L1 ⚛ L1-207 ⊙ Testnet

Linear Elasticity — Navier-Cauchy 3D solid mechanics

Structural Mechanics · Infinitesimal-strain 3D linear elasticity

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⬇ principle.md ⬇ principle.json

Fork L1-207

⬇ L1-207.md ⬇ L1-207.json

Prompt — copy into your LLM

You are helping me submit a MODIFICATION of L1-207 (Linear Elasticity — Navier-Cauchy 3D solid mechanics) to the PWM Protocol — a Principle (L1) artifact.

I will paste a Markdown template (or the current L1-207.md).
1. Rewrite the Markdown so the science is correct and clearly explained for my change.
2. Regenerate the sibling L1-207.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.solve` `L.constitutive` `L.fem_assemble` x + n,    n ~ 𝒩(0, σ²)

world state x (3D displacement field)

→

L · fem assemble `L.fem_assemble`

→

L · constitutive `L.constitutive`

→

L · solve `L.solve`

→

Spatial integration ∫_A dA

→

observation y (mechanical detector)

Noise: additive Gaussian noise

Markdown — human-readable source of truth

⚙ auto-generated

⬇ L1-207.md

# ⚛ L1 Principle — Linear Elasticity — Navier-Cauchy 3D solid mechanics **ID:** `L1-207` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Structural Mechanics — *Infinitesimal-strain 3D linear elasticity* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D displacement field > **📡 Carrier:** mechanical · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554009 --- ## 🧠 1. Introduction **Linear Elasticity — Navier-Cauchy 3D solid mechanics** is a **linear inverse problem** whose unknown lives in **3D displacement field** space, within the **Infinitesimal-strain 3D linear elasticity** sub-domain of **Structural Mechanics**. Measurements consist of mechanical vibrations or strain signals via a **strain gauges dic** sensing mechanism. The forward operator applies, in order: L · fem assemble operator; L · constitutive operator; L · solve 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 · fem assemble → L · constitutive → L · solve → Spatial integration → **y** (detector). ``` y = ∫_A dA `L.solve` `L.constitutive` `L.fem_assemble` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.fem_assemble` | L · fem assemble operator | | `L.constitutive` | L · constitutive operator | | `L.solve` | L · solve operator | | `int.spatial` | Pixel-level spatial averaging on the detector | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Structural Mechanics | | Sub domain | Infinitesimal-strain 3D linear elasticity | | Carrier | mechanical | | Problem class | linear_inverse | | Solution space | 3D_displacement_field | | Noise model | gaussian | | Integration axis | spatial | | Difficulty delta | 3 | | L dag | 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:** `linear_elasticity` · **Forward operator:** `linear_elasticity_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | N dof | — | 10000 | | Snr db | dB | 30 | | E error | — | 0 | | Nu error | — | 0 | | N sensors | — | 32 | | Load error | — | 0 | | Boundary error | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | N dof | — | 100 – 1000000 | | Snr db | dB | 0 – 40 | | E error | — | 0.0 – 0.1 | | Nu error | — | 0.0 – 0.05 | | N sensors | — | 4 – 4096 | | Load error | — | 0.0 – 0.1 | | Boundary error | — | 0.0 – 0.2 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 26.0 | Metric | Range | |---|---| | Psnr db | 10.0 – 45.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `240` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x47b3232750d828d3cb44e609e076325904fc4d5284ffb63a27c5198bfbcf0baa` - **Chain tx hash:** `0xe8ecc4d396d026d5f9a2bfdffcc59c6c07a39c2e2a2184aa71720b8c698e2656` - **Chain block:** `41554009` --- ## File Mapping This bundle consists of: `L1-207.md`, `L1-207.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-207.md` | Source of truth — edit this | Human or LLM | | `L1-207.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 `L1-207`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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L2-207-001 Linear Elasticity — Navier-Cauchy 3D solid mechanics Mismatch-Only Spec

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📋 JSON metadata

{
  "artifact_id": "L1-207",
  "chain_block": 41554009,
  "chain_hash": "0x47b3232750d828d3cb44e609e076325904fc4d5284ffb63a27c5198bfbcf0baa",
  "chain_tx_hash": "0xe8ecc4d396d026d5f9a2bfdffcc59c6c07a39c2e2a2184aa71720b8c698e2656",
  "domain": "Structural Mechanics",
  "hardness_fn": {
    "delta": 3,
    "kappa": 240,
    "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": 3.0,
    "carrier": "mechanical",
    "difficulty_delta": 3,
    "domain": "Structural Mechanics",
    "integration_axis": "spatial",
    "noise_model": "gaussian",
    "primitives": [
      "L.fem_assemble",
      "L.constitutive",
      "L.solve",
      "int.spatial"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "strain_gauges_dic",
    "solution_space": "3D_displacement_field",
    "sub_domain": "Infinitesimal-strain 3D linear elasticity",
    "title": "Linear Elasticity \u2014 Navier-Cauchy 3D solid mechanics"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "linear_elasticity_forward"
    ],
    "allowed_omega_dimensions": [
      "N_DOF",
      "N_sensors",
      "SNR_dB",
      "E_error",
      "nu_error",
      "boundary_error",
      "load_error",
      "E_error",
      "nu_error",
      "boundary_error",
      "load_error"
    ],
    "allowed_problem_classes": [
      "linear_elasticity"
    ],
    "center_spec": {
      "epsilon_fn_center": "26.0",
      "forward_operator": "linear_elasticity_forward",
      "input_format": "measurement_only",
      "omega": {
        "E_error": 0.0,
        "N_DOF": 10000,
        "N_sensors": 32,
        "SNR_dB": 30,
        "boundary_error": 0.0,
        "load_error": 0.0,
        "nu_error": 0.0
      },
      "problem_class": "linear_elasticity"
    },
    "epsilon_bounds": {
      "psnr_db": [
        10.0,
        45.0
      ]
    },
    "omega_bounds": {
      "E_error": [
        0.0,
        0.1
      ],
      "N_DOF": [
        100,
        1000000
      ],
      "N_sensors": [
        4,
        4096
      ],
      "SNR_dB": [
        0,
        40
      ],
      "boundary_error": [
        0.0,
        0.2
      ],
      "load_error": [
        0.0,
        0.1
      ],
      "nu_error": [
        0.0,
        0.05
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Infinitesimal-strain 3D linear elasticity",
  "title": "Linear Elasticity \u2014 Navier-Cauchy 3D solid mechanics"
}

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