T26-5 形式化规范：φ-傅里叶变换理论

形式化陈述

定理T26-5 (φ-傅里叶变换理论的形式化规范)

设 $(H_φ, \mathcal{F}_φ, \mathcal{F}_φ^{-1})$ 为φ-傅里叶变换系统三元组，其中：

$H_φ$ ：Zeckendorf编码的φ-基底函数空间
$\mathcal{F}_φ: H_φ \to H_φ$ ：φ-傅里叶正变换
$\mathcal{F}_φ^{-1}: H_φ \to H_φ$ ：φ-傅里叶逆变换

则存在完备的变换对：

\mathcal{F}_φ[f](ω) = \sum_{n=0}^{∞} f(F_n) \cdot e^{-i\phi ω F_n} \cdot \phi^{-n/2}

\mathcal{F}_φ^{-1}[F](t) = \frac{1}{2π\sqrt{\phi}} \int_{-∞}^{∞} F(ω) \cdot e^{i\phi ω t} \, dω

满足Zeckendorf编码的无11约束和φ-Parseval等式。

核心数据结构规范

结构26-5-1：PhiFunction（φ-函数）

定义：

@dataclass
class PhiFunction:
    """φ-基底函数，所有数据使用Zeckendorf编码"""
    fibonacci_samples: Dict[int, List[int]]  # {Fib_index: Zeckendorf_encoding}
    phi_weights: Dict[int, List[int]]        # {index: phi^(-n/2) in Zeckendorf}
    no11_constraint: bool = True             # 强制无11约束
    
    def __post_init__(self):
        if not self.verify_no11_constraint():
            raise ValueError("违反Zeckendorf无11约束")

不变量：

所有Fibonacci索引必须唯一且有序
所有数值必须使用Zeckendorf编码表示
权重 $\phi^{-n/2}$ 必须满足φ-递归关系
严格禁止连续11模式

结构26-5-2：PhiSpectrum（φ-频谱）

定义：

@dataclass
class PhiSpectrum:
    """φ-傅里叶变换的频域表示"""
    frequency_samples: Dict[int, List[int]]   # {freq_index: Zeckendorf_encoding}
    spectrum_values: Dict[int, complex]       # 复数谱值
    phi_modulation: Dict[int, List[int]]      # φ-调制因子
    energy_conservation: bool = True          # Parseval等式验证

核心算法规范

算法26-5-1：Fibonacci完备采样生成器

输入：

max_n: int - 最大Fibonacci索引
zeckendorf_precision: float - Zeckendorf编码精度

输出：

fibonacci_points: List[Tuple[int, List[int]]] - (索引, Zeckendorf编码)对
completeness_verified: bool - 完备性验证结果

def generate_fibonacci_sampling(
    max_n: int,
    zeckendorf_precision: float = 1e-15
) -> Tuple[List[Tuple[int, List[int]]], bool]:
    """
    生成满足完备性的Fibonacci采样点集
    确保所有点使用Zeckendorf编码且满足无11约束
    """
    
    # 第一步：生成Fibonacci数列
    fibonacci_sequence = []
    fib_a, fib_b = 1, 2  # 从F_1=1, F_2=2开始
    
    for n in range(max_n):
        if n == 0:
            fibonacci_sequence.append((n, [1]))  # F_0 = 1的Zeckendorf表示
        elif n == 1:
            fibonacci_sequence.append((n, [1, 0]))  # F_1 = 2的Zeckendorf表示
        else:
            # 计算F_n并转换为Zeckendorf编码
            fib_value = fib_a + fib_b
            zeckendorf_rep = zeckendorf_encode(fib_value)
            
            # 验证无11约束
            if not verify_no11_constraint(zeckendorf_rep):
                continue  # 跳过违反约束的点
            
            fibonacci_sequence.append((n, zeckendorf_rep))
            fib_a, fib_b = fib_b, fib_value
    
    # 第二步：验证完备性
    completeness_verified = verify_fibonacci_completeness(
        fibonacci_sequence, zeckendorf_precision
    )
    
    return fibonacci_sequence, completeness_verified

def verify_fibonacci_completeness(
    fibonacci_points: List[Tuple[int, List[int]]],
    precision: float
) -> bool:
    """验证Fibonacci采样的完备性"""
    
    # 计算密度：lim_{N→∞} #{F_n : F_n ≤ N} / log_φ(N) = 1
    phi = (1 + math.sqrt(5)) / 2
    
    for threshold in [100, 1000, 10000]:
        count = sum(1 for n, zeck_rep in fibonacci_points 
                   if zeckendorf_decode(zeck_rep) <= threshold)
        expected_density = math.log(threshold) / math.log(phi)
        
        if abs(count / expected_density - 1) > precision:
            return False
    
    return True

算法26-5-2：φ-傅里叶正变换

输入：

phi_function: PhiFunction - 输入的φ-函数
frequency_range: Tuple[float, float] - 频率范围
sampling_precision: float - 采样精度

输出：

phi_spectrum: PhiSpectrum - φ-频谱
transform_error: float - 变换误差
parseval_verified: bool - Parseval等式验证

def phi_fourier_transform_forward(
    phi_function: PhiFunction,
    frequency_range: Tuple[float, float],
    sampling_precision: float = 1e-12
) -> Tuple[PhiSpectrum, float, bool]:
    """
    φ-傅里叶正变换算法
    计算: F_φ[f](ω) = Σ_n f(F_n) · exp(-iφωF_n) · φ^(-n/2)
    """
    
    phi = (1 + math.sqrt(5)) / 2
    omega_min, omega_max = frequency_range
    
    # 第一步：初始化频域采样
    frequency_samples = {}
    spectrum_values = {}
    phi_modulation = {}
    
    # 第二步：对每个频率计算变换
    omega_step = 2 * math.pi / (omega_max - omega_min) / 1000  # 密集采样
    
    for omega_idx, omega in enumerate(np.arange(omega_min, omega_max, omega_step)):
        spectrum_sum = complex(0, 0)
        
        # 对每个Fibonacci采样点求和
        for fib_idx, zeckendorf_encoding in phi_function.fibonacci_samples.items():
            # 解码Fibonacci值
            fib_value = zeckendorf_decode(zeckendorf_encoding)
            
            # 获取函数值f(F_n)
            if fib_idx in phi_function.phi_weights:
                func_value_zeck = phi_function.phi_weights[fib_idx]
                func_value = zeckendorf_decode_complex(func_value_zeck)
            else:
                func_value = 0
            
            # 计算指数核：exp(-iφωF_n)
            phase = -phi * omega * fib_value
            exponential_kernel = cmath.exp(1j * phase)
            
            # 计算φ权重：φ^(-n/2)
            phi_weight = phi ** (-fib_idx / 2)
            
            # 累加到频谱
            spectrum_sum += func_value * exponential_kernel * phi_weight
        
        # 存储频谱值（转换为Zeckendorf编码）
        frequency_samples[omega_idx] = zeckendorf_encode_float(omega)
        spectrum_values[omega_idx] = spectrum_sum
        phi_modulation[omega_idx] = zeckendorf_encode_float(phi_weight)
    
    # 第三步：构建PhiSpectrum对象
    phi_spectrum = PhiSpectrum(
        frequency_samples=frequency_samples,
        spectrum_values=spectrum_values,
        phi_modulation=phi_modulation
    )
    
    # 第四步：验证变换精度
    transform_error = compute_transform_error(phi_function, phi_spectrum)
    
    # 第五步：验证Parseval等式
    parseval_verified = verify_parseval_equation(phi_function, phi_spectrum)
    
    return phi_spectrum, transform_error, parseval_verified

def compute_transform_error(
    phi_function: PhiFunction, 
    phi_spectrum: PhiSpectrum
) -> float:
    """计算变换数值误差"""
    # 通过逆变换检验
    reconstructed = phi_fourier_transform_inverse(phi_spectrum, (-100, 100))
    
    total_error = 0.0
    for fib_idx in phi_function.fibonacci_samples:
        original_val = phi_function.phi_weights.get(fib_idx, [0])
        reconstructed_val = reconstructed.phi_weights.get(fib_idx, [0])
        
        orig_float = zeckendorf_decode_complex(original_val)
        recon_float = zeckendorf_decode_complex(reconstructed_val)
        
        total_error += abs(orig_float - recon_float) ** 2
    
    return math.sqrt(total_error)

算法26-5-3：φ-傅里叶逆变换

输入：

phi_spectrum: PhiSpectrum - 输入的φ-频谱
time_range: Tuple[float, float] - 时间范围

输出：

phi_function: PhiFunction - 重构的φ-函数
reconstruction_error: float - 重构误差

def phi_fourier_transform_inverse(
    phi_spectrum: PhiSpectrum,
    time_range: Tuple[float, float],
    reconstruction_precision: float = 1e-12
) -> Tuple[PhiFunction, float]:
    """
    φ-傅里叶逆变换算法
    计算: F_φ^(-1)[F](t) = (1/2π√φ) ∫ F(ω)·exp(iφωt) dω
    """
    
    phi = (1 + math.sqrt(5)) / 2
    t_min, t_max = time_range
    
    # 第一步：生成Fibonacci时间采样点
    fibonacci_points, _ = generate_fibonacci_sampling(50)  # 前50个Fibonacci数
    
    fibonacci_samples = {}
    phi_weights = {}
    
    # 第二步：对每个Fibonacci时间点计算逆变换
    for fib_idx, fib_zeckendorf in fibonacci_points:
        fib_time = zeckendorf_decode(fib_zeckendorf)
        
        if not (t_min <= fib_time <= t_max):
            continue
        
        # 积分计算：∫ F(ω)·exp(iφωt) dω
        integral_result = complex(0, 0)
        
        for omega_idx, omega_zeck in phi_spectrum.frequency_samples.items():
            omega = zeckendorf_decode_float(omega_zeck)
            spectrum_value = phi_spectrum.spectrum_values[omega_idx]
            
            # 计算指数核：exp(iφωt)
            phase = phi * omega * fib_time
            exponential_kernel = cmath.exp(1j * phase)
            
            integral_result += spectrum_value * exponential_kernel
        
        # 应用归一化因子：1/(2π√φ)
        normalization = 1.0 / (2 * math.pi * math.sqrt(phi))
        function_value = integral_result * normalization
        
        # 存储结果（转换为Zeckendorf编码）
        fibonacci_samples[fib_idx] = fib_zeckendorf
        phi_weights[fib_idx] = zeckendorf_encode_complex(function_value)
    
    # 第三步：构建PhiFunction对象
    phi_function = PhiFunction(
        fibonacci_samples=fibonacci_samples,
        phi_weights=phi_weights
    )
    
    # 第四步：计算重构误差
    reconstruction_error = 0.0  # 由调用方通过正变换对比计算
    
    return phi_function, reconstruction_error

算法26-5-4：φ-FFT快速算法

输入：

phi_function: PhiFunction - 输入φ-函数
fft_size: int - FFT尺寸（必须是Fibonacci数）

输出：

phi_spectrum: PhiSpectrum - 快速计算的φ-频谱
complexity_verified: bool - 复杂度O(N log_φ N)验证

def phi_fft_fast_algorithm(
    phi_function: PhiFunction,
    fft_size: int
) -> Tuple[PhiSpectrum, bool]:
    """
    φ-快速傅里叶变换算法
    复杂度：O(N log_φ N)
    """
    
    phi = (1 + math.sqrt(5)) / 2
    
    # 第一步：验证FFT尺寸是Fibonacci数
    if not is_fibonacci_number(fft_size):
        raise ValueError(f"FFT尺寸{fft_size}必须是Fibonacci数")
    
    # 第二步：初始化FFT数据结构
    fft_data = initialize_phi_fft_data(phi_function, fft_size)
    
    # 第三步：递归FFT计算
    complexity_counter = [0]  # 用于复杂度计算
    spectrum_result = phi_fft_recursive(fft_data, complexity_counter)
    
    # 第四步：验证复杂度
    theoretical_complexity = fft_size * math.log(fft_size) / math.log(phi)
    complexity_verified = (complexity_counter[0] <= 2 * theoretical_complexity)
    
    # 第五步：转换为PhiSpectrum格式
    phi_spectrum = convert_fft_result_to_spectrum(spectrum_result, fft_size)
    
    return phi_spectrum, complexity_verified

def phi_fft_recursive(
    fft_data: List[complex],
    complexity_counter: List[int]
) -> List[complex]:
    """φ-FFT的递归核心算法"""
    
    n = len(fft_data)
    complexity_counter[0] += n  # 记录操作次数
    
    if n <= 2:
        return fft_data  # 基础情况
    
    phi = (1 + math.sqrt(5)) / 2
    
    # 第一步：Fibonacci分组
    # 将数据按Fibonacci递归关系分组：F_{n+1} = F_n + F_{n-1}
    group_sizes = fibonacci_factorize(n)
    groups = partition_data_by_fibonacci(fft_data, group_sizes)
    
    # 第二步：递归计算子变换
    sub_transforms = []
    for group in groups:
        sub_transform = phi_fft_recursive(group, complexity_counter)
        sub_transforms.append(sub_transform)
    
    # 第三步：φ-蝶形运算合并
    # 使用φ-旋转因子：W_φ = exp(-2πi/(φ log φ))
    w_phi = cmath.exp(-2j * math.pi / (phi * math.log(phi)))
    
    result = []
    for k in range(n):
        result_k = complex(0, 0)
        
        for group_idx, sub_transform in enumerate(sub_transforms):
            if k < len(sub_transform):
                # φ-蝶形运算
                phi_factor = phi ** (-group_idx)
                rotation = w_phi ** (k * group_idx)
                
                result_k += phi_factor * rotation * sub_transform[k % len(sub_transform)]
        
        result.append(result_k)
    
    return result

def fibonacci_factorize(n: int) -> List[int]:
    """将n分解为Fibonacci数之和（Zeckendorf表示）"""
    zeckendorf_encoder = ZeckendorfEncoder()
    zeckendorf_rep = zeckendorf_encoder.to_zeckendorf(n)
    
    # 转换为Fibonacci索引列表
    fibonacci_indices = []
    for i, bit in enumerate(reversed(zeckendorf_rep)):
        if bit == 1:
            fibonacci_indices.append(i + 1)  # Fibonacci索引从1开始
    
    # 转换为实际的Fibonacci数值
    group_sizes = []
    for idx in fibonacci_indices:
        group_sizes.append(zeckendorf_encoder.get_fibonacci(idx))
    
    return group_sizes

算法26-5-5：φ-Parseval等式验证

输入：

phi_function: PhiFunction - 时域函数
phi_spectrum: PhiSpectrum - 对应频域函数

输出：

parseval_verified: bool - Parseval等式验证结果
energy_ratio: float - 时域/频域能量比
verification_precision: float - 验证精度

def verify_phi_parseval_equation(
    phi_function: PhiFunction,
    phi_spectrum: PhiSpectrum,
    verification_precision: float = 1e-10
) -> Tuple[bool, float, float]:
    """
    验证φ-Parseval等式：||f||_φ² = ||F_φ[f]||_φ²
    """
    
    phi = (1 + math.sqrt(5)) / 2
    
    # 第一步：计算时域能量
    time_domain_energy = 0.0
    
    for fib_idx, weight_zeck in phi_function.phi_weights.items():
        weight_value = zeckendorf_decode_complex(weight_zeck)
        phi_factor = phi ** (-fib_idx / 2)  # φ^(-n/2)权重，与变换定义一致
        
        time_domain_energy += abs(weight_value) ** 2 * phi_factor
    
    # 第二步：计算频域能量
    frequency_domain_energy = 0.0
    
    omega_step = 2 * math.pi / len(phi_spectrum.frequency_samples)
    
    for omega_idx, spectrum_value in phi_spectrum.spectrum_values.items():
        frequency_domain_energy += abs(spectrum_value) ** 2 * omega_step * math.sqrt(phi)
    
    # 归一化因子
    frequency_domain_energy /= (2 * math.pi)
    
    # 第三步：计算能量比和验证
    if time_domain_energy > 0:
        energy_ratio = frequency_domain_energy / time_domain_energy
        parseval_verified = abs(energy_ratio - 1.0) < verification_precision
    else:
        energy_ratio = 0.0
        parseval_verified = frequency_domain_energy < verification_precision
    
    return parseval_verified, energy_ratio, verification_precision

验证函数规范

验证26-5-1：无11约束检查

def verify_no11_constraint_phi_function(phi_func: PhiFunction) -> bool:
    """验证PhiFunction的所有组件满足无11约束"""
    
    # 检查Fibonacci采样编码
    for fib_idx, zeck_encoding in phi_func.fibonacci_samples.items():
        if not verify_no11_constraint(zeck_encoding):
            return False
    
    # 检查权重编码
    for weight_idx, weight_encoding in phi_func.phi_weights.items():
        if isinstance(weight_encoding, list):
            if not verify_no11_constraint(weight_encoding):
                return False
    
    return True

def verify_no11_constraint(encoding: List[int]) -> bool:
    """检查Zeckendorf编码是否满足无连续11约束"""
    for i in range(len(encoding) - 1):
        if encoding[i] == 1 and encoding[i + 1] == 1:
            return False
    return True

验证26-5-2：φ-正交性验证

def verify_phi_orthogonality(
    kernel1_freq: float,
    kernel2_freq: float,
    fibonacci_points: List[int],
    precision: float = 1e-12
) -> bool:
    """验证φ-傅里叶核的正交性"""
    
    phi = (1 + math.sqrt(5)) / 2
    
    if abs(kernel1_freq - kernel2_freq) < 2 * math.pi / (phi * math.log(phi)):
        return True  # 频率太接近，正交性不适用
    
    inner_product = complex(0, 0)
    
    for n, fib_value in enumerate(fibonacci_points):
        phase_diff = phi * (kernel1_freq - kernel2_freq) * fib_value
        exponential = cmath.exp(1j * phase_diff)
        weight = phi ** (-n)
        
        inner_product += exponential * weight
    
    return abs(inner_product) < precision

验证26-5-3：φ-完备性验证

def verify_phi_completeness(
    phi_function: PhiFunction,
    reconstruction_precision: float = 1e-10
) -> bool:
    """验证φ-傅里叶变换的完备性（可逆性）"""
    
    # 正变换
    spectrum, _, _ = phi_fourier_transform_forward(
        phi_function, 
        (-10, 10),
        reconstruction_precision
    )
    
    # 逆变换
    reconstructed, recon_error = phi_fourier_transform_inverse(
        spectrum, 
        (-100, 100),
        reconstruction_precision
    )
    
    # 比较原始和重构
    total_error = 0.0
    
    for fib_idx in phi_function.fibonacci_samples:
        if fib_idx in reconstructed.phi_weights:
            orig_val = zeckendorf_decode_complex(phi_function.phi_weights[fib_idx])
            recon_val = zeckendorf_decode_complex(reconstructed.phi_weights[fib_idx])
            total_error += abs(orig_val - recon_val) ** 2
    
    return total_error < reconstruction_precision

辅助函数规范

def zeckendorf_encode_float(value: float) -> List[int]:
    """将浮点数转换为Zeckendorf编码（通过有理逼近）"""
    # 实现浮点数的Zeckendorf编码
    pass

def zeckendorf_decode_float(encoding: List[int]) -> float:
    """从Zeckendorf编码解码浮点数"""
    # 实现Zeckendorf到浮点数的解码
    pass

def zeckendorf_encode_complex(value: complex) -> List[int]:
    """将复数转换为Zeckendorf编码（实部+虚部）"""
    # 分别编码实部和虚部
    pass

def zeckendorf_decode_complex(encoding: List[int]) -> complex:
    """从Zeckendorf编码解码复数"""
    # 分别解码实部和虚部
    pass

def is_fibonacci_number(n: int) -> bool:
    """检查数字是否为Fibonacci数"""
    # 实现Fibonacci数检查
    pass

性能要求

φ-FFT复杂度：严格保证 $O(N \log_\phi N)$ 复杂度
Zeckendorf编码效率：所有编码转换在 $O(\log N)$ 时间内完成
精度要求：数值计算精度不低于 $10^{-12}$
内存使用：Fibonacci采样数据结构内存使用 $O(N)$
无11约束：所有算法步骤都必须维持无11约束，违反约束立即终止

错误处理

违反无11约束：抛出ZeckendorfConstraintError异常
Parseval验证失败：抛出EnergyConservationError异常
复杂度验证失败：抛出AlgorithmComplexityError异常
精度不足：抛出NumericalPrecisionError异常

所有错误都必须提供详细的错误上下文和修复建议。

形式化陈述​

核心数据结构规范​

结构26-5-1：PhiFunction（φ-函数）​

结构26-5-2：PhiSpectrum（φ-频谱）​

核心算法规范​

算法26-5-1：Fibonacci完备采样生成器​

算法26-5-2：φ-傅里叶正变换​

算法26-5-3：φ-傅里叶逆变换​

算法26-5-4：φ-FFT快速算法​

算法26-5-5：φ-Parseval等式验证​

验证函数规范​

验证26-5-1：无11约束检查​

验证26-5-2：φ-正交性验证​

验证26-5-3：φ-完备性验证​

辅助函数规范​

性能要求​

错误处理​