# ⚛  L1 Principle — Drug-Target Binding Affinity Classification (PWDR)

**ID:** `L1-521` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Chemistry — *Quantum-chemistry binding-energy estimation with affinity-class categorical readout*
> **🎯 Problem class:** nonlinear inverse with categorical readout · **🧮 Solution space:** 1D affinity class
> **📡 Carrier:** atomic_potential · **🌫 Noise:** ensemble variance
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554143

---

## 🧠 1. Introduction

**Drug-Target Binding Affinity Classification (PWDR)** is a **nonlinear inverse with categorical readout** whose unknown lives in **1D affinity class** space, within the **Quantum-chemistry binding-energy estimation with affinity-class categorical readout** sub-domain of **Computational Chemistry**.

Measurements consist of atomic potential via a **fep with affinity threshold** sensing mechanism.

The forward operator applies, in order: L · molecular topology operator; L · dft charge calculation operator; L · force field parameterization operator; L · molecular dynamics operator; L · fep thermodynamic cycle operator; L · binding free energy operator; L · affinity threshold classifier operator; int · ensemble operator.

Observations are corrupted by ensemble variance. Existence guaranteed within Omega bounds. Uniqueness conditional on adequate sampling (typically ≥100 ns of FEP for druglike ligands) and convergent thermodynamic cycle. Stability dominated by force_field_uncertainty (~1-2 kcal/mol systematic error) and sampling_convergence_error. Joint Hadamard well-posedness for the coupled DFT + MD + FEP + threshold forward established by Wang 2015 (FEP+ benchmark), Mey 2020 (best practices for FEP), Cournia 2020 (relative binding free energy review), Lipinski 1997 (Rule of 5), Veber 2002 (drug-likeness rules), Hopkins 2004 (ligand efficiency).

## ⚙ 2. Forward Model

Physical chain: **x** → L · molecular topology → L · dft charge calculation → L · force field parameterization → L · molecular dynamics → L · fep thermodynamic cycle → L · binding free energy → L · affinity threshold classifier → int · ensemble → **y** (detector).

```
y = `int.ensemble` `L.affinity_threshold_classifier` `L.binding_free_energy` `L.fep_thermodynamic_cycle` `L.molecular_dynamics` `L.force_field_parameterization` `L.dft_charge_calculation` `L.molecular_topology` x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.molecular_topology` | L · molecular topology operator |
| `L.dft_charge_calculation` | L · dft charge calculation operator |
| `L.force_field_parameterization` | L · force field parameterization operator |
| `L.molecular_dynamics` | L · molecular dynamics operator |
| `L.fep_thermodynamic_cycle` | L · fep thermodynamic cycle operator |
| `L.binding_free_energy` | L · binding free energy operator |
| `L.affinity_threshold_classifier` | L · affinity threshold classifier operator |
| `int.ensemble` | Int · ensemble operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Chemistry |
| Sub domain | Quantum-chemistry binding-energy estimation with affinity-class categorical readout |
| Carrier | atomic_potential |
| Problem class | nonlinear_inverse_with_categorical_readout |
| Solution space | 1D_affinity_class |
| Noise model | ensemble_variance |
| Integration axis | ensemble |
| Difficulty delta | 5 |
| L dag | 7 |

## 📡 4. Measurement Model

Existence guaranteed within Omega bounds. Uniqueness conditional on adequate sampling (typically ≥100 ns of FEP for druglike ligands) and convergent thermodynamic cycle. Stability dominated by force_field_uncertainty (~1-2 kcal/mol systematic error) and sampling_convergence_error. Joint Hadamard well-posedness for the coupled DFT + MD + FEP + threshold forward established by Wang 2015 (FEP+ benchmark), Mey 2020 (best practices for FEP), Cournia 2020 (relative binding free energy review), Lipinski 1997 (Rule of 5), Veber 2002 (drug-likeness rules), Hopkins 2004 (ligand efficiency).

| Metric | Value |
|---|---|
| Metric | categorical_accuracy |
| Secondary | RMSE_log10_Kd |

## 📏 5. Operating Range (Ω)

**Center problem class:** `drug_target_affinity_pwdr` · **Forward operator:** `fep_affinity_pwdr_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Force field | — | AMBER_FF14SB |
| Water model | — | TIP3P |
| N atoms drug | — | 50 |
| Temperature k | — | 298 |
| N atoms target | — | 5000 |
| Simulation time ns | ns | 100 |
| Salt concentration m | m | 0.15 |
| Drug protonation state | — | 0 |
| Force field uncertainty | — | 0 |
| Water model uncertainty | — | 0 |
| Sampling convergence error | — | 0 |
| Protein flexibility truncation | — | 0 |
| Conformational search incompleteness | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N atoms drug | — | 10 – 200 |
| Temperature k | — | 273 – 320 |
| N atoms target | — | 500 – 50000 |
| Simulation time ns | ns | 1 – 10000 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.70_accuracy

| Metric | Range |
|---|---|
| Categorical accuracy | 0.3 – 0.95 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **categorical_accuracy**, with κ = `100` and δ = `5`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0x361fddfc11128102f96777208aa7734516caaa5086b4f3878351ed4f1fd3073a`
- **Chain tx hash:** `0x5ff82741bdf6b9f8b5e4db2a02645c615150199e7bc19e96ee2f2beb96d2233f`
- **Chain block:** `41554143`

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## File Mapping

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| File | Role | How to regenerate |
|------|------|-------------------|
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| `L1-521.json` | Structured metadata for the registry | LLM regenerates from the sections above |

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