L1 ⚛ L1-435 ⊙ Testnet

4D-Var Data Assimilation

Control Theory · Variational data assimilation

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⬇ principle.md ⬇ principle.json

Fork L1-435

⬇ L1-435.md ⬇ L1-435.json

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You are helping me submit a MODIFICATION of L1-435 (4D-Var Data Assimilation) to the PWM Protocol — a Principle (L1) artifact.

I will paste a Markdown template (or the current L1-435.md).
1. Rewrite the Markdown so the science is correct and clearly explained for my change.
2. Regenerate the sibling L1-435.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.cost_function.4dvar` `S.adjoint.gradient_4dvar` ∇ x + n,    n ~ 𝒩(0, σ²)

world state x (initial condition vector)

→

Spatial derivative ∇

→

S · adjoint · gradient 4dvar `S.adjoint.gradient_4dvar`

→

O · cost function · 4dvar `O.cost_function.4dvar`

→

observation y (N/A detector)

Noise: additive Gaussian noise

Markdown — human-readable source of truth

⚙ auto-generated

⬇ L1-435.md

# ⚛ L1 Principle — 4D-Var Data Assimilation **ID:** `L1-435` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Control Theory — *Variational data assimilation* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** initial condition vector > **📡 Carrier:** N/A · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41555258 --- ## 🧠 1. Introduction **4D-Var Data Assimilation** is a **nonlinear inverse problem** whose unknown lives in **initial condition vector** space, within the **Variational data assimilation** sub-domain of **Control Theory**. Measurements consist of N/A via a **variational state estimation** sensing mechanism. The forward operator applies, in order: gradient / divergence with respect to position; S · adjoint · gradient 4dvar operator; O · cost function · 4dvar operator. Observations are corrupted by additive Gaussian noise. Existence of the recovered initial_condition_vector 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 ~= 10000.0); model_error_within_window dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor. ## ⚙ 2. Forward Model Physical chain: **x** → Spatial derivative → S · adjoint · gradient 4dvar → O · cost function · 4dvar → **y** (detector). ``` y = `O.cost_function.4dvar` `S.adjoint.gradient_4dvar` ∇ x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `D.space` | Gradient / divergence with respect to position | | `S.adjoint.gradient_4dvar` | S · adjoint · gradient 4dvar operator | | `O.cost_function.4dvar` | O · cost function · 4dvar operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Control Theory | | Sub domain | Variational data assimilation | | Carrier | N/A | | Problem class | nonlinear_inverse | | Solution space | initial_condition_vector | | Noise model | gaussian | | Integration axis | assimilation_window | | Difficulty delta | 5 | | L dag | 4.5 | ## 📡 4. Measurement Model Existence of the recovered initial_condition_vector 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 ~= 10000.0); model_error_within_window dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor. | Metric | Value | |---|---| | Metric | analysis_RMSE_vs_background | | Secondary | cost_function_reduction | ## 📏 5. Operating Range (Ω) **Center problem class:** `nonlinear_inverse` · **Forward operator:** `variational_state_estimation` **Center point:** | Parameter | Unit | Value | |---|---|---| | Window hours | — | 6 | | N observations | — | 5000 | | State dimension n | N | 10000 | | Observation density | — | 1 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | Window hours | — | 1 – 24 | | N observations | — | 100 – 10000000 | | State dimension n | N | 100 – 1000000 | | Observation density | — | 0.1 – 10.0 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 0.5 analysis_RMSE_vs_background | Metric | Range | |---|---| | Analysis rmse vs background | 0.1 – 1.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **analysis_RMSE_vs_background**, with κ = `1000000.0` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x9641f2631b1a13749ead5c81801872e13fe277403e72819087cd6bde2c95aba7` - **Chain tx hash:** `0xa9c24989164e6d9a5bbb4ff8b516e1a5b4e444b0a32ad763440d8352438b5d91` - **Chain block:** `41555258` --- ## File Mapping This bundle consists of: `L1-435.md`, `L1-435.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-435.md` | Source of truth — edit this | Human or LLM | | `L1-435.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 `L1-435`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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L2-435-001 4D-Var Data Assimilation — Nominal + Mismatch Specs

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📋 JSON metadata

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  "artifact_id": "L1-435",
  "chain_block": 41555258,
  "chain_hash": "0x9641f2631b1a13749ead5c81801872e13fe277403e72819087cd6bde2c95aba7",
  "chain_tx_hash": "0xa9c24989164e6d9a5bbb4ff8b516e1a5b4e444b0a32ad763440d8352438b5d91",
  "domain": "Control Theory",
  "hardness_fn": {
    "delta": 5,
    "kappa": 1000000.0,
    "metric": "analysis_RMSE_vs_background",
    "type": "epsilon_fn"
  },
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      "license_hash": null,
      "name": "primary",
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  "layer": "L1",
  "observable_profile": {
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    "regime": "Existence of the recovered initial_condition_vector 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 ~= 10000.0); model_error_within_window dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.",
    "secondary": "cost_function_reduction"
  },
  "physics_fingerprint": {
    "L_DAG": 4.5,
    "carrier": "N/A",
    "difficulty_delta": 5,
    "domain": "Control Theory",
    "integration_axis": "assimilation_window",
    "noise_model": "gaussian",
    "primitives": [
      "D.space",
      "S.adjoint.gradient_4dvar",
      "O.cost_function.4dvar"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "variational_state_estimation",
    "solution_space": "initial_condition_vector",
    "sub_domain": "Variational data assimilation",
    "title": "4D-Var Data Assimilation"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "variational_state_estimation"
    ],
    "allowed_omega_dimensions": [
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      "N_observations",
      "window_hours",
      "observation_density"
    ],
    "allowed_problem_classes": [
      "nonlinear_inverse"
    ],
    "center_spec": {
      "epsilon_fn_center": "0.5 analysis_RMSE_vs_background",
      "forward_operator": "variational_state_estimation",
      "input_format": "measurement_only",
      "omega": {
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        "observation_density": 1.0,
        "state_dimension_n": 10000,
        "window_hours": 6
      },
      "problem_class": "nonlinear_inverse"
    },
    "epsilon_bounds": {
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        0.1,
        1.0
      ]
    },
    "omega_bounds": {
      "N_observations": [
        100,
        10000000
      ],
      "observation_density": [
        0.1,
        10.0
      ],
      "state_dimension_n": [
        100,
        1000000
      ],
      "window_hours": [
        1,
        24
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Variational data assimilation",
  "title": "4D-Var Data Assimilation"
}

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