PT mathematics / sieve and primes

PT mechanics of gaps: from the sieve to prime numbers

The idea is simple: let a sieve act, some numbers survive, and the distances between survivors form a mechanics. At the right depth, those survivors become exactly the prime numbers.

PT does not treat prime gaps as a mysterious sequence handed in from outside. It starts from a survival dynamics: points persistent under sieve constraints form gaps, and the window $y=\lfloor\sqrt{x}\rfloor$ reduces those survivors exactly to primes.

Picture

A gap first means a distance between survivors

Remove the multiples of 2, 3, and 5. The positions still alive modulo 30 are 1, 7, 11, 13, 17, 19, 23, and 29. The gaps 6, 4, 2, 4, 2, 4, 6, 2 are not added by hand: they are the shape left by the sieve.

This is what PT generalizes: the discrete layer is not arbitrary input, but the set of points that persist through a mechanics of constraints.

Survivors modulo 30

exact

Sieve gaps

The coprime residues of a primorial give a circular sequence of survivors; their differences give the sieve gaps.

exact

Reduction to primes

At the $\sqrt{x}$ window, every composite has already been removed. The remaining survivors are therefore $1$ and the primes.

open

Exact prime gap

This is not yet a closed law producing every $p_{n+1}-p_n$ without already knowing the primes. It is a mechanics of reduction and transport.

Standard

The key point: the square-root window

Let $S(x;y)$ count integers $n\le x$ divisible by no prime $\le y$. If $y=\lfloor\sqrt{x}\rfloor$, then no composite can survive: it necessarily has a prime divisor $\le\sqrt{n}\le\sqrt{x}$.

The survivors are exactly $1$ and the primes in $(\sqrt{x},x]. Hence:

$\pi(x)=S(x;\lfloor\sqrt{x}\rfloor)+\pi(\lfloor\sqrt{x}\rfloor)-1$

This changes the problem: understanding prime distribution becomes understanding a finite-window survival law.

What PT shifts

1. First build the sieve survivors.
2. Read their gaps as persistence gaps.
3. Move to the $\sqrt{x}$ window.
4. Recover primes as ultimate survivors.

Technical

Formulas of the mechanics

Finite-torus density

exact

$\varphi(M_A)=M_A\prod_{p\in A}(1-1/p)$

Over a full sieve period, each active prime removes exactly its local collapse direction.

Square-root window

exact

$S(x;\lfloor\sqrt{x}\rfloor)=1+\pi(x)-\pi(\lfloor\sqrt{x}\rfloor)$

At this depth, to survive means to be 1 or prime: every composite already has a visible divisor.

Survivor transport

exact

$\Phi(x,a)=\Phi(x,a-1)-\Phi(\lfloor x/p_a\rfloor,a-1)$

Adding a prime does not merely multiply a density: it transports the problem to a contracted window.

PT-Buchstab kernel

candidate

$K_{PT}(u)=e^\gamma\,\omega(u),\quad u=\log x/\log y$

Natural candidate for the universal rough-survivor law, with the square-root endpoint $K_{PT}(2)=e^\gamma/2$.

The cubic threshold

In the exact transport equation, the contracted window $\lfloor x/p_a\rfloor$ has its own native depth. Once $p_a^3>x$, this window is already sieved beyond its native square-root threshold. The transported term then becomes essentially a prime-boundary term.

$x^{1/3}<p\le x^{1/2}$: natural window/boundary distortion band.

What is closed

Finite-torus density, reconstruction identity for $\pi(x)$, exact survivor transport, cubic oversieve threshold.

What remains open

Close the window law independently, control correction channels, and only then target a complete generative law for prime gaps.