# On the Discrete Nature of the Casimir Effect Residue?

Cédric Gueguen

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Cédric Gueguen. On the Discrete Nature of the Casimir Effect Residue?. 2026. hal-05470710v2

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# On the Discrete Nature of the Casimir Effect Residue?

Cédric Gueguen

IRISA, University of Rennes, France.

The theoretical calculation of the Casimir effect conventionally relies on the regularization of divergent series, specifically employing the value $\zeta(-1) = -1/12$ to yield a finite force consistent with experimental observations. This work proposes to reframe this regularization term by drawing upon the Euler-Maclaurin formula and the analysis of smoothed sums as elucidated by Tao. We suggest that this value corresponds to the mathematical residue generated by the transition from a finite discrete substrate to an infinite continuous model. Consequently, we propose an interpretation wherein the experimental adequacy of this residue is consistent with the hypothesis that spacetime may not be a continuum, but a granular structure composed of finite resolution units.

**Keywords:** Casimir effect, Euler-Maclaurin formula, Zeta function regularization, Discrete spacetime, Computational finiteness

## I. INTRODUCTION

The fundamental nature of spacetime, whether it is an infinitely divisible continuum or a discrete structure composed of elementary units, remains one of the most profound questions in contemporary physics. Although General Relativity [1] and Quantum Field Theory [2, 3] rely on the hypothesis of a continuous differential manifold, the recurrent appearance of infinite divergences in calculations suggests a limit to the validity of this model. The Casimir effect [4] offers a privileged field of study in this regard, as it directly links vacuum fluctuations to a measurable macroscopic force.

The theoretical calculation of this force, however, encounters a major mathematical difficulty: the summation of vacuum vibration modes results in a divergent series. To obtain a finite value in line with high-precision experimental observations [5], physics conventionally employs Riemann Zeta function regularization. In this work, we adopt a principle of computational finiteness [6, 7], posit-ing that physical processes are equivalent to effective calculations performed with finite resources. Within this framework, we propose that this regularization term represents a mathematical residue necessitated by the transition from a finite discrete substrate to an infinite continuous model, suggesting that the empirical success of the Casimir effect constitutes a compelling argument for the discretization of spacetime.

The remainder of this work is organized as follows. Section II highlights the mathematical origin of the discretization residue by examining the structural discrepancy between finite discrete sums and infinite continuous models through the Euler-Maclaurin formula and the work of Terence Tao [8]. Section III details the Casimir experiment and the decomposition of its predictive formula, emphasizing its reliance on this specific residue to achieve finite results. Section IV presents our proposed interpretation, suggesting that the empirical adequacy of this finite-to-infinite correction points toward a granular physical substrate. Finally, Section V summarizes our findings.

## II. ANALYSIS OF THE DISCRETIZATION RESIDUE: FROM FINITE SUM TO INFINITE CONTINUOUS MODEL

### A. Approximation by the integral

In mathematical analysis, the integral is conventionally used as a smoothing tool to estimate the sum of a discrete sequence. For a sequence with increasing terms, such as the series of natural integers $\sum_{n=1}^{N} n$, the transition to a continuous model via the integral $\int_{0}^{N} x dx$ simplifies the calculation of accumulation. However, for a divergent series where $N \to \infty$, this approach leads to an infinite divergence, rendering the result physically unusable without a specific regularization procedure.

### B. The Euler-Maclaurin Formula: Measuring Granularity

The Euler-Maclaurin formula [9, 10] provides the formal framework to precisely quantify the gap between a discrete summation and its continuous integral approximation. It establishes that for a sufficiently differentiable function $f$, the difference between the sum and the integral is not zero but is expressed by a series of corrective terms:

$$ \sum_{i=a}^{b} f(i) \approx \int_{a}^{b} f(x) dx + \frac{f(a) + f(b)}{2} + \sum_{k=1}^{p} \frac{B_{2k}}{(2k)!} \left( f^{(2k-1)}(b) - f^{(2k-1)}(a) \right) \quad (1) $$

where $B_{2k}$ represents the Bernoulli numbers. These corrective terms measure the granular structure—the "staircase steps"—that integral smoothing omits by definition.
### C. The Constant Term: Bridging Finite and Infinite Models

The structural origin of the regularization values used in physics is explicitly elucidated by Terence Tao [8]. He demonstrates that the values assigned to divergent series through analytic continuation, such as $\sum_{n=1}^{\infty} n = -1/12$, are “nothing more than the constant terms of the asymptotic expansion of the smoothed partial sums” [8, p. 93].

For the series of integers, this constant term is directly linked to the second Bernoulli number ($B_2 = 1/6$). The calculation of this residue, which persists independently of the smoothing scale, yields:

$$C = -\frac{B_2}{2} = -\frac{1}{12} \qquad (2)$$

This result represents the mathematical signature of discretization, acting as the inevitable remnant that persists when attempting to project a finite discrete structure into an infinite continuous formalism. By identifying this, Tao provides a purely real-variable interpretation of these sums, characterizing the regularization constant as a structural property of the transition between discrete and continuous models.

## III. PHYSICAL EVIDENCE (THE CASIMIR EFFECT)

The Casimir effect is the most direct macroscopic manifestation of the zero-point fluctuations of the electromagnetic field. Predicted by Hendrik Casimir in 1948 [4], it describes the appearance of an attractive force between two neutral, parallel conducting plates placed in a vacuum. This attraction results from a pressure imbalance: the density of authorized vibration modes between the plates is lower than that of the external vacuum, creating a net force tending to bring the walls together.

### A. The divergence of the continuous model

Within the framework of a supposedly continuous spacetime, the vacuum energy $E_0$ contained between two plates separated by a distance $d$ is calculated by the sum of the energies of all possible vibration modes. For each mode $n$, the energy is proportional to its frequency $\omega_n$. The calculation results in a summation over all natural integers:

$$E_0(d) = \frac{\hbar\pi c}{2d} \sum_{n=1}^{\infty} n \qquad (3)$$

In a continuum, this expression diverges to infinity, illustrating the failure of the model to yield a finite physical value without further intervention.

### B. Regularization and the 3D residue

To obtain a prediction consistent with experimental measurements, theoretical physics substitutes the divergent sum with the regularized value $\zeta(-1) = -1/12$. However, in three-dimensional space, the density of states between the plates must be accounted for. When integrating over the transverse dimensions, the effective energy sum transforms into a sum of cubes $\sum_{n=1}^{\infty} n^3$ [11].

Applying the Euler-Maclaurin formula to this cubic summation involves the fourth Bernoulli number ($B_4 = -1/30$). The constant residue for this 3D effective sum is given by $-B_4/4 = 1/120$. In this framework, the terms of the expansion depending on the cutoff scale $N$ (such as $N^4$ or $N^2$) represent the constant background energy of the substrate. Following the standard renormalization procedure established by Casimir [4], these scale-dependent terms cancel out during the subtraction of the reference vacuum energy, as they are independent of the distance $d$.

This leaves the constant residue as the sole observable structural signature of the transition from a finite discrete structure to an infinite continuous model [8]. When accounting for the two polarization states of the photon, the denominator of the final force formula integrates a factor of $2 \times 120 = 240$. The resulting predictive formula for the force per unit area, which serves as the standard for experimental verification [5], is:

$$F_{Casimir} = -\frac{\pi^2 \hbar c}{240d^4} \qquad (4)$$

This equation's predictive success structurally relies on the identification of this discretization residue, which provides the finite contribution that aligns the theoretical model with experimental observations.

## IV. INTERPRETATION — THE SIGNATURE OF THE GRANULAR SUBSTRATE

In this section, we confront the mathematical and physical results previously discussed to derive a new perspective on the fundamental nature of spacetime.

### A. Identification of the structural residue

The cross-analysis of Sections II and III highlights a major structural correlation. On one hand, mathematics demonstrates that the transition from a finite discrete summation to an infinite continuous approximation generates a systematic residue of $-1/12$ (or $1/120$ in three dimensions), identified as the measure of the error. On the other hand, the physical Casimir experiment validates that adequacy with reality is only achieved by precisely integrating this same residue into the equations.
## B. On the possibility of an involuntary method correction?

In standard continuum Quantum Field Theory (QFT), Casimir forces are derived using regulators that preserve the continuity hypothesis, where the appearance of Euler-Maclaurin-type constants is attributed to spectral discreteness imposed by boundary conditions. Our objective is not to dispute the validity of these derivations, but to point out that this same structural constant is precisely what is expected if the vacuum sum were fundamentally finite. This convergence motivates a research program focused on identifying discriminants between these two frameworks.

The perfect match between the mathematical discretization residue and the corrector required for the physical experiment suggests that the use of the Riemann Zeta function by physicists may not be a mere calculation trick, but rather an involuntary method correction. By injecting the value $-1/12$, the formalism may compensate for the error induced by the initial hypothesis of a continuous spacetime. If the integral model fails to describe reality without the contribution of a term linked to granularity (finite between Casimir's plates), logic dictates questioning the validity of the continuity postulate. By identifying the discretization residue as the source of the regularization constant, this approach moves from a descriptive tool to a causal explanation rooted in the discrete/granular nature of the physical substrate.

## C. Spacetime as a sum of Planck Blocks

The predictive success of the Casimir effect, interpreted in light of Tao's work [8] on the Euler-Maclaurin formula, constitutes a compelling argument for a minimum resolution unit of spacetime. In this perspective, vacuum vibrations would not be infinite but physically bounded by the discrete structure of the substrate.

* If the universe were a true continuum, the summation of modes would not produce this specific residue during measurement.

* Since the physical model relies on the $-1/12$ regularization term—precisely the value required if vibration modes were limited by a finite number of discrete states between the Casimir plates—we suggest considering that this may indeed be the case. The sum of vibrations could be physically finite and stop at a very large but finite number $N$ of states, a direct consequence of a discretization of space consisting of elementary units or “Planck-scale cells,” possibly of Planck volume $V_P$. While the mathematical residue is independent of the specific cutoff scale, the Planck scale emerges as the most consistent physical candidate for this fundamental unit of resolution, aligning the

mathematical requirement for a finite sum with the known limits of physical measurement.

This interpretation aligns with recent research exploring the Casimir effect within discrete spacetime frameworks [12]. The fact that models assuming a minimal length and the structural analysis of discretization residues converge toward the same conclusion—namely, that vacuum energy properties are intrinsically linked to a finite resolution—suggests that this should not be a mere formal coincidence. Such methodological convergence from distinct starting points reinforces the hypothesis that a discrete substrate provides a consistent framework for the physical description of spacetime discretization.

## D. Conclusion of the interpretation

The Casimir effect could therefore be reframed: beyond proof of vacuum energy, it may represent the direct measurement of the universal substrate's granularity. The use of the value $-1/12$ would not be a metaphysical property of infinity, but the indelible signature of a spacetime composed of discrete units. Furthermore, a discrete approach avoids the necessity of handling infinite intermediate energy densities, providing a more parsimonious framework where the observable force is derived from finite sums rather than the subtraction of divergent terms.

## V. CONCLUSION

This paper highlights the structural alignment between the analysis of divergent series and experimental measurements of vacuum energy. By identifying the regularization term $-1/12$ (or $1/120$ in three dimensions) not as a metaphysical value of infinity, but as the discretization residue derived from the Euler-Maclaurin formula, we propose a new reading of the Casimir effect. The adequacy between this logical residue and the measured physical force suggests that the success of the current formalism could rely on an involuntary compensation for the error induced by the continuum hypothesis and its assumption of an infinitely divisible spacetime. This interpretation invites considering spacetime as an intrinsically granular structure, composed of finite resolution units or “Planck Blocks.” Such a reframing offers a logical exit from the paradox of infinite divergences and sets the stage for a physics where reality is the manifestation of a finite substrate. In subsequent works, we will endeavor to apply this structural approach to other cosmological anomalies. We also aim to characterize the “spatial shot noise”—a statistical signature of granularity—and its potential role in explaining dark matter as a structural tension of the finite substrate.
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