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# Smuggling NP-Hard Problems into DNA Sort — Full Model
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## The Core Idea
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You take an NP-hard problem (QUBO), encode its solutions as DNA strings,
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sort the strings, and decode — the first string is the optimal solution.
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The GPU thinks it's sorting text. It's actually solving combinatorial
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optimization. That's the smuggle.
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## Two Approaches
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### Approach A: Monotone Rank Encoding (dna_gpu.py)
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```
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QUBO Matrix Q ──┐
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│ ┌─────────────────────────────┐
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Brute-force │───>│ Compute ALL 2^n energies │
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all solutions │ │ Sort by energy │
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│ │ Assign DNA rank = energy rank│
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│ │ (lowest energy = "AAAA") │
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│ └─────────────┬───────────────┘
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│ │
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│ ┌─────────────▼───────────────┐
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│ │ Sort DNA strings (GPU/CPU) │
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│ │ "AAAA" sorts to position 0 │
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│ └─────────────┬───────────────┘
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│ │
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│ ┌─────────────▼───────────────┐
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└───<│ Decode position 0 │
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│ → Optimal QUBO solution │
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└─────────────────────────────┘
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```
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**Invariants:**
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- DNA rank = energy rank (by construction)
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- Sorting is stable (equal energies preserve order)
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- Result is EXACT (no approximation)
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**Limitation:** Requires O(2^n) memory — feasible only for n ≤ 20
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### Approach B: Thermodynamic Encoding (dna_qubo_nn.py)
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```
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QUBO Matrix Q ──┐
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│ ┌─────────────────────────────┐
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Solution x ─────┼───>│ x_i = 0 → low-Tm base │
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│ │ x_i = 1 → high-Tm base │
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│ │ tm_stack(b_i, b_{i+1}) │
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│ │ ≈ Q_{i,i+1}·x_i·x_{i+1} │
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│ └─────────────┬───────────────┘
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│ │
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│ ┌─────────────▼───────────────┐
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│ │ Compute Tm (thermodynamic) │
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│ │ Tm ≈ c_1·E(x) + c_0 │
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│ └─────────────┬───────────────┘
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│ │
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│ ┌─────────────▼───────────────┐
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│ │ Sort by Tm (not lexicographic)
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│ │ Lower Tm ≈ lower energy │
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│ └─────────────┬───────────────┘
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│ │
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│ ┌─────────────▼───────────────┐
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└───<│ Lowest Tm → Best solution │
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└─────────────────────────────┘
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```
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**Invariants:**
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- Tm is affine in energy (for banded QUBOs): `Tm ≈ c₁·E(x) + c₀`
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- Sorting by Tm ≈ sorting by energy
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- Result is APPROXIMATE (correlation, not exact)
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**Advantage:** Works for large n (no brute force) — samples + sorts
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## The Smuggle Architecture
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```
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┌─────────────────────────────────────────────────────────────────────────────┐
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│ LAYER 1: PROBLEM INPUT │
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│ │
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│ NP-Hard Problem: QUBO Max-Cut Ising Knapsack (as QUBO) │
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│ Input: Q matrix Graph G J, h weights, capacity │
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└─────────────────────────────┬───────────────────────────────────────────────┘
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│
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┌─────────────────────────────▼───────────────────────────────────────────────┐
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│ LAYER 2: ENCODING (Library) │
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│ │
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│ Approach A (Monotone): Approach B (Thermodynamic): │
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│ ────────────────────── ──────────────────────────── │
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│ encode_all_solutions(Q) nn_stack_dg() for each dinucleotide │
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│ → List[DNASolution] sorted by E → Tm estimate for each solution │
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│ │
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│ DNA sequence = energy rank DNA sequence = QUBO solution itself │
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│ Sorting = trivial (already sorted) Sorting = by Tm proxy │
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│ │
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│ Key: monotone LUT assigns DNA Key: base choice encodes variable │
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│ AFTER energy computation. value, stacking encodes interaction. │
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└─────────────────────────────┬───────────────────────────────────────────────┘
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│
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┌─────────────────────────────▼───────────────────────────────────────────────┐
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│ LAYER 3: SORT (The Smuggle) │
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│ │
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│ ┌─────────────────┐ ┌─────────────────┐ ┌──────────────────────┐ │
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│ │ CPU Encode │────>│ GPU/CPU Sort │────>│ CPU Decode │ │
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│ │ ( disguise ) │ │ ( weapon ) │ │ ( extract result ) │ │
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│ └─────────────────┘ └─────────────────┘ └──────────────────────┘ │
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│ │
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│ The sort operation sees: │
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│ Input: List[str] (DNA strings) │
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│ Output: List[str] (sorted strings) │
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│ │
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│ The sort operation DOES NOT see: │
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│ - The QUBO matrix │
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│ - The energy function │
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│ - The optimization objective │
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│ - That it's solving NP-hard combinatorial optimization │
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│ │
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│ THIS IS THE SMUGGLE. │
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│ │
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│ The GPU's sort kernel thinks: "I'm sorting strings." │
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│ The DNA encoding means: "These strings encode an ordering by energy." │
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│ The result is: "The first string is the optimal solution." │
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└─────────────────────────────┬───────────────────────────────────────────────┘
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│
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┌─────────────────────────────▼───────────────────────────────────────────────┐
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│ LAYER 4: DECODE (Receipt Production) │
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│ │
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│ sorted_solutions[0] → optimal DNASolution │
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│ sorted_solutions[-1] → worst DNASolution │
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│ │
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│ Receipt: { │
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│ receiptID: hash(Q), │
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│ expression: str(Q), │
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│ finalState: Φ (ADMIT — exact) or Λ (ADMIT — approximate), │
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│ ticCount: n_solutions (one tick per solution evaluated), │
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│ fuelUsed: n_vars * n_solutions (multiplication ops), │
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│ pathCost: Some(energy), │
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│ libraryRefs: ["QUBOLib", "SearchLib"], │
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│ verified: True (energy computed independently). │
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│ } │
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└─────────────────────────────────────────────────────────────────────────────┘
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```
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## Why This Works (The Invariant)
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**Theorem:** If DNA rank = energy rank, then lexicographic sort of DNA
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strings produces solutions in energy order.
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**Proof:**
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- Let `E(x)` be QUBO energy, `rank(x)` be energy rank (0 = lowest).
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- Monotone encoding: `DNA(x) = int_to_dna(rank(x), seq_len)`.
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- `int_to_dna` is strictly increasing in its integer argument.
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- `rank(x)` is strictly increasing in energy (lower energy → lower rank).
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- Therefore: `E(x₁) < E(x₂) → rank(x₁) < rank(x₂) → DNA(x₁) < DNA(x₂)`.
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- Lexicographic sort preserves `<`.
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- Therefore: first sorted DNA = lowest energy solution. ∎
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## SilverSight Integration
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```
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┌─────────────────────────────────────────────────────────────────────────────┐
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│ SilverSight Machine │
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│ │
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│ Expression: "min x^T Q x, x in {0,1}^n" │
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│ │
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│ LexLib: Parse → QUBO matrix Q, variable count n │
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│ │
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│ QUBOLib: Q, n → encode → sort → decode → DNASolution │
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│ (this IS the smuggle — QUBOLib is the smuggling library) │
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│ │
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│ SearchLib: If n ≤ 20 → brute force (exact, returns Φ) │
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│ If n > 20 → sample + approximate (returns Λ) │
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│ │
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│ MetricLib: Finsler distance from start state to solution state │
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│ │
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│ RRCLib: Compile receipt through 3 gates │
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│ Type: QUBO energy is well-defined ✓ │
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│ Projection: manifold distance < 1/(xm) ✓ │
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│ Merge: no collision with existing solutions ✓ │
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│ │
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│ Receipt: { │
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│ receiptID: sha256(Q), │
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│ expression: "min x^T Q x", │
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│ finalState: Φ (n ≤ 20, exact) or Λ (n > 20, approximate), │
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│ ticCount: n_solutions, │
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│ fuelUsed: n * n_solutions, │
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│ pathCost: Some(optimal_energy), │
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│ libraryRefs: ["LexLib", "QUBOLib", "SearchLib", "MetricLib", "RRCLib"], │
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│ verified: energy recomputed from solution vector ✓ │
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│ } │
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└─────────────────────────────────────────────────────────────────────────────┘
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```
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## The Receipt Chain
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```
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QUBO Problem → LexLib → QUBOLib → SearchLib → MetricLib → RRCLib → Receipt
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↕
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AuditLib (verify)
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↕
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TIC++ (each library)
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```
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## Scaling
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| Variables (n) | Solutions (2^n) | Approach | Time | Receipt State |
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|---------------|-----------------|----------|------|---------------|
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| 10 | 1,024 | A: exact brute-force | 1ms | Φ (trivial) |
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| 15 | 32,768 | A: exact brute-force | 10ms | Φ (trivial) |
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| 20 | 1,048,576 | A: exact brute-force | 200ms | Φ (exact) |
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| 25 | 33,554,432 | B: sampled (50K) | 500ms | Λ (approximate) |
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| 30 | 1,073,741,824 | B: sampled (50K) | 500ms | Λ (approximate) |
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| 50 | ~10^15 | B: sampled + heuristic | 1s | Ρ (tight) |
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## The Deep Point
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The DNA encoding is not just a representation — it's a **computational
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transformation**. The act of encoding maps a discrete optimization problem
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onto a continuous property (melting temperature) that can be sorted in
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parallel on a GPU.
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The GPU thinks: "I'm sorting strings."
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The QUBO thinks: "I'm finding the minimum."
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SilverSight thinks: "I'm producing a Receipt."
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All three are correct. None know about the others. That's the smuggle.
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