L1 ⚛ L1-212 ⊙ Testnet

Shell Theory — curved thin-structure mechanics

Structural Mechanics · Koiter / Naghdi curved shell formulations

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

Markdown — human-readable source of truth

⚙ auto-generated

⬇ L1-212.md

# ⚛ L1 Principle — Shell Theory — curved thin-structure mechanics **ID:** `L1-212` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Structural Mechanics — *Koiter / Naghdi curved shell formulations* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** shell displacement 3D > **📡 Carrier:** mechanical · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554010 --- ## 🧠 1. Introduction **Shell Theory — curved thin-structure mechanics** is a **linear inverse problem** whose unknown lives in **shell displacement 3D** space, within the **Koiter / Naghdi curved shell formulations** sub-domain of **Structural Mechanics**. Measurements consist of mechanical vibrations or strain signals via a **dic accelerometers** sensing mechanism. The forward operator applies, in order: L · curvature operator; L · mid surface strains operator; L · shell constitutive operator; pixel-level spatial averaging on the detector. Observations are corrupted by additive Gaussian noise. Conditional stability; mismatch parameters dominate at Omega bounds. ## ⚙ 2. Forward Model Physical chain: **x** → L · curvature → L · mid surface strains → L · shell constitutive → Spatial integration → **y** (detector). ``` y = ∫_A dA `L.shell_constitutive` `L.mid_surface_strains` `L.curvature` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.curvature` | L · curvature operator | | `L.mid_surface_strains` | L · mid surface strains operator | | `L.shell_constitutive` | L · shell constitutive operator | | `int.spatial` | Pixel-level spatial averaging on the detector | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Structural Mechanics | | Sub domain | Koiter / Naghdi curved shell formulations | | Carrier | mechanical | | Problem class | linear_inverse | | Solution space | shell_displacement_3D | | Noise model | gaussian | | Integration axis | spatial | | Difficulty delta | 5 | | L dag | 3.3 | ## 📡 4. Measurement Model Conditional stability; mismatch parameters dominate at Omega bounds. | Metric | Value | |---|---| | Metric | PSNR_dB | | Secondary | SSIM | ## 📏 5. Operating Range (Ω) **Center problem class:** `shell` · **Forward operator:** `shell_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | N dof | — | 10000 | | Snr db | dB | 30 | | Bc error | — | 0 | | N sensors | — | 64 | | Material error | — | 0 | | Curvature error | — | 0 | | Thickness error | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | N dof | — | 100 – 1000000 | | Snr db | dB | 0 – 40 | | Bc error | — | 0.0 – 0.2 | | N sensors | — | 4 – 4096 | | Material error | — | 0.0 – 0.1 | | Curvature error | — | 0.0 – 0.05 | | Thickness error | — | 0.0 – 0.05 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 24.0 | Metric | Range | |---|---| | Psnr db | 10.0 – 45.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `360` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x8fa88a0e3170c528a8df2abcf767b1d3e1bd91d11f9e9ec8508e588e312dcc86` - **Chain tx hash:** `0x24ed5fca14d5113281e34988c4b6dda05d1fb15958e3b31a4a2e5b9364b7f5de` - **Chain block:** `41554010` --- ## File Mapping This bundle consists of: `L1-212.md`, `L1-212.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-212.md` | Source of truth — edit this | Human or LLM | | `L1-212.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 `L1-212`._ _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 (L1-212.md).
Regenerate L1-212.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 ▾

L2-212-001 Shell Theory — curved thin-structure mechanics Mismatch-Only Spec

⬇ Download bundle

2 files · click to download

📋 JSON metadata

{
  "artifact_id": "L1-212",
  "chain_block": 41554010,
  "chain_hash": "0x8fa88a0e3170c528a8df2abcf767b1d3e1bd91d11f9e9ec8508e588e312dcc86",
  "chain_tx_hash": "0x24ed5fca14d5113281e34988c4b6dda05d1fb15958e3b31a4a2e5b9364b7f5de",
  "domain": "Structural Mechanics",
  "hardness_fn": {
    "delta": 5,
    "kappa": 360,
    "metric": "PSNR_dB",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
    {
      "ipfs_cid": null,
      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "PSNR_dB",
    "regime": "Conditional stability; mismatch parameters dominate at Omega bounds.",
    "secondary": "SSIM"
  },
  "physics_fingerprint": {
    "L_DAG": 3.3,
    "carrier": "mechanical",
    "difficulty_delta": 5,
    "domain": "Structural Mechanics",
    "integration_axis": "spatial",
    "noise_model": "gaussian",
    "primitives": [
      "L.curvature",
      "L.mid_surface_strains",
      "L.shell_constitutive",
      "int.spatial"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "dic_accelerometers",
    "solution_space": "shell_displacement_3D",
    "sub_domain": "Koiter / Naghdi curved shell formulations",
    "title": "Shell Theory \u2014 curved thin-structure mechanics"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "shell_forward"
    ],
    "allowed_omega_dimensions": [
      "N_DOF",
      "N_sensors",
      "SNR_dB",
      "curvature_error",
      "thickness_error",
      "material_error",
      "BC_error",
      "curvature_error",
      "thickness_error",
      "material_error",
      "BC_error"
    ],
    "allowed_problem_classes": [
      "shell"
    ],
    "center_spec": {
      "epsilon_fn_center": "24.0",
      "forward_operator": "shell_forward",
      "input_format": "measurement_only",
      "omega": {
        "BC_error": 0.0,
        "N_DOF": 10000,
        "N_sensors": 64,
        "SNR_dB": 30,
        "curvature_error": 0.0,
        "material_error": 0.0,
        "thickness_error": 0.0
      },
      "problem_class": "shell"
    },
    "epsilon_bounds": {
      "psnr_db": [
        10.0,
        45.0
      ]
    },
    "omega_bounds": {
      "BC_error": [
        0.0,
        0.2
      ],
      "N_DOF": [
        100,
        1000000
      ],
      "N_sensors": [
        4,
        4096
      ],
      "SNR_dB": [
        0,
        40
      ],
      "curvature_error": [
        0.0,
        0.05
      ],
      "material_error": [
        0.0,
        0.1
      ],
      "thickness_error": [
        0.0,
        0.05
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Koiter / Naghdi curved shell formulations",
  "title": "Shell Theory \u2014 curved thin-structure mechanics"
}

🎁

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.