L1 ⚛ L1-325 ⊙ Testnet

Electron-Phonon Coupling and Eliashberg Theory

Quantum Mechanics · Phonon-mediated superconductivity

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⬇ L1-325.md

# ⚛ L1 Principle — Electron-Phonon Coupling and Eliashberg Theory **ID:** `L1-325` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Quantum Mechanics — *Phonon-mediated superconductivity* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** eliashberg spectral function > **📡 Carrier:** electron · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41554081 --- ## 🧠 1. Introduction **Electron-Phonon Coupling and Eliashberg Theory** is a **nonlinear inverse problem** whose unknown lives in **eliashberg spectral function** space, within the **Phonon-mediated superconductivity** sub-domain of **Quantum Mechanics**. Measurements consist of electrons collected by an electron detector via a **Tc measurement IXS** sensing mechanism. The forward operator applies, in order: E · phonon dispersion operator; E · electron phonon operator; O · composite method operator; O · Tc operator. Observations are corrupted by additive Gaussian noise. Non-linear; T_c sensitive to spectral function shape; inversion ill-conditioned. ## ⚙ 2. Forward Model Physical chain: **x** → E · phonon dispersion → E · electron phonon → O · composite method → O · Tc → **y** (detector). ``` y = `O.Tc` `O.composite_method` `E.electron_phonon` `E.phonon_dispersion` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `E.phonon_dispersion` | E · phonon dispersion operator | | `E.electron_phonon` | E · electron phonon operator | | `O.composite_method` | O · composite method operator | | `O.Tc` | O · tc operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Quantum Mechanics | | Sub domain | Phonon-mediated superconductivity | | Carrier | electron | | Problem class | nonlinear_inverse | | Solution space | eliashberg_spectral_function | | Noise model | gaussian | | Integration axis | energy | | Difficulty delta | 10 | | L dag | 3.9 | ## 📡 4. Measurement Model Non-linear; T_c sensitive to spectral function shape; inversion ill-conditioned. | Metric | Value | |---|---| | Metric | Tc_relative_error | | Secondary | alpha2F_L2_error | ## 📏 5. Operating Range (Ω) **Center problem class:** `electron_phonon_coupling` · **Forward operator:** `eliashberg_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | N bands | bands | 20 | | Mu star | — | 0.1 | | Species | — | Pb | | N kpoints | — | 32 | | N qpoints | — | 16 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | N bands | bands | 5 – 200 | | Mu star | — | 0.05 – 0.3 | | N kpoints | — | 8 – 1000 | | N qpoints | — | 4 – 256 | ## 🎯 6. Tolerance (ε) **Center tolerance:** T_c rel error <= 0.05 | Metric | Range | |---|---| | Tc relative error | 0.01 – 0.5 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **Tc_relative_error**, with κ = `1500` and δ = `10`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xa3bfae629bea0395757305e590a06d442b84dec6abc40dcea12dc9c9cea588f0` - **Chain tx hash:** `0xbc5c6d23ecdd75360cee6049d543746a9d626ff56173749251831649a7996d5e` - **Chain block:** `41554081` --- ## File Mapping This bundle consists of: `L1-325.md`, `L1-325.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-325.md` | Source of truth — edit this | Human or LLM | | `L1-325.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 `L1-325`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-325.md).
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📋 JSON metadata

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  "chain_block": 41554081,
  "chain_hash": "0xa3bfae629bea0395757305e590a06d442b84dec6abc40dcea12dc9c9cea588f0",
  "chain_tx_hash": "0xbc5c6d23ecdd75360cee6049d543746a9d626ff56173749251831649a7996d5e",
  "domain": "Quantum Mechanics",
  "hardness_fn": {
    "delta": 10,
    "kappa": 1500,
    "metric": "Tc_relative_error",
    "type": "epsilon_fn"
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  "observable_profile": {
    "metric": "Tc_relative_error",
    "regime": "Non-linear; T_c sensitive to spectral function shape; inversion ill-conditioned.",
    "secondary": "alpha2F_L2_error"
  },
  "physics_fingerprint": {
    "L_DAG": 3.9,
    "carrier": "electron",
    "difficulty_delta": 10,
    "domain": "Quantum Mechanics",
    "integration_axis": "energy",
    "noise_model": "gaussian",
    "primitives": [
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      "E.electron_phonon",
      "O.composite_method",
      "O.Tc"
    ],
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    "sensing_mechanism": "Tc_measurement_IXS",
    "solution_space": "eliashberg_spectral_function",
    "sub_domain": "Phonon-mediated superconductivity",
    "title": "Electron-Phonon Coupling and Eliashberg Theory"
  },
  "size_tiers": {
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      "eliashberg_forward"
    ],
    "allowed_omega_dimensions": [
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      "N_qpoints",
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      "mu_star",
      "species"
    ],
    "allowed_problem_classes": [
      "electron_phonon_coupling"
    ],
    "center_spec": {
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      "forward_operator": "eliashberg_forward",
      "input_format": "measurement_only",
      "omega": {
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        "N_kpoints": 32,
        "N_qpoints": 16,
        "mu_star": 0.1,
        "species": "Pb"
      },
      "problem_class": "electron_phonon_coupling"
    },
    "epsilon_bounds": {
      "Tc_relative_error": [
        0.01,
        0.5
      ]
    },
    "omega_bounds": {
      "N_bands": [
        5,
        200
      ],
      "N_kpoints": [
        8,
        1000
      ],
      "N_qpoints": [
        4,
        256
      ],
      "mu_star": [
        0.05,
        0.3
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Phonon-mediated superconductivity",
  "title": "Electron-Phonon Coupling and Eliashberg Theory"
}

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