L1 ⚛ L1-326 ⊙ Testnet

k.p Perturbation Theory

Quantum Mechanics · Near-gap band structure expansion

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⬇ principle.md ⬇ principle.json

Fork L1-326

⬇ L1-326.md ⬇ L1-326.json

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You are helping me submit a MODIFICATION of L1-326 (k.p Perturbation Theory) to the PWM Protocol — a Principle (L1) artifact.

I will paste a Markdown template (or the current L1-326.md).
1. Rewrite the Markdown so the science is correct and clearly explained for my change.
2. Regenerate the sibling L1-326.json so EVERY field matches the Markdown.
3. Keep the schema in the "File Mapping" footer at the bottom of the MD.
4. Keep the parent reference unchanged unless I ask otherwise.
Rules: the Markdown is the source of truth; use SI units; do NOT invent benchmark scores.
Output each file in its own fenced code block tagged with the filename.

Here is my template:
[PASTE THE .md HERE]

⚙ Forward Model

y = `O.effective_mass` `E.perturbative_expansion` `E.momentum_matrix` x + n,    n ~ 𝒩(0, σ²)

world state x (effective mass tensor)

→

E · momentum matrix `E.momentum_matrix`

→

E · perturbative expansion `E.perturbative_expansion`

→

O · effective mass `O.effective_mass`

→

observation y (electron detector)

Noise: additive Gaussian noise

Markdown — human-readable source of truth

⚙ auto-generated

⬇ L1-326.md

# ⚛ L1 Principle — k.p Perturbation Theory **ID:** `L1-326` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Quantum Mechanics — *Near-gap band structure expansion* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** effective mass tensor > **📡 Carrier:** electron · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554081 --- ## 🧠 1. Introduction **k.p Perturbation Theory** is a **linear inverse problem** whose unknown lives in **effective mass tensor** space, within the **Near-gap band structure expansion** sub-domain of **Quantum Mechanics**. Measurements consist of electrons collected by an electron detector via a **cyclotron resonance** sensing mechanism. The forward operator applies, in order: E · momentum matrix operator; E · perturbative expansion operator; O · effective mass operator. Observations are corrupted by additive Gaussian noise. Well-posed near k_0; fails far from band edge. ## ⚙ 2. Forward Model Physical chain: **x** → E · momentum matrix → E · perturbative expansion → O · effective mass → **y** (detector). ``` y = `O.effective_mass` `E.perturbative_expansion` `E.momentum_matrix` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `E.momentum_matrix` | E · momentum matrix operator | | `E.perturbative_expansion` | E · perturbative expansion operator | | `O.effective_mass` | O · effective mass operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Quantum Mechanics | | Sub domain | Near-gap band structure expansion | | Carrier | electron | | Problem class | linear_inverse | | Solution space | effective_mass_tensor | | Noise model | gaussian | | Integration axis | k_space | | Difficulty delta | 3 | | L dag | 3.1 | ## 📡 4. Measurement Model Well-posed near k_0; fails far from band edge. | Metric | Value | |---|---| | Metric | effective_mass_relative_error | | Secondary | g_factor_error | ## 📏 5. Operating Range (Ω) **Center problem class:** `kp_perturbation` · **Forward operator:** `kp_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | Species | — | GaAs | | N bands kp | — | 8 | | Include soc | — | True | | K range a inv | — | 0.1 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | N bands kp | — | 4 – 30 | | Include soc | — | True – False | | K range a inv | — | 0.01 – 1.0 | ## 🎯 6. Tolerance (ε) **Center tolerance:** m* rel error <= 0.03 | Metric | Range | |---|---| | Effective mass relative error | 0.01 – 0.3 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **effective_mass_relative_error**, with κ = `200` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xcc48c3e2dc2376bee128bd0198bf23cfe1b34b8d39a5f43977df072efb1f7298` - **Chain tx hash:** `0x940afd71a48c82fcf0988da2986bc7b5e2a2f5dbd57dcc7ca0912bfb4b32f6ca` - **Chain block:** `41554081` --- ## File Mapping This bundle consists of: `L1-326.md`, `L1-326.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-326.md` | Source of truth — edit this | Human or LLM | | `L1-326.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 `L1-326`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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L2-326-001 k.p Perturbation Theory Mismatch-Only Spec

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📋 JSON metadata

{
  "artifact_id": "L1-326",
  "chain_block": 41554081,
  "chain_hash": "0xcc48c3e2dc2376bee128bd0198bf23cfe1b34b8d39a5f43977df072efb1f7298",
  "chain_tx_hash": "0x940afd71a48c82fcf0988da2986bc7b5e2a2f5dbd57dcc7ca0912bfb4b32f6ca",
  "domain": "Quantum Mechanics",
  "hardness_fn": {
    "delta": 3,
    "kappa": 200,
    "metric": "effective_mass_relative_error",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
    {
      "ipfs_cid": null,
      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "effective_mass_relative_error",
    "regime": "Well-posed near k_0; fails far from band edge.",
    "secondary": "g_factor_error"
  },
  "physics_fingerprint": {
    "L_DAG": 3.1,
    "carrier": "electron",
    "difficulty_delta": 3,
    "domain": "Quantum Mechanics",
    "integration_axis": "k_space",
    "noise_model": "gaussian",
    "primitives": [
      "E.momentum_matrix",
      "E.perturbative_expansion",
      "O.effective_mass"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "cyclotron_resonance",
    "solution_space": "effective_mass_tensor",
    "sub_domain": "Near-gap band structure expansion",
    "title": "k.p Perturbation Theory"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "kp_forward"
    ],
    "allowed_omega_dimensions": [
      "N_bands_kp",
      "k_range_A_inv",
      "species",
      "include_SOC"
    ],
    "allowed_problem_classes": [
      "kp_perturbation"
    ],
    "center_spec": {
      "epsilon_fn_center": "m* rel error \u003c= 0.03",
      "forward_operator": "kp_forward",
      "input_format": "measurement_only",
      "omega": {
        "N_bands_kp": 8,
        "include_SOC": true,
        "k_range_A_inv": 0.1,
        "species": "GaAs"
      },
      "problem_class": "kp_perturbation"
    },
    "epsilon_bounds": {
      "effective_mass_relative_error": [
        0.01,
        0.3
      ]
    },
    "omega_bounds": {
      "N_bands_kp": [
        4,
        30
      ],
      "include_SOC": [
        true,
        false
      ],
      "k_range_A_inv": [
        0.01,
        1.0
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Near-gap band structure expansion",
  "title": "k.p Perturbation Theory"
}

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