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The Representability Gap: Massive-Graviton Physics Cannot Be Held by Virtual-World Physics Engines but Survives the Low-Precision Number Formats of Artificial Intelligence

This paper demonstrates that while the computational precision required to represent massive-graviton physics exceeds the capabilities of virtual-world physics engines, causing significant numerical artifacts, the same effects remain representable and learnable within the lower-precision formats used by artificial intelligence systems.

Original authors: Alfredo Merlet

Published 2026-10-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alfredo Merlet

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Gravity is the invisible force that keeps our feet on the ground and the planets in their orbits, but in the deepest theories of physics, it might also be made of tiny, invisible particles called gravitons. Scientists have never actually seen a single graviton, and many believe it is impossible to detect one directly because the force it carries is so incredibly weak. However, if these particles do exist and have even a tiny amount of mass, it would change how gravity works over vast distances. Instead of stretching out forever like a rubber band that never snaps, gravity would have a limit, and ripples in the fabric of space-time, known as gravitational waves, would travel at slightly different speeds depending on their pitch. This idea is not just a thought experiment for physicists; it is a question that now touches our digital lives. As we build massive virtual worlds and train artificial intelligence to understand the universe, we rely on computers to simulate these forces. But computers do not think in infinite precision; they use a system of numbers that can only hold a certain amount of detail. If the effect of a massive graviton is smaller than the smallest detail a computer can store, then that effect simply does not exist inside the machine, no matter how powerful the simulation is.

A researcher named Alfredo Merlet set out to test exactly this boundary between the real laws of physics and the limits of computer numbers. He asked a simple but profound question: if the universe contains a massive graviton at the very edge of what current science allows, could a computer running a virtual world or an artificial intelligence actually see it? To find the answer, he ran two distinct sets of experiments using the most up-to-date bounds on graviton mass, which come from observations of gravitational waves by the LIGO, Virgo, and KAGRA collaborations. The first experiment focused on the physics engines that power video games and virtual reality. These engines simulate gravity by calculating how objects move and interact. Merlet tested whether the tiny correction to gravity caused by a massive graviton could be stored in the standard number formats used by these engines, which range from high-precision to low-precision formats designed for speed. He found that for any scale a virtual world would ever use, from a small room to the size of a planet, the effect is too small to be held by the computer's memory. In fact, the effect only becomes large enough to be stored in the most precise format at a distance of about 540 million meters, which is roughly one and a half times the distance from the Earth to the Moon. In the less precise formats used for faster processing, the effect only becomes visible at a distance of 1.25 × 10^13 meters, which is about 84 times the distance from the Earth to the Sun.

The most surprising discovery in this first experiment was not just that the effect was invisible, but that the computer sometimes invented a fake version of it. When the physics engine tried to calculate the correction using standard math, the tiny numbers involved canceled each other out in a way that created a large, false signal. In some cases, the computer reported a change in gravity that was up to 9 × 10^16 times larger than the true effect, or it claimed the effect existed when it was actually zero. This means that a virtual world running on standard hardware would not just fail to show the subtle influence of a massive graviton; it would actively generate phantom forces that look real but are entirely made up by the limitations of its own number system. The engine would alternate between silence and fabrication as the scale of the world changed, creating a chaotic and unfaithful simulation of the universe.

The second experiment turned the question toward artificial intelligence, which is increasingly used to analyze gravitational waves and learn the laws of physics from data. Unlike the physics engines that simulate motion, AI systems often use very low-precision number formats to process information quickly and efficiently. Merlet tested whether the unique signature of a massive graviton, which appears as a specific distortion in the pattern of gravitational waves, could survive the rounding process of these low-precision formats. He found that the opposite of the virtual world scenario occurred here. Even the most compressed number formats, including 8-bit formats used in modern AI hardware, were able to hold the tiny distortion caused by the graviton. The signal remained clear and distinct from the noise of rounding errors. When he trained a neural network to distinguish between waves with a massive graviton and those without, the accuracy of the AI did not drop when it used these low-precision numbers. The system could still learn the difference perfectly well.

The study concludes that the ability to represent massive-graviton physics depends entirely on the type of digital system being used. For the engines that build virtual worlds, the precision of the numbers is the main barrier; the effect is simply too small to fit, and the attempt to calculate it often leads to dangerous errors. For artificial intelligence, however, the precision of the numbers is not the problem. The signal is strong enough to be stored and learned, even in the most compressed formats. The real limits for AI are not the hardware's ability to hold the number, but the presence of noise in the data and the difficulty of teaching the system to recognize the pattern. This research provides a clear warning for anyone building digital twins or simulated universes: before claiming that a simulated world exhibits a subtle physical effect, one must first check if the computer can actually hold that effect in its memory. Without this check, the simulation may be reporting a ghost rather than a law of nature.

Technical Summary: The Representability Gap in Massive-Graviton Physics

Problem Statement
The paper addresses a fundamental question regarding the intersection of theoretical physics and computational systems: Can the physical effects of a massive graviton (specifically, the Yukawa correction to gravity and the dispersion of gravitational waves) be represented within the arithmetic constraints of modern virtual-world physics engines and artificial intelligence (AI) hardware? While the graviton remains hypothetical and undetectable as a single particle, current bounds on its mass ( $m_g \le 2.42 \times 10^{-23}$ eV/ $c^2$ ) imply calculable deviations from General Relativity. The study investigates whether these deviations are "representable"—i.e., storable and computable without being lost to rounding errors or generating spurious artifacts—within the floating-point formats used by physics engines (typically float32/float64) and AI accelerators (float16, bfloat16, FP8).

Methodology
The author conducted two computational experiments using known truths (synthetic data with 60-digit precision references) to test three pre-specified hypotheses:

H1: The Yukawa correction is not representable in standard physics engine formats at virtual-world scales.
H2: Low-precision AI formats erase the massive-graviton dispersion residual.
H3: Neural discriminators trained on low-precision inputs lose accuracy compared to high-precision inputs.

Experiment 1 (Static Yukawa Correction):
- Scope: Evaluated the fractional force deficit $\delta(x) = 1 - e^{-x}(1+x)$ across six floating-point formats (float64, float32, float16, bfloat16, FP8 E4M3, FP8 E5M2).
- Tests:
  - Representability: Calculated the threshold distance ( $r^*$ ) where the correction first exceeds the machine epsilon ( $\epsilon$ ) and becomes storable.
  - Direct Evaluation: Simulated how a physics engine evaluates the expression $1 - e^{-x}(1+x)$ with rounding after every operation to detect cancellation errors.
  - Orbit Integration: Used a velocity-Verlet integrator (common in game engines) to simulate orbits under Newtonian vs. Yukawa gravity at various scales (room to 1,000 AU) to measure displacement errors.
Experiment 2 (Dispersion Signature in AI):
- Scope: Analyzed gravitational wave inspiral waveforms (modeled after GW150914) containing a massive-graviton phase term.
- Tests:
  - Representational Floor: Measured the Signal-to-Noise Ratio (SNR) of the dispersion residual after rounding waveforms to AI formats. Determined the graviton mass floor below which the signal is lost to rounding noise.
  - Learned Discriminator: Trained a Multi-Layer Perceptron (MLP) to distinguish between General Relativity and massive-graviton waveforms. Tested classifier accuracy (AUC) when inputs were rounded to various formats, using noise-free data to isolate precision limits from detector noise.

Key Results

Virtual-World Physics Engines (Experiment 1):
- Unrepresentability: At the current mass bound, the Yukawa correction is unrepresentable in float64 for distances up to $\approx 5.4 \times 10^8$ m (1.4 Earth-Moon distances) and in float32 up to $\approx 1.25 \times 10^{13}$ m (84 AU). Even at planetary scales, the effect is orders of magnitude below the representability threshold.
- Phantom Corrections: Direct evaluation of the Yukawa formula in floating-point arithmetic generates "phantom" corrections due to catastrophic cancellation. In float32, at 1 AU, the engine reported a displacement of 834 km where the true effect is 8 m (an amplification of $10^5$ ). The engine alternates between silence (no effect) and fabrication (false large effects) as scale increases.
- Conclusion: H1 is supported. Virtual-world scales cannot hold the correction; direct implementations do not merely fail to show the effect; they generate "phantom" physics that is orders of magnitude larger than the true effect and indistinguishable in sign.
Artificial Intelligence Formats (Experiment 2):
- Preservation of Signal: Contrary to H2, the dispersion residual survives rounding in all tested formats, including 8-bit FP8. The representational SNR ranges from 9.7 (FP8 E5M2) to $2.0 \times 10^7$ (float32) at the published bound.
- Representational Floors: The mass floor (where SNR=1) is 3 to 4,500 times below the current bound for formats ranging from FP8 to float32.
- Learnability: H3 is not supported. Classifier accuracy did not systematically degrade with lower precision. At the bound, accuracy was limited by the learnability of the signal under nuisance parameters (phase/time shifts), not by numerical precision. AUC values remained high (often 1.0) for masses above the bound across all formats.
- Caveat: Raw astrophysical strain ( $10^{-21}$ ) is flushed to zero in float16 and FP8 without prior normalization, but standard ML pipelines normalize data, preserving the signal.

Significance and Claims
The paper establishes a "Representability Gap" that is domain-specific:

For Virtual Worlds: Number representation is the binding constraint. Massive-graviton physics cannot be simulated by current physics engines at any relevant scale. Furthermore, direct implementations do not merely fail to show the effect; they generate "phantom" physics that is orders of magnitude larger than the true effect and indistinguishable in sign.
For AI: Number representation is not the binding constraint. Even low-precision formats (FP8) can store the dispersion signature. The limits for AI are detector noise and the learnability of the signal, not the hardware's floating-point precision.

Governance Implication
The author proposes a formal analogy for the governance of simulated worlds and digital twins: any claim that a simulated environment exhibits a sub-precision physical effect (like a graviton-scale deviation) must be accompanied by an explicit representability check. This check should verify that the effect size exceeds the format's rounding threshold (specifically $\epsilon/4$ ) at the relevant scale before the effect is reported or used for training. The paper does not claim that graviton physics is currently detectable in these systems, but rather that the arithmetic of current systems is insufficient to represent the theory without generating artifacts.

Technical Summary: The Representability Gap in Massive-Graviton Physics

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