The Theory of Persistence
Start here In a few words Results
Mathematics
Fundamental principleMathematical atlasTheoremsPrime gapsChainScripts
Physics
43 observablesTimeRelativityQuantum gravityNuclearCosmologyCalculatorsAudit
Chemistry
Periodic tableInteractive proofIonization energiesElectron affinities
MonographResult statusHard to dismissLimitsFAQGlossaryAboutContact Français
Théorème

N4 — First cascade level

In the canonical sieve ordering, $p = 3$ is the first dynamical level. $p = 2$ has a structurally distinct role (info / anti-info).

Statement

In the canonically-ordered Eratosthenes sieve, the prime p=3p = 3 is the first dynamical cascade level: it produces the first non-trivial transfer matrix and imposes the fundamental symmetry s=1/2s = 1/2.

The prime p=2p = 2 has a structurally distinct role: it is the info / anti-info operator creating the GFT partition log⁡2m=DKL+H\log_2 m = D_{\rm KL} + H. It is active in the raw sense and carries the spin/parity dynamics, but it is not a dynamical cascade face: T2T_2 is 1×11 \times 1.

Théorème

Plain reading. Why does the cascade start at p=3p = 3 and not p=2p = 2? Because 2 first supplies parity: two channels, even/odd, info/anti-info. The first transition dynamics begins at 3, which creates the first non-trivial rule (forbidden mod-3 transitions, T1).

Why it matters

N4 justifies the special role of p=2p = 2 throughout the PT cascade. Without this distinction, one would try to include p=2p = 2 in the αEM\alpha_{\rm EM} computation (BA5) — and obtain wrong numbers. The p=2p = 2 factor enters elsewhere: via the informational leakage F(2)≈0.758F(2) \approx 0.758 that dresses αbare\alpha_{\rm bare} to give αEM=1/137.036\alpha_{\rm EM} = 1/137.036.

It is also N4 that justifies U4 (excluding p=2p = 2 as a dynamical prime) in T0.

Proof — outline

  1. p=2p = 2 is infrastructure. The even/odd partition is a Boolean projection; it supplies the proto-bifurcation, but not a non-trivial cascade matrix.
  2. p=3p = 3 is dynamical. The transfer matrix T3T_3 has two distinct states ({1,2}(mod3)\{1, 2\} \pmod 3) with a non-trivial elimination rule.
  3. Forced symmetry. T3=antidiag(1,1)T_3 = \mathrm{antidiag}(1,1) has stationary distribution (1/2,1/2)(1/2, 1/2), hence s=1/2s = 1/2.

Detailed proof

Part 1 — p=2p = 2 is spin infrastructure

The mod-2 sieve partitions N\mathbb{N} into evens {2,4,6,…}\{2, 4, 6, \ldots\} and odds {1,3,5,7,…}\{1, 3, 5, 7, \ldots\}. This partition is:

  • Boolean: an integer is even or odd, no intermediate state.
  • Without internal cascade transitions: an integer’s class is trivially determined by its last bit.

The mod-2 transfer matrix on survivors (odds) is trivial: all survivors are in the same class (1 mod 2). No inter-class dynamics — no analogue of T1.

This is what makes PT call p=2p = 2 a partition operator rather than a cascade actor. Its role appears in GFT: log⁡2m=DKL+H\log_2 m = D_{\rm KL} + H, where the factor 2 separates persistence from entropy.

Part 2 — p=3p = 3 is the first dynamical

The mod-3 sieve on 6-rough survivors (integers neither multiples of 2 nor of 3) gives two non-trivial classes: {1,2}(mod3)\{1, 2\} \pmod 3. The transfer matrix T3T_3 (theorem T1) has an explicit elimination rule: T3=antidiag(1,1)T_3 = \mathrm{antidiag}(1, 1).

This is the first non-trivial transfer matrix of the sieve — the distinction between T2T_2 (spin/infrastructure) and T3T_3 (cascade) is structural.

Part 3 — The symmetry s=1/2s = 1/2

The stationary distribution of T3T_3 is computed by diagonalisation: eigenvalues ±1\pm 1, stationary eigenvector (1,1)/2(1, 1)/\sqrt{2}. Hence weight 1/21/2 on each class — that is the value of ss.

This symmetry could not emerge at p=2p = 2 (which has only one non-trivial state). It emerges at p=3p = 3 for the first time, and stays constant for all later active primes (by CRT).

Consequence: U4 exclusion in T0

T0 (BA0 closure) includes condition U4: exclusion of p=2p = 2 as a dynamical prime. N4 justifies this condition at a structural level.

This exclusion is not lost: p=2p = 2 returns at another level via the F(2)F(2) dressing of αEM\alpha_{\rm EM} (binary channel). But its dynamical contribution is filtered out of the main cascade.

For the complete derivation, see chapter 2 of the monograph.

See also