T27-1 形式化规范：纯二进制Zeckendorf数学体系
形式化陈述
定理T27-1 (纯二进制Zeckendorf数学体系的形式化规范)
设 为纯Zeckendorf数学体系七元组，其中：
- ：无11约束的Zeckendorf数字空间
- ：Fibonacci加法运算
- ：Fibonacci乘法运算
- ：黄金比例变换算子
- ：圆周率旋转算子
- ：自然底数增长算子
- ：Fibonacci分析运算集
则该体系满足完备性条件：
核心算法规范
算法27-1-1：Zeckendorf编码器
输入：
real_number: 实数输入值max_fibonacci_index: 最大Fibonacci索引Nprecision: 精度要求ε
输出：
zeckendorf_encoding: 无11约束的Zeckendorf编码encoding_error: 编码误差constraint_satisfied: 无11约束验证结果
def encode_to_zeckendorf(
real_number: float,
max_fibonacci_index: int = 50,
precision: float = 1e-12
) -> Tuple[List[int], float, bool]:
"""
将实数编码为Zeckendorf表示，严格满足无11约束
"""
if abs(real_number) < precision:
return [0] * max_fibonacci_index, 0.0, True
# 处理符号
sign = 1 if real_number >= 0 else -1
abs_value = abs(real_number)
# 生成Fibonacci序列：F₁=1, F₂=2, F₃=3, F₄=5, F₅=8...
fibonacci_sequence = generate_fibonacci_sequence(max_fibonacci_index)
# 贪心算法编码
encoding = [0] * max_fibonacci_index
remaining = abs_value
# 从大到小选择Fibonacci数
for i in range(max_fibonacci_index - 1, -1, -1):
if remaining >= fibonacci_sequence[i] - precision:
encoding[i] = 1
remaining -= fibonacci_sequence[i]
if remaining < precision:
break
# 强制执行无11约束
encoding = enforce_no_consecutive_ones(encoding, fibonacci_sequence)
# 计算编码误差
decoded_value = sum(encoding[i] * fibonacci_sequence[i] for i in range(max_fibonacci_index))
encoding_error = abs(abs_value - decoded_value)
# 验证约束
constraint_satisfied = verify_no_consecutive_ones(encoding)
# 添加符号信息
if sign == -1:
encoding = [-1] + encoding
return encoding, encoding_error, constraint_satisfied
def generate_fibonacci_sequence(n: int) -> List[int]:
"""生成标准Fibonacci序列 F₁=1, F₂=2, F₃=3, F₄=5, ..."""
if n <= 0:
return []
fib = [1, 2] # F₁=1, F₂=2
for i in range(2, n):
fib.append(fib[i-1] + fib[i-2])
return fib
def enforce_no_consecutive_ones(encoding: List[int], fib_seq: List[int]) -> List[int]:
"""
使用Fibonacci恒等式强制执行无11约束
恒等式：F_n + F_{n+1} = F_{n+2}
"""
result = encoding.copy()
changed = True
while changed:
changed = False
for i in range(len(result) - 1):
if result[i] == 1 and result[i + 1] == 1:
# 应用恒等式 F_i + F_{i+1} = F_{i+2}
result[i] = 0
result[i + 1] = 0
if i + 2 < len(result):
result[i + 2] = 1
changed = True
break
return result
def verify_no_consecutive_ones(encoding: List[int]) -> bool:
"""验证编码是否满足无11约束"""
# 跳过符号位
start_idx = 1 if encoding[0] == -1 else 0
for i in range(start_idx, len(encoding) - 1):
if encoding[i] == 1 and encoding[i + 1] == 1:
return False
return True
算法27-1-2：Fibonacci加法运算
输入：
zeckendorf_a: 第一个Zeckendorf编码zeckendorf_b: 第二个Zeckendorf编码normalization_enabled: 是否启用规范化
输出：
result_encoding: 加法结果的Zeckendorf编码operation_overflow: 运算溢出标志constraint_maintained: 约束维护状态
def fibonacci_addition(
zeckendorf_a: List[int],
zeckendorf_b: List[int],
normalization_enabled: bool = True
) -> Tuple[List[int], bool, bool]:
"""
Fibonacci加法：a ⊕ b
"""
# 处理符号
sign_a, encoding_a = extract_sign_and_encoding(zeckendorf_a)
sign_b, encoding_b = extract_sign_and_encoding(zeckendorf_b)
# 确保编码长度一致
max_len = max(len(encoding_a), len(encoding_b))
encoding_a = pad_encoding(encoding_a, max_len)
encoding_b = pad_encoding(encoding_b, max_len)
# 根据符号决定运算类型
if sign_a == sign_b:
# 同号：执行加法
result_encoding, overflow = perform_fibonacci_add(encoding_a, encoding_b, max_len)
result_sign = sign_a
else:
# 异号：执行减法
result_encoding, result_sign, overflow = perform_fibonacci_subtract(
encoding_a, encoding_b, sign_a, sign_b, max_len
)
# 规范化结果
if normalization_enabled:
result_encoding = fibonacci_normalize(result_encoding)
# 验证约束
constraint_maintained = verify_no_consecutive_ones(result_encoding)
# 添加符号
if result_sign == -1 and any(result_encoding):
result_encoding = [-1] + result_encoding
return result_encoding, overflow, constraint_maintained
def perform_fibonacci_add(
encoding_a: List[int],
encoding_b: List[int],
max_len: int
) -> Tuple[List[int], bool]:
"""执行Fibonacci位级加法"""
result = [0] * (max_len + 2) # 额外空间防止溢出
carry = 0
for i in range(max_len):
total = encoding_a[i] + encoding_b[i] + carry
if total == 0:
result[i] = 0
carry = 0
elif total == 1:
result[i] = 1
carry = 0
elif total == 2:
# 使用Fibonacci恒等式：2×F_i = F_{i-1} + F_{i+1}
result[i] = 0
if i > 0:
result[i-1] += 1
if i + 1 < len(result):
result[i+1] += 1
carry = 0
else: # total >= 3
# 递归应用Fibonacci恒等式
result[i] = total % 2
carry = total // 2
# 处理最终carry
overflow = carry > 0
if carry > 0 and max_len + 1 < len(result):
result[max_len + 1] = carry
return result, overflow
def fibonacci_normalize(encoding: List[int]) -> List[int]:
"""
规范化Fibonacci编码，确保满足所有约束
"""
result = encoding.copy()
# 第一阶段：处理大于1的系数
changed = True
while changed:
changed = False
for i in range(len(result)):
if result[i] > 1:
if result[i] == 2:
# 2×F_i = F_{i-1} + F_{i+1}
result[i] = 0
if i > 0:
result[i-1] += 1
if i + 1 < len(result):
result[i+1] += 1
else:
# result[i] > 2的情况
quotient = result[i] // 2
remainder = result[i] % 2
result[i] = remainder
if i > 0:
result[i-1] += quotient
if i + 1 < len(result):
result[i+1] += quotient
changed = True
break
# 第二阶段：强制执行无11约束
result = enforce_no_consecutive_ones(result, generate_fibonacci_sequence(len(result)))
return result
算法27-1-3：Fibonacci乘法运算
输入：
zeckendorf_a: 第一个Zeckendorf编码zeckendorf_b: 第二个Zeckendorf编码lucas_coefficients: Lucas数系数表
输出：
result_encoding: 乘法结果的Zeckendorf编码computation_precision: 计算精度估计constraint_validated: 约束验证结果
def fibonacci_multiplication(
zeckendorf_a: List[int],
zeckendorf_b: List[int],
lucas_coefficients: Optional[Dict] = None
) -> Tuple[List[int], float, bool]:
"""
Fibonacci乘法：a ⊗ b
使用Lucas数恒等式：F_m × F_n = (L_m × φⁿ + (-1)ⁿ × L_m × φ⁻ⁿ) / 5
"""
# 处理符号
sign_a, encoding_a = extract_sign_and_encoding(zeckendorf_a)
sign_b, encoding_b = extract_sign_and_encoding(zeckendorf_b)
result_sign = sign_a * sign_b
# 预计算Lucas数表
if lucas_coefficients is None:
lucas_coefficients = precompute_lucas_coefficients(max(len(encoding_a), len(encoding_b)))
# 执行分布乘法：∑ᵢ∑ⱼ aᵢbⱼ (Fᵢ × Fⱼ)
partial_products = []
max_result_length = len(encoding_a) + len(encoding_b) + 10 # 额外空间
for i, a_coeff in enumerate(encoding_a):
if a_coeff == 0:
continue
for j, b_coeff in enumerate(encoding_b):
if b_coeff == 0:
continue
# 计算 aᵢ × bⱼ × (Fᵢ × Fⱼ)
fibonacci_product = compute_fibonacci_product(i, j, lucas_coefficients)
scaled_product = scale_fibonacci_encoding(
fibonacci_product, a_coeff * b_coeff, max_result_length
)
partial_products.append(scaled_product)
# 累加所有部分乘积
result_encoding = [0] * max_result_length
for product in partial_products:
result_encoding = fibonacci_add_encodings(result_encoding, product)
# 规范化结果
result_encoding = fibonacci_normalize(result_encoding)
# 估计计算精度
computation_precision = estimate_multiplication_precision(
encoding_a, encoding_b, lucas_coefficients
)
# 验证约束
constraint_validated = verify_no_consecutive_ones(result_encoding)
# 添加符号
if result_sign == -1 and any(result_encoding):
result_encoding = [-1] + result_encoding
return result_encoding, computation_precision, constraint_validated
def compute_fibonacci_product(i: int, j: int, lucas_coeff: Dict) -> List[int]:
"""
计算两个Fibonacci数的乘积：Fᵢ × Fⱼ
使用Lucas恒等式的Zeckendorf展开
"""
if i == 0 or j == 0:
return [0]
# Lucas恒等式：F_m × F_n = (L_m × φⁿ + (-1)ⁿ × L_m × φ⁻ⁿ) / 5
# 在Zeckendorf空间中的近似实现
# 简化版本：使用预计算的乘积表
product_key = (i, j)
if product_key in lucas_coeff:
return lucas_coeff[product_key]
# 递归计算
if i < j:
i, j = j, i # 确保i >= j
if j == 1:
# F_n × F_1 = F_n
result = [0] * (i + 1)
result[i] = 1
return result
elif j == 2:
# F_n × F_2 = F_n × 2 = F_{n-1} + F_{n+1}
result = [0] * (i + 2)
if i > 0:
result[i-1] = 1
result[i+1] = 1
return result
else:
# 使用递推关系：F_m × F_n = F_{m-1} × F_n + F_{m-2} × F_n
term1 = compute_fibonacci_product(i-1, j, lucas_coeff)
term2 = compute_fibonacci_product(i-2, j, lucas_coeff)
return fibonacci_add_encodings(term1, term2)
def precompute_lucas_coefficients(max_index: int) -> Dict:
"""预计算Lucas数和Fibonacci乘积系数"""
coefficients = {}
# 基础情况
coefficients[(0, 0)] = [0]
coefficients[(1, 1)] = [0, 1] # F_1 × F_1 = 1 = F_1
coefficients[(2, 2)] = [0, 0, 0, 1] # F_2 × F_2 = 4 = F_3 + F_1 = 3 + 1
# 递推计算
for i in range(1, max_index + 1):
for j in range(1, i + 1):
if (i, j) not in coefficients:
# 使用恒等式计算
product_value = fibonacci_multiply_integers(i, j)
coefficients[(i, j)] = encode_to_zeckendorf(
product_value, max_index * 2, 1e-12
)[0]
coefficients[(j, i)] = coefficients[(i, j)] # 对称性
return coefficients
def fibonacci_multiply_integers(i: int, j: int) -> float:
"""计算两个Fibonacci数的数值乘积（用于预计算）"""
phi = (1 + (5 ** 0.5)) / 2
# Binet公式：F_n = (φⁿ - (-φ)⁻ⁿ) / √5
fi = (phi**i - ((-1/phi)**i)) / (5**0.5)
fj = (phi**j - ((-1/phi)**j)) / (5**0.5)
return fi * fj
算法27-1-4：数学常数运算符
输入：
zeckendorf_input: Zeckendorf编码输入operator_type: 运算符类型 ('phi', 'pi', 'e')operation_precision: 运算精度
输出：
transformed_encoding: 变换后的Zeckendorf编码operator_eigenvalue: 运算符特征值（如适用）convergence_verified: 收敛性验证结果
def apply_mathematical_operator(
zeckendorf_input: List[int],
operator_type: str,
operation_precision: float = 1e-12
) -> Tuple[List[int], Optional[float], bool]:
"""
应用数学常数运算符：φ_op, π_op, e_op
"""
if operator_type == 'phi':
return apply_phi_operator(zeckendorf_input, operation_precision)
elif operator_type == 'pi':
return apply_pi_operator(zeckendorf_input, operation_precision)
elif operator_type == 'e':
return apply_e_operator(zeckendorf_input, operation_precision)
else:
raise ValueError(f"Unknown operator type: {operator_type}")
def apply_phi_operator(
zeckendorf_input: List[int],
precision: float
) -> Tuple[List[int], float, bool]:
"""
φ运算符：实现黄金比例变换
φ_op: [a₀, a₁, a₂, ...] → [a₁, a₀+a₁, a₁+a₂, a₂+a₃, ...]
"""
sign, encoding = extract_sign_and_encoding(zeckendorf_input)
if not any(encoding):
return zeckendorf_input, 1.618033988749895, True
# 应用φ变换
result_length = len(encoding) + 2
result = [0] * result_length
# φ变换规则：φ × F_n = F_{n+1}
for i, coeff in enumerate(encoding):
if coeff == 1 and i + 1 < result_length:
result[i + 1] = 1
# 规范化
result = fibonacci_normalize(result)
# 计算特征值（φ的近似值）
phi_eigenvalue = (1 + 5**0.5) / 2
# 验证收敛性
convergence_verified = verify_phi_operator_convergence(encoding, result, precision)
# 恢复符号
if sign == -1:
result = [-1] + result
return result, phi_eigenvalue, convergence_verified
def apply_pi_operator(
zeckendorf_input: List[int],
precision: float
) -> Tuple[List[int], None, bool]:
"""
π运算符：实现Fibonacci空间旋转
π_op: 循环位移操作，模拟π的几何旋转性质
"""
sign, encoding = extract_sign_and_encoding(zeckendorf_input)
if not any(encoding):
return zeckendorf_input, None, True
# π旋转：实现为特定的循环位移
# 基于π ≈ 3.14159... 选择位移量
shift_amount = 3 # 对应π的整数部分
result = fibonacci_circular_shift(encoding, shift_amount)
# 验证旋转的周期性
convergence_verified = verify_pi_operator_periodicity(encoding, result, precision)
# 恢复符号
if sign == -1:
result = [-1] + result
return result, None, convergence_verified
def apply_e_operator(
zeckendorf_input: List[int],
precision: float
) -> Tuple[List[int], float, bool]:
"""
e运算符：实现指数增长变换
e_op: 基于Fibonacci递推的指数展开
"""
sign, encoding = extract_sign_and_encoding(zeckendorf_input)
if not any(encoding):
# e^0 = 1 在Zeckendorf中是 [1, 0, 0, ...]
return [1] + [0] * (len(encoding) - 1), 2.718281828459045, True
# e变换：实现Fibonacci指数级数
result = fibonacci_exponential_series(encoding, precision)
# 计算特征值（e的近似值）
e_eigenvalue = 2.718281828459045
# 验证指数性质
convergence_verified = verify_e_operator_exponential_property(encoding, result, precision)
# 恢复符号
if sign == -1:
result = [-1] + result
return result, e_eigenvalue, convergence_verified
def fibonacci_exponential_series(
encoding: List[int],
precision: float,
max_terms: int = 20
) -> List[int]:
"""
计算Fibonacci指数级数：e^x = ∑(x^n / n!)
其中x和所有运算都在Zeckendorf空间中进行
"""
# 级数第0项：e^0 = 1
result = [1] + [0] * (len(encoding) * max_terms)
# 级数的后续项：x^n / n!
x_power = encoding.copy() # x^1
factorial = [1] + [0] * (len(encoding) * max_terms) # 1!
for n in range(1, max_terms):
# 计算当前项：x^n / n!
term_numerator = x_power.copy()
term = fibonacci_division(term_numerator, factorial, precision)
# 累加到结果
result = fibonacci_add_encodings(result, term)
# 为下一次迭代准备
# x^{n+1} = x^n × x
x_power = fibonacci_multiplication(x_power, encoding, None)[0]
# (n+1)! = n! × (n+1)
n_plus_1_zeck = encode_to_zeckendorf(n + 1, len(factorial), precision)[0]
factorial = fibonacci_multiplication(factorial, n_plus_1_zeck, None)[0]
# 检查收敛性
if all(t == 0 for t in term):
break
return fibonacci_normalize(result)
算法27-1-5：Fibonacci微积分运算
输入：
function_encoding: 函数的Zeckendorf编码表示variable_point: 求值点的Zeckendorf编码operation_type: 运算类型 ('derivative', 'integral')differential_precision: 微分精度
输出：
result_encoding: 运算结果的Zeckendorf编码convergence_rate: 收敛速率估计accuracy_bound: 精度边界
def fibonacci_calculus_operation(
function_encoding: List[List[int]], # 函数系数的Zeckendorf编码
variable_point: List[int],
operation_type: str,
differential_precision: float = 1e-10
) -> Tuple[List[int], float, float]:
"""
Fibonacci微积分运算
"""
if operation_type == 'derivative':
return fibonacci_derivative(function_encoding, variable_point, differential_precision)
elif operation_type == 'integral':
return fibonacci_integral(function_encoding, variable_point, differential_precision)
else:
raise ValueError(f"Unknown calculus operation: {operation_type}")
def fibonacci_derivative(
function_coeffs: List[List[int]],
point: List[int],
h_precision: float
) -> Tuple[List[int], float, float]:
"""
Fibonacci导数：df/dx = lim_{h→0} [f(x+h) - f(x)] / h
"""
# 选择Fibonacci差分步长
h_zeck = compute_fibonacci_epsilon(len(point), h_precision)
# 计算 f(x + h)
x_plus_h = fibonacci_addition(point, h_zeck)[0]
f_x_plus_h = evaluate_fibonacci_function(function_coeffs, x_plus_h)
# 计算 f(x)
f_x = evaluate_fibonacci_function(function_coeffs, point)
# 计算分子：f(x+h) - f(x)
numerator = fibonacci_subtraction(f_x_plus_h, f_x)
# 计算分母：h
denominator = h_zeck
# 执行Fibonacci除法
derivative_result = fibonacci_division(numerator, denominator, h_precision)
# 估计收敛速率
convergence_rate = estimate_derivative_convergence(function_coeffs, point, h_precision)
# 计算精度边界
accuracy_bound = h_precision * convergence_rate
return derivative_result, convergence_rate, accuracy_bound
def fibonacci_integral(
function_coeffs: List[List[int]],
upper_limit: List[int],
integration_precision: float
) -> Tuple[List[int], float, float]:
"""
Fibonacci积分：∫f(x)dx 使用Fibonacci求和规则
"""
# Fibonacci积分：∑_{n=0}^∞ f(F_n × x) / F_n
result = [0] * (len(upper_limit) * 10)
fibonacci_seq = generate_fibonacci_sequence(50)
convergence_rate = 0.0
for n in range(len(fibonacci_seq)):
# 计算积分点：F_n × upper_limit
integration_point = fibonacci_scalar_multiply(upper_limit, fibonacci_seq[n])
# 计算函数值：f(F_n × upper_limit)
function_value = evaluate_fibonacci_function(function_coeffs, integration_point)
# 计算权重：1 / F_n
weight = fibonacci_reciprocal([0] * n + [1], integration_precision)
# 计算当前项：f(F_n × x) / F_n
current_term = fibonacci_multiplication(function_value, weight)[0]
# 累加
result = fibonacci_addition(result, current_term)[0]
# 检查收敛性
term_magnitude = estimate_encoding_magnitude(current_term)
if term_magnitude < integration_precision:
convergence_rate = (n + 1) / len(fibonacci_seq)
break
# 计算精度边界
accuracy_bound = integration_precision * (1 - convergence_rate)
return fibonacci_normalize(result), convergence_rate, accuracy_bound
def evaluate_fibonacci_function(
coeffs: List[List[int]],
point: List[int]
) -> List[int]:
"""
计算Fibonacci多项式在给定点的值
f(x) = ∑ᵢ aᵢ × x^i (所有运算都在Zeckendorf空间中)
"""
if not coeffs:
return [0]
result = coeffs[0].copy() # 常数项
x_power = [1] # x^0 = 1
for i in range(1, len(coeffs)):
# x^i = x^{i-1} × x
x_power = fibonacci_multiplication(x_power, point)[0]
# aᵢ × x^i
term = fibonacci_multiplication(coeffs[i], x_power)[0]
# 累加
result = fibonacci_addition(result, term)[0]
return fibonacci_normalize(result)
算法27-1-6：自指完备性验证
输入：
system_state: 系统当前状态编码verification_depth: 验证深度entropy_threshold: 熵增阈值
输出：
self_consistency: 自一致性验证结果entropy_increase: 熵增量测量completeness_proof: 完备性证明状态
def verify_self_referential_completeness(
system_state: Dict[str, List[int]],
verification_depth: int = 10,
entropy_threshold: float = 1e-6
) -> Tuple[bool, float, bool]:
"""
验证纯Zeckendorf数学体系的自指完备性
根据唯一公理：自指完备的系统必然熵增
"""
# 第一步：验证系统可以描述自身
self_description = system_describes_itself(system_state, verification_depth)
# 第二步：测量熵增
initial_entropy = compute_system_entropy(system_state)
evolved_state = evolve_system_one_step(system_state)
final_entropy = compute_system_entropy(evolved_state)
entropy_increase = final_entropy - initial_entropy
# 第三步：验证完备性
completeness_proof = verify_mathematical_completeness(system_state, verification_depth)
# 自一致性：系统描述自身 ∧ 熵增 > 阈值
self_consistency = self_description and (entropy_increase > entropy_threshold)
return self_consistency, entropy_increase, completeness_proof
def system_describes_itself(
state: Dict[str, List[int]],
depth: int
) -> bool:
"""
验证系统是否能够描述自身的数学结构
"""
# 系统必须包含描述Zeckendorf编码规则的函数
if 'zeckendorf_rules' not in state:
return False
# 系统必须包含描述无11约束的函数
if 'no_11_constraint' not in state:
return False
# 递归验证：系统描述的规则是否能够重新生成系统本身
for level in range(depth):
# 使用系统的规则重新构建系统
reconstructed_state = reconstruct_system_using_own_rules(state, level + 1)
# 检查重构的系统是否与原系统一致
if not states_are_equivalent(state, reconstructed_state, 1e-10):
return False
return True
def compute_system_entropy(state: Dict[str, List[int]]) -> float:
"""
计算Zeckendorf系统的熵
熵定义为系统中非零Fibonacci系数的信息含量
"""
total_entropy = 0.0
for component_name, encoding in state.items():
# 计算每个组件的熵
non_zero_count = sum(1 for x in encoding if x != 0)
if non_zero_count > 0:
# Shannon熵的Fibonacci版本
component_entropy = -sum(
(1/non_zero_count) * math.log2(1/non_zero_count)
for _ in range(non_zero_count)
)
total_entropy += component_entropy
return total_entropy
def evolve_system_one_step(state: Dict[str, List[int]]) -> Dict[str, List[int]]:
"""
演化系统一步：应用自指运算符
"""
evolved_state = {}
for component_name, encoding in state.items():
# 对每个组件应用适当的演化规则
if 'phi' in component_name:
evolved_encoding = apply_phi_operator(encoding, 1e-12)[0]
elif 'pi' in component_name:
evolved_encoding = apply_pi_operator(encoding, 1e-12)[0]
elif 'e' in component_name:
evolved_encoding = apply_e_operator(encoding, 1e-12)[0]
else:
# 默认演化：Fibonacci左移（模拟时间演化）
evolved_encoding = fibonacci_left_shift(encoding, 1)
evolved_state[component_name] = evolved_encoding
return evolved_state
def verify_mathematical_completeness(
state: Dict[str, List[int]],
verification_depth: int
) -> bool:
"""
验证数学完备性：系统是否包含所有必要的数学运算
"""
required_operations = [
'fibonacci_addition',
'fibonacci_multiplication',
'fibonacci_division',
'phi_operator',
'pi_operator',
'e_operator',
'fibonacci_derivative',
'fibonacci_integral'
]
# 检查所有必需运算是否在系统中可实现
for operation in required_operations:
if not operation_is_implementable(operation, state, verification_depth):
return False
# 检查运算的封闭性
closure_verified = verify_operation_closure(state, verification_depth)
# 检查一致性公理
axioms_satisfied = verify_mathematical_axioms(state, verification_depth)
return closure_verified and axioms_satisfied
def verify_operation_closure(state: Dict[str, List[int]], depth: int) -> bool:
"""验证运算的封闭性：Zeckendorf运算的结果仍在Zeckendorf空间中"""
test_cases = generate_test_encodings(10) # 生成10个测试用例
for a, b in test_cases:
# 测试加法封闭性
sum_result = fibonacci_addition(a, b)[0]
if not is_valid_zeckendorf_encoding(sum_result):
return False
# 测试乘法封闭性
product_result = fibonacci_multiplication(a, b)[0]
if not is_valid_zeckendorf_encoding(product_result):
return False
# 测试运算符封闭性
for op_type in ['phi', 'pi', 'e']:
op_result = apply_mathematical_operator(a, op_type)[0]
if not is_valid_zeckendorf_encoding(op_result):
return False
return True
def is_valid_zeckendorf_encoding(encoding: List[int]) -> bool:
"""验证编码是否为有效的Zeckendorf表示"""
# 检查无11约束
if not verify_no_consecutive_ones(encoding):
return False
# 检查系数范围
sign_offset = 1 if encoding[0] == -1 else 0
for i in range(sign_offset, len(encoding)):
if encoding[i] not in {0, 1}:
return False
return True
性能基准与优化要求
计算复杂度边界
| 运算类型 | 时间复杂度 | 空间复杂度 | 精度保证 |
|---|---|---|---|
| Zeckendorf编码 | O(N log φ) | O(N) | φ^(-N) |
| Fibonacci加法 | O(N) | O(N) | 精确 |
| Fibonacci乘法 | O(N²) | O(N²) | φ^(-N) |
| 数学常数运算符 | O(N log N) | O(N) | φ^(-N) |
| Fibonacci微积分 | O(N³) | O(N²) | φ^(-N/2) |
| 自指完备性验证 | O(N^d) | O(N²) | 递归深度d |
数值稳定性要求
- 无11约束维护：所有中间结果必须满足无11约束
- 溢出控制：自动扩展编码长度防止信息丢失
- 精度累积：误差传播控制在φ^(-N)范围内
- 收敛性保证：级数运算必须验证收敛条件
测试验证标准
必需测试用例
- 编码一致性测试：验证编码-解码的无损性
- 运算封闭性测试：确保所有运算结果仍在Zeckendorf空间
- 数学公理测试：验证交换律、结合律、分配律
- 常数运算符测试：验证φ、π、e运算符的数学性质
- 微积分基本定理测试：验证微积分运算的一致性
- 自指完备性测试：验证系统的自描述能力
- 熵增验证测试：确认系统演化符合唯一公理
边界条件处理
- 零值处理：正确处理Zeckendorf零编码
- 大数值处理：自适应扩展编码长度
- 精度极限：处理接近φ^(-N)的数值
- 运算符特征值：验证运算符的谱性质
与经典数学的兼容性验证
- 连续极限定理：验证N→∞时与经典数学的收敛性
- 精度分析：量化有限N下的逼近误差
- 函数逼近：验证复杂函数的Fibonacci级数展开
这个形式化规范确保了T27-1理论的完整实现和严格验证。所有算法都必须在纯二进制Zeckendorf宇宙中运行，严格维护无11约束，并提供可验证的数学完备性。