L1 ⚛ L1-316 ⊙ Testnet

Electronic Band Structure

Quantum Mechanics · Crystal momentum eigenproblem

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⬇ L1-316.md

# ⚛ L1 Principle — Electronic Band Structure **ID:** `L1-316` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Quantum Mechanics — *Crystal momentum eigenproblem* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** band structure E n k > **📡 Carrier:** electron · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554079 --- ## 🧠 1. Introduction **Electronic Band Structure** is a **nonlinear inverse problem** whose unknown lives in **band structure E n k** space, within the **Crystal momentum eigenproblem** sub-domain of **Quantum Mechanics**. Measurements consist of electrons collected by an electron detector via a **arpes optical absorption** sensing mechanism. The forward operator applies, in order: E · crystal potential operator; computes eigen-pairs of a linear operator; O · band structure operator. Observations are corrupted by additive Gaussian noise. Well-posed; DFT gap underestimate (Kohn-Sham gap problem); inverse structure→band mapping requires phase info. ## ⚙ 2. Forward Model Physical chain: **x** → E · crystal potential → O · band structure → **y** (detector). ``` y = `O.band_structure` `E.crystal_potential` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `E.crystal_potential` | E · crystal potential operator | | `O.band_structure` | O · band structure operator | **🛠 Solver components** _(used inside the solver, not in the forward equation)_: | Primitive | What it does | |---|---| | `E.eigensolve` | Computes eigen-pairs of a linear operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Quantum Mechanics | | Sub domain | Crystal momentum eigenproblem | | Carrier | electron | | Problem class | nonlinear_inverse | | Solution space | band_structure_E_n_k | | Noise model | gaussian | | Integration axis | k_space | | Difficulty delta | 5 | | L dag | 3.2 | ## 📡 4. Measurement Model Well-posed; DFT gap underestimate (Kohn-Sham gap problem); inverse structure→band mapping requires phase info. | Metric | Value | |---|---| | Metric | band_gap_error_eV | | Secondary | effective_mass_error | ## 📏 5. Operating Range (Ω) **Center problem class:** `band_structure` · **Forward operator:** `dft_band_structure_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | Xc | — | PBE | | Ecut ev | — | 400 | | N atoms | — | 2 | | Species | — | Si | | N kpoints | — | 100 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | Xc | — | LDA, PBE, HSE06 | | Ecut ev | — | 100 – 2000 | | N atoms | — | 1 – 100 | | N kpoints | — | 10 – 10000 | ## 🎯 6. Tolerance (ε) **Center tolerance:** band gap error <= 0.1 eV | Metric | Range | |---|---| | Band gap error ev | 0.02 – 2.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **band_gap_error_eV**, with κ = `600` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x2e034436660d1d9528fcf4496770ad9a044d166f70d9fa0099c908850700c36c` - **Chain tx hash:** `0xa8a4570fb71fc030771d010711633f48b6ae995429fa695257eb502fd1bb2135` - **Chain block:** `41554079` --- ## File Mapping This bundle consists of: `L1-316.md`, `L1-316.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-316.md` | Source of truth — edit this | Human or LLM | | `L1-316.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 `L1-316`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-316.md).
Regenerate L1-316.json so every field matches the Markdown.
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L2-316-001 Electronic Band Structure Mismatch-Only Spec

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📋 JSON metadata

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  "chain_block": 41554079,
  "chain_hash": "0x2e034436660d1d9528fcf4496770ad9a044d166f70d9fa0099c908850700c36c",
  "chain_tx_hash": "0xa8a4570fb71fc030771d010711633f48b6ae995429fa695257eb502fd1bb2135",
  "domain": "Quantum Mechanics",
  "hardness_fn": {
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    "kappa": 600,
    "metric": "band_gap_error_eV",
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    "secondary": "effective_mass_error"
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  "physics_fingerprint": {
    "L_DAG": 3.2,
    "carrier": "electron",
    "difficulty_delta": 5,
    "domain": "Quantum Mechanics",
    "integration_axis": "k_space",
    "noise_model": "gaussian",
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    "sensing_mechanism": "arpes_optical_absorption",
    "solution_space": "band_structure_E_n_k",
    "sub_domain": "Crystal momentum eigenproblem",
    "title": "Electronic Band Structure"
  },
  "size_tiers": {
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    "allowed_omega_dimensions": [
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      "Ecut_eV",
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    "allowed_problem_classes": [
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    "center_spec": {
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      "omega": {
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        "N_atoms": 2,
        "N_kpoints": 100,
        "species": "Si",
        "xc": "PBE"
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      "problem_class": "band_structure"
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    "epsilon_bounds": {
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    "omega_bounds": {
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        2000
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      "N_atoms": [
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      "N_kpoints": [
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      "xc": [
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        "PBE",
        "HSE06"
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Crystal momentum eigenproblem",
  "title": "Electronic Band Structure"
}

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