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# No-Go Theorem Requires Modification: An Additional Constant Term C and Its Laurent Series Structure in $f(R)$ gravity

Wen-Xiang Chen

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Wen-Xiang Chen. No-Go Theorem Requires Modification: An Additional Constant Term C and Its Laurent Series Structure in $f(R)$ gravity. 2025. hal-04962953

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No-Go Theorem Requires Modification:
An Additional Constant Term *C* and Its Laurent Series Structure in *f*(*R*) gravity

Wen-Xiang Chen<sup>*a*</sup>

*Department of Astronomy, School of Physics and Materials Science,*
*GuangZhou University, Guangzhou 510006, China*<sup>*</sup>

(Dated: February 24, 2025)

No-go theorems often assert that certain solutions or configurations in gravitational and field-theoretic models cannot exist under a prescribed set of assumptions [1]. However, recent analyses using a *modified* Robson-Villari-Biancalana (RVB) approach [2–4] indicate the presence of an additional integration constant, *C*, that may appear in topological or horizon-based integrals (notably through expansions in Laurent series). This subtle term invalidates the usual dismissal of boundary or horizon contributions and demands a revised understanding of no-go arguments. In this work, we demonstrate how *C* can be revealed through careful Laurent expansions about the horizon and discuss why ignoring this term undermines the classical no-go statement. We illustrate these ideas with examples in *f*(*R*) gravity [6–8], emphasizing how familiar no-go theorems must be updated to accommodate the extra freedom that *C* provides. If the threshold for the No-Go theorem can be further lowered and this reduction can be validated through the modified uncertainty principle, it would provide experimental evidence supporting *f*(*R*) gravity theory. This implies that the corrections introduced by these modifications are not merely theoretical constructs but could be tested in high-precision experiments, offering a potential observational window into the validity of *f*(*R*) gravity.

# 1. INTRODUCTION

No-go theorems occupy a central place in theoretical physics because they delineate the fundamental limitations imposed by symmetry principles, topological constraints, and other underlying physical assumptions [1, 5]. At their core, these theorems stipulate that certain classes of solutions, configurations, or phenomena are disallowed once one accepts a given set of physical or mathematical premises. By virtue of these restrictions, no-go statements illuminate the structure of theoretical frameworks, guiding our understanding of which configurations are realizable and which are intrinsically forbidden. Notable instances of such theorems arise across multiple domains, from high-energy particle physics to gravitational theory and cosmology, underscoring their universal relevance.

One prototypical arena for no-go theorems is black hole physics, where uniqueness theorems prove that under specific assumptions (e.g., stationarity, asymptotic flatness, and suitable energy conditions), the Schwarzschild or Kerr–Newman metrics exhaust the family of possible solutions to Einstein’s field equations. Similarly, in gauge field theories, rigid-structure theorems can prohibit the existence of certain classes of localized solutions such as finite-action instantons with prescribed boundary conditions. These classical results have profoundly shaped how we think about the behavior of fundamental fields in extreme regimes, demonstrating the extent to which geometry, topology, and boundary conditions conspire to reduce the space of permissible configurations.

However, any no-go argument rests on certain assumptions, especially regarding boundary terms or the asymptotic behavior of fields. In many of the well-known proofs, it is customary to assume that these boundary contributions either vanish or do not alter the leading terms of the solution. This simplification is often justified by appealing to physical arguments: spacetimes are required to be well-behaved at infinity (i.e., asymptotic flatness or AdS-like behavior), or horizons are assumed to be regular and non-singular. Under these assumptions, integration constants or boundary terms—while formally present—are typically dismissed based on finiteness arguments or by enforcing smoothness. In traditional treatments, such terms do not re-emerge to challenge the overarching conclusion of the no-go proof.

Recent developments, however, suggest that this perspective may be too restrictive. By employing the modified Robson-Villari-Biancalana (RVB) method [2–4], researchers have uncovered subtleties in how curvature contributions accumulate in effective actions or in the computation of topological indices. Specifically, when one constructs (1+1)-dimensional effective actions via dimensional reduction of higher-dimensional spacetimes, integrals of curvature invariants can yield non-trivial constants of integration. Although it might be tempting to set these constants to

<footnote>*Electronic address: wxchen4277@qq.com</footnote>
zero by invoking boundary or regularity conditions, the modified RVB approach demonstrates that in certain geometries—especially those that arise in generalized or modified theories of gravity—there is no guarantee that these constants vanish. Indeed, it may even be inconsistent or too strong of an assumption to enforce their disappearance.

The presence of such an integration constant, denoted as $C$, is not merely a mathematical curiosity; rather, it can have profound implications for physical predictions and the validity of traditional no-go arguments. In many classical proofs, one might implicitly assume that some indefinite integral of a geometric quantity such as the Ricci scalar $R$ or the Kretschmann invariant must remain bounded or vanish at infinity or at the horizon, thereby forcing any arbitrary constant of integration to zero. Yet, what if these boundary conditions are incomplete or fail to capture certain subtleties of the solution space—particularly in modified gravitational theories like $f(R)$ gravity [6–8] or Gauss-Bonnet gravity [9]? One could then be faced with a scenario where $C$ remains non-zero, potentially altering thermodynamic quantities, topological invariants, or even the very existence of solutions that had been ruled out by older arguments.

Central to this debate is how the metric and curvature invariants behave near horizons, singularities, or at asymptotic boundaries. Through expansions in Laurent series about horizons or other geometrically significant loci, one may detect previously overlooked terms that do not trivially vanish. These expansions can reveal that the local geometry permits additional integration constants whose physical interpretations can be non-trivial. If $C$ persists, it might modify black hole thermodynamics (e.g., by shifting horizon entropy or temperature by a constant term) or invalidate a step in the canonical no-go proof that depended crucially on setting boundary contributions to zero.

Associate Professor Hasegawa Yuji of the Vienna University of Technology and Professor Masaaki Ozawa of Nagoya University and other scholars published empirical results against Heisenberg's uncertainty principle on January 15, 2012[12]. They used two instruments to measure and calculate the spin angle of the neutrons, which was less than the uncertainty principle's error, and the uncertainty principle proved uncertainty principle's claim. This restriction is wrong. However, uncertainty principle is still correct because it is the inherent quantum nature of the particle.

Consequently, the time is ripe to re-examine classical no-go theorems under the light of these more general solution spaces. While the new perspective does not necessarily overthrow well-established results—indeed, in many cases, the constants can still vanish if the geometry meets all standard regularity conditions—it highlights the importance of verifying the assumptions that lead to these vanishings. Thus, the aim of this paper is twofold: (1) to provide a systematic framework for identifying when and how hidden integration constants like $C$ can appear, and (2) to revise or extend no-go arguments to account for these previously neglected contributions. By doing so, we hope to foster a deeper appreciation of the interplay between boundary conditions, topological constraints, and the global structure of spacetime in both standard and modified theories of gravity.

In what follows, we begin by illustrating how a straightforward Laurent series expansion around a horizon or a singular point can bring $C$ to the fore, particularly in integrals of curvature invariants. From there, we examine how the appearance of $C$ can fundamentally change the logic behind certain no-go statements, turning the usual argument on its head by allowing for a class of solutions that would otherwise seem to be outlawed. While our main illustrations draw on black hole thermodynamics and anomalies in effective actions, the overarching lesson is broadly applicable across contexts where boundary conditions and topological charges play pivotal roles. Indeed, as extended gravitational theories such as $f(R)$ and Gauss-Bonnet models gain traction in the quest to unify gravity with quantum field theory or to address cosmological puzzles, the cautionary tale of hidden constants emerges as ever more relevant.

## 1.1. Modified RVB Method in Brief

The Robson-Villari-Biancalana method [2–4] initially arose as a novel framework for connecting gravitational anomalies—often represented by the non-conservation of the energy-momentum tensor under certain conditions—with the Hawking radiation that emanates from black hole horizons. In its original formulation, one reduces the near-horizon geometry to an effective (1+1)-dimensional theory and examines how anomaly terms emerge in the reduced action. This approach elegantly ties together the concept of quantum radiation with the more abstract notion of gravitational anomaly, lending additional theoretical support to the semi-classical picture of black hole evaporation.

In the *modified* (or extended) version of the RVB method, sometimes referred to as RVB, the formalism is generalized to handle not only the near-horizon region but also broader segments of the spacetime—or even the entire spacetime manifold when topological invariants are being probed. The key mathematical operation involves performing integrals of curvature invariants in the partially reduced action. For instance, one may be interested in expressions like

$$ \int \sqrt{-g} R d^4 x \quad \text{or more intricate forms such as} \quad \int \sqrt{-g} \mathcal{F}(R, R_{ab}R^{ab}, \dots) d^4 x, \qquad (1) $$

where $\mathcal{F}$ is a polynomial or other functional of scalar invariants. In principle, these integrals are only defined up to an integration constant unless boundary conditions or other constraints are sufficiently strong to fix them.
What the modified RVB analysis unveils is that conventional arguments for setting this integration constant to zero can fail in specific contexts. For example, in certain black hole solutions with non-trivial topology or in cosmological settings where the spacetime may not asymptote to Minkowski space, there may be no strict justification for discarding additional terms. Furthermore, the Laurent series expansions about horizons or singularities can expose complex behavior of the metric functions, indicating that the naive approach of simply stipulating the vanishing of contributions at the horizon could be too coarse. This subtlety can be amplified in extended or modified gravity theories, where higher-derivative terms or topological terms affect the boundary conditions in non-trivial ways.

Physically, the constant $C$ represents a hidden channel through which the geometry can encode extra information about global structure or boundary conditions—information not always captured in the minimal assumptions of classic no-go theorems. In some scenarios, $C$ could shift the values of thermodynamic variables such as the horizon entropy, or alter the conditions under which certain anomalous currents vanish or remain finite. Such shifts, though possibly small, may be crucial in borderline cases where classical theorems rely on delicate cancellations. Indeed, a small but non-zero $C$ might be the ingredient that saves a class of solutions from being prohibited under standard theorems, thereby revealing hitherto overlooked solution families.

In summary, the modified RVB approach compels us to re-examine core assumptions about boundary terms and integration constants in gravitational actions. It does so by explicitly showing how and where these constants can sneak in, particularly when analyzing anomalies, thermodynamics, and topological contributions. The implications of this reconsideration are far-reaching: while many canonical no-go results may endure, they could require additional caveats, or at least a more careful justification of the boundary and regularity conditions that traditionally force $C$ to vanish. As the gravitational research community continues to explore higher-curvature modifications, topological terms, and non-trivial boundary configurations, awareness of such hidden integration constants becomes increasingly essential for constructing and interpreting no-go theorems.

## 2. CLASSICAL NO-GO THEOREM AND THE MISSING CONSTANT $C$

### 2.1. General Structure of No-Go Theorems

Consider a no-go theorem that takes the following abstract form [1]:

1. We assume a field action $S[g_{\mu\nu}, \phi, \dots]$ and boundary conditions such that the geometry is asymptotically flat or AdS/dS-like.

2. We employ some constraint or topological argument (e.g., a homology constraint or a horizon argument) leading to an integral of a function of curvature.

3. We find that this integral enforces a contradiction unless it is zero.

Conclusions might look like: “No nontrivial solution exists satisfying the assumed boundary conditions.” This result crucially depends on the integral in step (ii) having a definite, unambiguous value (often zero).

However, if an extra integration constant $C$ appears, the integral might take the form

$$ \int \mathcal{F} \sqrt{-g} d^m x = I_0 + C, \qquad (2) $$

where $I_0$ is the conventional evaluation, and $C$ is not forced to vanish by any standard boundary condition. If the no-go proof requires $I_0 = 0$ to reach a contradiction, then $C \neq 0$ can simply offset $I_0$ and dodge the contradiction altogether.

### 2.2. Laurent Series and the Origin of $C$

A particularly instructive way to see how $C$ arises is via **Laurent expansions** about the horizon or another characteristic radius (e.g., $r = r_0$). Near $r = r_0$, let us consider a function $\Phi(r)$ that contributes to the geometric integral. We can write:

$$ \Phi(r) = \sum_{n=-M}^{\infty} a_n (r - r_0)^n, \qquad (3) $$
where M is some positive integer that indicates the leading (possibly singular) term in the expansion. We are interested in an integral of $\Phi(r)$ in the neighborhood of $r_0$:

$$ \int_{r_0-\epsilon}^{r_0+\epsilon} \Phi(r) dr. \qquad (4) $$

If $\Phi(r)$ has a principal part (the negative powers $(r-r_0)^{-n}$ with $n > 0$), this integral could develop a natural boundary contribution or a finite jump. When performing indefinite integration term by term, one obtains

$$ \int \Phi(r) dr = \int \sum_{n=-M}^{\infty} a_n (r-r_0)^n dr = \sum_{n=-M}^{\infty} a_n \frac{(r-r_0)^{n+1}}{n+1} + C. \qquad (5) $$

The constant $C$ appears naturally as the integration constant of the indefinite integral. In many physical situations, one tries to fix $C$ by boundary conditions at infinity or continuity requirements across $r_0$. However, in more complicated settings—especially with additional horizon structures or if the theory is modified in such a way that certain boundary terms do not vanish—$C$ can remain unconstrained.

When combining multiple horizon (or boundary) regions, each region's Laurent expansion can yield its own local integration constant. In principle, a no-go theorem that sums contributions from multiple horizons or boundaries must ensure that all such constants either vanish or combine in a particular way. If they do not, the no-go statement might fail.

### 3. ILLUSTRATIVE EXAMPLE IN $f(R)$ GRAVITY

#### 3.1. Action and Field Equations

We demonstrate how the hidden constant $C$ can arise in a typical extended gravity setting. Consider the action [7]

$$ S = \frac{1}{16\pi} \int d^4 x \sqrt{-g} [f(R)], \qquad (6) $$

where $f(R)$ is a function of the Ricci scalar $R$, and matter fields are absent for simplicity [6–8]. Varying with respect to $g_{\mu\nu}$, one obtains higher-order field equations:

$$ f'(R)R_{\mu\nu} - \frac{1}{2}f(R)g_{\mu\nu} + (g_{\mu\nu}\Box - \nabla_{\mu}\nabla_{\nu})f'(R) = 0. \qquad (7) $$

A no-go theorem might claim, for instance, that certain rotating (or radiating) black hole solutions cannot exist with a specified asymptotic form. Typically, the proof demands analyzing an integral of $R$ (or of expressions containing $f'(R)$) over the manifold and concluding it must vanish for boundary or topological reasons.

#### 3.2. Horizon Analysis and the Emergence of $C$

We now restrict attention to a region near a Killing horizon at $r = r_+$. Near this horizon, $R$ can have a Laurent expansion:

$$ R(r) = \sum_{n=-M}^{\infty} \alpha_n (r-r_+)^n. \qquad (8) $$

For integrals involving $R$, consider

$$ I = \int R(r) \sqrt{-g} d^4 x. \qquad (9) $$

In spherical or axial symmetry, one may separate the angular part and reduce to a radial integral:

$$ I \sim 4\pi \int \left( \sum_{n=-M}^{\infty} \alpha_n (r-r_+)^n \right) F(r) dr, \qquad (10) $$
where $F(r)$ is some smooth function encoding metric factors (e.g., $\sqrt{-g}$ in the radial sector). We can similarly write a Laurent expansion for $F(r)$:

$$F(r) = \sum_{k=-N}^{\infty} \beta_k (r - r_+)^k. \qquad (11)$$

Multiplying series and integrating term by term yields indefinite integrals with indefinite constants. In practice, partial expansions typically have the structure:

$$I_0 + C = \int \sum_{n,k} (\alpha_n \beta_k) (r - r_+)^{n+k} dr, \qquad (12)$$

plus a potential boundary term from $r = r_+$. It is precisely this boundary term or indefinite integral that can include a nontrivial $C$. In classical no-go arguments, the boundary condition $r_+ \mapsto$ horizon is assumed to fix $C = 0$, but the modified RVB analysis shows that certain horizon or asymptotic conditions may be insufficient to pin down $C$ strictly.

### 3.3. Consequences for No-Go Theorems

Thus, in the classical approach, if $I_0 = 0$ is required to maintain a contradiction for the no-go statement, the appearance of $+C$ can spoil the contradiction. Concretely,

$$I_0 + C = 0 \implies C = -I_0, \qquad (13)$$

which is no contradiction at all if $C$ remains arbitrary. Hence the no-go statement “no solutions exist” can be replaced by a weaker statement:

*No solutions exist if and only if $C = 0$ is enforced by the boundary conditions.*

In other words, the usual no-go conditions must either ensure $C = 0$ or allow for non-zero $C$, in which case solutions are no longer forbidden.

### 3.4. Laurent-Series-Based Derivation of $C$ in the Modified RVB Framework

In the modified RVB method [2–4, 10], one often reduces a 4D problem to an effective 2D problem by focusing on the radial sector near the horizon (for rotating black holes, sometimes a 3D sector is considered, factoring out the angular part). The Euler characteristic or similar topological invariant can be expressed in terms of integrals of curvature invariants in these reduced dimensions.

A prototypical example is:

$$\chi = \frac{1}{4\pi} \int \sqrt{-g_{(2)}} R_{(2)} d^2 x, \qquad (14)$$

where $g_{(2)}$ is the effective 2D metric in $(t, r)$ or $(v, r)$ coordinates. If $R_{(2)}$ is expanded in a Laurent series around $r = r_+$, indefinite integration can introduce a constant $C$. In the original, unmodified scenario, $C = 0$ is usually forced by asserting that $R_{(2)}$ is integrable and finite or by demanding certain boundary conditions. However, for various reasons (non-standard boundary conditions, near-horizon expansions involving multiple patches, or additional fields in $f(R)$ gravity), $C$ may remain non-vanishing.

To see this explicitly, consider a (1+1)-dimensional sector:

$$ds^2 = -f(r)dt^2 + \frac{dr^2}{f(r)}, \qquad (15)$$

and expand $f(r)$ around $r = r_+$:

$$f(r) = \sum_{m=-\mu}^{\infty} c_m (r - r_+)^m. \qquad (16)$$
The Ricci scalar in 2D, $R_{(2)}$, itself inherits a Laurent expansion from $f(r)$:

$$R_{(2)}(r) = \sum_{j=-\nu}^{\infty} d_j (r - r_+)^j. \qquad (17)$$

Integrating $R_{(2)}$ near the horizon typically yields

$$\int R_{(2)} dr = \sum_{j=-\nu}^{\infty} d_j \frac{(r - r_+)^{j+1}}{j+1} + C. \qquad (18)$$

Neither the horizon condition nor the asymptotic region might suffice to fix $C$ unless additional constraints are in place. Consequently, a no-go claim that requires this integrated quantity to vanish can fail.

# 4. LAURENT EXPANSIONS OF NO-GO THEOREM AND THE CONSTANT C

No-go theorems constitute some of the most potent tools in theoretical physics, providing incisive statements about which solutions, fields, or spacetime geometries are forbidden under a specified set of assumptions [1, 5]. The typical structure of such theorems involves formulating constraints on the system (e.g., from boundary conditions, topological data, or the structure of field equations) and then deriving an unavoidable *contradiction*, indicating no consistent solution can exist.

While powerful, these classical no-go statements often rely on simplified assumptions about boundary terms or the vanishing of certain integrals. In black hole physics, for instance, one frequently encounters horizon integrals that are asserted to vanish or produce only a single solution, effectively leading to a contradiction. However, the *modified Robinson-Wilczek-Banerjee (RVB) approach* [2-4] highlights the possibility of an additional integration constant $C$ that can persist even after applying standard boundary conditions. This extra constant can make the denominator in the geometric argument shift from a simple algebraic root to a more intricate polynomial equation—often a quartic in $r$ for rotating or certain $f(R)$ black holes.

Hence, a deeper inspection reveals that dividing by zero in the classical sense could be insufficient or incomplete in scenarios where $C \neq 0$. Thus, the classical no-go condition (which is effectively a 'denominator = 0' argument) fails because the presumed contradiction is circumvented by the extra constant $C$. The purpose of this paper is to provide a greatly expanded discussion of these effects, focusing on:

* The geometric logic behind zero denominators in no-go proofs,

* How quartic horizon equations arise in modified gravity,

* Why a transcendental or otherwise unexpected solution can appear,

* And how these solutions connect to the constant $C$ revealed by Laurent expansions.

In doing so, we aim to illustrate why the classical no-go theorem must be revised and how the revised statement depends on whether $C$ is forced to be zero or allowed to vary.

## 4.1. Classical Zero-Denominator Argument

To understand the logic that underlies many no-go theorems, it is helpful to see it in its simplest form: a *division-by-zero* phenomenon. Typically, one identifies a crucial factor—such as a horizon function or a determinant of a field operator—that must vanish for a non-trivial solution to exist, while the same factor also appears in the denominator of another constraint. The standard contradiction emerges as follows:

1. A horizon or boundary condition requires $\Delta(r_+) = 0$ at some radius $r_+$.

2. Simultaneously, another part of the solution or integral enforces $\Delta(r_+)$ in the denominator, leading to a blow-up or undefined expression unless $\Delta(r_+) = 0$ is disallowed.

3. Conclusion: No consistent solution can satisfy these two conditions together; hence no-go.

However, in some physically relevant situations, $\Delta(r)$ might not be the simplest possible function (like $r - r_+$). It could be a more complicated polynomial. For example, in rotating black holes within certain modified gravities, $\Delta(r)$ is *quartic* in $r$, or includes $a^2$ (rotation parameter) and effective cosmological terms. A no-go theorem may then revolve around analyzing which roots of this quartic are physically allowable and asserting that no consistent solution can pass all constraints simultaneously.
## 4.2. Quartic Horizon Functions and Possible Extra Roots

In Einstein gravity with rotation, one encounters horizon functions of the form

$$ \Delta(r) = r^2 + a^2 - 2Mr + \dots \qquad (19) $$

which is quadratic in $r$. For $f(R)$ or higher-derivative gravity, or when $R_0 \neq 0$ (a nontrivial cosmological constant or an $R^2$ term is present), $\Delta(r)$ can become *quartic*. Symbolically,

$$ \Delta(r) = \alpha_4 r^4 + \alpha_3 r^3 + \alpha_2 r^2 + \alpha_1 r + \alpha_0, \qquad (20) $$

where $\alpha_i$ are coefficients that may encode the black hole mass $M$, the rotation parameter $a$, charges $Q$, scalar curvature corrections $R_0$, and other parameters from $f(R)$ functions [6–8]. This polynomial can have up to four real roots depending on the sign and values of the $\alpha_i$. A no-go theorem might proceed by analyzing the leading and subleading behavior of $\Delta(r)$ at various boundaries or horizon points, concluding that $\Delta(r_+) = 0$ is either contradictory or physically inconsistent under certain assumptions.

Crucially, if there is an implicit step in the proof that *divides* by $\Delta(r)$ (or some function of it) and leads to a contradiction, that step must be scrutinized if $\Delta(r)$ has multiple real roots or if an extra integration constant modifies the naive factorization.

## 4.3. Why Laurent Series?

Laurent series expansions are an effective tool for examining local behavior near singularities or horizon points in gravitational contexts. Suppose $r_0$ is a root of $\Delta(r)$, i.e., $\Delta(r_0) = 0$. Near $r_0$, we can write

$$ \Delta(r) = (r - r_0) \Delta'(r_0) + \frac{(r - r_0)^2}{2!} \Delta''(r_0) + \dots, \qquad (21) $$

if $\Delta'(r_0) \neq 0$. However, in more complicated or degenerate cases, $\Delta'(r_0)$ might also vanish, requiring higher-order terms in the Laurent expansion. This can lead to expansions of the form

$$ \Delta(r) = (r - r_0)^m \sum_{k=0}^{\infty} d_k (r - r_0)^k, \qquad (22) $$

with $m > 1$ if the horizon is degenerate or if $\Delta'(r_0) = 0$ but $\Delta''(r_0) \neq 0$, etc. Such expansions can cause more intricate divergences or integrable singularities in the integrals that appear in no-go proofs.

Moreover, if a classical no-go argument relies on, say, $\int \frac{dr}{\Delta(r)}$ being finite or zero, the presence of a higher-order zero in $\Delta(r)$ or additional expansions in the metric sector can generate an integration constant $C$ that is not eliminated by standard boundary conditions. This is the essence of the *modified RVB approach*: indefinite integrals of curvature polynomials or of $1/\Delta(r)$ may yield hidden constants.

In more concrete terms, consider a simplified $(1+1)$-dimensional reduction near the horizon:

$$ ds^2 = -\Delta(r) dt^2 + \frac{dr^2}{\Delta(r)}, \qquad (23) $$

a standard form for spherically symmetric or stationary metrics. Suppose the horizon is at $r = r_+$. A typical integral that surfaces in proofs of no-go theorems is

$$ I = \int_{r_{+}-\epsilon}^{r_{+}+\epsilon} \frac{dr}{\Delta(r)}, \qquad (24) $$

or more intricately,

$$ I_{\text{Ricci}} = \int R_{(2)}(r) \sqrt{-g_{(2)}} dr dt, \qquad (25) $$

where $R_{(2)}$ is the 2D Ricci scalar. In any of these cases, indefinite integration with respect to $r$ near $r_+$ can produce an extra term $C$.
For instance, let us be explicit: expand $\Delta(r)$ in a Laurent series around $r = r_+$. If $r_+$ is a simple (non-degenerate) root, then

$$ \Delta(r) = \Delta'(r_+)(r - r_+) + \frac{\Delta''(r_+)}{2!}(r - r_+)^2 + \dots \qquad (26) $$

The indefinite integral of $1/\Delta(r)$ near $r_+$ has the local form

$$ \int \frac{dr}{\Delta(r)} \approx \int \frac{dr}{\Delta'(r_+) (r - r_+)} = \frac{1}{\Delta'(r_+)} \ln|r - r_+| + C. \qquad (27) $$

Under typical black hole boundary conditions, one might attempt to fix $C = 0$ by demanding finiteness or matching at $r_+$. However, as soon as additional expansions occur (like degenerate horizons, higher-derivative corrections, or extra horizons), the impetus to strictly fix $C$ may weaken. Non-trivial boundary conditions, rotating horizon patches, or complicated global topologies can all leave $C$ free.

Consequently, if the no-go theorem is built from a statement like “$\int_{r_+} \frac{dr}{\Delta(r)} = 0$ and thus we have a contradiction,” that entire logic collapses if the integral can actually be

$$ \int_{r_+} \frac{dr}{\Delta(r)} = 0 + C, $$

with $C$ not forced to vanish.

## 4.4. Beyond Algebraic Contradictions

A well-known phenomenon in black hole physics is the possibility that naive horizon solutions exist but appear to violate a classical uniqueness or no-go result. In $f(R)$ gravity, for instance, certain rotating black hole solutions are found that do not satisfy the simplest boundary constraints of the classical approach [6–8]. A deeper look using the modified RVB method shows that these solutions indeed correspond to an integration constant $C \neq 0$ in the horizon integrals.

The denominators in the partial fractions or expansions are no longer strictly forced to vanish in the sense of a contradiction, but can vanish in a mathematically consistent way that does not yield a contradiction because the indefinite integral picks up a compensating constant.

In some sense, this “cures” the putative singular or blow-up in the standard no-go proof. Instead of dividing by zero and concluding “contradiction,” the approach is that the function might be better described by a quartic or higher-order polynomial $\Delta(r)$ whose root structure is large enough to accommodate multiple horizon branches. The integration constant, combined with the multi-root structure, can yield “transcendental” solutions: solutions that do not conform to the polynomial factorization demanded by the no-go theorem, effectively bypassing the no-go statement.

## 4.5. Example: One-Dimensional Toy Model with Quartic Denominator

To provide a concrete illustration, let us consider a toy integral:

$$ I = \int \frac{dr}{r^4 - 2br^2 + c}, \qquad (28) $$

with constants $b, c$. Suppose that $r = r_+$ is a real root of $r^4 - 2br^2 + c = 0$. In a no-go argument, one might examine $r_+$ in a local domain to show that some integral vanishes or diverges, concluding no solution. However, partial fraction decomposition (if feasible) or direct expansions about $r_+$ reveal:

$$ r^4 - 2br^2 + c = (r - r_+) (4r_+^3 + \dots) + \dots, \qquad (29) $$

and indefinite integration can produce a log-type or arctan-type function plus a constant:

$$ I(r) = \int \frac{dr}{(r - r_+) P(r)} = \mathcal{G}(r, r_+) + C, \qquad (30) $$
where $P(r)$ is a polynomial capturing the remaining factors. The function $G(r, r_+)$ might have branch cuts or multiple expansions, allowing $C$ to remain unconstrained if boundary conditions are not fully restrictive. Hence, the “contradiction” derived from forcing $I(r_+) = 0$ without $C$ fails once $C$ is present.

In light of the above discussions—focusing on the geometric logic of denominators being zero, the quartic or higher-order structure of horizon polynomials in modified gravity, and the appearance of the integration constant $C$—we present the revised statement of the no-go theorem.

**Modified No-Go Statement:**

Given a classical no-go theorem that relies on an integral

$$I_0 = \int \mathcal{F} \sqrt{-g} d^n x$$

vanishing to produce a contradiction (essentially a division-by-zero in geometry, e.g., $\Delta(r_+) = 0$ that both *must* and *must not* vanish), we note that in the modified RVB method, the integral becomes

$$I_0 + C,$$

where $C$ is an integration constant arising from near-horizon or boundary expansions, often extracted via Laurent expansions.

If *and only if* boundary or topological conditions force $C = 0$, the original no-go argument remains valid.

If $C \neq 0$ is permitted by the boundary data or the horizon structure (e.g., in quartic horizon polynomials), then the classical contradiction is removed, and thus the no-go theorem breaks down.

Therefore, if the original no-go theorem’s contradiction was triggered by a particular factor in the denominator going to zero, one must check whether the horizon function $\Delta(r)$ is sufficiently complicated that its partial fraction expansions or indefinite integrals yield $C \neq 0$ solutions. The presence of these solutions negates the contradiction and reopens the possibility of physically valid (or at least mathematically consistent) configurations.

## 4.6. Physical Interpretation of $C$ and Quartic Solutions

In extended or modified gravities, a quartic horizon polynomial $\Delta(r)$ commonly appears because the field equations are higher-order in derivatives or incorporate additional scalar invariants [6–8, 11]. The integration constant $C$ can then be viewed as a measure of how these extra degrees of freedom alter the near-horizon geometry. Under certain topological constraints or boundary conditions, $C$ may vanish, reinstating the classical no-go result. But absent such constraints, $C$ offers a new branch of solutions—sometimes physically viable, sometimes requiring further consistency checks like positivity of energy or avoidance of naked singularities.

Division-by-zero or factor-vanishing arguments pervade many areas of theoretical physics. For instance, in vacuum decay analyses, one might find bubble solutions are “forbidden” unless a polynomial barrier in the effective potential has real roots. The logic mirrors a no-go argument that can be circumvented when extra parameters shift the position of those roots. Similarly, in renormalization group (RG) flows, certain “fixed points” might appear impossible under standard assumptions but become feasible once an extra boundary term or integration constant is admitted. Our extended discussion thus resonates with a wide array of problems beyond just black hole physics.

Several open questions arise naturally:

* **Stability of $C \neq 0$ Solutions:** Do these new solutions remain stable under perturbations? Might the naive no-go theorem reappear as an instability argument?

* **Global Topology vs. Local Horizons:** In multi-horizon spacetimes, each horizon can contribute its own local integration constant. Do separate horizons fix each other’s constants or do they remain independent?

* **Matching Conditions at Infinity:** Asymptotic flatness or AdS/dS boundary conditions might partially constrain $C$. Under what precise conditions do they pin $C = 0$ vs. allow $C$ arbitrary?

* **Extensions to Non-Stationary Systems:** For time-dependent geometries (like radiating black holes), how do we track the evolution of $C$ in time?

Addressing these points will cement our understanding of how the no-go logic changes in dynamic or more complicated gravitational contexts.
We have greatly expanded upon the role of zero denominators (and their geometry-based logic) in classical no-go theorems, showing that for quartic (or higher-order) horizon functions in modified gravities, the standard division-by-zero contradiction can be bypassed by a non-zero integration constant $C$. Laurent expansions near horizons highlight precisely how this $C$ arises in indefinite integrals. Consequently, the classical no-go theorem must be revised: one must explicitly check if boundary or topological conditions *truly* fix $C = 0$. If not, the “forbidden” solutions re-enter the picture, effectively invalidating the classical no-go statement.

# 5. BROADER PHYSICAL IMPLICATIONS

## 5.1. Extended Solutions in Modified Gravity

In $f(R)$ and other higher-derivative theories [6–9], solutions with non-trivial near-horizon expansions can exist, circumventing classical no-go results. Such solutions may include rotating or charged black holes, wormhole-like spacetimes, or geometries with unusual asymptotics (for instance, non-Kaluza-Klein circle fibrations). The presence of a non-zero $C$ modifies the naive boundary cancellation, possibly allowing these geometries to thrive.

## 5.2. Interpretation of $C$ in Terms of Fluxes and Conserved Charges

In some instances, $C$ can be interpreted in analogy to a flux or a charge. For example, in the context of two-dimensional anomalies, integration constants can be related to the net flux of the anomaly across boundaries [2–4, 10]. One might interpret $C$ as an additional “degree of freedom” that arises from the interplay of topology and geometry. If no classical boundary condition kills this “flux,” $C$ remains, and the no-go claim is bypassed.

## 5.3. Relation to Stability and Physical Realizability

A logical next step is to ask whether $C \neq 0$ solutions are stable or physically admissible. It could happen that certain $C$ values lead to pathological spacetimes, e.g., naked singularities or negative-energy states, so the mere mathematical possibility of $C \neq 0$ does not necessarily guarantee physically meaningful solutions. However, the crucial point is that classical no-go theorems that neglected $C$ are incomplete from a purely theoretical standpoint, and physical viability must be checked on a case-by-case basis.

# 6. CONCLUSIONS

We have provided a comprehensive argument showing how an extra integration constant $C$ emerges under more careful integration of curvature-based quantities, particularly through Laurent series expansions in the modified RVB framework [2–4, 10]. This $C$ has been overlooked by many classical no-go arguments, which habitually assume boundary terms vanish or integrals are unambiguously zero. Once we allow for a non-zero $C$, the no-go conclusion is rendered invalid unless further conditions are introduced to explicitly fix $C = 0$.

Our findings motivate a systematic re-examination of a variety of no-go theorems, especially in extended or modified gravities like $f(R)$ models. Future directions include:

* Examining whether $C$ can be interpreted as a new physical charge or flux, and under what conditions it remains non-zero.

* Performing explicit stability analyses to see which $C \neq 0$ solutions are physically viable.

* Investigating whether $C$ can couple to quantum corrections or anomaly-induced boundary terms in ways that open new solution branches.

Overall, the necessity of revising no-go theorems by allowing for $C$ highlights the interplay between boundary conditions, topology, and the intricacies of higher-curvature/modified theories of gravity.

If the threshold for the No-Go theorem can be further reduced, and such a reduction can be verified through the modified uncertainty principle, then it indeed serves as experimental evidence for $f(R)$ gravity theory. This suggests
that the corrections introduced by these modifications are not merely theoretical constructs but could be probed in
high-precision experiments, thereby offering a potential observational window into the validity of $f(R)$ gravity.

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