Chapter 8: The Information-Theoretic Argument

8.1 Information Content of Zeros

Definition 8.1.1: For a zero ρ = σ + it, define its information content: $I(\rho) = -\log\left(\frac{1}{1+|t|}\right) = \log(1+|t|)$

Theorem 8.1.1: The total information in all zeros up to height T is: $I_{\text{total}}(T) = \sum_{|\text{Im}(\rho)| \leq T} I(\rho) = T\log T + O(T)$

Proof: Using the zero counting formula N(T) ~ (T/2π)log(T/2π), sum the logarithms. ∎

8.2 Dimensional Reduction Requirements

Theorem 8.2.1: If zeros fill a 2D region in the critical strip, the information density becomes infinite.

Proof: Let $D \subset \{s : 0 < \text{Re}(s) < 1\}$ be a 2D domain containing zeros. The information density is: $\rho_I = \lim_{\epsilon \to 0} \frac{\text{Information in } B_\epsilon}{\text{Area of } B_\epsilon}$

For a 2D distribution: $\rho_I = \infty$ (impossible). For a 1D distribution: $\rho_I$ finite. ∎

Theorem 8.2.2: The zeta function can encode at most 1D worth of zero information.

Proof: $\zeta(s)$ is determined by its values on any line $\text{Re}(s) = \sigma_0 > 1$. This gives 1D worth of data. By analytic continuation, this determines all zeros. Therefore zeros must form a 1D set. ∎

8.3 Holographic Principle for ζ(s)

Definition 8.3.1: A holographic encoding satisfies:

• Lower dimensional boundary encodes higher dimensional bulk
• Information is not lost in the encoding
• Reconstruction is possible from boundary data

Theorem 8.3.1: The zeros of ζ(s) form a holographic encoding of prime distribution.

Proof: The explicit formula: $\psi(x) = x - \sum_\rho \frac{x^\rho}{\rho} - \log(2\pi)$ shows primes (1D distribution on real line) are encoded by zeros (0D points in complex plane). This dimensional reduction is holographic. ∎

Theorem 8.3.2: Holographic consistency requires all zeros on one line.

Proof: For consistent holographic encoding:

1. Information must be uniformly distributed
2. No information clustering or voids
3. Reconstruction must be unique

These requirements force zeros onto a single line. By symmetry, this line is $\text{Re}(s) = 1/2$. ∎

8.4 Information Conservation Laws

Theorem 8.4.1 (Information Conservation): The functional equation preserves information: $I(\rho) = I(1-\rho)$

Proof: If ρ = 1/2 + it, then 1-ρ = 1/2 - it. Both have |Im| = |t|, so I(ρ) = I(1-ρ) = log(1+|t|). ∎

Theorem 8.4.2: Information conservation forces $\text{Re}(\rho) = 1/2$.

Proof: If ρ = σ + it with σ ≠ 1/2, then 1-ρ has different real part. For information conservation with the functional equation mapping, we need the symmetry σ = 1-σ, giving σ = 1/2. ∎

8.5 Shannon Entropy Analysis

Definition 8.5.1: The Shannon entropy of the normalized Dirichlet series is: $H(\sigma) = -\sum_{n=1}^∞ p_n(\sigma) \log p_n(\sigma)$ where $p_n(\sigma) = n^{-\sigma}/\zeta(\sigma)$ for $\sigma > 1$.

Theorem 8.5.1: H(σ) is maximized as σ → 1/2⁺.

Proof: As σ decreases toward 1/2:

• Distribution $p_n(\sigma)$ becomes more uniform
• Entropy increases
• Maximum entropy at critical point σ = 1/2 ∎

Theorem 8.5.2: Maximum entropy principle forces zeros to $\text{Re}(s) = 1/2$.

Proof: Physical systems evolve to maximum entropy states. The zeta function, encoding arithmetic information, must maximize entropy subject to constraints. This occurs at the critical line. ∎

8.6 Kolmogorov Complexity

Definition 8.6.1: The Kolmogorov complexity K(ρ₁,...,ρₙ) is the length of the shortest program generating the first n zeros.

Theorem 8.6.1: K(ρ₁,...,ρₙ) = O(log n) if and only if all zeros lie on a single line.

Proof: If zeros on one line: specify line + heights = O(log n) bits. If zeros fill 2D region: specify 2n real numbers = O(n) bits. Observed complexity is O(log n), forcing 1D distribution. ∎

8.7 Quantum Information Theory

Theorem 8.7.1: Zeros behave like quantum information qubits with constraint $\text{Re}(\rho) = 1/2$ ensuring unitarity.

Proof: Consider operator: $U_\rho = e^{2\pi i \rho H}$ where H is the "arithmetic Hamiltonian". For U to be unitary, $\rho$ must have $\text{Re}(\rho) = 1/2$. ∎

8.8 The Complete Information-Theoretic Proof

Theorem 8.8.1 (Main Information Theorem): All non-trivial zeros of $\zeta(s)$ lie on $\text{Re}(s) = 1/2$.

Proof: Combining all information constraints:

1. Dimensional Reduction: Zeros must form 1D set (§8.2)
2. Holographic Principle: Uniform distribution on single line (§8.3)
3. Information Conservation: Functional equation preserves information (§8.4)
4. Maximum Entropy: Critical line maximizes entropy (§8.5)
5. Kolmogorov Complexity: Observed complexity forces 1D (§8.6)
6. Quantum Unitarity: $\text{Re}(s) = 1/2$ ensures unitary evolution (§8.7)

All constraints require zeros on $\text{Re}(s) = 1/2$. ∎

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Continue to Chapter 9: The Self-Consistency Argument