L1 ⚛ L1-411 ⊙ Testnet

Oxygen Transport (Krogh Cylinder)

Computational Biology · Physiology

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⬇ L1-411.md

# ⚛ L1 Principle — Oxygen Transport (Krogh Cylinder) **ID:** `L1-411` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Computational Biology — *Physiology* > **🎯 Problem class:** parameter estimation · **🧮 Solution space:** oxygen distribution field > **📡 Carrier:** N/A · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555219 --- ## 🧠 1. Introduction **Oxygen Transport (Krogh Cylinder)** is a **parameter-estimation problem** whose unknown lives in **oxygen distribution field** space, within the **Physiology** sub-domain of **Computational Biology**. Measurements consist of N/A via a **po2 measurement fiber optic** sensing mechanism. The forward operator applies, in order: gradient / divergence with respect to position; S · fdm · oxygen diffusion operator; O · chi2 · po2 profile operator. Observations are corrupted by additive Gaussian noise. Existence of the recovered oxygen_distribution_field 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 ~= 50); non_uniform_capillary_spacing 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 · fdm · oxygen diffusion → O · chi2 · po2 profile → **y** (detector). ``` y = `O.chi2.po2_profile` `S.fdm.oxygen_diffusion` ∇ x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `D.space` | Gradient / divergence with respect to position | | `S.fdm.oxygen_diffusion` | S · fdm · oxygen diffusion operator | | `O.chi2.po2_profile` | O · chi2 · po2 profile operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Computational Biology | | Sub domain | Physiology | | Carrier | N/A | | Problem class | parameter_estimation | | Solution space | oxygen_distribution_field | | Noise model | gaussian | | Integration axis | radial_tissue | | Difficulty delta | 3 | | L dag | 2.8 | ## 📡 4. Measurement Model Existence of the recovered oxygen_distribution_field 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 ~= 50); non_uniform_capillary_spacing dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor. | Metric | Value | |---|---| | Metric | pO2_profile_RMSE_mmHg | | Secondary | anoxic_fraction_error | ## 📏 5. Operating Range (Ω) **Center problem class:** `parameter_estimation` · **Forward operator:** `po2_measurement_fiber_optic` **Center point:** | Parameter | Unit | Value | |---|---|---| | R t tissue um | µm | 50 | | Q max mlo2 ml s | s | 0.0005 | | Arterial po2 mmhg | — | 100 | | Capillary radius um | µm | 5 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | R t tissue um | µm | 10 – 200 | | Q max mlo2 ml s | s | 0.0001 – 0.005 | | Arterial po2 mmhg | — | 50 – 150 | | Capillary radius um | µm | 2 – 20 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 5 pO2_profile_RMSE_mmHg | Metric | Range | |---|---| | Po2 profile rmse mmhg | 0.5 – 30 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **pO2_profile_RMSE_mmHg**, with κ = `1000` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x04892ea35760c3889fa934f728182804e3f053623ababf6b45d343a17885cbca` - **Chain tx hash:** `0x5e5eb665acbb6f52a4a75a7ab90a69309f68eafff1619f8289bc502746e887ac` - **Chain block:** `41555219` --- ## File Mapping This bundle consists of: `L1-411.md`, `L1-411.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-411.md` | Source of truth — edit this | Human or LLM | | `L1-411.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 `L1-411`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-411.md).
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L2-411-001 Oxygen Transport (Krogh Cylinder) — Nominal + Mismatch Specs

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📋 JSON metadata

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  "chain_block": 41555219,
  "chain_hash": "0x04892ea35760c3889fa934f728182804e3f053623ababf6b45d343a17885cbca",
  "chain_tx_hash": "0x5e5eb665acbb6f52a4a75a7ab90a69309f68eafff1619f8289bc502746e887ac",
  "domain": "Computational Biology",
  "hardness_fn": {
    "delta": 3,
    "kappa": 1000,
    "metric": "pO2_profile_RMSE_mmHg",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
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      "ipfs_cid": null,
      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
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  "layer": "L1",
  "observable_profile": {
    "metric": "pO2_profile_RMSE_mmHg",
    "regime": "Existence of the recovered oxygen_distribution_field 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 ~= 50); non_uniform_capillary_spacing dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.",
    "secondary": "anoxic_fraction_error"
  },
  "physics_fingerprint": {
    "L_DAG": 2.8,
    "carrier": "N/A",
    "difficulty_delta": 3,
    "domain": "Computational Biology",
    "integration_axis": "radial_tissue",
    "noise_model": "gaussian",
    "primitives": [
      "D.space",
      "S.fdm.oxygen_diffusion",
      "O.chi2.po2_profile"
    ],
    "problem_class": "parameter_estimation",
    "sensing_mechanism": "po2_measurement_fiber_optic",
    "solution_space": "oxygen_distribution_field",
    "sub_domain": "Physiology",
    "title": "Oxygen Transport (Krogh Cylinder)"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "po2_measurement_fiber_optic"
    ],
    "allowed_omega_dimensions": [
      "capillary_radius_um",
      "R_t_tissue_um",
      "Q_max_mlO2_ml_s",
      "arterial_pO2_mmHg"
    ],
    "allowed_problem_classes": [
      "parameter_estimation"
    ],
    "center_spec": {
      "epsilon_fn_center": "5 pO2_profile_RMSE_mmHg",
      "forward_operator": "po2_measurement_fiber_optic",
      "input_format": "measurement_only",
      "omega": {
        "Q_max_mlO2_ml_s": 0.0005,
        "R_t_tissue_um": 50,
        "arterial_pO2_mmHg": 100,
        "capillary_radius_um": 5
      },
      "problem_class": "parameter_estimation"
    },
    "epsilon_bounds": {
      "pO2_profile_RMSE_mmHg": [
        0.5,
        30
      ]
    },
    "omega_bounds": {
      "Q_max_mlO2_ml_s": [
        0.0001,
        0.005
      ],
      "R_t_tissue_um": [
        10,
        200
      ],
      "arterial_pO2_mmHg": [
        50,
        150
      ],
      "capillary_radius_um": [
        2,
        20
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Physiology",
  "title": "Oxygen Transport (Krogh Cylinder)"
}

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