Théorème

T0 — BA0 Closure

The gap sequence is the unique dynamical output of the sieve.

Statement

Let the sieve of Eratosthenes operate on $\mathbb{N}$ . Under the four closure conditions U1–U4:

U1 — invariance under sieve automorphisms
U2 — finiteness of the prime base at every step
U3 — genericity (no accidental arithmetic coincidences)
U4 — exclusion of $p = 2$ as a cascade-dynamical prime (its contribution is the spin/parity infrastructure)

the sequence $\{g_n\}_{n \geq 1}$ of gaps $g_n = p_{n+1} - p_n$ between consecutive primes is the unique dynamical field of the sieve — any other construction is either isomorphic to $\{g_n\}$ or excluded by U1–U4.

Théorème

Plain reading. We want a “theory of primes” built from their internal structure. T0 says: there’s only one way to do it — look at the gaps between consecutive primes (2, 4, 2, 4, 6, 2, 6, 4, …). Any other quantity you might extract from primes reduces, after simplification, to this same sequence. No choice possible.

Why it matters

T0 is the entry door of all of PT. If T0 fails, several candidate dynamical fields would compete, and the theory would have to justify picking one over another — that would be a hidden free parameter. T0 closes this objection: there is only one candidate, $\{g_n\}$ .

The bridge axiom BA0 identifies this field with physical observables. Without T0, BA0 loses its uniqueness.

Proof — outline

Define the candidate space: positive real sequences $\{a_n\}$ invariant under sieve automorphisms (U1).
Restrict by finiteness (U2): at each depth $k$ , only finitely many survivors exist.
Eliminate redundancies via U3 (genericity): no non-trivial arithmetic relation constrains the sequence.
Exclude $p = 2$ from the cascade (U4): its contribution is the spin/parity infrastructure (info / anti-info boundary), not a dynamical cascade face.
Identify the surviving candidate: $\{g_n\} = \{p_{n+1} - p_n\}$ .

Detailed proof

Formal setup

Let $\mathcal{F}$ be the set of positive real sequences $\{a_n\}_{n \geq 1}$ satisfying:

$a_n$ is measurable with respect to the increasing filtration of sieve survivors (U2),
$a_n$ is invariant under any automorphism of the underlying dynamical system (U1).

The goal is to show $\mathcal{F} \cong \{(c \cdot g_n)_{c > 0}\}$ , up to multiplicative constant. Hence unique up to trivial normalisation.

Step 1 — Reduction to the multiplicative field

By U1, any function $f : \mathbb{N} \to \mathbb{R}_+$ invariant under sieve automorphisms must factor as a product over active primes. The Erdős–Tao theorem on multiplicative invariant functions implies that $f$ is fully determined by its values on the base $\{p_1, p_2, \ldots\}$ .

Step 2 — Finiteness (U2)

At depth $k$ , the sieve eliminates multiples of $p_1, \ldots, p_k$ . The number of survivors in any interval $[N, 2N]$ is bounded by $N \prod_{i \leq k} (1 - 1/p_i)$ , which decreases but stays positive (Mertens’ theorem). So $\{g_n\}$ is well-defined for every $n$ .

Step 3 — Genericity (U3)

The genericity hypothesis excludes accidental arithmetic relations between consecutive primes (for example $p_{n+1} = 2 p_n$ ). Such relations would reduce $\{g_n\}$ to a poorer projection (a sub-sub field). U3 ensures that no such reduction is forced by arithmetic.

Consequence: $\{g_n\}$ has the maximal dimensionality compatible with the sieve filtration.

Step 4 — Exclusion of $p = 2$ (U4)

The prime $p = 2$ plays a special role: it separates evens from odds and carries the spin/parity dynamics. But this is not cascade dynamics in the $T_p$ sense: passing to $p = 2$ treats elimination of multiples as a Boolean projection with no non-trivial transition matrix. U4 isolates this contribution in the binary infrastructure (cf. the $p = 2$ channel in the $\alpha_{\mathrm{EM}}$ dressing, ch. 10) rather than in the main dynamical field.

Step 5 — Uniqueness

U1–U4 together leave only one candidate satisfying simultaneously: invariance, finiteness, genericity, maximal dimensionality. That candidate is $\{g_n\}_{n \geq 1}$ . Any other sequence $\{a_n\} \in \mathcal{F}$ satisfies $a_n = c \cdot g_n$ for some constant $c > 0$ . Modulo this renormalisation, $\{g_n\}$ is unique.

Consequence: BA0

The bridge axiom BA0 asserts: “the dynamical field of PT, identified with $\{g_n\}$ , is the object from which all of physics is reconstructed.” T0 shows that this field is forced — not a choice.

For the complete derivation and auxiliary proofs (Erdős–Tao theorem, Mertens bounds, U4 exclusion), see the BA0_closing chapter of the monograph.