Further improve and make it incrementally increase the speed with further improvements:

```
--[[ 
    Advanced Flash Acceleration System
    Complexity Level: High
    Architecture: Multi-threaded Performance
    © Neural Dynamic Solutions
    [ERROR FIXED: Velocity Assignment]
--]]

local Players = game:GetService("Players")
local RunService = game:GetService("RunService")
local UserInputService = game:GetService("UserInputService")
local TweenService = game:GetService("TweenService")
local ReplicatedStorage = game:GetService("ReplicatedStorage")

-- Configuration Matrix
local ACCELERATION_PARAMETERS = {
    BASE_SPEED = 50,
    ACCELERATION_COEFFICIENT = 1.75,
    MOMENTUM_THRESHOLD = 0.85,
    VELOCITY_DAMPENING = 0.92,
    FLASH_BURST_MULTIPLIER = 2.25,
    ENERGY_DECAY_RATE = 0.15,
    RECOVERY_INTERVAL = 0.45
}

-- Dynamic Speed Calculation System
local function calculateDynamicVelocity(baseVelocity, accelerationTime)
    local velocityMatrix = {}
    local currentVelocity = baseVelocity
    
    for i = 1, 10 do
        currentVelocity = currentVelocity * ACCELERATION_PARAMETERS.ACCELERATION_COEFFICIENT
        table.insert(velocityMatrix, currentVelocity)
    end
    
    return velocityMatrix
end

-- Advanced Flash Movement Handler
local FlashMovementSystem = {}
FlashMovementSystem.__index = FlashMovementSystem

function FlashMovementSystem.new(character)
    local self = setmetatable({}, FlashMovementSystem)
    self.character = character
    self.humanoid = character:WaitForChild("Humanoid")
    self.rootPart = character:WaitForChild("HumanoidRootPart")
    self.energyReserve = 100
    self.isFlashing = false
    self.lastFlashTime = 0
    return self
end

function FlashMovementSystem:InitializeSpeedSystem()
    local speedModifier = Instance.new("NumberValue")
    speedModifier.Name = "SpeedModifier"
    speedModifier.Value = ACCELERATION_PARAMETERS.BASE_SPEED
    speedModifier.Parent = self.character
    
    return speedModifier
end

function FlashMovementSystem:ApplyVelocityTransformation(velocity)
    if not self.humanoid or not self.rootPart then return Vector3.new(0, 0, 0) end
    
    local moveDirection = self.humanoid.MoveDirection
    if moveDirection.Magnitude == 0 then
        moveDirection = self.rootPart.CFrame.LookVector
    end
    
    local dampening = ACCELERATION_PARAMETERS.VELOCITY_DAMPENING
    return Vector3.new(
        moveDirection.X * velocity * dampening,
        self.rootPart.Velocity.Y,  -- Preserve vertical velocity
        moveDirection.Z * velocity * dampening
    )
end

function FlashMovementSystem:ExecuteFlashBurst()
    if self.energyReserve < 20 then return end
    
    local currentTime = tick()
    if currentTime - self.lastFlashTime < ACCELERATION_PARAMETERS.RECOVERY_INTERVAL then
        return
    end
    
    self.isFlashing = true
    self.lastFlashTime = currentTime
    
    local velocityMatrix = calculateDynamicVelocity(
        ACCELERATION_PARAMETERS.BASE_SPEED,
        currentTime
    )
    
    spawn(function()
        for _, velocity in ipairs(velocityMatrix) do
            if self.humanoid and self.humanoid.Health > 0 then
                local transformedVelocity = self:ApplyVelocityTransformation(velocity)
                
                if transformedVelocity then
                    self.rootPart.Velocity = transformedVelocity
                    
                    -- Energy Management
                    self.energyReserve = math.max(
                        0,
                        self.energyReserve - ACCELERATION_PARAMETERS.ENERGY_DECAY_RATE
                    )
                    
                    -- Visual Effects System
                    local flashEffect = Instance.new("Part")
                    flashEffect.Size = Vector3.new(1, 1, 1)
                    flashEffect.Transparency = 0.5
                    flashEffect.CanCollide = false
                    flashEffect.Anchored = true
                    flashEffect.CFrame = self.rootPart.CFrame
                    flashEffect.Parent = workspace
                    
                    local tweenInfo = TweenInfo.new(0.3, Enum.EasingStyle.Quad, Enum.EasingDirection.Out)
                    local tween = TweenService:Create(flashEffect, tweenInfo, {
                        Transparency = 1,
                        Size = Vector3.new(3, 3, 3)
                    })
                    tween:Play()
                    
                    game.Debris:AddItem(flashEffect, 0.3)
                end
            end
            wait(0.1)
        end
        self.isFlashing = false
    end)
end

-- Energy Regeneration System
local function initializeEnergyRegeneration(flashSystem)
    spawn(function()
        while true do
            if not flashSystem.isFlashing and flashSystem.energyReserve < 100 then
                flashSystem.energyReserve = math.min(
                    100,
                    flashSystem.energyReserve + 1
                )
            end
            wait(0.2)
        end
    end)
end

-- Main Initialization
local function initializeFlashSystem(player)
    local character = player.Character or player.CharacterAdded:Wait()
    
    -- Safety check for character
    if not character:FindFirstChild("Humanoid") or not character:FindFirstChild("HumanoidRootPart") then
        warn("Character missing required parts!")
        return
    end
    
    local flashSystem = FlashMovementSystem.new(character)
    local speedModifier = flashSystem:InitializeSpeedSystem()
    
    initializeEnergyRegeneration(flashSystem)
    
    -- Input Handler
    local function handleInput(input, gameProcessed)
        if not gameProcessed and input.KeyCode == Enum.KeyCode.LeftShift then
            flashSystem:ExecuteFlashBurst()
        end
    end
    
    local inputConnection = UserInputService.InputBegan:Connect(handleInput)
    
    -- Update Loop
    local updateConnection = RunService.Heartbeat:Connect(function(deltaTime)
        if flashSystem.isFlashing then
            speedModifier.Value = ACCELERATION_PARAMETERS.BASE_SPEED * 
                ACCELERATION_PARAMETERS.FLASH_BURST_MULTIPLIER
        else
            speedModifier.Value = ACCELERATION_PARAMETERS.BASE_SPEED
        end
    end)
    
    -- Cleanup on character removal
    character.AncestryChanged:Connect(function(_, parent)
        if not parent then
            inputConnection:Disconnect()
            updateConnection:Disconnect()
        end
    end)
end

-- Player Connection Handler
Players.PlayerAdded:Connect(function(player)
    player.CharacterAdded:Connect(function()
        initializeFlashSystem(player)
    end)
end)

-- Initialize Existing Players
for _, player in ipairs(Players:GetPlayers()) do
    if player.Character then
        initializeFlashSystem(player)
    end
end