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zeekayandHanzo Dev
0b87d2a9f2
Bylines, paper authorship, correspondence addresses and the founder blurb under each post named an individual. They name the organisation that published the work instead: the brand on posts and pages, the research byline on papers, and research@ in place of a personal address. Both are what these repositories already used elsewhere — nothing new was coined. Co-authors are untouched. Where a paper named someone else alongside, they stay named; only the one byline changed. Bibliography entries, running heads, PDF metadata and affiliations move with it, so a citation still resolves to whoever the cited paper names. Sample and fixture data uses the placeholder name these tests already use elsewhere, so no assertion changed meaning. Co-authored-by: Hanzo Dev <[email protected]>
107 lines
4.8 KiB
Lean4
107 lines
4.8 KiB
Lean4
/-
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GPU DEX SCALING LAWS
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Formal verification of GPU-accelerated decentralized exchange
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operations: orderbook matching, AMM routing, and settlement.
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A native DEX VM is the third core component of any modern chain.
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GPU acceleration enables sub-millisecond matching:
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1. Orderbook matching parallelizes across independent pairs
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2. AMM routing: multi-pool pathfinding on GPU (parallel BFS)
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3. Batch settlement: GPU processes all fills simultaneously
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4. Price oracle aggregation: median computation on GPU
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5. ZAP binary protocol: GPU-native serialization
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Authors: Lux Research, Woo Bin
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Zoo Labs Foundation
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-/
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import Mathlib.Data.Nat.Defs
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import Mathlib.Tactic
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namespace Zoo.GPU.DEX
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/-! ## Model
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A DEX processes orders across `m` trading pairs.
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Each pair has an independent orderbook.
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GPU parallelizes matching across pairs and within pairs
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(bid-ask sweep is sequential per pair but pairs are independent). -/
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/-- Independent pair parallelism: `m` trading pairs matched on
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`p` cores. Each pair's orderbook is independent.
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Time = max(time_per_pair) when p ≥ m. -/
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theorem pair_parallel (m p : Nat) (_hp : 0 < p) :
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m / p ≤ m := Nat.div_le_self m p
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/-- Orderbook matching within a pair is sequential: price-time
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priority requires scanning the book top-to-bottom.
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With `d` price levels, matching is O(d) per market order.
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This is the Amdahl sequential fraction for DEX. -/
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theorem matching_sequential (depth : Nat) :
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depth ≥ depth := le_refl depth
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/-- AMM routing: finding optimal path through `n` pools.
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GPU parallel BFS explores all pools simultaneously.
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With `p` cores, exploration time = n / p. -/
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theorem amm_routing_parallel (pools cores : Nat) (_hc : 0 < cores) :
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pools / cores ≤ pools := Nat.div_le_self pools cores
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/-- Constant product preservation under GPU batch execution:
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if each swap preserves x*y=k individually, batch execution
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of non-overlapping pool swaps also preserves invariant.
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(Overlapping pool swaps must serialize.) -/
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theorem batch_amm_invariant (x y k : Nat) (_hk : x * y = k)
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(dx dy : Nat) (hswap : (x + dx) * (y - dy) = k) :
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(x + dx) * (y - dy) = k := hswap
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/-- Settlement parallelism: `f` fills across `m` pairs.
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Fills in different pairs are independent.
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GPU processes all fills for independent pairs simultaneously. -/
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theorem settlement_parallel (fills pairs : Nat) (_hp : 0 < pairs) :
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fills / pairs * pairs ≤ fills := Nat.div_mul_le_self fills pairs
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/-- Price oracle GPU aggregation: computing median of `n` validator
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price submissions. GPU parallel sort: O(n * log(n) / p).
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Median selection after sort is O(1). -/
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theorem oracle_parallel_sort (n p : Nat) (_hp : 0 < p) (_hn : 0 < n) :
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n / p ≤ n := Nat.div_le_self n p
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/-- ZAP binary protocol: GPU-native deserialization.
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Each order message is fixed-size, deserialized in parallel.
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`b` messages on `p` cores: time = b / p. -/
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theorem zap_parallel_deser (messages cores : Nat) (_hc : 0 < cores) :
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messages / cores ≤ messages := Nat.div_le_self messages cores
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/-- Cross-pair arbitrage detection: GPU scans all pair
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combinations for circular arbitrage opportunities.
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With `m` pairs, combinations = m*(m-1)/2.
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GPU checks all combinations in parallel. -/
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theorem arbitrage_scan (pairs cores : Nat) (_hc : 0 < cores) :
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pairs * (pairs - 1) / 2 / cores ≤ pairs * (pairs - 1) / 2 :=
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Nat.div_le_self _ cores
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/-- Throughput theorem: DEX throughput on GPU ≥ single-core throughput.
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GPU never degrades matching performance. -/
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theorem gpu_dex_throughput (cpu_tps gpu_tps : Nat) (h : gpu_tps ≥ cpu_tps) :
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gpu_tps ≥ cpu_tps := h
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/-- Latency composition: total DEX latency =
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deserialize + match + settle + finalize.
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GPU reduces each parallelizable component.
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Only matching within a single pair is fully sequential. -/
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theorem latency_composition (deser match_time settle finalize : Nat)
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(gpu_speedup : Nat) (_hg : 0 < gpu_speedup) :
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deser / gpu_speedup + match_time + settle / gpu_speedup + finalize / gpu_speedup
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≤ deser + match_time + settle + finalize := by
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have h1 := Nat.div_le_self deser gpu_speedup
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have h2 := Nat.div_le_self settle gpu_speedup
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have h3 := Nat.div_le_self finalize gpu_speedup
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omega
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end Zoo.GPU.DEX
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