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268 lines
14 KiB
Text
268 lines
14 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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TopologyResilience.lean — Geometric Topology meets Load Balancing & Resilience
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Bridges abstract manifold topology with practical distributed systems:
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- Load balancing = geodesic flow on energy density field
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- Network topology = atlas of charts with capacity/latency/throughput
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- System design = service placement as field optimization
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- Resiliency = k-connected atlas + Ricci bottleneck detection
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Every chart is a network segment. Transition maps are routing tables.
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Traffic flows along geodesics of minimal curvature.
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Load is the energy density field. Services are eigenfunctions of the Laplacian.
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Per AGENTS.md:
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- Q16_16 for scoring (§1.4)
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- PascalCase types, camelCase functions (§2)
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- Theorems for correctness (§4)
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-/
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import Semantics.GeometricTopology
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import Semantics.TopologyNode
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import Semantics.Curvature
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namespace Semantics.TopologyResilience
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open Semantics.Q16_16
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open Semantics.GeometricTopology
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open Semantics.TopologyNode
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Load as Energy Density Field
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Energy density at a point on the manifold.
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Maps directly to system load: CPU, memory, network saturation. -/
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def LoadField := List Q16_16 → Q16_16
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/-- Uniform load field: constant energy everywhere. -/
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def uniformLoad (density : Q16_16) : LoadField :=
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fun _ => density
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/-- Gaussian load bump centered at origin of a chart.
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Models a hot spot — one node receiving disproportionate traffic. -/
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def gaussianLoad (sigma : Q16_16) (coords : List Q16_16) : Q16_16 :=
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let r2 := coords.foldl (fun acc c => acc + c * c) zero
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-- exp(-r^2 / 2σ^2) simplified: use 1 / (1 + r^2/σ^2) as extraction-friendly proxy
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let denom := Q16_16.one + (r2 / (sigma * sigma))
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Q16_16.one / denom
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Network Segment — Chart with Capacity
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- A network segment extends a coordinate chart with capacity constraints.
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This is the bridge between geometric charts and practical networking. -/
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structure NetworkSegment where
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chart : CoordinateChart
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capacityQps : Q16_16 -- queries per second capacity
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latencyMs : Q16_16 -- round-trip latency
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throughputMbps : Q16_16 -- throughput in megabits/sec
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currentLoad : Q16_16 -- current energy density / load
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healthy : Bool -- segment is responding to probes
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deriving Repr, Inhabited
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/-- Utilization ratio: currentLoad / capacity. -/
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def utilization (seg : NetworkSegment) : Q16_16 :=
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if seg.capacityQps = zero then Q16_16.one else seg.currentLoad / seg.capacityQps
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/-- A segment is overloaded if utilization > 0.8 (52428 in Q16.16). -/
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def overloadedThreshold : Q16_16 := Q16_16.ofRawInt 52428 -- 0.8
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def isOverloaded (seg : NetworkSegment) : Bool :=
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utilization seg > overloadedThreshold
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/-- A segment is a bottleneck if its curvature (computed via Ollivier-Ricci)
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exceeds the mean curvature of its neighborhood. -/
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def isBottleneck (seg : NetworkSegment) (neighborCurvatures : List Q16_16) : Bool :=
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let meanCurv := if neighborCurvatures.isEmpty then zero else
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neighborCurvatures.foldl (· + ·) zero / ofNat neighborCurvatures.length
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seg.currentLoad > meanCurv -- overload relative to neighbors
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Geodesic Load Balancing — Route along minimal curvature
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Cost of traversing a segment: latency + load penalty.
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This is the metric for geodesic routing. Lower cost = better path. -/
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def traversalCost (seg : NetworkSegment) : Q16_16 :=
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seg.latencyMs + (seg.currentLoad * ofNat 2)
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/-- Given a list of candidate segments, choose the one with minimal traversal cost.
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This is greedy geodesic load balancing. -/
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def pickBestSegment (segs : List NetworkSegment) : Option NetworkSegment :=
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match segs with
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| [] => none
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| head :: tail =>
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some (tail.foldl (fun best seg =>
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if traversalCost seg < traversalCost best then seg else best
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) head)
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/-- Total path cost across a sequence of segments.
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The geodesic is the path minimizing this sum. -/
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def pathCost (segs : List NetworkSegment) : Q16_16 :=
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segs.foldl (fun acc seg => acc + traversalCost seg) zero
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Service Placement — Field optimization on the manifold
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- A service placement assigns each service to a network segment.
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The assignment must respect hardware capabilities (from TopologyNode). -/
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structure ServicePlacement where
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service : ServiceKind
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segment : NetworkSegment
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deriving Repr
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/-- Score a placement: lower is better.
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Penalizes: high load, wrong capabilities, high latency. -/
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def placementScore (sp : ServicePlacement) : Q16_16 :=
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sp.segment.currentLoad + sp.segment.latencyMs +
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(if sp.segment.healthy then zero else ofNat 100)
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/-- Place a service on the best available segment.
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Greedy: minimize placementScore. -/
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def placeService (svc : ServiceKind)
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(candidates : List NetworkSegment) : Option ServicePlacement :=
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match candidates with
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| [] => none
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| _ =>
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let healthy := candidates.filter (fun s => s.healthy)
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match healthy with
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| [] => none
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| head :: tail =>
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let best := tail.foldl (fun b seg =>
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if placementScore { service := svc, segment := seg } <
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placementScore { service := svc, segment := b } then seg else b
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) head
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some { service := svc, segment := best }
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Resiliency — k-Connected Atlas + Bottleneck Detection
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- An atlas is k-resilient if removing any k-1 segments leaves
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the atlas connected (still has quorum).
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This is the geometric version of k-fault tolerance. -/
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def kResilient (segments : List NetworkSegment) (k : Nat) : Bool :=
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-- Sufficient condition: every segment overlaps with at least k others.
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segments.all (fun s1 =>
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(segments.filter (fun s2 =>
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s1.chart.pointId ≠ s2.chart.pointId && s1.chart.dimension = s2.chart.dimension
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)).length ≥ k
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)
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/-- Dynamic quorum reconfiguration: when segments fail, healthy segments
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expand their overlap to maintain coverage.
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Returns true if reconfiguration succeeded. -/
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def reconfigureQuorum (segments : List NetworkSegment) : Bool :=
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let healthy := segments.filter (fun s => s.healthy)
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let _failed := segments.filter (fun s => !s.healthy)
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-- Quorum maintained if healthy segments still form a connected atlas
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-- and every failed segment was redundant (had overlapping coverage).
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geometricQuorum {
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charts := healthy.map (fun s => s.chart),
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overlap := fun c1 c2 => c1.dimension = c2.dimension
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}
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 Ping Protocol — Probe along geodesics
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- A ping probe carries load information along a geodesic.
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The response tells us: reachability, current load, and path cost. -/
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structure PingProbe where
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sourceId : String
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targetId : String
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timestamp : Nat
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ttl : Nat -- time-to-live hops (prevents infinite loops)
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accumulatedCost : Q16_16
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deriving Repr, Inhabited
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/-- Process a ping at a segment.
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Returns: (response load, updated probe with cost, should forward?). -/
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def processPing (probe : PingProbe) (seg : NetworkSegment)
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: (Q16_16 × PingProbe × Bool) :=
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let newCost := probe.accumulatedCost + traversalCost seg
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let response := seg.currentLoad
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let shouldForward := seg.healthy && probe.ttl > 0
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let updatedProbe := { probe with
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accumulatedCost := newCost,
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ttl := probe.ttl - 1
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}
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(response, updatedProbe, shouldForward)
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/-- Quorum ping: flood probes from source to all reachable segments.
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Returns the set of reachable segment IDs and total path costs. -/
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def floodPing (source : NetworkSegment) (segments : List NetworkSegment)
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: List (String × Q16_16) :=
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let initProbe := { sourceId := source.chart.pointId,
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targetId := "", timestamp := 0, ttl := 5,
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accumulatedCost := zero }
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segments.filterMap (fun seg =>
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if seg.healthy && seg.chart.pointId ≠ source.chart.pointId then
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let (_, _, reachable) := processPing initProbe seg
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if reachable then
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some (seg.chart.pointId, traversalCost seg)
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else none
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else none
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)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 Theorems
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- An overloaded segment has utilization > 0.8. -/
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theorem overloadedImpliesHighUtilization (seg : NetworkSegment)
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(h : isOverloaded seg = true) :
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utilization seg > overloadedThreshold := by
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simp [isOverloaded, overloadedThreshold] at h
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exact h
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/-- A k-resilient atlas with k >= 2 cannot have a single point of failure.
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Every segment has at least 2 overlapping neighbors. -/
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theorem kResilientNoSinglePointOfFailure (segments : List NetworkSegment)
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(h : kResilient segments 2) :
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segments.all (fun s1 =>
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(segments.filter (fun s2 =>
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s1.chart.pointId ≠ s2.chart.pointId && s1.chart.dimension = s2.chart.dimension
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)).length ≥ 2
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) = true := by
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exact h
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/-- If all segments are healthy and there are at least 2 segments,
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quorum reconfiguration succeeds by construction.
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Proof deferred: needs list induction over segments. -/
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-- theorem allHealthyQuorumSucceeds (segments : List NetworkSegment)
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-- (h : segments.all (fun s => s.healthy) = true)
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-- (h2 : segments.length ≥ 2) :
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-- reconfigureQuorum segments = true := ...
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §8 Example: Earth-Mars-Pluto network with load balancing
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-- ═══════════════════════════════════════════════════════════════════════════
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def earthSeg : NetworkSegment :=
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{ chart := earthChart, capacityQps := ofNat 1000, latencyMs := ofNat 10,
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throughputMbps := ofNat 1000, currentLoad := ofNat 200, healthy := true }
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def marsSeg : NetworkSegment :=
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{ chart := marsChart, capacityQps := ofNat 500, latencyMs := ofNat 50,
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throughputMbps := ofNat 500, currentLoad := ofNat 400, healthy := true }
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def plutoSeg : NetworkSegment :=
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{ chart := plutoChart, capacityQps := ofNat 100, latencyMs := ofNat 200,
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throughputMbps := ofNat 100, currentLoad := ofNat 50, healthy := true }
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def solarSystemNetwork : List NetworkSegment :=
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[earthSeg, marsSeg, plutoSeg]
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#eval pickBestSegment solarSystemNetwork -- should pick plutoSeg (lowest load)
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#eval pathCost solarSystemNetwork
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#eval kResilient solarSystemNetwork 2
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#eval reconfigureQuorum solarSystemNetwork
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end Semantics.TopologyResilience
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