L1 ⚛ L1-212 ⊙ Testnet

Shell Theory — curved thin-structure mechanics

Structural Mechanics · Koiter / Naghdi curved shell formulations

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⬇ principle.md ⬇ principle.json

Fork L1-212

⬇ L1-212.md ⬇ L1-212.json

Prompt — copy into your LLM

You are helping me submit a MODIFICATION of L1-212 (Shell Theory — curved thin-structure mechanics) to the PWM Protocol — a Principle (L1) artifact.

I will paste a Markdown template (or the current L1-212.md).
1. Rewrite the Markdown so the science is correct and clearly explained for my change.
2. Regenerate the sibling L1-212.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 = ∫_A dA `L.shell_constitutive` `L.mid_surface_strains` `L.curvature` x + n,    n ~ 𝒩(0, σ²)

world state x (shell displacement 3D)

→

L · curvature `L.curvature`

→

L · mid surface strains `L.mid_surface_strains`

→

L · shell constitutive `L.shell_constitutive`

→

Spatial integration ∫_A dA

→

observation y (mechanical detector)

Noise: additive Gaussian noise

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._

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

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📋 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": [
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      "license_hash": null,
      "name": "primary",
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  ],
  "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": {
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    "allowed_omega_dimensions": [
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    "allowed_problem_classes": [
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    "center_spec": {
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      "input_format": "measurement_only",
      "omega": {
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    "epsilon_bounds": {
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    "omega_bounds": {
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      "N_sensors": [
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      "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"
}

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