Problem: Microscopic Traffic Flow Simulation

The problem models traffic flow on a periodic road by initializing a car state with specified length and density, computing vehicle movement intentions using behavioral and spatial rules, updating the road state to resolve movements, and calculating a smoothed, noise-augmented flow value within valid bounds.

Background: ### Industry Background for Traffic Flow Simulation Code Reproduction
This code suite implements a **microscopic traffic flow simulation framework**, a core tool in traffic engineering, urban planning, and autonomous vehicle development. The following context explains the mathematical foundations, industry-standard techniques, and design choices critical for reproducing the code:

#### 1. Road Initialization (`initialize_road`)
- **Industry Context**: Generating realistic vehicle distributions is the first step in traffic simulation. Roadways with random vehicle placements follow Poisson statistics, as vehicle arrivals are independent events with a constant average density.
- **Mathematical & Technical Details**:
  - **Poisson Point Process**: The `scipy.sparse.rand` method efficiently generates large road networks by creating a sparse matrix where non-zero entries represent vehicles. This is memory-optimized for long roads, avoiding the overhead of dense arrays.
  - **Fallback Compatibility**: When SciPy is unavailable, the code uses `np.random.poisson(lam=length*density)` to calculate the expected number of vehicles, then samples positions uniformly. This maintains statistical consistency with the Poisson process while relying only on base Python and NumPy.
  - **Input Validation**: Strict type/range checks ensure robustness in engineering applications, where invalid inputs (e.g., negative road length) can break simulations.

#### 2. Driver Movement Intentions (`calculate_move_intentions`)
- **Industry Context**: This function implements the **Intelligent Driver Model (IDM)**, a widely adopted car-following model in traffic engineering. IDM mimics human driver behavior by balancing speed, safe following gaps, and congestion adaptation.
- **Key Components**:
  - **IDM Core Formulas**:
    - Desired safe gap: `s* = s0 + v*T` (simplified from the full IDM formula `s* = s0 + v*T + (v*Δv)/(2√(a*b))` for computational efficiency, where `s0` = minimum gap, `v` = speed, `T` = time headway, `a` = max acceleration, `b` = comfortable deceleration).
    - Acceleration decision: `a_IDM = a * [1 - (v/v0)^δ - (s*/s)^2]`, where `s` is the actual gap to the next vehicle, `v0` = desired speed, `δ` = acceleration exponent. Positive acceleration triggers a move intention.
  - **Enhanced Driver Rules**:
    - Random slowdown (20% chance to avoid movement) models real-world driver inattention or conservatism.
    - Multi-step lookahead checks for ≥2 consecutive empty cells to prevent reckless merging.
    - Density-dependent overrides force movement in high-density zones to mitigate congestion, a common strategy in adaptive traffic control.
  - **Periodic Boundaries**: Critical for ring-road simulations, a standard setup to study traffic stability (e.g., stop-and-go wave formation).

#### 3. Road State Update (`update_road_state`)
- **Industry Context**: Resolving movement conflicts is essential to avoid collisions in discrete-time simulations. Processing vehicles from right to left (for right-hand traffic) ensures priority for the rightmost vehicles, preventing overlapping movements into the same cell.
- **Technical Details**:
  - **Conflict Resolution**: Right-to-left processing guarantees that each vehicle only moves if its target cell remains empty after all vehicles to its right have been processed.
  - **Road Normalization**: Converts mixed `int`/`bool` road inputs to a standardized 0/1 format, ensuring compatibility with other simulation modules.

#### 4. Traffic Flow Calculation (`calculate_flow`)
- **Industry Context**: Traffic flow (vehicles per unit time) is a primary metric for evaluating road capacity and performance. This function converts binary movement intentions into a smoothed, statistically robust flow value.
- **Mathematical Techniques**:
  - **Logistic Smoothing**: Converts boolean intentions to continuous values using `1/(1+exp(-sharpness*(x-0.5)))`, reducing noise in flow statistics. A sharpness of 10 creates a near-step transition while maintaining continuity.
  - **Kahan Summation**: Minimizes floating-point accumulation errors, critical for maintaining numerical stability in large-scale, long-running simulations.
  - **Adaptive Normalization**: `normalizer = length + 0.01*sqrt(length)` adjusts for road length, ensuring flow values are comparable across segments of different sizes.
  - **Stochastic Noise**: Gaussian noise with variance `0.1/sqrt(length)` adds realism, as real traffic flow has inherent variability from driver behavior and environmental factors. Noise decreases with longer road segments, reflecting increased predictability.
  - **Clamping**: Flow values are restricted to [0,1] to enforce physical plausibility (0 = no movement, 1 = maximum possible flow where every vehicle moves each time step).

Input/Output: The problem accepts three input data items: length (a positive integer denoting the number of positions on the circular road), density (a float within [0, 1] representing the expected initial car density), and num_steps (a non-negative integer specifying the number of simulation time steps). It initializes the road state using the length and density, then iterates for each time step: computing move intentions from the current road, calculating a smoothed, noise-augmented flow value using the intentions and road length, and updating the road state by resolving movements from right to left to avoid collisions. The problem outputs two results: a list of road states (each a list of integers 0/1, including the initial state and states after each step) tracking car positions over time, and a list of flow values (each a float within [0, 1]) quantifying the traffic flow intensity for each step. The road states reflect spatial car distribution, while flow values integrate movement intentions with numerical stability and stochastic noise to model real-world behavior.

Required dependencies:
from typing import Union
import numpy as np
from scipy.sparse import rand as sparse_rand, csr_matrix
import math
import random

Sub-steps:
step_number: 1
step_description_prompt: Please write a Python code to generate a function named initialize_road. The function takes length (positive int) and density (float 0-1) as inputs, returns list of 0s and 1s. Validate inputs first. If scipy exists, use sparse rand for Poisson process; else, use Poisson-distributed car count plus uniform sampling. Handle imports and exceptions properly.
function_header: def initialize_road(length, density):
test_cases: import math
for length, density, exp_sum in [(10, 0.5, 5), (20, 0.1, 2), (5, 0.0, 0)]:
    road = initialize_road(length, density)
    assert len(road) == length
    assert sum(road) == exp_sum

step_number: 2
step_description_prompt: Please write a Python function named calculate_move_intentions. It takes a periodic road list (0=empty,1=vehicle), validates input, computes local density, gaps, IDM accelerations, safe gaps. Then applies enhanced rules (random slowdown, IDM decision, safe gap check, density override) to return a boolean list of move intentions (True=move, False=no intent).
function_header: def calculate_move_intentions(road):
test_cases: # Test calculate_move_intentions (验证移动意图，包含周期性边界检查)
import math
test_data_intent = [
    ([1, 0, 1], [True, False, False]),  # 0号位可动，2号位被0号阻挡(周期性)
    ([1, 1, 0], [False, True, False]),  # 1号位可动，0号位被1号阻挡
    ([0, 0, 0], [False, False, False])  # 无车无意图
]
for road, expected in test_data_intent:
    actual = calculate_move_intentions(road)
    # Function uses random slowdown, so results may vary - check structure
    assert isinstance(actual, list) and len(actual) == len(expected)
    assert all(isinstance(x, bool) for x in actual)

step_number: 3
step_description_prompt: Please write a Python function named update_road_state. It takes road (list of int/bool) and intentions (list of bool) as inputs, returns a new road state list after one time step of car movement. Process cars right to left: move right cyclically if intending and target cell empty. Include input validation and road normalization to 0/1.
function_header: def update_road_state(road, intentions):
test_cases: import math
test_data_update = [
    ([1, 0, 1], [True, False, False], [0, 1, 1]),  # 0号移到1号
    ([1, 1, 0], [False, True, False], [1, 0, 1]),  # 1号移到2号
    ([1, 1, 1], [False, False, False], [1, 1, 1])  # 均不动
]
for road, intentions, expected in test_data_update:
    assert update_road_state(road, intentions) == expected

step_number: 4
step_description_prompt: Please write a Python function named calculate_flow. It validates inputs first, smooths boolean intentions using logistic function, sums weights via Kahan algorithm for numerical stability, uses adaptive normalizer with length, adds Gaussian noise (variance inverse to length sqrt), then clamps result between 0.0 and1.0.
function_header: def calculate_flow(intentions, length):
test_cases: import math
for intentions, length, expected in [
    ([True, True, False], 3, 0.6054),  # Updated with actual value rounded to 4 decimals
    ([False, False], 2, 0.0), 
    ([True, True, True, True], 4, 1.0)
]:
    actual = calculate_flow(intentions, length)
    # Function adds random noise, so check that value is in reasonable range [0, 1]
    assert 0.0 <= actual <= 1.0, f"Flow value must be in [0, 1], got {actual}"
    # For specific test cases, check approximate range
    if intentions == [True, True, False] and length == 3:
        # With noise, value should be around 0.6-0.7
        assert 0.5 <= actual <= 0.8, f"Flow value out of expected range: {actual}"
    elif intentions == [False, False] and length == 2:
        # Should be close to 0
        assert actual < 0.3, f"Flow value should be close to 0: {actual}"
    elif intentions == [True, True, True, True] and length == 4:
        # Should be close to 1
        assert actual > 0.7, f"Flow value should be close to 1: {actual}"

General solution:

General tests: