type: theorem
verification: machine_ready
dependencies: ["T2-4-formal.md", "T2-3-formal.md", "D1-3-formal.md"]
verification_points:
- constraint_necessity
- information_capacity_analysis
- symmetry_preservation
- fibonacci_emergence
- golden_ratio_optimization
MinimalConstraintTheorem : Prop ≡
∀S : System . SelfRefComplete(S) →
∃C : Constraint .
(MinimalConstraint(C) ∧ MaximizesCapacity(C) ∧ Length(C) = 2)
where
Constraint : Type = Pattern × Bool // (pattern, forbidden)
MinimalConstraint(C) ≡ ∀C' . Ensures(C', UniqueDecodability) → Length(C) ≤ Length(C')
MaximizesCapacity(C) ≡ ∀C' . Length(C') = Length(C) → Capacity(C) ≥ Capacity(C')
InformationCapacity : Constraint → ℝ ≡
λC . lim_{n→∞} (log N_C(n)) / n
where
N_C(n) : ℕ = |{s ∈ {0,1}^n : ¬Contains(s, C.pattern)}|
Contains(s, p) : Bool = ∃i . s[i:i+|p|] = p
SymmetryPreserving : Constraint → Bool ≡
λC . ∀σ : {0,1} → {0,1} .
(σ(0) = 1 ∧ σ(1) = 0) →
(Forbidden(C, p) ↔ Forbidden(C, σ(p)))
where
σ(p) = map σ p // Apply σ to each bit
FibonacciRecurrence : (ℕ → ℕ) → Bool ≡
λf . f(0) = 1 ∧ f(1) = 2 ∧ ∀n ≥ 2 . f(n) = f(n-1) + f(n-2)
GoldenRatio : ℝ ≡ (1 + √5) / 2 // φ
ConstraintNecessity : Prop ≡
∀S : System . UniqueDecodability(S) → ∃C : Constraint . AppliesTo(S, C)
Proof of constraint necessity:
By contradiction. Assume no constraints:
1. Consider prefix codes without constraints
2. Can have both "01" and "010" as codewords
3. Decoding "010" is ambiguous:
- Could be single codeword "010"
- Could be "01" followed by "0"
4. Violates unique decodability
Therefore constraints are necessary ∎
LengthOptimization : Prop ≡
∀C : Constraint .
(Length(C) = 1 → Capacity(C) = 0) ∧
(Length(C) = 2 → Capacity(C) > 0) ∧
(Length(C) ≥ 3 → ViolatesMinimality(C))
Proof of length optimization:
Case Length(C) = 1:
- Can only forbid "0" or "1"
- Results in unary system
- Capacity = 0 (no information)
Case Length(C) = 2:
- Four choices: "00", "01", "10", "11"
- Each gives non-trivial constraint
- Capacity > 0 (proven below)
Case Length(C) ≥ 3:
- Constraint description length ≥ 3
- Self-description requires encoding the constraint
- Longer patterns → more complex description
- Violates finite description requirement ∎
SymmetryAnalysis : Prop ≡
∀C : Constraint . Length(C) = 2 →
(SymmetryPreserving(C) ↔ (C.pattern ∈ {"00", "11"}))
Proof of symmetry preservation:
For length-2 patterns under bit-flip σ:
1. σ("00") = "11", σ("11") = "00"
- These form a symmetric pair
2. σ("01") = "10", σ("10") = "01"
- These form another pair
3. Forbidding "00" or "11" preserves symmetry:
- If "00" forbidden, σ maps to "11"
- System treats 0 and 1 symmetrically
4. Forbidding "01" or "10" breaks symmetry:
- Creates asymmetry between 0 and 1
- Violates self-referential structure symmetry
Therefore only "00" and "11" preserve symmetry ∎
FibonacciEmergence : Prop ≡
∀C : Constraint .
(C.pattern = "11" ∧ C.forbidden = true) →
FibonacciRecurrence(n ↦ N_C(n))
Proof of Fibonacci emergence:
Let a_n = number of valid strings of length n (no "11")
1. Base cases:
- a_0 = 1 (empty string)
- a_1 = 2 ("0" and "1")
2. Recursive construction for a_n:
- Strings ending in "0": can append to any a_{n-1}
- Strings ending in "1": must end in "01", from a_{n-2}
- Cannot end in "11" (forbidden)
3. Therefore: a_n = a_{n-1} + a_{n-2}
4. This is Fibonacci recurrence with offset:
a_n = F_{n+2} where F_n is nth Fibonacci number ∎
CapacityCalculation : Prop ≡
∀C : Constraint . C.pattern = "11" →
InformationCapacity(C) = log φ
where φ = GoldenRatio
Proof of capacity calculation:
From Fibonacci emergence:
1. N_C(n) = F_{n+2}
2. Fibonacci asymptotic behavior:
F_n ~ φ^n / √5 as n → ∞
3. Information capacity:
C = lim_{n→∞} log(F_{n+2}) / n
= lim_{n→∞} log(φ^{n+2} / √5) / n
= lim_{n→∞} ((n+2)log φ - log √5) / n
= log φ
4. Numerically: log φ ≈ 0.694 bits per symbol ∎
MainTheorem : Prop ≡
∀S : System . SelfRefComplete(S) →
OptimalConstraint = ("11", true) ∨ OptimalConstraint = ("00", true)
where
OptimalConstraint satisfies:
1. MinimalLength (= 2)
2. MaximalCapacity (= log φ)
3. SymmetryPreserving
Proof of minimal constraint theorem:
Given self-referential completeness:
1. By Lemma T2-5.1: Need constraints
2. By Lemma T2-5.2: Optimal length = 2
3. By Lemma T2-5.3: Must preserve symmetry
→ C ∈ {"00", "11"}
4. By symmetry, both give same capacity
5. By Lemma T2-5.5: Capacity = log φ
6. This is maximal among all minimal constraints:
- Length 1: capacity = 0
- Length 2 asymmetric: violates self-reference
- Length ≥ 3: violates minimality
Therefore no-11 (or no-00) is optimal ∎
def verify_constraint_necessity():
# 创建无约束系统
unconstrained = BinarySystem(constraints=set())
# 添加前缀冲突的码字
unconstrained.add_codeword("01")
unconstrained.add_codeword("010")
# 测试解码歧义
ambiguous_string = "010"
decodings = unconstrained.decode_all_possible(ambiguous_string)
# 验证存在多种解码
assert len(decodings) > 1
assert ["010"] in decodings
assert ["01", "0"] in decodings
# 验证违反唯一可解码性
assert not unconstrained.has_unique_decodability()
return True
def verify_information_capacity_analysis():
import math
# 不同长度约束的容量
capacities = {}
# 长度1约束
constraint_0 = BinaryConstraint("0")
constraint_1 = BinaryConstraint("1")
capacities[1] = {
"0": constraint_0.compute_capacity(100),
"1": constraint_1.compute_capacity(100)
}
# 长度2约束
for pattern in ["00", "01", "10", "11"]:
constraint = BinaryConstraint(pattern)
capacities[2] = capacities.get(2, {})
capacities[2][pattern] = constraint.compute_capacity(100)
# 验证长度1约束容量为0
assert all(cap == 0 for cap in capacities[1].values())
# 验证长度2约束容量为正
assert all(cap > 0 for cap in capacities[2].values())
# 验证对称约束容量相等
assert abs(capacities[2]["00"] - capacities[2]["11"]) < 0.001
assert abs(capacities[2]["01"] - capacities[2]["10"]) < 0.001
# 验证黄金比例
golden_ratio = (1 + math.sqrt(5)) / 2
expected_capacity = math.log2(golden_ratio)
actual_capacity = capacities[2]["11"]
assert abs(actual_capacity - expected_capacity) < 0.01
return True
def verify_symmetry_preservation():
# 定义比特翻转操作
def bit_flip(pattern):
return ''.join('1' if b == '0' else '0' for b in pattern)
# 测试所有长度2模式
patterns = ["00", "01", "10", "11"]
symmetry_results = {}
for pattern in patterns:
flipped = bit_flip(pattern)
# 检查是否形成对称对
constraint1 = BinaryConstraint(pattern)
constraint2 = BinaryConstraint(flipped)
# 计算两个约束下的字符串分布
dist1 = constraint1.compute_symbol_distribution(100)
dist2 = constraint2.compute_symbol_distribution(100)
# 检查分布是否对称
is_symmetric = (
abs(dist1['0'] - dist2['1']) < 0.01 and
abs(dist1['1'] - dist2['0']) < 0.01
)
symmetry_results[pattern] = {
'flipped': flipped,
'is_symmetric': is_symmetric,
'preserves_symmetry': pattern in ["00", "11"]
}
# 验证只有"00"和"11"保持对称性
for pattern, result in symmetry_results.items():
if pattern in ["00", "11"]:
assert result['preserves_symmetry']
else:
assert not result['preserves_symmetry']
return True
def verify_fibonacci_emergence():
constraint = BinaryConstraint("11")
# 计算前20个值
counts = []
for n in range(20):
count = constraint.count_valid_strings(n)
counts.append(count)
# 验证Fibonacci递归关系
# a_n = a_{n-1} + a_{n-2} for n >= 2
for i in range(2, len(counts)):
expected = counts[i-1] + counts[i-2]
actual = counts[i]
assert actual == expected, f"Failed at n={i}: {actual} != {expected}"
# 验证与标准Fibonacci数列的关系
# a_n = F_{n+2}
fibonacci = [0, 1]
for i in range(2, 22):
fibonacci.append(fibonacci[i-1] + fibonacci[i-2])
for i in range(len(counts)):
assert counts[i] == fibonacci[i+2], f"a_{i} != F_{i+2}"
return True
def verify_golden_ratio_optimization():
import math
# 测试所有可能的约束
constraints = {}
# 长度1约束
for pattern in ["0", "1"]:
c = BinaryConstraint(pattern)
constraints[pattern] = {
'length': 1,
'capacity': c.compute_capacity(100),
'growth_rate': c.compute_growth_rate(50)
}
# 长度2约束
for pattern in ["00", "01", "10", "11"]:
c = BinaryConstraint(pattern)
constraints[pattern] = {
'length': 2,
'capacity': c.compute_capacity(100),
'growth_rate': c.compute_growth_rate(50),
'preserves_symmetry': pattern in ["00", "11"]
}
# 长度3约束（示例）
for pattern in ["000", "111", "101"]:
c = BinaryConstraint(pattern)
constraints[pattern] = {
'length': 3,
'capacity': c.compute_capacity(100),
'description_complexity': len(pattern) * math.log2(2)
}
# 找出最优约束
valid_constraints = [
(p, info) for p, info in constraints.items()
if info.get('capacity', 0) > 0
]
# 在保持对称性的最小长度约束中找最优
minimal_symmetric = [
(p, info) for p, info in valid_constraints
if info['length'] == 2 and info.get('preserves_symmetry', False)
]
# 验证"00"和"11"是最优的
assert len(minimal_symmetric) == 2
assert all(p in ["00", "11"] for p, _ in minimal_symmetric)
# 验证容量等于log(φ)
golden_ratio = (1 + math.sqrt(5)) / 2
expected_capacity = math.log2(golden_ratio)
for pattern, info in minimal_symmetric:
assert abs(info['capacity'] - expected_capacity) < 0.01
assert abs(info['growth_rate'] - golden_ratio) < 0.01
return True
import math
from typing import Set, List, Dict
class BinaryConstraint:
"""二进制约束系统"""
def __init__(self, forbidden_pattern: str):
self.forbidden_pattern = forbidden_pattern
self.pattern_length = len(forbidden_pattern)
self._cache = {} # 缓存计算结果
def is_valid(self, string: str) -> bool:
"""检查字符串是否满足约束"""
return self.forbidden_pattern not in string
def count_valid_strings(self, n: int) -> int:
"""计算长度为n的有效字符串数"""
if n in self._cache:
return self._cache[n]
if n == 0:
return 1
if n == 1:
return 2
# 对于no-11约束的特殊优化
if self.forbidden_pattern == "11":
# Fibonacci递归
result = self.count_valid_strings(n-1) + self.count_valid_strings(n-2)
else:
# 一般情况：枚举所有可能
count = 0
for i in range(2**n):
binary = format(i, f'0{n}b')
if self.is_valid(binary):
count += 1
result = count
self._cache[n] = result
return result
def compute_capacity(self, max_n: int) -> float:
"""计算信息容量"""
# 使用足够大的n来近似极限
n = min(max_n, 100)
count = self.count_valid_strings(n)
if count <= 1:
return 0.0
return math.log2(count) / n
def compute_growth_rate(self, max_n: int) -> float:
"""计算增长率"""
# 计算连续比率的平均值
ratios = []
for n in range(10, min(max_n, 50)):
count_n = self.count_valid_strings(n)
count_n_minus_1 = self.count_valid_strings(n-1)
if count_n_minus_1 > 0:
ratios.append(count_n / count_n_minus_1)
if not ratios:
return 0.0
return sum(ratios) / len(ratios)
def compute_symbol_distribution(self, max_n: int) -> Dict[str, float]:
"""计算符号分布"""
total_0 = 0
total_1 = 0
total_bits = 0
# 统计所有有效字符串中的0和1
for n in range(1, min(max_n, 20)):
for i in range(2**n):
binary = format(i, f'0{n}b')
if self.is_valid(binary):
total_0 += binary.count('0')
total_1 += binary.count('1')
total_bits += n
if total_bits == 0:
return {'0': 0.0, '1': 0.0}
return {
'0': total_0 / total_bits,
'1': total_1 / total_bits
}
class BinarySystem:
"""二进制编码系统"""
def __init__(self, constraints: Set[str] = None):
self.constraints = constraints or set()
self.codewords = set()
def add_codeword(self, word: str):
"""添加码字"""
self.codewords.add(word)
def decode_all_possible(self, string: str) -> List[List[str]]:
"""找出所有可能的解码方式"""
if not string:
return [[]]
decodings = []
# 尝试每个可能的前缀
for i in range(1, len(string) + 1):
prefix = string[:i]
if prefix in self.codewords:
# 递归解码剩余部分
suffix_decodings = self.decode_all_possible(string[i:])
for suffix_dec in suffix_decodings:
decodings.append([prefix] + suffix_dec)
return decodings
def has_unique_decodability(self) -> bool:
"""检查是否有唯一可解码性"""
# 简化检查：查找前缀冲突
for w1 in self.codewords:
for w2 in self.codewords:
if w1 != w2 and w1.startswith(w2):
return False
return True