/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Research Stack Team

TopologyResilience.lean — Geometric Topology meets Load Balancing & Resilience

Bridges abstract manifold topology with practical distributed systems:
- Load balancing      = geodesic flow on energy density field
- Network topology    = atlas of charts with capacity/latency/throughput
- System design       = service placement as field optimization
- Resiliency          = k-connected atlas + Ricci bottleneck detection

Every chart is a network segment. Transition maps are routing tables.
Traffic flows along geodesics of minimal curvature.
Load is the energy density field. Services are eigenfunctions of the Laplacian.

Per AGENTS.md:
  - Q16_16 for scoring (§1.4)
  - PascalCase types, camelCase functions (§2)
  - Theorems for correctness (§4)
-/

import Semantics.GeometricTopology
import Semantics.TopologyNode
import Semantics.Curvature

namespace Semantics.TopologyResilience

open Semantics.Q16_16
open Semantics.GeometricTopology
open Semantics.TopologyNode

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Load as Energy Density Field
-- ═══════════════════════════════════════════════════════════════════════════

/-- Energy density at a point on the manifold.
    Maps directly to system load: CPU, memory, network saturation. -/
def LoadField := List Q16_16 → Q16_16

/-- Uniform load field: constant energy everywhere. -/
def uniformLoad (density : Q16_16) : LoadField :=
  fun _ => density

/-- Gaussian load bump centered at origin of a chart.
    Models a hot spot — one node receiving disproportionate traffic. -/
def gaussianLoad (sigma : Q16_16) (coords : List Q16_16) : Q16_16 :=
  let r2 := coords.foldl (fun acc c => acc + c * c) zero
  -- exp(-r^2 / 2σ^2) simplified: use 1 / (1 + r^2/σ^2) as extraction-friendly proxy
  let denom := Q16_16.one + (r2 / (sigma * sigma))
  Q16_16.one / denom

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Network Segment — Chart with Capacity
-- ═══════════════════════════════════════════════════════════════════════════

/-- A network segment extends a coordinate chart with capacity constraints.
    This is the bridge between geometric charts and practical networking. -/
structure NetworkSegment where
  chart : CoordinateChart
  capacityQps : Q16_16      -- queries per second capacity
  latencyMs : Q16_16        -- round-trip latency
  throughputMbps : Q16_16   -- throughput in megabits/sec
  currentLoad : Q16_16      -- current energy density / load
  healthy : Bool            -- segment is responding to probes
  deriving Repr, Inhabited

/-- Utilization ratio: currentLoad / capacity. -/
def utilization (seg : NetworkSegment) : Q16_16 :=
  if seg.capacityQps = zero then Q16_16.one else seg.currentLoad / seg.capacityQps

/-- A segment is overloaded if utilization > 0.8 (52428 in Q16.16). -/
def overloadedThreshold : Q16_16 := Q16_16.ofRawInt 52428  -- 0.8

def isOverloaded (seg : NetworkSegment) : Bool :=
  utilization seg > overloadedThreshold

/-- A segment is a bottleneck if its curvature (computed via Ollivier-Ricci)
    exceeds the mean curvature of its neighborhood. -/
def isBottleneck (seg : NetworkSegment) (neighborCurvatures : List Q16_16) : Bool :=
  let meanCurv := if neighborCurvatures.isEmpty then zero else
    neighborCurvatures.foldl (· + ·) zero / ofNat neighborCurvatures.length
  seg.currentLoad > meanCurv  -- overload relative to neighbors

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Geodesic Load Balancing — Route along minimal curvature
-- ═══════════════════════════════════════════════════════════════════════════

/-- Cost of traversing a segment: latency + load penalty.
    This is the metric for geodesic routing. Lower cost = better path. -/
def traversalCost (seg : NetworkSegment) : Q16_16 :=
  seg.latencyMs + (seg.currentLoad * ofNat 2)

/-- Given a list of candidate segments, choose the one with minimal traversal cost.
    This is greedy geodesic load balancing. -/
def pickBestSegment (segs : List NetworkSegment) : Option NetworkSegment :=
  match segs with
  | [] => none
  | head :: tail =>
      some (tail.foldl (fun best seg =>
        if traversalCost seg < traversalCost best then seg else best
      ) head)

/-- Total path cost across a sequence of segments.
    The geodesic is the path minimizing this sum. -/
def pathCost (segs : List NetworkSegment) : Q16_16 :=
  segs.foldl (fun acc seg => acc + traversalCost seg) zero

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  Service Placement — Field optimization on the manifold
-- ═══════════════════════════════════════════════════════════════════════════

/-- A service placement assigns each service to a network segment.
    The assignment must respect hardware capabilities (from TopologyNode). -/
structure ServicePlacement where
  service : ServiceKind
  segment : NetworkSegment
  deriving Repr

/-- Score a placement: lower is better.
    Penalizes: high load, wrong capabilities, high latency. -/
def placementScore (sp : ServicePlacement) : Q16_16 :=
  sp.segment.currentLoad + sp.segment.latencyMs +
  (if sp.segment.healthy then zero else ofNat 100)

/-- Place a service on the best available segment.
    Greedy: minimize placementScore. -/
def placeService (svc : ServiceKind)
    (candidates : List NetworkSegment) : Option ServicePlacement :=
  match candidates with
  | [] => none
  | _ =>
      let healthy := candidates.filter (fun s => s.healthy)
      match healthy with
      | [] => none
      | head :: tail =>
          let best := tail.foldl (fun b seg =>
            if placementScore { service := svc, segment := seg } <
               placementScore { service := svc, segment := b } then seg else b
          ) head
          some { service := svc, segment := best }

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  Resiliency — k-Connected Atlas + Bottleneck Detection
-- ═══════════════════════════════════════════════════════════════════════════

/-- An atlas is k-resilient if removing any k-1 segments leaves
    the atlas connected (still has quorum).
    This is the geometric version of k-fault tolerance. -/
def kResilient (segments : List NetworkSegment) (k : Nat) : Bool :=
  -- Sufficient condition: every segment overlaps with at least k others.
  segments.all (fun s1 =>
    (segments.filter (fun s2 =>
      s1.chart.pointId ≠ s2.chart.pointId && s1.chart.dimension = s2.chart.dimension
    )).length ≥ k
  )

/-- Dynamic quorum reconfiguration: when segments fail, healthy segments
    expand their overlap to maintain coverage.
    Returns true if reconfiguration succeeded. -/
def reconfigureQuorum (segments : List NetworkSegment) : Bool :=
  let healthy := segments.filter (fun s => s.healthy)
  let _failed := segments.filter (fun s => !s.healthy)
  -- Quorum maintained if healthy segments still form a connected atlas
  -- and every failed segment was redundant (had overlapping coverage).
  geometricQuorum {
    charts := healthy.map (fun s => s.chart),
    overlap := fun c1 c2 => c1.dimension = c2.dimension
  }

-- ═══════════════════════════════════════════════════════════════════════════
-- §6  Ping Protocol — Probe along geodesics
-- ═══════════════════════════════════════════════════════════════════════════

/-- A ping probe carries load information along a geodesic.
    The response tells us: reachability, current load, and path cost. -/
structure PingProbe where
  sourceId : String
  targetId : String
  timestamp : Nat
  ttl : Nat  -- time-to-live hops (prevents infinite loops)
  accumulatedCost : Q16_16
  deriving Repr, Inhabited

/-- Process a ping at a segment.
    Returns: (response load, updated probe with cost, should forward?). -/
def processPing (probe : PingProbe) (seg : NetworkSegment)
    : (Q16_16 × PingProbe × Bool) :=
  let newCost := probe.accumulatedCost + traversalCost seg
  let response := seg.currentLoad
  let shouldForward := seg.healthy && probe.ttl > 0
  let updatedProbe := { probe with
    accumulatedCost := newCost,
    ttl := probe.ttl - 1
  }
  (response, updatedProbe, shouldForward)

/-- Quorum ping: flood probes from source to all reachable segments.
    Returns the set of reachable segment IDs and total path costs. -/
def floodPing (source : NetworkSegment) (segments : List NetworkSegment)
    : List (String × Q16_16) :=
  let initProbe := { sourceId := source.chart.pointId,
                     targetId := "", timestamp := 0, ttl := 5,
                     accumulatedCost := zero }
  segments.filterMap (fun seg =>
    if seg.healthy && seg.chart.pointId ≠ source.chart.pointId then
      let (_, _, reachable) := processPing initProbe seg
      if reachable then
        some (seg.chart.pointId, traversalCost seg)
      else none
    else none
  )

-- ═══════════════════════════════════════════════════════════════════════════
-- §7  Theorems
-- ═══════════════════════════════════════════════════════════════════════════

/-- An overloaded segment has utilization > 0.8. -/
theorem overloadedImpliesHighUtilization (seg : NetworkSegment)
    (h : isOverloaded seg = true) :
    utilization seg > overloadedThreshold := by
  simp [isOverloaded, overloadedThreshold] at h
  exact h

/-- A k-resilient atlas with k >= 2 cannot have a single point of failure.
    Every segment has at least 2 overlapping neighbors. -/
theorem kResilientNoSinglePointOfFailure (segments : List NetworkSegment)
    (h : kResilient segments 2) :
    segments.all (fun s1 =>
      (segments.filter (fun s2 =>
        s1.chart.pointId ≠ s2.chart.pointId && s1.chart.dimension = s2.chart.dimension
      )).length ≥ 2
    ) = true := by
  exact h

/-- If all segments are healthy and there are at least 2 segments,
    quorum reconfiguration succeeds by construction.
    Proof deferred: needs list induction over segments. -/
-- theorem allHealthyQuorumSucceeds (segments : List NetworkSegment)
--     (h : segments.all (fun s => s.healthy) = true)
--     (h2 : segments.length ≥ 2) :
--     reconfigureQuorum segments = true := ...

-- ═══════════════════════════════════════════════════════════════════════════
-- §8  Example: Earth-Mars-Pluto network with load balancing
-- ═══════════════════════════════════════════════════════════════════════════

def earthSeg : NetworkSegment :=
  { chart := earthChart, capacityQps := ofNat 1000, latencyMs := ofNat 10,
    throughputMbps := ofNat 1000, currentLoad := ofNat 200, healthy := true }

def marsSeg : NetworkSegment :=
  { chart := marsChart, capacityQps := ofNat 500, latencyMs := ofNat 50,
    throughputMbps := ofNat 500, currentLoad := ofNat 400, healthy := true }

def plutoSeg : NetworkSegment :=
  { chart := plutoChart, capacityQps := ofNat 100, latencyMs := ofNat 200,
    throughputMbps := ofNat 100, currentLoad := ofNat 50, healthy := true }

def solarSystemNetwork : List NetworkSegment :=
  [earthSeg, marsSeg, plutoSeg]

#eval pickBestSegment solarSystemNetwork  -- should pick plutoSeg (lowest load)
#eval pathCost solarSystemNetwork
#eval kResilient solarSystemNetwork 2
#eval reconfigureQuorum solarSystemNetwork

end Semantics.TopologyResilience