "mri_fingerprinting",
    "solution_space": "4D_parameter_maps",
    "sub_domain": "Pseudo-random acquisition with dictionary matching for T1/T2/rho",
    "title": "MR Fingerprinting (MRF)"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "mrf_forward"
    ],
    "allowed_omega_dimensions": [
      "H",
      "W",
      "Z",
      "N_frames",
      "TR_ms_variable",
      "SNR_dB",
      "B0_inhomogeneity",
      "undersampling_artifacts",
      "partial_volume",
      "motion",
      "B0_inhomogeneity",
      "undersampling_artifacts",
      "partial_volume",
      "motion"
    ],
    "allowed_problem_classes": [
      "mrf"
    ],
    "center_spec": {
      "epsilon_fn_center": "18.0",
      "forward_operator": "mrf_forward",
      "input_format": "measurement_only",
      "omega": {
        "B0_inhomogeneity": 0.0,
        "H": 256,
        "N_frames": 1000,
        "SNR_dB": 18,
        "TR_ms_variable": true,
        "W": 256,
        "Z": 32,
        "motion": 0.0,
        "partial_volume": 0.0,
        "undersampling_artifacts": 0.0
      },
      "problem_class": "mrf"
    },
    "epsilon_bounds": {
      "psnr_db": [
        5.0,
        45.0
      ]
    },
    "omega_bounds": {
      "B0_inhomogeneity": [
        0.0,
        0.3
      ],
      "H": 64,
      "N_frames": [
        100,
        5000
      ],
      "SNR_dB": [
        0.0,
        30.0
      ],
      "W": 64,
      "Z": [
        4,
        64
      ],
      "motion": [
        0.0,
        0.3
      ],
      "partial_volume": [
        0.0,
        0.3
      ],
      "undersampling_artifacts": [
        0.0,
        0.5
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Pseudo-random acquisition with dictionary matching for T1/T2/rho",
  "title": "MR Fingerprinting (MRF)"
}

🎁

Earn PWM by promoting this to mainnet

• 100 PWM one-time grant on mainnet promotion
• 5% upstream royalty on every downstream L4 mining event, forever
• Your wallet is enrolled the moment the founder accepts

🚀 Submit to mainnet →

How: Download the bundle above → polish the Markdown in ChatGPT / Claude / Gemini → submit your version → founder reviews → mainnet.