L1 ⚛ L1-037 ⊙ Testnet

Dual-Energy X-ray Absorptiometry (DEXA)

Medical Imaging · Bone mineral density via two-energy decomposition

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

Markdown — human-readable source of truth

⚙ auto-generated

⬇ L1-037.md

# ⚛ L1 Principle — Dual-Energy X-ray Absorptiometry (DEXA) **ID:** `L1-037` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Medical Imaging — *Bone mineral density via two-energy decomposition* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** 2D bone mineral density > **📡 Carrier:** x_ray · **🌫 Noise:** shot poisson > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41552301 --- ## 🧠 1. Introduction **Dual-Energy X-ray Absorptiometry (DEXA)** is a **linear inverse problem** whose unknown lives in **2D bone mineral density** space, within the **Bone mineral density via two-energy decomposition** sub-domain of **Medical Imaging**. Measurements consist of X-ray photons transmitted through (or scattered by) the sample via a **dual energy xray** sensing mechanism. The forward operator applies, in order: L · xray source dual operator; exponential attenuation along the propagation path; L · two material decompose operator; pixel-level spatial averaging on the detector. Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered 2D bone mineral density is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by the declared priors. Stability is moderately conditioned (kappa_eff ~= 10); beam_hardening dominates the stability cliff; patient_alignment and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while mild Tikhonov or analytic inversion is sufficient at the nominal Omega point. ## ⚙ 2. Forward Model Physical chain: **x** → L · xray source dual → Beer-Lambert attenuation → L · two material decompose → Spatial integration → **y** (detector). ``` y = ∫_A dA `L.two_material_decompose` exp(-∫µ dl) `L.xray_source_dual` x, measurements ~ Poisson(αy) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.xray_source_dual` | L · xray source dual operator | | `L.beer_lambert` | Exponential attenuation along the propagation path | | `L.two_material_decompose` | L · two material decompose operator | | `int.spatial` | Pixel-level spatial averaging on the detector | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Medical Imaging | | Sub domain | Bone mineral density via two-energy decomposition | | Carrier | x_ray | | Problem class | linear_inverse | | Solution space | 2D_bone_mineral_density | | Noise model | shot_poisson | | Integration axis | spatial | | Difficulty delta | 3 | | L dag | 3.2 | ## 📡 4. Measurement Model Existence of the recovered 2D bone mineral density is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by the declared priors. Stability is moderately conditioned (kappa_eff ~= 10); beam_hardening dominates the stability cliff; patient_alignment and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while mild Tikhonov or analytic inversion is sufficient at the nominal Omega point. | Metric | Value | |---|---| | Metric | PSNR_dB | | Secondary | SSIM | ## 📏 5. Operating Range (Ω) **Center problem class:** `dexa` · **Forward operator:** `dexa_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 1024 | | W | px | 512 | | Mas | — | 10 | | Kvp hi | — | 140 | | Kvp lo | — | 70 | | Pixel mm | mm | 0.5 | | Beam hardening | — | 0.05 | | Patient alignment | — | 0 | | Fat fraction variation | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 512 | | W | px | 256 | | Mas | — | 1 – 100 | | Kvp hi | — | 100 – 150 | | Kvp lo | — | 60 – 80 | | Pixel mm | mm | 0.1 – 2.0 | | Beam hardening | — | 0.0 – 0.2 | | Calibration drift | — | 0.0 – 0.05 | | Patient alignment | — | 0.0 – 2.0 | | Fat fraction variation | — | 0.0 – 0.3 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 30.0 | Metric | Range | |---|---| | Psnr db | 5.0 – 45.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `200` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x1b180345715c22e98dce5616f472c3135d9d037edf910135ff5401bfb049afa0` - **Chain tx hash:** `0xa2c4275ad933053a705d24233547692e6480513eda72372e3e28e98e515152cb` - **Chain block:** `41552301` --- ## File Mapping This bundle consists of: `L1-037.md`, `L1-037.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-037.md` | Source of truth — edit this | Human or LLM | | `L1-037.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 `L1-037`._ _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 (L1-037.md).
Regenerate L1-037.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 ▾

L2-037-001 Dual-Energy X-ray Absorptiometry (DEXA) Mismatch-Only Spec

⬇ Download bundle

2 files · click to download

📋 JSON metadata

{
  "artifact_id": "L1-037",
  "chain_block": 41552301,
  "chain_hash": "0x1b180345715c22e98dce5616f472c3135d9d037edf910135ff5401bfb049afa0",
  "chain_tx_hash": "0xa2c4275ad933053a705d24233547692e6480513eda72372e3e28e98e515152cb",
  "domain": "Medical Imaging",
  "hardness_fn": {
    "delta": 3,
    "kappa": 200,
    "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 2D bone mineral density is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by the declared priors. Stability is moderately conditioned (kappa_eff ~= 10); beam_hardening dominates the stability cliff; patient_alignment and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while mild Tikhonov or analytic inversion is sufficient at the nominal Omega point.",
    "secondary": "SSIM"
  },
  "physics_fingerprint": {
    "L_DAG": 3.2,
    "carrier": "x_ray",
    "difficulty_delta": 3,
    "domain": "Medical Imaging",
    "integration_axis": "spatial",
    "noise_model": "shot_poisson",
    "primitives": [
      "L.xray_source_dual",
      "L.beer_lambert",
      "L.two_material_decompose",
      "int.spatial"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "dual_energy_xray",
    "solution_space": "2D_bone_mineral_density",
    "sub_domain": "Bone mineral density via two-energy decomposition",
    "title": "Dual-Energy X-ray Absorptiometry (DEXA)"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "dexa_forward"
    ],
    "allowed_omega_dimensions": [
      "H",
      "W",
      "kVp_hi",
      "kVp_lo",
      "mAs",
      "pixel_mm",
      "beam_hardening",
      "patient_alignment",
      "fat_fraction_variation",
      "beam_hardening",
      "patient_alignment",
      "fat_fraction_variation",
      "calibration_drift"
    ],
    "allowed_problem_classes": [
      "dexa"
    ],
    "center_spec": {
      "epsilon_fn_center": "30.0",
      "forward_operator": "dexa_forward",
      "input_format": "measurement_only",
      "omega": {
        "H": 1024,
        "W": 512,
        "beam_hardening": 0.05,
        "fat_fraction_variation": 0.0,
        "kVp_hi": 140,
        "kVp_lo": 70,
        "mAs": 10,
        "patient_alignment": 0.0,
        "pixel_mm": 0.5
      },
      "problem_class": "dexa"
    },
    "epsilon_bounds": {
      "psnr_db": [
        5.0,
        45.0
      ]
    },
    "omega_bounds": {
      "H": 512,
      "W": 256,
      "beam_hardening": [
        0.0,
        0.2
      ],
      "calibration_drift": [
        0.0,
        0.05
      ],
      "fat_fraction_variation": [
        0.0,
        0.3
      ],
      "kVp_hi": [
        100,
        150
      ],
      "kVp_lo": [
        60,
        80
      ],
      "mAs": [
        1,
        100
      ],
      "patient_alignment": [
        0.0,
        2.0
      ],
      "pixel_mm": [
        0.1,
        2.0
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Bone mineral density via two-energy decomposition",
  "title": "Dual-Energy X-ray Absorptiometry (DEXA)"
}

🎁

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.