L1 ⚛ L1-499 ⊙ Testnet

Shape Optimization

Optimization · Boundary shape

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⬇ principle.md ⬇ principle.json

Fork L1-499

⬇ L1-499.md ⬇ L1-499.json

Prompt — copy into your LLM

You are helping me submit a MODIFICATION of L1-499 (Shape Optimization) to the PWM Protocol — a Principle (L1) artifact.

I will paste a Markdown template (or the current L1-499.md).
1. Rewrite the Markdown so the science is correct and clearly explained for my change.
2. Regenerate the sibling L1-499.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.gradient.descent` `S.shape.derivative` x    (deterministic)

world state x (2D boundary curve)

→

S · shape · derivative `S.shape.derivative`

→

O · gradient · descent `O.gradient.descent`

→

observation y (N/A detector)

Noise: no stochastic noise (deterministic measurement)

Solver / auxiliary components (1)

O.iter — a fixed-point or gradient iteration on the unknown

Markdown — human-readable source of truth

⚙ auto-generated

⬇ L1-499.md

# ⚛ L1 Principle — Shape Optimization **ID:** `L1-499` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Optimization — *Boundary shape* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 2D boundary curve > **📡 Carrier:** N/A · **🌫 Noise:** deterministic > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555318 --- ## 🧠 1. Introduction **Shape Optimization** is a **nonlinear inverse problem** whose unknown lives in **2D boundary curve** space, within the **Boundary shape** sub-domain of **Optimization**. Measurements consist of N/A via a **shape sensitivity analysis** sensing mechanism. The forward operator applies, in order: S · shape · derivative operator; a fixed-point or gradient iteration on the unknown; O · gradient · descent operator. Observations are corrupted by no stochastic noise (deterministic measurement). Existence of the recovered 2D_boundary_curve is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 100); mesh_quality_degradation dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Deterministic sets the irreducible data-fidelity floor. ## ⚙ 2. Forward Model Physical chain: **x** → S · shape · derivative → O · gradient · descent → **y** (detector). ``` y = `O.gradient.descent` `S.shape.derivative` x (deterministic) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `S.shape.derivative` | S · shape · derivative operator | | `O.gradient.descent` | O · gradient · descent operator | **🛠 Solver components** _(used inside the solver, not in the forward equation)_: | Primitive | What it does | |---|---| | `O.iter` | A fixed-point or gradient iteration on the unknown | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Optimization | | Sub domain | Boundary shape | | Carrier | N/A | | Problem class | nonlinear_inverse | | Solution space | 2D_boundary_curve | | Noise model | deterministic | | Integration axis | boundary_surface | | Difficulty delta | 3 | | L dag | 3 | ## 📡 4. Measurement Model Existence of the recovered 2D_boundary_curve is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 100); mesh_quality_degradation dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Deterministic sets the irreducible data-fidelity floor. | Metric | Value | |---|---| | Metric | objective_reduction_ratio | | Secondary | constraint_violation | ## 📏 5. Operating Range (Ω) **Center problem class:** `nonlinear_inverse` · **Forward operator:** `shape_sensitivity_analysis` **Center point:** | Parameter | Unit | Value | |---|---|---| | Step size dt | — | 0.1 | | N control pts | — | 50 | | Regularization alpha | — | 0.01 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | Step size dt | — | 0.01 – 1.0 | | N control pts | — | 10 – 200 | | Regularization alpha | — | 0.001 – 0.1 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 0.30 objective_reduction_ratio | Metric | Range | |---|---| | Objective reduction ratio | 0.1 – 0.6 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **objective_reduction_ratio**, with κ = `5000` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xdc96f40d3e84826813b0ddb0c1843e6922da9104742e19ffb27457113e3af136` - **Chain tx hash:** `0xd945e1c325827dfc3d0806d148f0b1ef9e34bcf195e2811b702c7eea212b49bc` - **Chain block:** `41555318` --- ## File Mapping This bundle consists of: `L1-499.md`, `L1-499.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-499.md` | Source of truth — edit this | Human or LLM | | `L1-499.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 `L1-499`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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L2-499-001 Shape Optimization — Nominal + Mismatch Specs

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📋 JSON metadata

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  "artifact_id": "L1-499",
  "chain_block": 41555318,
  "chain_hash": "0xdc96f40d3e84826813b0ddb0c1843e6922da9104742e19ffb27457113e3af136",
  "chain_tx_hash": "0xd945e1c325827dfc3d0806d148f0b1ef9e34bcf195e2811b702c7eea212b49bc",
  "domain": "Optimization",
  "hardness_fn": {
    "delta": 3,
    "kappa": 5000,
    "metric": "objective_reduction_ratio",
    "type": "epsilon_fn"
  },
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      "license_hash": null,
      "name": "primary",
      "weight": 1.0
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  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "objective_reduction_ratio",
    "regime": "Existence of the recovered 2D_boundary_curve is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 100); mesh_quality_degradation dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Deterministic sets the irreducible data-fidelity floor.",
    "secondary": "constraint_violation"
  },
  "physics_fingerprint": {
    "L_DAG": 3.0,
    "carrier": "N/A",
    "difficulty_delta": 3,
    "domain": "Optimization",
    "integration_axis": "boundary_surface",
    "noise_model": "deterministic",
    "primitives": [
      "S.shape.derivative",
      "O.iter",
      "O.gradient.descent"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "shape_sensitivity_analysis",
    "solution_space": "2D_boundary_curve",
    "sub_domain": "Boundary shape",
    "title": "Shape Optimization"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "shape_sensitivity_analysis"
    ],
    "allowed_omega_dimensions": [
      "N_control_pts",
      "regularization_alpha",
      "step_size_dt"
    ],
    "allowed_problem_classes": [
      "nonlinear_inverse"
    ],
    "center_spec": {
      "epsilon_fn_center": "0.30 objective_reduction_ratio",
      "forward_operator": "shape_sensitivity_analysis",
      "input_format": "measurement_only",
      "omega": {
        "N_control_pts": 50,
        "regularization_alpha": 0.01,
        "step_size_dt": 0.1
      },
      "problem_class": "nonlinear_inverse"
    },
    "epsilon_bounds": {
      "objective_reduction_ratio": [
        0.1,
        0.6
      ]
    },
    "omega_bounds": {
      "N_control_pts": [
        10,
        200
      ],
      "regularization_alpha": [
        0.001,
        0.1
      ],
      "step_size_dt": [
        0.01,
        1.0
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Boundary shape",
  "title": "Shape Optimization"
}

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