Where does gravity come from?

Newton, Einstein, and then?

Newton said: gravity is a force pulling masses together. An apple falls because Earth pulls it.

Einstein said: gravity is not a force, it is the curvature of spacetime. The apple does not fall — it follows the slope of the fabric of the universe, which Earth has hollowed out.

Both work. But neither says where it comes from. Why is there a slope? What hollows it out? Persistence Theory offers an answer one level deeper. But to understand it, you need to follow three words in order: information, landscape, relief.

1. What is an information?

Information is not what you store on a hard drive. It is simpler, and it runs deeper.

Picture a sieve. You pour in a mix of sand and pebbles. You shake. The sand falls through; the pebbles stay.

The pebbles carry an information: “I am too big to pass.” The sand, on the other hand, carries nothing remarkable — it slipped through without resisting, without leaving a trace.

The information is not in the matter. The pebbles are just stone. It lives in the distinction between what passes and what stays.

The more a thing resists a filter, the more information it carries. The more easily it dissolves, the less. Information is what survives.

2. What is an informational landscape?

Now imagine walking through the universe with a small device that tells you, at each spot you stop, how much information persists here. It displays a number.

You write the number on a map. At each point in space, a value. You get a map of numbers.

That is an informational landscape — exactly like a weather map, but instead of temperature, it is the amount of structure that survives the filter.

• In the intergalactic vacuum, almost nothing persists. The map is flat.
• Near a planet, much persists (matter is highly organized, very distinct from randomness). The map “rises”.
• Inside a star, even more.

You walk, you watch the map change. You measure nothing material — just “how much distinction here”.

3. What is an informational relief?

Now take your map of numbers and turn it into altitudes. Like a topographic chart: where the number is large, the ground is low (it has been hollowed out); where the number is small, the ground is flat or high.

You have just created a relief. Not a mountain in the geological sense — a relief of distinction.

Earth is a region where a great deal of information persists. Its structure is highly organized, very distinct from randomness. So it digs a deep valley in the landscape.

The Moon digs a smaller one. A lead bead in vacuum digs a tiny one. But all of them dig. And the denser, the deeper.

4. So, gravity?

Gravity is simply the slope of that relief.

Place a marble at the rim of the valley Earth has dug. It rolls downward. Not because it is pulled by some mysterious force at a distance — but because it sits on a slope, and a slope makes things slide.

An apple falling from a tree does the very same thing: it slides down the slope of the landscape. It does not fall toward Earth — it falls toward the bottom of the valley.

A planet orbiting the Sun is doing nothing more than rolling in a circle along the rim of the great valley the Sun has dug. It does not get closer to the Sun because it has the right speed — it stays mid-slope, indefinitely.

Why this is not a metaphor

Here is the important point: in PT, this informational landscape is not an image that helps comprehension. It is exactly what the theory computes.

The Einstein field equations of general relativity — the ones taught in every university — actually say, word for word: “the curvature of spacetime equals the way the persistence landscape hollows out, point by point”.

No one had phrased it that way before PT. But that is exactly what the formulas say when you read them in the right language.

Two surprising things

(a) Gravity is the circumference of another force.

The universe has another fundamental interaction, far better known: electromagnetism, which makes magnets attract, wires carry current, light exist.

PT proves that gravity and electromagnetism are linked by an exact number: gravity is the circumference of the circle ($2\pi$) multiplied by the strength of electromagnetism. Not a rough factor. The exact formula. Confirmed by measurement to within 0.29 %.

It is as if you discovered that the length of a road and the cost of a plane ticket were, mysteriously, the same thing multiplied by π. There is no reason for this to be true — unless they both come from the same place.

(b) Time itself is born from the landscape.

Before a certain threshold in the arithmetic cascade of the sieve, the persistence landscape is too flat for time to exist. Not “time hasn’t started yet” — but the very notion of time is not yet possible. The universe is frozen in a geometry where everything is “space”, with no distinguished direction for flow.

Then, at a precise moment, the landscape becomes hollow enough for one direction to stand out: that direction is what we call time. Time was not added to the theory. It emerged from the relief.

This is the exact analogue of an old cosmological hypothesis (Hartle–Hawking: “the universe has no temporal boundary”) — except that in PT, it is not a hypothesis. It is computed.

What about gravitational waves?

When two black holes merge hundreds of millions of light-years away, their motion ripples the landscape all around, just as two ships passing close make the surface of the sea ripple. These landscape waves travel at the speed of light (which is also derived in PT) and eventually reach detectors on Earth — exactly what LIGO and Virgo have measured since 2015.

PT predicts the same thing as Einstein for these waves, because its equations are the same. It does add one discrete signature at very high frequency: a particular background noise, detectable one day with much more precise interferometers, that would carry the imprint of the underlying arithmetic sieve.

What about gravitons?

Light has a “grain”: the photon. A natural question follows: does gravity also have a grain? A “particle of gravity”, baptised the graviton?

PT’s answer: the question is ill-posed. Gravity is not a force — it is the shape of a landscape. Asking “what is the grain of gravity” is a bit like asking “what is the grain of a mountain’s slope”: the mountain does have a microscopic structure (atoms, molecules), but its slope is not made of small elementary slopes.

The “grain” exists in PT — it lives in the arithmetic structure itself (the primes, the sieve) — but it does not look like a graviton in the classical sense. It is a different, deeper answer to the problem that has prevented quantum mechanics and general relativity from coexisting for fifty years.

The summary in three lines

1. An information is what survives a filter.
2. An informational landscape is the map of that persistence, point by point, across the entire universe.
3. The slope of that landscape is what we call gravity.

The universe does not contain gravity the way a bag contains an apple. The universe is the informational relief — and gravity is just the word we use for its slopes.

PT reformulation

“Where does gravity come from?” is reformulated in PT as: why $g_{ab} = \mathrm{Hess}(S)$ with $S = -\ln \alpha_{\text{persistence}}$, and why $G_{\text{Newton}} = 2\pi \cdot \alpha_{\text{EM}}$? The answer is built in three stages: information, landscape, metric.

1. Information: $D_{KL}$ as persistence measure

In PT, information is not a substantial quantity but a distance from uniformity. For a distribution $P$ on $m$ states, we write the Kullback–Leibler divergence:

where $H(P)$ is the Shannon entropy. The Gallagher Fluctuation Theorem (GFT, [T2]) guarantees the exact algebraic identity $\log_2 m = D_{KL} + H$: the total budget of distinctions splits between persistence ($D_{KL}$) and dispersion ($H$).

When $P$ is identified with the stationary distribution of consecutive prime gaps filtered by the sieve, $D_{KL}$ becomes a computable arithmetic quantity, conserved along the cascade T0 → L0 → T6, and capped at $\log_2 2 = 1$ bit per active CRT channel (PT Shannon cap, [T6]).

Physical information then reads as what has not dispersed under the sieve.

2. Landscape: $S = -\ln \alpha_{\text{persistence}}$

At each cascade level $\mu$, the sieve produces a stationarity coefficient $\alpha_{\text{persistence}}(\mu)$ that converges to $1/2 = s^2$ (theorem T5, double Mertens law). The persistence potential is defined as

It is a scalar function computable from the unique input $s = 1/2$, with no fitted parameter. The variables are $\mu$ (cascade parameter) and the three directions $x_p$ associated with the active primes $\{3, 5, 7\}$ at the reduced attractor $\mu^* = 15$.

Notation convention: throughout, we distinguish $\alpha_{\text{persistence}}$ (PT coefficient, converges to $1/2$) from $\alpha_{\text{EM}} \approx 1/137$ (fine-structure constant). The latter is derived as a product of $\sin^2(\theta_p, q_{\text{stat}})$ over $\{3, 5, 7\}$, but they are two distinct objects.

3. Metric: $g_{ab} = \mathrm{Hess}(S)$

Central result [D31, PROVED, B2 ch07]:

The spacetime metric is the Hessian of the persistence potential. Not by analogy: by derived identity. The associated Bianchi I metric reads explicitly [D10, D11]:

where $\gamma_p$ is the RG dimension per prime (theorem T6). The three spatial directions are the three active channels; 3+1D is not postulated.

Epistemic refinement: there are two reading levels of the same landscape. The gradient $\nabla S$ gives the gravitational force in Newton’s sense (weak-field limit, $a \approx -\nabla \Phi$ with $\Phi \approx (g_{00} + 1)/2$). The Hessian $\mathrm{Hess}(S)$ gives the metric in Einstein’s sense. They are two orders of differentiation of the same object; the Newtonian sense is not wrong, it is the weak-field approximation of the Einsteinian sense.

4. Verifiable consequences

(a) $G_{\text{Newton}} = 2\pi \cdot \alpha_{\text{EM}}$

Status: [DERIVED, D12, 0 fitted parameters]. The formula is exact; the prediction is confirmed by measurement to $0.29\%$. The factor $2\pi$ is the perimeter of $S^1$, a continuous structure already contained in the modular cascade $\mathbb{Z}/p\mathbb{Z}$ — the “informational circumference” of the coupling, in the spin foam language (theorem T6, $\alpha' = 2\pi$ derived from the same structure).

(b) Exact Bianchi identity: $G_{\text{trace}} = -R$

Status: [PROVED, D12]. The Einstein equation $G_{\mu\nu} = 8\pi G \, T_{\mu\nu}$ holds exactly in the PT Bianchi I formulation, with no externally added stress-energy tensor (everything derives from the potential $S$).

(c) Lorentzian emergence

• For $\mu < \mu_c \approx 6.97$: $g_{00} > 0$, Euclidean signature (4 spatial directions, no time).
• For $\mu > \mu_c$: $g_{00} < 0$, Lorentzian signature $(-,+,+,+)$, time emerges.

This is the exact analogue of the Hartle–Hawking (“no-boundary”) transition: same mathematical structure of the Euclidean → Lorentzian passage, but here it is computed, not postulated as an ansatz on the wavefunction of the universe.

(d) Anisotropic equation of state

On the three directions $\{3, 5, 7\}$ one computes [D12, NEC PASS, Raychaudhuri PASS]:

Three regimes coexist: tension close to dark energy ($p=3$), matter close to dust ($p=5$), radiation close to $w = 1/3$ ($p=7$). The dark sector is not an add-on — it is a direction of the same landscape.

5. Gravitational waves and gravitons

(a) Gravitational waves. Linearised perturbations of the Bianchi I metric propagate at the speed of light (itself derived in PT, [D11]). The entire LIGO/Virgo regime is predicted standardly because Einstein’s equations are derived exactly [D12]. PT adds two specific predictions:

• High-frequency stochastic background [D27, D28]: spectral density $S_L(f) = (\ell_0 \, c / \pi^2) \cdot \ln(f_c/f) / f^2$, slope $-2$ (and not $-1$ as proposed by Hogan). The logarithmic factor $\ln(f_c/f)$ is the signature of the number-theoretic $\mu = \ln N$ inside the geometry.
• No Lorentz invariance violation: $\ell_{\text{PT}} = 2$ is a coherence length (the deterministic/stochastic boundary), not a spatial lattice. PT effects manifest as non-local correlations, not as frequency-dependent dispersion.

(b) Gravitons: distinct ontology. PT contains no perturbative spin-2 quantum in the QFT sense. Discreteness is already inscribed in the arithmetic cascade through $\mathbb{Z}/p\mathbb{Z}$ — the continuous $S^1$ is already present as an intrinsic property of the modular structure, not postulated. The “PT graviton” is an excitation of the $U(1)^3$ spin foam [D13]:

Direct consequence: the perturbative non-renormalisability of quantum gravity disappears, because the theory is never perturbative beyond its underlying discrete structure. PT is closer to a non-critical string on a discrete spin foam [D14, GFT = Ruelle = Polyakov = Regge].

6. Caveat on the valley image

The “gravitational valley” image (2D rubber sheet embedded in a 3rd dimension) is a useful visual simplification but geometrically wrong: spacetime is not embedded in any external dimension, it is intrinsically curved. The Hessian $g_{ab}$ encodes intrinsic curvature (Riemann, Ricci), accessible without embedding.

7. Epistemic status recap

In one sentence

Gravity is the Hessian of the persistence potential, time emerges from its convexity, and the numerical agreement with $\alpha_{\text{EM}}$ via $2\pi$ betrays the unique underlying structure: an arithmetic sieve at the fixed point $\mu^* = 15$, with no fitted parameter.