Proof of DistinguishableEncodable:
1. Given: x ≠ y with different descriptions
2. By T2-2: Desc(x) and Desc(y) are finite strings
3. Finite strings can be mapped to distinct naturals
4. Define e(z) = StringToNat(Desc(z))
5. Since Desc(x) ≠ Desc(y), we have e(x) ≠ e(y) ∎
Construction of encoding chain:
Step 1: Info(x) → Desc(x)
- By definition of Info, x has distinguishing description
Step 2: Desc(x) → n ∈ ℕ
- By T2-2, finite descriptions map to naturals
- Injection guaranteed by distinguishability
Step 3: n → φ_repr
- By Zeckendorf theorem
- Unique representation for each n
Each step preserves information ∎
Proof of self-encoding:
1. φ-representation rules are finite formal specifications
2. Finite specifications have descriptions in Desc
3. Descriptions map to naturals
4. Naturals have φ-representations
5. Therefore, the system can encode its own rules ∎
Proof:
1. Continuous objects in self-referential systems exist as algorithms
2. Algorithms are finite descriptions
3. Apply standard encoding chain
4. Result: φ-representation of the generating algorithm ∎
Proof of completeness:
Given: x ∈ S with Info(x)
1. By Lemma T2-10.1: x can be distinguished
2. By Lemma T2-10.3: Complete encoding chain exists
3. Apply chain: Info(x) → Desc(x) → n → φ_repr
4. By Lemma T2-10.2: φ_repr exists and is unique
5. By Lemma T2-10.4: System maintains self-reference
Therefore: ∀x . Info(x) → ∃ φ_repr ∎
def verify_zeckendorf_representation():
# 测试Zeckendorf定理
fibonacci = [1, 2, 3, 5, 8, 13, 21, 34, 55, 89]
for n in range(100):
phi_repr = compute_zeckendorf(n)
# 验证唯一性
assert is_unique_representation(n, phi_repr)
# 验证no-11约束
assert no_adjacent_ones(phi_repr)
# 验证值的正确性
value = sum(fibonacci[i] for i, bit in enumerate(phi_repr) if bit)
assert value == n
return True
def is_valid_phi_repr(repr_list):
"""检查是否是有效的φ-表示"""
# 检查no-11约束
for i in range(len(repr_list) - 1):
if repr_list[i] == 1 and repr_list[i+1] == 1:
return False
return True
def compute_zeckendorf(n):
"""计算n的Zeckendorf表示"""
if n == 0:
return []
# 生成Fibonacci数列
fibs = [1, 2]
while fibs[-1] < n:
fibs.append(fibs[-1] + fibs[-2])
# 贪心算法
result = []
for f in reversed(fibs):
if f <= n:
result.append(1)
n -= f
else:
result.append(0)
# 移除前导零
while result and result[0] == 0:
result.pop(0)
return result
def encode_to_phi_system(information):
"""将信息编码到φ-表示系统"""
# Step 1: 提取描述
description = extract_description(information)
# Step 2: 转换为自然数
n = string_to_natural(description)
# Step 3: 计算φ-表示
return compute_zeckendorf(n)