L1 ⚛ L1-170 ⊙ Testnet

Incompressible Navier-Stokes — low-Mach viscous flow

Fluid Dynamics · Incompressible NS at moderate Re

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⬇ principle.md ⬇ principle.json

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⬇ L1-170.md ⬇ L1-170.json

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You are helping me submit a MODIFICATION of L1-170 (Incompressible Navier-Stokes — low-Mach viscous flow) to the PWM Protocol — a Principle (L1) artifact.

I will paste a Markdown template (or the current L1-170.md).
1. Rewrite the Markdown so the science is correct and clearly explained for my change.
2. Regenerate the sibling L1-170.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 = ∫_t dt `L.pressure_projection` ∂_t ρ + ∇·(ρv) ∂_t(ρv) + ∇·(ρv⊗v) x + n,    n ~ 𝒩(0, σ²)

world state x (3D velocity pressure fields)

→

Momentum equation ∂_t(ρv) + ∇·(ρv⊗v)

→

Continuity equation ∂_t ρ + ∇·(ρv)

→

L · pressure projection `L.pressure_projection`

→

Temporal integration ∫_t dt

→

observation y (mechanical detector)

Noise: additive Gaussian noise

Markdown — human-readable source of truth

⚙ auto-generated

⬇ L1-170.md

# ⚛ L1 Principle — Incompressible Navier-Stokes — low-Mach viscous flow **ID:** `L1-170` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Fluid Dynamics — *Incompressible NS at moderate Re* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 3D velocity pressure fields > **📡 Carrier:** mechanical · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553965 --- ## 🧠 1. Introduction **Incompressible Navier-Stokes — low-Mach viscous flow** is a **nonlinear inverse problem** whose unknown lives in **3D velocity pressure fields** space, within the **Incompressible NS at moderate Re** sub-domain of **Fluid Dynamics**. Measurements consist of mechanical vibrations or strain signals via a **piv pressure probes** sensing mechanism. The forward operator applies, in order: momentum balance (Navier-Stokes / Euler); mass-balance: ∂_t ρ + ∇·(ρv) = 0; L · pressure projection operator; detector accumulates flux over the exposure window. Observations are corrupted by additive Gaussian noise. Conditional stability; mismatch parameters dominate at Omega bounds. ## ⚙ 2. Forward Model Physical chain: **x** → Momentum equation → Continuity equation → L · pressure projection → Temporal integration → **y** (detector). ``` y = ∫_t dt `L.pressure_projection` ∂_t ρ + ∇·(ρv) ∂_t(ρv) + ∇·(ρv⊗v) x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.momentum` | Momentum balance (navier-stokes / euler) | | `L.continuity` | Mass-balance: ∂_t ρ + ∇·(ρv) = 0 | | `L.pressure_projection` | L · pressure projection operator | | `int.temporal` | Detector accumulates flux over the exposure window | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Fluid Dynamics | | Sub domain | Incompressible NS at moderate Re | | Carrier | mechanical | | Problem class | nonlinear_inverse | | Solution space | 3D_velocity_pressure_fields | | Noise model | gaussian | | Integration axis | temporal | | Difficulty delta | 5 | | L dag | 3.5 | ## 📡 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:** `incomp_ns` · **Forward operator:** `incomp_ns_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 256 | | W | px | 256 | | Z | — | 128 | | Re | — | 1000 | | Snr db | dB | 30 | | N times | — | 500 | | Bc error | — | 0 | | Nu error | — | 0 | | Initial condition error | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 32 – 4096 | | W | px | 32 – 4096 | | Z | — | 16 – 1024 | | Re | — | 1 – 100000000.0 | | Snr db | dB | 0 – 40 | | N times | — | 50 – 100000 | | Bc error | — | 0.0 – 0.2 | | Nu error | — | 0.0 – 0.1 | | Initial condition error | — | 0.0 – 0.3 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 22.0 | Metric | Range | |---|---| | Psnr db | 10.0 – 45.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `600` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xeac424843de2c14dd8b64269029a8866cfe3d7db8510c686570778618df557d0` - **Chain tx hash:** `0xf77b9f0ff6ab90d53e466e80e4466a541d856deab3c2cd4982eb8ae594f1444c` - **Chain block:** `41553965` --- ## File Mapping This bundle consists of: `L1-170.md`, `L1-170.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-170.md` | Source of truth — edit this | Human or LLM | | `L1-170.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 `L1-170`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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L2-170-001 Incompressible Navier-Stokes — low-Mach viscous flow Mismatch-Only Spec

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📋 JSON metadata

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  "artifact_id": "L1-170",
  "chain_block": 41553965,
  "chain_hash": "0xeac424843de2c14dd8b64269029a8866cfe3d7db8510c686570778618df557d0",
  "chain_tx_hash": "0xf77b9f0ff6ab90d53e466e80e4466a541d856deab3c2cd4982eb8ae594f1444c",
  "domain": "Fluid Dynamics",
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    "secondary": "SSIM"
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  "physics_fingerprint": {
    "L_DAG": 3.5,
    "carrier": "mechanical",
    "difficulty_delta": 5,
    "domain": "Fluid Dynamics",
    "integration_axis": "temporal",
    "noise_model": "gaussian",
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    "sensing_mechanism": "piv_pressure_probes",
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    "sub_domain": "Incompressible NS at moderate Re",
    "title": "Incompressible Navier-Stokes \u2014 low-Mach viscous flow"
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  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Incompressible NS at moderate Re",
  "title": "Incompressible Navier-Stokes \u2014 low-Mach viscous flow"
}

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