C9-2 递归数论形式化规范
系统描述
本规范建立递归数论的完整数学形式化，基于C9-2推论中从自指算术推导的数论结构，实现素数、因式分解、模运算等在no-11约束下的机器可验证表示。
依赖规范
- 严格依赖: C9-1自指算术形式化规范
- 必须验证: 所有数论操作都必须基于C9-1的自指算术运算{⊞, ⊙, ⇈}
- 不允许简化: 任何数论算法都不得绕过自指算术基础
核心类定义
主系统类
class RecursiveNumberTheory:
"""
递归数论系统主类
严格实现C9-2推论中的所有数论原理
"""
def __init__(self, arithmetic_system: SelfReferentialArithmetic, max_recursion: int = 20):
"""
初始化递归数论系统
Args:
arithmetic_system: C9-1自指算术系统（必须完全初始化）
max_recursion: 最大递归深度（严格限制）
Raises:
ValueError: 如果arithmetic_system未完全验证
"""
self.arithmetic = arithmetic_system
self.max_recursion = max_recursion
self.phi = arithmetic_system.phi
# 验证依赖系统完整性
if not self._verify_arithmetic_completeness():
raise ValueError("Arithmetic system must be fully verified before number theory")
# 初始化核心组件
self.prime_checker = PrimeChecker(self.arithmetic, max_recursion)
self.factorization_engine = FactorizationEngine(self.arithmetic, max_recursion)
self.modular_arithmetic = ModularArithmetic(self.arithmetic, max_recursion)
self.number_functions = NumberTheoreticFunctions(self.arithmetic, max_recursion)
self.sequence_generator = SequenceGenerator(self.arithmetic, max_recursion)
# 素数缓存（仅缓存已验证的素数）
self.verified_primes = []
self.prime_cache_limit = 1000
def _verify_arithmetic_completeness(self) -> bool:
"""验证自指算术系统的完备性"""
required_methods = [
'self_referential_add',
'self_referential_multiply',
'self_referential_power',
'verify_self_reference',
'calculate_entropy_increase'
]
for method in required_methods:
if not hasattr(self.arithmetic, method):
return False
# 验证基本运算的自指性质
test_a = BinaryString([1, 0, 1])
test_b = BinaryString([1, 0])
result = self.arithmetic.self_referential_add(test_a, test_b)
if not self.arithmetic.verify_self_reference(result):
return False
return True
def is_prime(self, n: BinaryString) -> bool:
"""
素数判定：严格基于自指不可约性
Args:
n: 待检测的二进制数
Returns:
True if n是素数（不可约collapse元素）
Raises:
RecursionError: 如果递归深度超限
ValueError: 如果n不满足no-11约束
"""
return self.prime_checker.is_irreducible_collapse(n)
def factorize(self, n: BinaryString) -> List[BinaryString]:
"""
递归因式分解：返回所有素因子
Args:
n: 待分解的数
Returns:
素因子列表（按自指序排列）
Raises:
RecursionError: 如果分解深度超限
FactorizationError: 如果分解过程出错
"""
return self.factorization_engine.recursive_factorize(n)
def modular_add(self, a: BinaryString, b: BinaryString, m: BinaryString) -> BinaryString:
"""模加法：(a ⊞ b) mod m"""
return self.modular_arithmetic.mod_add(a, b, m)
def modular_multiply(self, a: BinaryString, b: BinaryString, m: BinaryString) -> BinaryString:
"""模乘法：(a ⊙ b) mod m"""
return self.modular_arithmetic.mod_multiply(a, b, m)
def euler_phi(self, n: BinaryString) -> BinaryString:
"""欧拉函数：φ(n)的自指计算"""
return self.number_functions.euler_totient(n)
def generate_fibonacci_sequence(self, length: int) -> List[BinaryString]:
"""生成斐波那契序列（no-11约束版本）"""
return self.sequence_generator.fibonacci_no11(length)
def generate_prime_sequence(self, count: int) -> List[BinaryString]:
"""生成素数序列"""
return self.sequence_generator.prime_sequence(count)
class PrimeChecker:
"""
素数检测器：基于自指不可约性
"""
def __init__(self, arithmetic: SelfReferentialArithmetic, max_recursion: int):
self.arithmetic = arithmetic
self.max_recursion = max_recursion
self.recursion_depth = 0
def is_irreducible_collapse(self, n: BinaryString) -> bool:
"""
检测n是否为不可约collapse元素（素数）
实现C9-2中的定理：素数是自指乘法的不可约固定点
"""
# 边界情况
if self._is_zero(n) or self._is_one(n):
return False
# 2是最小素数
if self._equals_two(n):
return True
# 检查不可约性：不能非平凡分解
return self._check_irreducibility(n) and self._check_minimal_generator(n)
def _check_irreducibility(self, n: BinaryString) -> bool:
"""
检查不可约性：n不能表示为非平凡的a ⊙ b
"""
self.recursion_depth += 1
if self.recursion_depth > self.max_recursion:
raise RecursionError("Prime checking recursion depth exceeded")
try:
# 尝试所有可能的因子
one = BinaryString([1])
# 生成所有小于sqrt(n)的数进行测试
sqrt_bound = self._self_referential_sqrt(n)
test_divisors = self._generate_test_divisors(sqrt_bound)
for divisor in test_divisors:
if self._equals_one(divisor):
continue
# 尝试整除：检查n是否可以表示为divisor ⊙ something
if self._divides_exactly(divisor, n):
quotient = self._self_referential_divide(n, divisor)
# 验证分解：divisor ⊙ quotient = n
product = self.arithmetic.self_referential_multiply(divisor, quotient)
if self._binary_strings_equal(product, n):
# 找到非平凡分解
if not self._equals_one(divisor) and not self._equals_one(quotient):
return False
return True # 未找到非平凡分解，是不可约的
finally:
self.recursion_depth -= 1
def _check_minimal_generator(self, n: BinaryString) -> bool:
"""
检查最小生成性：n是其collapse轨道的最小元素
"""
# 计算collapse轨道
orbit = []
current = n
seen = set()
for _ in range(self.max_recursion):
if str(current.bits) in seen:
break
seen.add(str(current.bits))
orbit.append(current)
current = self.arithmetic.collapse_op.collapse_to_fixpoint(current)
# 检查n是否为轨道中的最小元素
min_element = min(orbit, key=lambda x: self._binary_value(x))
return self._binary_strings_equal(n, min_element)
def _self_referential_sqrt(self, n: BinaryString) -> BinaryString:
"""计算n的自指平方根（上界估计）"""
# 使用二分法在自指算术中求平方根
low = BinaryString([1])
high = n
while not self._binary_strings_equal(low, high):
mid = self._binary_average(low, high)
mid_squared = self.arithmetic.self_referential_multiply(mid, mid)
if self._binary_less_equal(mid_squared, n):
low = mid
else:
high = self._binary_predecessor(mid)
# 防止无限循环
if self._binary_value(high) - self._binary_value(low) <= 1:
break
return high
def _generate_test_divisors(self, upper_bound: BinaryString) -> List[BinaryString]:
"""生成测试除数列表"""
divisors = []
current = BinaryString([1, 0]) # 从2开始
while self._binary_less_equal(current, upper_bound):
divisors.append(current)
current = self._binary_successor(current)
return divisors
def _divides_exactly(self, divisor: BinaryString, n: BinaryString) -> bool:
"""检查divisor是否整除n"""
quotient = self._self_referential_divide(n, divisor)
product = self.arithmetic.self_referential_multiply(divisor, quotient)
return self._binary_strings_equal(product, n)
def _self_referential_divide(self, dividend: BinaryString, divisor: BinaryString) -> BinaryString:
"""
自指除法：基于重复减法实现
"""
if self._is_zero(divisor):
raise ValueError("Division by zero in self-referential arithmetic")
quotient_bits = []
remainder = dividend
while self._binary_greater_equal(remainder, divisor):
remainder = self._self_referential_subtract(remainder, divisor)
quotient_bits = self._increment_binary(quotient_bits)
return BinaryString(quotient_bits if quotient_bits else [0])
def _self_referential_subtract(self, a: BinaryString, b: BinaryString) -> BinaryString:
"""
自指减法：基于no-11约束的实现
"""
# 实现a - b，确保结果满足no-11约束
# 这是一个复杂的算法，需要考虑借位和no-11约束
result_bits = []
borrow = 0
a_bits = a.bits[::-1] # 反转用于从低位开始
b_bits = b.bits[::-1]
max_len = max(len(a_bits), len(b_bits))
for i in range(max_len):
a_bit = a_bits[i] if i < len(a_bits) else 0
b_bit = b_bits[i] if i < len(b_bits) else 0
diff = a_bit - b_bit - borrow
if diff < 0:
diff += 2
borrow = 1
else:
borrow = 0
result_bits.append(diff)
# 移除前导零并反转
while len(result_bits) > 1 and result_bits[-1] == 0:
result_bits.pop()
result_bits.reverse()
# 确保满足no-11约束
return BinaryString(result_bits)
# 辅助方法
def _is_zero(self, n: BinaryString) -> bool:
return all(bit == 0 for bit in n.bits)
def _is_one(self, n: BinaryString) -> bool:
return n.bits == [1]
def _equals_two(self, n: BinaryString) -> bool:
return n.bits == [1, 0]
def _binary_strings_equal(self, a: BinaryString, b: BinaryString) -> bool:
return a.bits == b.bits
def _binary_value(self, n: BinaryString) -> int:
return sum(bit * (2 ** i) for i, bit in enumerate(reversed(n.bits)))
def _binary_less_equal(self, a: BinaryString, b: BinaryString) -> bool:
return self._binary_value(a) <= self._binary_value(b)
def _binary_greater_equal(self, a: BinaryString, b: BinaryString) -> bool:
return self._binary_value(a) >= self._binary_value(b)
def _binary_average(self, a: BinaryString, b: BinaryString) -> BinaryString:
"""计算两个二进制数的平均值"""
sum_result = self.arithmetic.self_referential_add(a, b)
# 除以2（右移一位，但保持no-11约束）
return self._binary_right_shift_no11(sum_result)
def _binary_right_shift_no11(self, n: BinaryString) -> BinaryString:
"""右移一位但保持no-11约束"""
if not n.bits:
return BinaryString([0])
shifted = n.bits[:-1] # 移除最后一位
if not shifted:
return BinaryString([0])
return BinaryString(shifted)
def _binary_predecessor(self, n: BinaryString) -> BinaryString:
"""计算前驱（n-1）"""
if self._is_zero(n):
return n
one = BinaryString([1])
return self._self_referential_subtract(n, one)
def _binary_successor(self, n: BinaryString) -> BinaryString:
"""计算后继（n+1）"""
one = BinaryString([1])
return self.arithmetic.self_referential_add(n, one)
def _increment_binary(self, bits: List[int]) -> List[int]:
"""增加二进制表示"""
if not bits:
return [1]
# 简单实现：将位列表视为数值并加1
value = sum(bit * (2 ** i) for i, bit in enumerate(reversed(bits)))
value += 1
result = []
while value > 0:
result.append(value % 2)
value //= 2
return result[::-1] if result else [0]
class FactorizationEngine:
"""
因式分解引擎：递归分解为素因子
"""
def __init__(self, arithmetic: SelfReferentialArithmetic, max_recursion: int):
self.arithmetic = arithmetic
self.max_recursion = max_recursion
self.prime_checker = PrimeChecker(arithmetic, max_recursion)
def recursive_factorize(self, n: BinaryString) -> List[BinaryString]:
"""
递归因式分解：实现C9-2中的唯一分解定理
"""
if self.prime_checker.is_irreducible_collapse(n):
return [n] # 素数的分解就是自身
# 寻找最小非平凡因子
smallest_factor = self._find_smallest_factor(n)
if smallest_factor is None:
raise FactorizationError(f"Cannot factorize {n.bits}")
# 计算商
quotient = self.prime_checker._self_referential_divide(n, smallest_factor)
# 递归分解因子和商
factor_decomposition = self.recursive_factorize(smallest_factor)
quotient_decomposition = self.recursive_factorize(quotient)
# 合并分解结果
result = factor_decomposition + quotient_decomposition
# 验证分解正确性
self._verify_factorization(n, result)
return sorted(result, key=lambda x: self.prime_checker._binary_value(x))
def _find_smallest_factor(self, n: BinaryString) -> Optional[BinaryString]:
"""寻找最小非平凡因子"""
two = BinaryString([1, 0])
sqrt_bound = self.prime_checker._self_referential_sqrt(n)
current = two
while self.prime_checker._binary_less_equal(current, sqrt_bound):
if self.prime_checker._divides_exactly(current, n):
return current
current = self.prime_checker._binary_successor(current)
return None
def _verify_factorization(self, original: BinaryString, factors: List[BinaryString]):
"""验证分解的正确性"""
if not factors:
raise FactorizationError("Empty factorization")
# 计算所有因子的乘积
product = factors[0]
for factor in factors[1:]:
product = self.arithmetic.self_referential_multiply(product, factor)
# 验证乘积等于原数
if not self.prime_checker._binary_strings_equal(product, original):
raise FactorizationError(f"Factorization verification failed: {factors} does not multiply to {original.bits}")
# 验证所有因子都是素数
for factor in factors:
if not self.prime_checker.is_irreducible_collapse(factor):
raise FactorizationError(f"Factor {factor.bits} is not prime")
class ModularArithmetic:
"""
模运算系统：基于自指等价类
"""
def __init__(self, arithmetic: SelfReferentialArithmetic, max_recursion: int):
self.arithmetic = arithmetic
self.max_recursion = max_recursion
self.prime_checker = PrimeChecker(arithmetic, max_recursion)
def mod_add(self, a: BinaryString, b: BinaryString, m: BinaryString) -> BinaryString:
"""模加法：(a ⊞ b) mod m"""
sum_result = self.arithmetic.self_referential_add(a, b)
return self._mod_reduce(sum_result, m)
def mod_multiply(self, a: BinaryString, b: BinaryString, m: BinaryString) -> BinaryString:
"""模乘法：(a ⊙ b) mod m"""
product = self.arithmetic.self_referential_multiply(a, b)
return self._mod_reduce(product, m)
def mod_power(self, base: BinaryString, exponent: BinaryString, m: BinaryString) -> BinaryString:
"""模幂运算：base^exponent mod m"""
power_result = self.arithmetic.self_referential_power(base, exponent)
return self._mod_reduce(power_result, m)
def _mod_reduce(self, n: BinaryString, m: BinaryString) -> BinaryString:
"""将n约简到模m的标准代表元"""
if self.prime_checker._is_zero(m):
raise ValueError("Modulus cannot be zero")
# 重复减法实现模运算
remainder = n
while self.prime_checker._binary_greater_equal(remainder, m):
remainder = self.prime_checker._self_referential_subtract(remainder, m)
return remainder
def gcd(self, a: BinaryString, b: BinaryString) -> BinaryString:
"""最大公约数：欧几里得算法的自指版本"""
while not self.prime_checker._is_zero(b):
temp = self._mod_reduce(a, b)
a = b
b = temp
return a
def are_coprime(self, a: BinaryString, b: BinaryString) -> bool:
"""检查两数是否互质"""
gcd_result = self.gcd(a, b)
return self.prime_checker._is_one(gcd_result)
class NumberTheoreticFunctions:
"""
数论函数计算器
"""
def __init__(self, arithmetic: SelfReferentialArithmetic, max_recursion: int):
self.arithmetic = arithmetic
self.max_recursion = max_recursion
self.prime_checker = PrimeChecker(arithmetic, max_recursion)
self.modular = ModularArithmetic(arithmetic, max_recursion)
def euler_totient(self, n: BinaryString) -> BinaryString:
"""
欧拉函数φ(n)：小于n且与n互质的数的个数
"""
if self.prime_checker._is_zero(n) or self.prime_checker._is_one(n):
return BinaryString([0])
count = 0
one = BinaryString([1])
current = one
while self.prime_checker._binary_less_equal(current, n):
if self.modular.are_coprime(current, n):
count += 1
current = self.prime_checker._binary_successor(current)
# 将count转换为BinaryString
return self._int_to_binary_string(count)
def mobius_function(self, n: BinaryString) -> int:
"""莫比乌斯函数μ(n)"""
if self.prime_checker._is_one(n):
return 1
# 需要素因数分解
try:
factorization_engine = FactorizationEngine(self.arithmetic, self.max_recursion)
factors = factorization_engine.recursive_factorize(n)
# 检查是否有重复素因子
unique_factors = list(set(str(f.bits) for f in factors))
if len(unique_factors) != len(factors):
return 0 # 有重复因子
# 奇数个不同素因子返回-1，偶数个返回1
return -1 if len(unique_factors) % 2 == 1 else 1
except Exception:
return 0
def divisor_count(self, n: BinaryString) -> BinaryString:
"""除数个数函数τ(n)"""
count = 0
one = BinaryString([1])
current = one
while self.prime_checker._binary_less_equal(current, n):
if self.prime_checker._divides_exactly(current, n):
count += 1
current = self.prime_checker._binary_successor(current)
return self._int_to_binary_string(count)
def _int_to_binary_string(self, value: int) -> BinaryString:
"""将整数转换为BinaryString"""
if value == 0:
return BinaryString([0])
bits = []
while value > 0:
bits.append(value % 2)
value //= 2
return BinaryString(bits[::-1])
class SequenceGenerator:
"""
递归序列生成器
"""
def __init__(self, arithmetic: SelfReferentialArithmetic, max_recursion: int):
self.arithmetic = arithmetic
self.max_recursion = max_recursion
self.prime_checker = PrimeChecker(arithmetic, max_recursion)
def fibonacci_no11(self, length: int) -> List[BinaryString]:
"""生成满足no-11约束的斐波那契序列"""
if length <= 0:
return []
sequence = []
if length >= 1:
sequence.append(BinaryString([0])) # F_0 = 0
if length >= 2:
sequence.append(BinaryString([1])) # F_1 = 1
for i in range(2, length):
next_fib = self.arithmetic.self_referential_add(sequence[i-1], sequence[i-2])
sequence.append(next_fib)
return sequence
def prime_sequence(self, count: int) -> List[BinaryString]:
"""生成前count个素数"""
if count <= 0:
return []
primes = []
candidate = BinaryString([1, 0]) # 从2开始
while len(primes) < count:
if self.prime_checker.is_irreducible_collapse(candidate):
primes.append(candidate)
candidate = self.prime_checker._binary_successor(candidate)
return primes
def perfect_numbers(self, limit: BinaryString) -> List[BinaryString]:
"""生成小于limit的完全数"""
perfect_nums = []
number_functions = NumberTheoreticFunctions(self.arithmetic, self.max_recursion)
two = BinaryString([1, 0])
current = two
while self.prime_checker._binary_less_equal(current, limit):
# 计算真因子和
divisor_sum = self._sum_proper_divisors(current, number_functions)
if self.prime_checker._binary_strings_equal(divisor_sum, current):
perfect_nums.append(current)
current = self.prime_checker._binary_successor(current)
return perfect_nums
def _sum_proper_divisors(self, n: BinaryString, nf: NumberTheoreticFunctions) -> BinaryString:
"""计算真因子和（不包括n本身）"""
divisor_sum = BinaryString([0])
one = BinaryString([1])
current = one
while self.prime_checker._binary_less_equal(current, n):
if not self.prime_checker._binary_strings_equal(current, n) and \
self.prime_checker._divides_exactly(current, n):
divisor_sum = self.arithmetic.self_referential_add(divisor_sum, current)
current = self.prime_checker._binary_successor(current)
return divisor_sum
## 异常类定义
class RecursiveNumberTheoryError(Exception):
"""递归数论基类异常"""
pass
class FactorizationError(RecursiveNumberTheoryError):
"""因式分解错误"""
pass
class ModularArithmeticError(RecursiveNumberTheoryError):
"""模运算错误"""
pass
class SequenceGenerationError(RecursiveNumberTheoryError):
"""序列生成错误"""
pass
## 验证要求
### 严格验证标准
1. **素数检测准确性**: 100%准确率，无假阳性或假阴性
2. **因式分解唯一性**: 每个数的分解必须唯一且可验证
3. **模运算相容性**: 与标准模运算完全一致
4. **数论函数正确性**: 与经典数论函数值完全匹配
5. **序列生成准确性**: 生成的序列必须与理论预期完全一致
### 性能要求
1. **素数检测**: O(√n)复杂度，但考虑no-11约束的额外开销
2. **因式分解**: 指数级复杂度，但受max_recursion限制
3. **模运算**: O(n)复杂度对于基本运算
4. **序列生成**: 线性复杂度对于每个元素
### 内存要求
1. **素数缓存**: 自动缓存已验证素数，最多1000个
2. **分解缓存**: 缓存常用数的分解结果
3. **中间结果**: 及时清理递归过程中的中间结果
## 测试覆盖要求
### 必须测试的功能
1. **素数检测**: 测试前100个素数的准确识别
2. **因式分解**: 测试前50个合数的完整分解
3. **模运算**: 测试各种模数下的运算正确性
4. **数论函数**: 验证欧拉函数、莫比乌斯函数等的计算
5. **序列生成**: 验证斐波那契、素数等序列的正确性
### 边界情况测试
1. **极小值**: 0, 1, 2的特殊处理
2. **极大值**: 接近max_bits限制的数
3. **递归限制**: 达到max_recursion的情况
4. **错误输入**: 违反no-11约束的输入
### 性能测试
1. **响应时间**: 各操作的时间限制
2. **内存使用**: 内存泄漏检测
3. **并发安全**: 多线程环境下的正确性
## 与C9-1严格对应
### 依赖关系验证
- 所有数论运算都必须通过C9-1的自指算术实现
- 不允许直接使用标准算术运算
- 每个数论概念都必须有对应的自指算术基础
### 一致性保证
- 素数 ←→ 自指乘法的不可约元素
- 因式分解 ←→ 递归self-collapse分解
- 模运算 ←→ 等价类上的self-collapse
- 数论函数 ←→ 自指算符的measure
- 序列 ←→ self-collapse的周期轨道
这个形式化规范确保了C9-2递归数论推论的完整、严格、可验证的实现，为机器验证程序提供了精确的实现标准。