L1 ⚛ L1-518 ⊙ Testnet

X-ray Diffraction Crystal Structure Recovery (analytical core)

Materials Science · Powder / single-crystal X-ray diffraction lattice and structure-factor recovery

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

Markdown — human-readable source of truth

⚙ auto-generated

⬇ L1-518.md

# ⚛ L1 Principle — X-ray Diffraction Crystal Structure Recovery (analytical core) **ID:** `L1-518` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Materials Science — *Powder / single-crystal X-ray diffraction lattice and structure-factor recovery* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** crystal structure parameters > **📡 Carrier:** x_ray · **🌫 Noise:** poisson > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554142 --- ## 🧠 1. Introduction **X-ray Diffraction Crystal Structure Recovery (analytical core)** is a **nonlinear inverse problem** whose unknown lives in **crystal structure parameters** space, within the **Powder / single-crystal X-ray diffraction lattice and structure-factor recovery** sub-domain of **Materials Science**. Measurements consist of X-ray photons transmitted through (or scattered by) the sample via a **bragg diffraction** sensing mechanism. The forward operator applies, in order: polyenergetic X-ray emission spectrum; L · bragg diffraction operator; L · structure factor operator; L · intensity calculation operator; L · profile function operator; L · detector response operator; integration over the solid angle of incidence/emission. Observations are corrupted by Poisson counting noise. Existence guaranteed within declared Omega bounds. Uniqueness conditional: for known atomic composition + Bravais lattice + sufficient resolution, structure refinement is unique up to centrosymmetric equivalence; phase problem requires Patterson methods (heavy-atom), direct methods (Karle-Hauptman maximum-entropy), or anomalous-scattering for ab-initio. Stability conditional with kappa raw ~ 600 (limited by phase problem and reflection-overlap in powder XRD); kappa_eff ~ 80 with Rietveld refinement under structural constraints. Joint Hadamard well-posedness for the XRD forward established by Rietveld 1969 (foundational), Hill-Howard 1986 (Rietveld profile refinement), Karle-Hauptman 1956 (direct methods, Nobel 1985), McCusker 1999 (Rietveld guidelines), Young 1993 (Rietveld method textbook), Coelho 2018 (TOPAS), Toby-Von Dreele 2013 (GSAS-II). ## ⚙ 2. Forward Model Physical chain: **x** → X-ray source → L · bragg diffraction → L · structure factor → L · intensity calculation → L · profile function → L · detector response → Angular integration → **y** (detector). ``` y = ∫dΩ `L.detector_response` `L.profile_function` `L.intensity_calculation` `L.structure_factor` `L.bragg_diffraction` I₀(E) x, measurements ~ Poisson(αy) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.xray_source` | Polyenergetic x-ray emission spectrum | | `L.bragg_diffraction` | L · bragg diffraction operator | | `L.structure_factor` | L · structure factor operator | | `L.intensity_calculation` | L · intensity calculation operator | | `L.profile_function` | L · profile function operator | | `L.detector_response` | L · detector response operator | | `int.angular` | Integration over the solid angle of incidence/emission | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Materials Science | | Sub domain | Powder / single-crystal X-ray diffraction lattice and structure-factor recovery | | Carrier | x_ray | | Problem class | nonlinear_inverse | | Solution space | crystal_structure_parameters | | Noise model | poisson | | Integration axis | angular_spatial | | Difficulty delta | 5 | | L dag | 6.8 | ## 📡 4. Measurement Model Existence guaranteed within declared Omega bounds. Uniqueness conditional: for known atomic composition + Bravais lattice + sufficient resolution, structure refinement is unique up to centrosymmetric equivalence; phase problem requires Patterson methods (heavy-atom), direct methods (Karle-Hauptman maximum-entropy), or anomalous-scattering for ab-initio. Stability conditional with kappa raw ~ 600 (limited by phase problem and reflection-overlap in powder XRD); kappa_eff ~ 80 with Rietveld refinement under structural constraints. Joint Hadamard well-posedness for the XRD forward established by Rietveld 1969 (foundational), Hill-Howard 1986 (Rietveld profile refinement), Karle-Hauptman 1956 (direct methods, Nobel 1985), McCusker 1999 (Rietveld guidelines), Young 1993 (Rietveld method textbook), Coelho 2018 (TOPAS), Toby-Von Dreele 2013 (GSAS-II). | Metric | Value | |---|---| | Metric | R_factor | | Secondary | GoF_chi2 | ## 📏 5. Operating Range (Ω) **Center problem class:** `powder_xrd_rietveld` · **Forward operator:** `powder_xrd_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | Snr db | dB | 30 | | Two theta max | — | 120 | | Two theta min | — | 5 | | Two theta step | — | 0.02 | | Beam divergence | — | 0 | | Lambda angstrom | — | 1.5418 | | Sample displacement | — | 0 | | Wavelength dispersion | — | 0 | | Instrumental broadening | — | 0.05 | | Monochromator misalignment | — | 0 | | Background subtraction error | — | 0 | | Preferred orientation powder | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | Snr db | dB | 10 – 50 | | Two theta max | — | 60 – 160 | | Two theta min | — | 1 – 30 | | Two theta step | — | 0.001 – 0.1 | | Lambda angstrom | — | 0.4 – 2.5 | | Instrumental broadening | — | 0.01 – 0.5 | | Preferred orientation powder | — | 0.0 – 0.5 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 0.05_R_factor | Metric | Range | |---|---| | R factor | 0.01 – 0.3 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **R_factor**, with κ = `600` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xd3a963f4f3a6f8f1d608a89e37a9410e205b38e2b7b8fbae0f02dd460668155f` - **Chain tx hash:** `0xb7ec20cef152165728154268692c108ecee957b28dc6a189658eb21ff9e110a2` - **Chain block:** `41554142` --- ## File Mapping This bundle consists of: `L1-518.md`, `L1-518.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-518.md` | Source of truth — edit this | Human or LLM | | `L1-518.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 `L1-518`._ _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 (L1-518.md).
Regenerate L1-518.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 ▾

L2-518-001 X-ray Diffraction Crystal Structure Recovery (analytical core) — Spec Skeleton

⬇ Download bundle

2 files · click to download

📋 JSON metadata

{
  "artifact_id": "L1-518",
  "chain_block": 41554142,
  "chain_hash": "0xd3a963f4f3a6f8f1d608a89e37a9410e205b38e2b7b8fbae0f02dd460668155f",
  "chain_tx_hash": "0xb7ec20cef152165728154268692c108ecee957b28dc6a189658eb21ff9e110a2",
  "domain": "Materials Science",
  "hardness_fn": {
    "delta": 5,
    "kappa": 600,
    "metric": "R_factor",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
    {
      "ipfs_cid": null,
      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "R_factor",
    "regime": "Existence guaranteed within declared Omega bounds. Uniqueness conditional: for known atomic composition + Bravais lattice + sufficient resolution, structure refinement is unique up to centrosymmetric equivalence; phase problem requires Patterson methods (heavy-atom), direct methods (Karle-Hauptman maximum-entropy), or anomalous-scattering for ab-initio. Stability conditional with kappa raw ~ 600 (limited by phase problem and reflection-overlap in powder XRD); kappa_eff ~ 80 with Rietveld refinement under structural constraints. Joint Hadamard well-posedness for the XRD forward established by Rietveld 1969 (foundational), Hill-Howard 1986 (Rietveld profile refinement), Karle-Hauptman 1956 (direct methods, Nobel 1985), McCusker 1999 (Rietveld guidelines), Young 1993 (Rietveld method textbook), Coelho 2018 (TOPAS), Toby-Von Dreele 2013 (GSAS-II).",
    "secondary": "GoF_chi2"
  },
  "physics_fingerprint": {
    "L_DAG": 6.8,
    "carrier": "x_ray",
    "difficulty_delta": 5,
    "domain": "Materials Science",
    "integration_axis": "angular_spatial",
    "noise_model": "poisson",
    "primitives": [
      "L.xray_source",
      "L.bragg_diffraction",
      "L.structure_factor",
      "L.intensity_calculation",
      "L.profile_function",
      "L.detector_response",
      "int.angular"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "bragg_diffraction",
    "solution_space": "crystal_structure_parameters",
    "sub_domain": "Powder / single-crystal X-ray diffraction lattice and structure-factor recovery",
    "title": "X-ray Diffraction Crystal Structure Recovery (analytical core)"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "powder_xrd_forward",
      "single_crystal_xrd_forward",
      "rietveld_refinement_forward",
      "direct_methods_phase_forward"
    ],
    "allowed_omega_dimensions": [
      "lambda_angstrom",
      "two_theta_min",
      "two_theta_max",
      "two_theta_step",
      "wavelength_dispersion",
      "instrumental_broadening",
      "preferred_orientation_powder",
      "sample_displacement",
      "beam_divergence",
      "monochromator_misalignment",
      "background_subtraction_error",
      "SNR_dB"
    ],
    "allowed_problem_classes": [
      "powder_xrd_rietveld",
      "single_crystal_structure_solution",
      "powder_indexing",
      "ab_initio_structure_determination"
    ],
    "center_spec": {
      "epsilon_fn_center": "0.05_R_factor",
      "forward_operator": "powder_xrd_forward",
      "input_format": "1d_diffraction_pattern",
      "omega": {
        "SNR_dB": 30,
        "background_subtraction_error": 0.0,
        "beam_divergence": 0.0,
        "instrumental_broadening": 0.05,
        "lambda_angstrom": 1.5418,
        "monochromator_misalignment": 0.0,
        "preferred_orientation_powder": 0.0,
        "sample_displacement": 0.0,
        "two_theta_max": 120,
        "two_theta_min": 5,
        "two_theta_step": 0.02,
        "wavelength_dispersion": 0.0
      },
      "problem_class": "powder_xrd_rietveld"
    },
    "epsilon_bounds": {
      "R_factor": [
        0.01,
        0.3
      ]
    },
    "omega_bounds": {
      "SNR_dB": [
        10,
        50
      ],
      "instrumental_broadening": [
        0.01,
        0.5
      ],
      "lambda_angstrom": [
        0.4,
        2.5
      ],
      "preferred_orientation_powder": [
        0.0,
        0.5
      ],
      "two_theta_max": [
        60,
        160
      ],
      "two_theta_min": [
        1,
        30
      ],
      "two_theta_step": [
        0.001,
        0.1
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Powder / single-crystal X-ray diffraction lattice and structure-factor recovery",
  "title": "X-ray Diffraction Crystal Structure Recovery (analytical core)"
}

🎁

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.