Skip to main content

Chapter 6: The Critical Line from First Principles

6.1 The Balance Theorem​

Theorem 6.1: Perfect cancellation in ∑n−s\sum n^{-s} requires Re(s) = 1/2.

Proof: For s = σ + it:

  • σ > 1/2: Early terms dominate, late terms vanish → no cancellation
  • σ < 1/2: Late terms explode → divergence
  • σ = 1/2: Perfect scale balance → cancellation possible ∎

6.2 Scale Invariance​

Theorem 6.2: Scale invariance occurs uniquely at σ = 1/2.

Proof: The ratio ∑n=2N4Nn−σ∑n=N2Nn−σ\frac{\sum_{n=2N}^{4N} n^{-\sigma}}{\sum_{n=N}^{2N} n^{-\sigma}} converges to 2^-0.5 only when σ = 1/2. ∎

6.3 The Uniqueness Theorem​

Theorem 6.3: Re(s) = 1/2 is the unique line where:

  1. Scale invariance holds
  2. Functional equation has fixed points
  3. Perfect interference possible
  4. Information encoding optimal

Proof: Each property independently requires σ = 1/2. Their intersection is unique. ∎

6.4 Why ZFC Cannot Derive This​

Critical Point: ZFC lacks the self-consistency framework to recognize that mathematical existence constrains zero locations. Without ψ = ψ(ψ), the connection between consistency and the critical line remains hidden.

Continue to Chapter 7: The Analytic Argument