L1 ⚛ L1-490 ⊙ Testnet

Carrier Boltzmann Transport Equation

Semiconductor Physics · Kinetic carrier transport

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⬇ principle.md ⬇ principle.json

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⬇ L1-490.md ⬇ L1-490.json

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You are helping me submit a MODIFICATION of L1-490 (Carrier Boltzmann Transport Equation) to the PWM Protocol — a Principle (L1) artifact.

I will paste a Markdown template (or the current L1-490.md).
1. Rewrite the Markdown so the science is correct and clearly explained for my change.
2. Regenerate the sibling L1-490.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.moments` `int.stochastic` `E.scattering_rates` `E.band_structure` x,    measurements ~ Poisson(αy)

world state x (distribution f k of r)

→

E · band structure `E.band_structure`

→

E · scattering rates `E.scattering_rates`

→

int · stochastic `int.stochastic`

→

O · moments `O.moments`

→

observation y (electron detector)

Noise: Poisson counting noise

Markdown — human-readable source of truth

⚙ auto-generated

⬇ L1-490.md

# ⚛ L1 Principle — Carrier Boltzmann Transport Equation **ID:** `L1-490` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Semiconductor Physics — *Kinetic carrier transport* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** distribution f k of r > **📡 Carrier:** electron · **🌫 Noise:** poisson > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554141 --- ## 🧠 1. Introduction **Carrier Boltzmann Transport Equation** is a **nonlinear inverse problem** whose unknown lives in **distribution f k of r** space, within the **Kinetic carrier transport** sub-domain of **Semiconductor Physics**. Measurements consist of electrons collected by an electron detector via a **mobility noise measurement** sensing mechanism. The forward operator applies, in order: E · band structure operator; E · scattering rates operator; int · stochastic operator; O · moments operator. Observations are corrupted by Poisson counting noise. Well-posed for given scattering rates; inversion for deformation potentials is ill-conditioned (multiple rates couple to same mobility). ## ⚙ 2. Forward Model Physical chain: **x** → E · band structure → E · scattering rates → int · stochastic → O · moments → **y** (detector). ``` y = `O.moments` `int.stochastic` `E.scattering_rates` `E.band_structure` x, measurements ~ Poisson(αy) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `E.band_structure` | E · band structure operator | | `E.scattering_rates` | E · scattering rates operator | | `int.stochastic` | Int · stochastic operator | | `O.moments` | O · moments operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Semiconductor Physics | | Sub domain | Kinetic carrier transport | | Carrier | electron | | Problem class | nonlinear_inverse | | Solution space | distribution_f_k_of_r | | Noise model | poisson | | Integration axis | time | | Difficulty delta | 5 | | L dag | 3.7 | ## 📡 4. Measurement Model Well-posed for given scattering rates; inversion for deformation potentials is ill-conditioned (multiple rates couple to same mobility). | Metric | Value | |---|---| | Metric | mobility_relative_error | | Secondary | noise_S_I_log_error | ## 📏 5. Operating Range (Ω) **Center problem class:** `kinetic_transport_inversion` · **Forward operator:** `boltzmann_mc_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | T k | — | 300 | | N d cm3 | — | 100000000000000000 | | Material | — | Si | | N particles | — | 100000 | | E field v cm | — | 10000 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | T k | — | 77 – 500 | | N d cm3 | — | 100000000000000.0 – 100000000000000000000 | | N particles | — | 10000 – 10000000 | | E field v cm | — | 10 – 1000000.0 | ## 🎯 6. Tolerance (ε) **Center tolerance:** mobility relative error <= 0.05 | Metric | Range | |---|---| | Mobility relative error | 0.01 – 0.5 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **mobility_relative_error**, with κ = `600` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x90b87c96eb25800edc6d30ddb99f6c0eac7c38b58d4bc07f49d884e55e13010b` - **Chain tx hash:** `0x3104ae52719cb42e7e16b4d1349748a0d9457e6d5551b72d641ca58c2f2b5461` - **Chain block:** `41554141` --- ## File Mapping This bundle consists of: `L1-490.md`, `L1-490.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-490.md` | Source of truth — edit this | Human or LLM | | `L1-490.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 `L1-490`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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L2-490-001 Carrier Boltzmann Transport Equation Mismatch-Only Spec

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📋 JSON metadata

{
  "artifact_id": "L1-490",
  "chain_block": 41554141,
  "chain_hash": "0x90b87c96eb25800edc6d30ddb99f6c0eac7c38b58d4bc07f49d884e55e13010b",
  "chain_tx_hash": "0x3104ae52719cb42e7e16b4d1349748a0d9457e6d5551b72d641ca58c2f2b5461",
  "domain": "Semiconductor Physics",
  "hardness_fn": {
    "delta": 5,
    "kappa": 600,
    "metric": "mobility_relative_error",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
    {
      "ipfs_cid": null,
      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "mobility_relative_error",
    "regime": "Well-posed for given scattering rates; inversion for deformation potentials is ill-conditioned (multiple rates couple to same mobility).",
    "secondary": "noise_S_I_log_error"
  },
  "physics_fingerprint": {
    "L_DAG": 3.7,
    "carrier": "electron",
    "difficulty_delta": 5,
    "domain": "Semiconductor Physics",
    "integration_axis": "time",
    "noise_model": "poisson",
    "primitives": [
      "E.band_structure",
      "E.scattering_rates",
      "int.stochastic",
      "O.moments"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "mobility_noise_measurement",
    "solution_space": "distribution_f_k_of_r",
    "sub_domain": "Kinetic carrier transport",
    "title": "Carrier Boltzmann Transport Equation"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "boltzmann_mc_forward"
    ],
    "allowed_omega_dimensions": [
      "E_field_V_cm",
      "T_K",
      "N_D_cm3",
      "material",
      "N_particles"
    ],
    "allowed_problem_classes": [
      "kinetic_transport_inversion"
    ],
    "center_spec": {
      "epsilon_fn_center": "mobility relative error \u003c= 0.05",
      "forward_operator": "boltzmann_mc_forward",
      "input_format": "measurement_only",
      "omega": {
        "E_field_V_cm": 10000.0,
        "N_D_cm3": 100000000000000000,
        "N_particles": 100000,
        "T_K": 300,
        "material": "Si"
      },
      "problem_class": "kinetic_transport_inversion"
    },
    "epsilon_bounds": {
      "mobility_relative_error": [
        0.01,
        0.5
      ]
    },
    "omega_bounds": {
      "E_field_V_cm": [
        10,
        1000000.0
      ],
      "N_D_cm3": [
        100000000000000.0,
        100000000000000000000
      ],
      "N_particles": [
        10000,
        10000000
      ],
      "T_K": [
        77,
        500
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Kinetic carrier transport",
  "title": "Carrier Boltzmann Transport Equation"
}

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