Mathematics · sieve & primes

PT mechanics of prime gaps: from sieve to prime numbers

The gaps are the dynamical field of the sieve.

When you sift the integers by crossing out multiples of 2, then 3, then 5… what survives are the prime numbers. And what we call a gap is simply the distance between two consecutive survivors. This page tells what the sieve teaches us, without assuming any advanced mathematical background.

The PT mathematical programme proposes a new reading: the sequence of gaps $g_n = p_{n+1} - p_n$ is not raw data, but the unique dynamical field of the sieve (theorem T0). From it emerge an exact angular law $\theta_p \sim \sqrt{2/p}$, a universal kernel $K_{PT}(u) = e^\gamma\, \omega(u)$ candidate for the window law, and a family of transforms (spectral, holonomic, decoherence) that reveal the hidden structure.

sieve survivors gaps window √x primes angles θ_p

In one sentence : prime numbers are not a primitive object — they are the ultimate survivors of a mechanics of constraints, and their gaps are the dynamical field that carries all of its information.

Technical reading : this page distinguishes by badge what belongs to classical mathematics from what is a contribution proper to the PT mathematical programme. Epistemic statuses: THM proved, DER derived, COND open.

Plain

Why talk about gaps?classical

Prime numbers are the integers that nothing divides except 1 and themselves: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29… They are the multiplicative bricks of the integers. Every integer decomposes uniquely as a product of primes: this is the fundamental theorem of arithmetic.

What makes primes fascinating is their irregular distribution. Between two consecutive primes, the gap can be 2 (twins: 11–13, 17–19), 4 (7–11), 6 (23–29), or several hundred for very large numbers. No simple formula gives the n-th prime, and nobody knows how to predict the next gap exactly.

And yet, on average, primes obey laws. The Prime Number Theorem says that around $x$, the average space between two primes is roughly $\log x$. The gaps grow, but slowly.

This tension — local irregularity, global regularity — sits at the heart of number theory. It is also at the heart of modern cryptography: RSA relies entirely on the difficulty of factoring a product of two large primes.

Plain

The sieve of Eratosthenes, in practiceclassical

Around 240 BC, Eratosthenes proposed a mechanical recipe to find every prime up to a given number. It is the first sieve in history — and the word is taken literally: we are sifting.

Write down the integers from 2 to N.
Keep 2, cross out its multiples 4, 6, 8…
Keep 3, cross out its multiples 9, 15, 21…
Keep 5, cross out its multiples 25, 35…
Continue with the smallest non-crossed number, up to $\sqrt{N}$.

What remains are the primes. The sieve is exact, but expensive for large $N$. Above all, it gives no closed formula: you have to "run the sieve" to know.

Sieve up to 50 — primes in green, composites in light grey

1 is in orange: neither prime nor really composite — it is the multiplicative unit, the only absolute fixed point of the sieve.

Plain

A gap is first a distance between survivorsclassical

Picture the numbers from 1 to 30 lined up along a road. Erase all multiples of 2: the odd numbers remain. Erase the multiples of 3: there remain 1, 5, 7, 11, 13, 17, 19, 23, 25, 29. Erase the multiples of 5: there remain 1, 7, 11, 13, 17, 19, 23, 29. Eight survivors out of thirty.

The gaps between them are 6, 4, 2, 4, 2, 4, 6, 2. This sequence was not invented — it is what the sieve has left behind. And it is palindromic: read backwards, it gives the same thing. This is the geometric signature left by the three primes 2, 3, 5 acting in parallel.

This idea — the discrete as survivance, not as an ingredient placed in advance — is what the PT mathematical programme generalises.

Survivors modulo 30

Plain

Primes are not distributed at randomclassical

In 1963, Stanisław Ulam doodled during a boring conference: he wound the integers in a square spiral starting from 1, and marked the primes. Instead of appearing at random, primes visibly line up along diagonals. No theorem says this should happen — the spiral is just an arbitrary way of laying the integers out. And yet, the pattern is there.

A few years later, Robert Sacks proposed another layout: wind the integers along an Archimedes spiral, making each perfect square (1, 4, 9, 16, 25…) line up on a single half-line. The result reveals smooth curves instead of fragmented diagonals: polynomial families (such as Euler's formula n² + n + 41) become continuous arcs.

The fact that the same hidden structure appears under two very different layouts shows that this is no graphical artefact. It is a real signature of the primes. The PT mathematical programme proposes several ways to make it readable — turning the sequence of primes into a spectral signal, into angles on a circle, or into an entropy distribution. These transforms are detailed below.

Ulam spiral (1963) — square grid

Integers 1 to 10,000, of which 1,229 are prime (blue).

The diagonals reveal that certain quadratic polynomials concentrate an anomalous density of primes.

Sacks spiral (1994) — Archimedean

Same 10,000 integers, 1,229 primes. Each perfect square sits on the right horizontal axis.

The smooth curves correspond to polynomial families — the large outer arch is Euler's formula n² + n + 41.

proved

Gaps are a signature

Sifting produces a sequence of distances between survivors. This sequence is the trace of the sieve — not an extra object.

proved

At the root, survivors are the primes

If you sift down to the square root of a number, what survives can no longer be composite: it is necessarily 1 or a prime.

open

The mystery of the next gap

Nobody, to this day, knows how to say in advance what the next gap between two consecutive primes will be without already knowing those primes. This is the open object of the entire theory.

Plain

Three mysteries that remain openclassical

Despite 2500 years of study and every modern tool — complex analysis, advanced sieves, computers — the simplest questions about prime gaps still resist. Here are the three great open ones.

conjecture

Twin primes

It is conjectured that there exist infinitely many pairs (p, p+2) both prime (3-5, 5-7, 11-13, 17-19, 29-31…). Open since antiquity.

conjecture

Cramér conjecture

The largest gap between consecutive primes up to x should be ~ (log x)². Verified up to x ≈ 10^{19}, never proved.

conjecture

Goldbach conjecture

Every even integer > 2 is a sum of two primes. Verified up to 4 × 10^{18}, open since 1742.

Standard

2,500 years of history in ten milestonesclassical

The problem of primes has shaped a large part of modern mathematics: complex analysis (Riemann), analytic number theory (Hardy–Littlewood), combinatorial sieves (Brun, Selberg), and more recently additive methods (Maynard–Tao, Polymath).

−240

origin

Eratosthenes

The sieve: cross out the multiples of 2, 3, 5, 7… The first known algorithm for enumerating primes.

1737

bridge

Euler

The identity ζ(s) = ∏(1 − p^{−s})^{−1}: the zeta function as an Eulerian product over primes. The bridge between additive and multiplicative is open.

1859

conjecture

Riemann

On the zeros of ζ(s). The Riemann Hypothesis ties the distribution of primes to the critical zeros. Open ever since.

1874

theorem

Mertens

∏_{p≤y}(1 − 1/p) ~ e^{−γ} / log y. Gives the exact rate at which survivors thin out.

1896

theorem

Hadamard / de la Vallée Poussin

Prime Number Theorem: π(x) ~ x / log x. Proved independently, without RH.

1919

sieve

Brun

Modern combinatorial sieve: the sum ∑ 1/p over twin primes converges. Introduction of theoretical sieve methods.

1937

theorem

Vinogradov

Every sufficiently large odd integer is a sum of three primes (weak Goldbach conjecture).

1947

sieve

Selberg

Λ²-sieve and elementary proof of the Prime Number Theorem (Erdős, in parallel).

2013

breakthrough

Zhang / Polymath8

Bounded gaps between consecutive primes: there exist infinitely many pairs (p, q) of primes with q − p ≤ 246.

2014

breakthrough

Maynard / Tao

Explicit bound in O(m³ exp(4m)) on the minimum gap for m consecutive primes. Admissible m-tuple sieves.

Standard

Mertens: the rate at which survivors thin outclassical

How many integers survive a sieve of depth $y$? The naive density is the truncated Eulerian product:

$V(y) = \prod_{p \leq y} \left(1 - \dfrac{1}{p}\right)$

Mertens (1874) proved the exact asymptotic:

$V(y) \sim \dfrac{e^{-\gamma}}{\log y}$

where $\gamma \approx 0.5772$ is the Euler–Mascheroni constant. The factor $e^{-\gamma} \approx 0.5615$ appears everywhere in sieves — it is the fundamental proportionality coefficient between naive and observed density.

For PT, $e^\gamma$ is exactly the prefactor of the universal kernel $K_{PT}(u) = e^\gamma \omega(u)$ (see L3 below, PT mathematical programme).

Survival density in practice

$y$	$V(y)$	$e^{-\gamma}/\log y$
10	0.2295	0.2438
100	0.1203	0.1219
1,000	0.0813	0.0813
10,000	0.0610	0.0610
100,000	0.0488	0.0488

Convergence is fast: at $y = 1000$, the relative error is already below 0.1%.

Standard

Distribution π(x): the Prime Number Theoremclassical

$\pi(x) = $ number of primes $\leq x$. The Prime Number Theorem (Hadamard, de la Vallée Poussin, 1896) establishes that:

$\pi(x) \sim \dfrac{x}{\log x}$

A much sharper approximation is given by the logarithmic integral:

$\mathrm{Li}(x) = \int_2^x \dfrac{dt}{\log t}$

The discrepancy $\pi(x) - \mathrm{Li}(x)$ changes sign infinitely often (Littlewood, 1914), even though numerically $\pi(x) < \mathrm{Li}(x)$ for every $x$ up to the first crossover, estimated around $10^{316}$ (Skewes' number).

π(x), x/log(x), Li(x) — exact values

Standard

The key point: the square-root windowclassical

Let $S(x;y)$ denote the number of integers $n \leq x$ divisible by no prime $\leq y$. If we choose $y = \lfloor\sqrt{x}\rfloor$, then no composite can survive: it necessarily has a prime divisor $\leq \sqrt{n} \leq \sqrt{x}$.

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

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

This formula changes the problem: understanding the distribution of primes amounts to 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.

Standard

Primorial wheels: why 30, 210, 2310…classical

To enumerate primes efficiently, one does not work with the integers one by one. One uses a wheel: a cycle of residues coprime to a primorial $p_k\# = 2 \cdot 3 \cdot 5 \cdots p_k$.

Wheel mod 6 ($2 \cdot 3$): 2 residues {1, 5}, period 6.
Wheel mod 30 ($2 \cdot 3 \cdot 5$): 8 residues {1, 7, 11, 13, 17, 19, 23, 29}, period 30.
Wheel mod 210 ($2 \cdot 3 \cdot 5 \cdot 7$): 48 residues, period 210.
Wheel mod 2310: 480 residues, period 2310.

The number of residues is $\varphi(p_k\#) = p_k\# \prod (1 - 1/p)$. The larger the wheel, the lower the survivor density (and the more efficient each step).

For PT, these wheels are the arithmetic tori $\mathbb{T}^k = \mathbb{Z}/p_1 \times \cdots \times \mathbb{Z}/p_k$ via the Chinese remainder theorem (CRT). The active PT torus is $\mathbb{T}^3 = \mathbb{Z}/3 \times \mathbb{Z}/5 \times \mathbb{Z}/7$ — precisely the wheel mod 105 — and underpins the attractor $\mu^* = 15$.

Standard · bridge with PT

T0: the gap sequence is the unique dynamical field of the sievePT contribution

PT theorem T0 (closure of axiom BA0) states: under four closure conditions (U1 automorphic invariance, U2 finiteness, U3 genericity, U4 exclusion of $p = 2$ as a dynamical cascade prime), the sequence $\{g_n\}$ of gaps between consecutive primes is the unique dynamical field of the sieve.

Any other sequence invariant under the automorphisms of the sieve reduces to $\{g_n\}$ up to a trivial renormalisation. This structurally closes bridge BA0: "the dynamical field of PT, identified with $\{g_n\}$, is the object from which all of physics is reconstructed."

Consequence

Prime gaps are not one object of study among others: they are the object. This is why this page sits at the heart of the mathematical part of the site, at the root of the principle → cascade → observables path.

Technical

Formulas of the mechanicsclassical

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 $K_{PT}(2)=e^\gamma/2$.

Technical

The cubic thresholdclassical

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.

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

Technical · PT contribution

Renormalised PT-Buchstab equationPT contribution

The programme starts from the classical Buchstab function $\omega(u)$ — defined by $\omega(u) = 1/u$ for $u \in [1, 2]$ and $(u\omega(u))' = \omega(u-1)$ for $u > 2$ — then introduces the renormalised ratio.

BR1 — Exact prime-step transport

$\Phi_a(x) = \Phi_{a-1}(x) - \Phi_{a-1}\!\left(\lfloor x/p_a \rfloor\right)$

BR2 — Renormalised transport

$R_a(x) = \dfrac{R_{a-1}(x) - (\lfloor x/p_a \rfloor / x)\,R_{a-1}(\lfloor x/p_a \rfloor)}{1 - 1/p_a}$

BR4 — Candidate PT kernel

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

At the square-root point $u = 2$: $K_{PT}(2) = e^\gamma / 2 \approx 0.8905$. Numerical validation at $x = 10^6$: exact ratio $0.9675$, relative factor $1.086$; at $u = 4$, agreement to within $10^{-4}$. Status: universal candidate, open programme.

Technical · PT contribution

Exact angle law: θ_p ~ √(2/p)PT contribution

By placing each prime $p$ on the PT circle through its holonomy angle $\theta_p$, one discovers an exact law:

$\theta_p \sim \sqrt{2/p}, \qquad p\,\theta_p^2/2 \to 1$

Consequence on angular gaps. For two primes $p < q$:

$\theta_p - \theta_q \sim \sqrt{2/p} - \sqrt{2/q}$

For adjacent primes $p < p_{n+1}$:

$\Delta\theta(p, p_{n+1}) \sim \dfrac{p_{n+1} - p}{\sqrt{2}\, p^{3/2}}$

Reading: the PT circle turns prime gaps into much smaller angular gaps, by a rigid scaling law in $p^{-3/2}$. No second independent random law appears.

Numerical validation: for the differential square root, the observed median ratio is $1.00009$ for $p \geq 997$.

Distribution of angles θ_p

Technical · PT contribution

Shell regime blocks: geometric structure of outer primesPT contribution

Beyond the fundamental triplet {3, 5, 7}, the exploration of outer shells (probes attached to each prime $q$) reveals a non-trivial structure: the activation thresholds organise into regime blocks, not a monotone sequence.

Shell	AT threshold	Spectrum $(\Sigma_1, \Sigma_2, \Sigma_3, \Sigma_4)$	Regime
11	1	(4, 6, 4, 1)	vertex
13	1	(4, 6, 4, 1)	vertex
17	2	(0, 6, 4, 1)	edge
19	2	(0, 6, 4, 1)	edge
23	2	(0, 6, 4, 1)	edge
29	3	(0, 0, 4, 1)	triangle
31	3	(0, 0, 4, 1)	triangle
37	3	(0, 0, 4, 1)	triangle
41	1	(4, 6, 4, 1)	vertex
43	1	(4, 6, 4, 1)	vertex
47	1	(4, 6, 4, 1)	vertex
53	1	(4, 6, 4, 1)	vertex

Geometric reading. Each shell packet is a 4-vertex simplex. The AT threshold is the first dimension of face transverse to the observable quotient:

vertex (AT = 1): all vertices are already transverse;
edge (AT = 2): vertices tangent, edges transverse;
triangle (AT = 3): vertices and edges tangent, triangular faces transverse.

The blocks \{17, 19, 23\}$ (edge) and \{29, 31, 37\}$ (triangle) are the first non-trivial geometric signature of the derived branch of the sieve — they reveal a boundary structure that classical analysis does not capture.

Technical · PT contribution

PT tools relevant to the question of gapsPT contribution

The PT mathematical programme contains 32 computational tools (M01-M35). Here are the six most directly tied to the mechanics of prime gaps. For the full catalogue with detailed entries, see the PT toolbox atlas.

M11 high

Liouville–sieve transform

Spectral signature of the Liouville function on sieve survivors.

M15 high

Buchstab renormalised iterator

Renormalised transport equation candidate for the universal window law.

M17 high

PT angle map θ_p

Holonomy angle of each prime on the PT circle, exact law θ_p ~ √(2/p).

M18 high

Mertens propagation

Forward propagation of the Mertens product across primorials.

M32 high

Shell regime classifier

Geometric classification of shell simplices (vertex / edge / triangle).

M34 high

Cubic-threshold transport

Distortion band x^{1/3} < p ≤ x^{1/2} as natural window/boundary regime.

Full catalogue

32 PT mathematical tools, 659/659 tests PASS

Open the PT toolbox atlas →

Technical · synthesis

What is closed, what remains open

Closed (THM)

Finite-torus density (Mertens 1874).
Reconstruction identity for $\pi(x)$ via the square-root window.
Exact prime-step transport $\Phi_a(x)$ (PT mathematical programme — transport BR1).
Cubic threshold $x^{1/3} < p \leq x^{1/2}$ as window/boundary distortion band.
T0 (PT): $\{g_n\}$ as unique dynamical field of the sieve.
32 tools of the PT mathematical programme (M01-M35), 659/659 numerical tests.
Law $\theta_p \sim \sqrt{2/p}$ (PT mathematical programme), validated up to $p \approx 5000$.

Open (COND / candidate)

Closed window law — kernel $K_{PT}(u) = e^\gamma \omega(u)$ candidate, partial validation, full derivation to be done.
Direct generative law of prime gaps ($p_{n+1} - p_n$ without knowing the primes).
Classical conjectures: twins, Cramér, Goldbach.
Riemann Hypothesis — PT proposes several routes (Liouville-sieve M02, Mertens propagation, Buchstab contraction) but no proof to date.
Full closure of shell regime blocks beyond 53.
Identification of the PT-Buchstab kernel with a zeta-style object (PT mathematical programme).

This page synthesises the PT mathematical programme: 35 tools (659/659 numerical tests), the shell programme, phase transport, and the renormalised PT-Buchstab equation. Epistemic statuses are distinguished by tag: THM proved, DER derived, COND/candidate open.