A Thermodynamic Derivation of the Cosmological Constant Λ and Resolution of the Vacuum Catastrophe from a New Quantum Scale

1. Introduction

The cosmological constant $\Lambda$ has undergone a remarkable transformation. Introduced by Einstein to stabilize the universe and later abandoned, it has returned as a cornerstone of modern cosmology with the discovery of accelerated expansion [1,2,3]. Yet its physical meaning remains obscure [4]. The “vacuum catastrophe”— quantum field theory’s (QFT) estimate of a vacuum energy density exceeding cosmological bounds by ${10}^{120}$—reveals a deep gap in how quantum mechanics, gravity, and thermodynamics interrelate.

Thermodynamic treatments of spacetime (e.g., Jacobson [5], Verlinde [6], Padmanabhan [7]) suggest that horizon entropy and temperature encode gravitational dynamics, but they do not pin down the observed vacuum density nor the role of $\Lambda$ as a fundamental constant. If $\Lambda$ is fundamental, physics must explain why it fixes the vacuum energy and how it enters the basic scales of nature [8,9]. This paper makes four contributions:

• Thermodynamic Derivation of the Vacuum Energy.
  Using the Gibbons–Hawking temperature of the de Sitter horizon [10], the Bekenstein–Hawking entropy [11,12], and the Clausius relation $E=TS$ (or $\delta Q=T\mathrm{d}S$) [5,13], we obtain directly the vacuum energy density in general relativity (GR),

  $$u_{\Lambda }={\displaystyle \frac{\Lambda c^4}{8\pi G}}.$$

  (1)

  identifying $\Lambda$ as a finite entropy bound of the universe (throughout, u denotes a generic energy density)[10,14,15].
• A New Quantum Scale—the $\Lambda$-Scale. As $\Lambda$ represents a finite information capacity of the universe, we argue the Planck scale is incomplete without it [16]. Combining $\{G,\hslash ,c,\Lambda \}$ selects a unique scale (Figure 1.) We introduce the dimensionless gravitational fine-structure constant (GFSC),

  $${\alpha }_{\Lambda }\equiv {\displaystyle \frac{c^3}{G\hslash \Lambda }}$$

  (2)

  an entropic bound that anchors quantum theory to cosmology and resolves the vacuum catastrophe without invoking Planck-scale physics. Details and the dimensional analysis are given in Appendix A. For historical context see Appendix B.
• Cross-Domain Validation of the $\Lambda$-scale.
  Applications— Force interactions in $\Lambda$ units (Section 4), electromagnetic radiation (Section 5), Casimir systems (Section 6) and boson/fermion gases (Section 7) — saturate the same bound, supporting the Law of Entropic Constraint (Section 9).
• A Quantum Derivation of $u_{\Lambda }$ from the Zero-Point Energy (ZPE): Crucially, we also provide an explicit zero-point (ZPE) density-of-states derivation of $u_{\Lambda }$ in Eq.(1), based on the $\Lambda$-scale in Figure 1 (see Section 8 and Appendix C), not via boson/fermion cancellations but through a finite mode budget. Without Planck-cutoff assumptions, ultraviolet divergences are eliminated, yielding a finite and radiatively stable vacuum energy [4,17].

2. Deriving the Vacuum Energy from Horizon Thermodynamics

2.1. Thermodynamics of Causal Event Horizons

The thermodynamic structure of causal event horizons — a defining feature of GR, partitions spacetime into accessible and inaccessible regions. This division gives rise to observable thermodynamic properties: temperature, entropy, and energy. The universality of their expressions and the deeper thermodynamic structure are governed by the four laws of black hole thermodynamics [11,12,18]. See Figure 2.

This viewpoint has been developed by many authors over the past decades [5,6,7,10,11,12,18,20]. The three archetypal cases are shown schematically in Figure 3, Figure 4, Figure 5 and Figure 6.

An accelerated observer in flat spacetime perceives a Rindler horizon and detects a thermal bath at the Unruh temperature [20]. See Figure 3.

A black hole horizon radiates at the Hawking temperature, its entropy fixed by the Bekenstein–Hawking area law [11,12] See Figure 4 and Figure 5.

Finally, de Sitter spacetime, relevant to our universe with positive $\Lambda$, possesses a cosmological horizon with Gibbons–Hawking temperature and entropy proportional to its area [10]. See Figure 6.

This principle - that causal event horizons are inextricably tied to an entropy and temperature - holds irrespective of the horizon’s origin — whether due to acceleration, mass, or vacuum energy — and it is the universality of this horizon thermodynamics that forms the basis for our derivation of the vacuum energy density. The common structure across these three spacetimes is illustrated in Table 1.

Because our universe is asymptotically approaching de Sitter space, its horizon provides the correct setting to apply the identity $E=TS$, yielding a finite and universal expression for the vacuum energy density $u_{\mathrm{vac}}$ in the context of horizon thermodynamics [19,21,22], and consistent with quantum–informational perspectives [23].

2.2. Deriving the Vacuum Energy $u_{vac}$ from Horizon Thermodynamics

The integrated form of the First Law of Thermodynamics $E=TS$, has its deeper roots in the Second Law of Thermodynamics, formalized by Rudolf Clausius in the mid-19th century [13,24],

$$\delta S={\displaystyle \frac{\delta Q}{T}}$$

(3)

We now make use of the thermodynamic identity, applied to the cosmological horizon of de Sitter spacetime.

2.3. Horizon Energy E

To obtain the total energy content E of the de Sitter universe, we begin with the entropy S of the de Sitter horizon:

$$S={\displaystyle \frac{k_B{c}^{3}A}{4G\hslash }},A=4\pi R^2,R=\sqrt{{\displaystyle \frac{3}{\Lambda }}}.$$

(4)

Substituting the de Sitter radius into this expression:

$$A=4\pi \left({\displaystyle \frac{3}{\Lambda }}\right)={\displaystyle \frac{12\pi }{\Lambda }}$$

(5)

$$S={\displaystyle \frac{k_B{c}^3}{4G\hslash }}·{\displaystyle \frac{12\pi }{\Lambda }}={\displaystyle \frac{3\pi k_B{c}^3}{G\hslash \Lambda }}$$

(6)

Note that Eq.(6) has the GFSC defined in Eq. (2) embedded within it, making this a maximum bound on entropy. Next, we use the Gibbons–Hawking temperature:

$$T={\displaystyle \frac{\hslash c}{2\pi k_{B}R}}={\displaystyle \frac{\hslash c}{2\pi k_B}}·\sqrt{{\displaystyle \frac{\Lambda }{3}}}.$$

(7)

Combining:

$$E=TS=\left({\displaystyle \frac{\hslash c}{2\pi k_B}}\sqrt{{\displaystyle \frac{\Lambda }{3}}}\right)\left({\displaystyle \frac{3\pi k_B{c}^3}{G\hslash \Lambda }}\right)={\displaystyle \frac{3}{2}}{\displaystyle \frac{c^4}{G}}·{\displaystyle \frac{1}{\sqrt{3\Lambda }}}$$

(8)

2.4. Horizon Volume and Energy Density

The spatial volume enclosed by the de Sitter horizon is:

$$V={\displaystyle \frac{4\pi }{3}}{R}^3={\displaystyle \frac{4\pi }{3}}{\left({\displaystyle \frac{3}{\Lambda }}\right)}^{3/2}.$$

(9)

The vacuum energy density is then:

$$u_{\Lambda }={\displaystyle \frac{E}{V}}=\left({\displaystyle \frac{3c^4}{2G}}·{\displaystyle \frac{1}{\sqrt{3\Lambda }}}\right)/\left({\displaystyle \frac{4\pi }{3}}{\left({\displaystyle \frac{3}{\Lambda }}\right)}^{3/2}\right)$$

(10)

Simplifying:

$$u_{\Lambda }={\displaystyle \frac{3c^4}{2G}}·{\displaystyle \frac{1}{\sqrt{3\Lambda }}}·{\displaystyle \frac{3}{4\pi }}·{\left({\displaystyle \frac{\Lambda }{3}}\right)}^{3/2}={\displaystyle \frac{\Lambda c^4}{8\pi G}}.$$

(11)

This elegant result is deeply significant. In GR Eq. (11) emerges only from dimensional consistency of the Einstein field equations, here the observed vacuum energy density is derived without invoking any matter fields or action principles. Instead, it arises from applying thermodynamics to the geometry of de Sitter spacetime.

3. From Thermodynamics to a New Quantum Scale in Nature

QFT has not yet produced a reliable prediction of the observed vacuum energy density when regulated at Planck scales because it is constructed without reference to $\Lambda$. Being blind to the large-scale curvature of spacetime, it is a miscalibrated scale. A new quantum scale is therefore needed, one that includes $\Lambda$ ab initio. Only then can we meaningfully measure the vacuum without triggering a theoretical breakdown [7].

Our thermodynamic derivation of $\Lambda$ fixes the vacuum energy density as a finite, horizon–regulated quantity. It is this that justifies why $\Lambda$ should be treated not as an arbitrary parameter but as a fundamental constant of nature, on the same footing as c and ℏ. Each of these constants encodes a fundamental limiting principle: c defines the maximum velocity of causal signals, ℏ defines the minimum quantum of action, and $\Lambda$ defines the maximum entropy – the finite information capacity of the universe. [10,11,12].

A dimensional analysis involving $\{c,\hslash ,G,\Lambda \}$ yields a new set of natural units in Appendix A, illustrated in Figure 1 and applied in Figure 7. For historical context see Appendix C.

This new quantum scale, unlike the Planck scale, avoids the break down when measuring the quantum vacuum. It signifies a new unity between GR, QFT and thermodynamics, in which the vacuum catastrophe is revealed as illusory, and nothing more than a physicist’s ghost. see Figure 7.

4. Forces in $\Lambda$-Units: How Interactions Respect the Entropy Bound

Within the $\Lambda$-framework, horizons defined by $\Lambda$ impose a finite entropy bound on spacetime (see Section 2.1); deviations from geodesic motion couple to this structure, producing an entropic response. Expressed in $\Lambda$-units, inertial $F_{\Lambda }^i$, gravitational $F_{\Lambda }^G$ and electromagnetic forces $F_{\Lambda }^E$, reduce to the scales shown in Figure 8:

We see that $\Lambda$ is present in all the force terms (Figure 8) and that Newton’s constant G, drops out of the expression for the gravitational force term, leaving a ’gravitational force’ without any need for G,

$$F={\displaystyle \frac{G{m}_1{m}_2}{r^2}}⟹F_{\Lambda }^{\left(G\right)}=\Lambda \hslash c.$$

(12)

One finds from the definition of the GFSC in Eq. (2),

$$G={\displaystyle \frac{1}{{\alpha }_{\Lambda }}}·{\displaystyle \frac{c^3}{\hslash \Lambda }}$$

(13)

substituting Eq. (13) into the inertial force term $F_{\Lambda }^i$ we find,

$$F_{\Lambda }^i={\left({\displaystyle \frac{\hslash \Lambda c^5}{G}}\right)}^{1/2}={\left({\alpha }_{\Lambda }{\hslash }^2{\Lambda }^2{c}^2\right)}^{1/2}$$

(14)

$$F_{\Lambda }^i=\sqrt{{\alpha }_{\Lambda }}·F_{\Lambda }^G$$

(15)

Thus inertial forces share the same form as gravitational ones but scaled by a factor $\sqrt{{\alpha }_{\Lambda }}$. It is remarkable that the electromagnetic sector (a gauge interaction) respects the same entropic bound, taking just a fraction, ${\alpha }_E$, of the inertial force.

Gravity looks weak at macroscales because the gravitational unit $F_{\Lambda }^{\left(G\right)}$ is suppressed relative to $F_{\Lambda }^{\left(i\right)}$ by ${\alpha }_{\Lambda }^{-1/2}$, while gauge forces scale with their own dimensionless couplings (e.g. ${\alpha }_E$). In all cases the common thread is that all forces and their resulting motions are constrained by a finite-information capacity of the vacuum.

5. Electromagnetic Radiation and the Vacuum Bound

Electromagnetism, like inertia and gravity, is constrained by the vacuum’s entropy bound. In $\Lambda$-units, the Coulomb force [25] is revealed not as arbitrarily strong but as a fraction of the vacuum’s maximal force, scaled by the fine-structure constant ${\alpha }_E$. This reframes electromagnetic interactions as vacuum-limited couplings.

Gauge invariance still permits shifting the zero of potential, but the $\Lambda$-framework fixes the maximum field strength by horizon thermodynamics. Thus, electromagnetism joins inertia and gravity as an expression of the same entropic constraint.

To illustrate this explicitly, we derive the Poynting flux [26,27] in $\Lambda$-units, showing how energy transport in electromagnetic waves saturates at the same thermodynamic limit set by $\Lambda$. See Table 2.

5.1. The Poynting Flux in $\Lambda$-Units

In standard notation the time–averaged Poynting flux for a plane electromagnetic wave in vacuum is

$$\langle S_{EM}\rangle =\frac{1}{2}{\epsilon }_{0}c{E}_0^2.$$

(16)

The electric field of a point charge at distance $r=L_{\Lambda }$ in $\Lambda$ units is [25,28]:

$$E_{\Lambda }={\displaystyle \frac{1}{4\pi {\epsilon }_0}}·{\displaystyle \frac{e}{L_{\Lambda }^2}}$$

(17)

The corresponding flux has a magnitude given by:

$$\begin{array}{cc}\hfill \langle {\mathbf{S}}_{\mathrm{EM}}\rangle & ={\displaystyle \frac{1}{2}}{\epsilon }_{0}c{E}_{\Lambda }^2\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{1}{2}}{\epsilon }_{0}c{\left({\displaystyle \frac{e}{4\pi {\epsilon }_0{L}_{\Lambda }^2}}\right)}^2={\displaystyle \frac{e^{2}c}{32{\pi }^2{\epsilon }_0{L}_{\Lambda }^4}}\hfill \end{array}$$

(19)

Expressing $e^2/{\epsilon }_0$ using the fine-structure constant [29]:

$${\alpha }_E={\displaystyle \frac{e^2}{4\pi {\epsilon }_0\hslash c}}\Rightarrow {\displaystyle \frac{e^2}{{\epsilon }_0}}=4\pi {\alpha }_E\hslash c$$

(20)

So:

$$\langle {\mathbf{S}}_{\mathrm{EM}}\rangle ={\displaystyle \frac{{\alpha }_E\hslash c^2}{8\pi L_{\Lambda }^4}}$$

(21)

Substituting for the Lambda length as,

$$L_{\Lambda }^4={\displaystyle \frac{\hslash G}{\Lambda c^3}}\Rightarrow {\displaystyle \frac{1}{L_{\Lambda }^4}}={\displaystyle \frac{\Lambda c^3}{\hslash G}}$$

(22)

into Eq.(21) for $\langle {\mathbf{S}}_{\mathrm{EM}}\rangle$ we obtain:

$$\langle {\mathbf{S}}_{\mathrm{EM}}\rangle ={\displaystyle \frac{{\alpha }_E\hslash c^2}{8\pi }}·{\displaystyle \frac{\Lambda c^3}{\hslash G}}={\displaystyle \frac{{\alpha }_E\Lambda c^5}{8\pi G}}={\alpha }_{E}c·u_{\Lambda }$$

(23)

The average radiative flux is thus:

$$\begin{array}{|c|}\hline \langle S_{\mathrm{EM}}\rangle ={\alpha }_{E}c{u}_{\Lambda }\\ \hline\end{array}$$

This is a striking result. It shows that the classical electromagnetic flux density is directly proportional to the vacuum energy density of spacetime — scaled by the fine-structure constant, a quantum measure of the coupling strength of light to matter.

Our Poynting analysis reveals the EM energy/flux, a gauge field, respects the bound set by $u_{\Lambda }$.

6. The Casimir Effect and the Quantum Vacuum

The Casimir force between conducting plates has long been taken as evidence of vacuum fluctuations arising from a modified mode spectrum between the plates [30,31]. However, the precise connection between such boundary–dependent zero-point energies and the homogeneous cosmological vacuum remains an open question in contemporary physics [4,32,33].

The standard expression for the pressure is,

$$P_{\mathrm{Casimir}}=-{\displaystyle \frac{{\pi }^2\hslash c}{240d^4}}$$

(24)

where d is the plate separation. The minus sign denotes vacuum tension (negative pressure): suppressed modes between the plates make the interior zero-point pressure lower than outside, producing an inward force. This fixes the sign (tension) of the vacuum in a bounded geometry. See Figure 9.

Introducing the $\Lambda$-length cutoff

$$L_{\Lambda }={\left({\displaystyle \frac{\hslash G}{\Lambda c^3}}\right)}^{1/4},$$

(25)

and substituting into Eq.(24), shows that the Casimir pressure is a fixed fraction of the vacuum bound,

$$P_{Casimir}=-{\displaystyle \frac{{\pi }^3}{30}}·{\displaystyle \frac{\Lambda c^4}{8\pi G}}=-{\displaystyle \frac{{\pi }^3}{30}}·u_{\Lambda }$$

(26)

Thus a laboratory-scale quantum effect validates the same vacuum limit that emerges thermodynamically from de Sitter horizons. It is precisely a negative pressure that gravitates in GR, yielding a cosmic repulsion and an accelerated universal expansion. See Figure 10.

6.1. From Quantum Fluctuations to Measurable Force

The $\Lambda$ length sits in the Casimir window. With CODATA values,

$$\begin{array}{cc}\hfill L_{\Lambda }& ={\left({\displaystyle \frac{\hslash G}{\Lambda c^3}}\right)}^{1/4}\approx 3.9\times {10}^{-5}\mathrm{m}=39\mu \mathrm{m},\hfill \\ \hfill {\nu }_{\Lambda }& \equiv {\displaystyle \frac{c}{L_{\Lambda }}}\approx 7.7\mathrm{THz}.\hfill \end{array}$$

(27)

which lies squarely in the mid–IR/THz, precisely the range where Casimir forces have been probed (e.g. Lamoreaux’s torsion–pendulum and subsequent torsional studies [31,34]). This coincidence is not incidental in our framework.

Evaluating the ideal ($T\to 0$) plate–plate pressure at the $\Lambda$ length yields a quartic correspondence with the GR vacuum density. This suggests, in principle, a laboratory route to infer $\Lambda$ using real materials, with corrections incorporated through the standard Lifshitz framework [35]. Figure 11 and Figure 12 outline how such a determination of $\Lambda$ could be made experimentally from micron–scale force measurements, far above the Planck length .

6.2. On the Origins of the Casimir Effect

The interpretation of Casimir forces has long been debated: zero–point fluctuations versus retarded van der Waals interactions. The Lifshitz framework [35] shows these are equivalent descriptions, with the Casimir expression emerging as the limiting case of Lifshitz theory for ideal reflectors ($T\to 0$).

7. Quantum Gases: Bosons and Fermions in Thermal Equilibrium with the Vacuum

Both bosonic and fermionic gases exhibit the same thermodynamic law: their equilibrium energy densities scale quartically with temperature, $u\left(T\right)\propto T^4$ [36,37]. This Stefan–Boltzmann scaling reflects a deeper geometric constraint imposed by causal horizons. For massless bosons such as photons, the cosmic microwave background (CMB) energy density is $u_{\gamma }$ where,

$$u_{\gamma }={\displaystyle \frac{{\pi }^2}{15}}{\displaystyle \frac{{\left(k_{B}T\right)}^4}{{\hslash }^3{c}^3}},$$

(27)

while fermionic gases such as relic neutrinos form the cosmic neutrino background (C$\nu$B), is denoted by $u_{\nu }$; $N_{\nu }$ denotes the number of light neutrino species (we take $N_{\nu }=3$),

$$u_{\nu }={\displaystyle \frac{7}{8}}{N}_{\nu }{\displaystyle \frac{{\pi }^2}{15}}{\displaystyle \frac{{\left(k_{B}T\right)}^4}{{\hslash }^3{c}^3}}.$$

(28)

Substituting the $\Lambda$–temperature,

$${\Theta }_{\Lambda }={\left({\displaystyle \frac{{\hslash }^3\Lambda c^7}{G{k}_B^4}}\right)}^{1/4},$$

(28)

into the boson and fermion energy densities, reveals both as fixed fractions of the vacuum energy, see Figure 13. Bosons saturate the bound rapidly, while fermions are Pauli-suppressed, but both approach finite values consistent with a capped vacuum.

The usual QFT bookkeeping—adding bosonic $+\frac{1}{2}\hslash \omega$ and fermionic $-\frac{1}{2}\hslash \omega$—does not capture the physics [4,38]. The vacuum density is not a delicate boson–fermion cancellation; it is a finite entropy–energy budget set by $\Lambda$.

8. Quantum Derivation of the Vacuum Density (ZPE with a $\Lambda$ cutoff)

We write the zero–point energy density in the standard (mode–count) form

$$u_{\mathrm{ZPE}}\left({\omega }_c\right)={\displaystyle \frac{\hslash }{2}}{\int }_0^{{\omega }_c}\omega g\left(\omega \right)d\omega ,$$

(29)

where $g\left(\omega \right)={\omega }^2/\left({\pi }^2{c}^3\right)$ for a relativistic linear dispersion $\omega =ck$. Evaluated with a Planck–scale ultraviolet cutoff ${\omega }_P=2\pi /t_P=2\pi \sqrt{c^5/(\hslash G)}$ this gives

$$u_{\mathrm{ZPE}}^{\left(\mathrm{Planck}\right)}={\displaystyle \frac{\hslash }{8{\pi }^2{c}^3}}{\omega }_P^4={\displaystyle \frac{2{\pi }^2{c}^7}{\hslash G^2}}(\mathrm{overestimates}\mathrm{by}\sim {10}^{120}).$$

(30)

The $\Lambda$ scale and the physical cutoff

The $\Lambda$ units fix a natural length, time, and (linear) frequency

$$L_{\Lambda }={\left({\displaystyle \frac{\hslash G}{\Lambda c^3}}\right)}^{1/4},T_{\Lambda }={\displaystyle \frac{L_{\Lambda }}{c}},{\nu }_{\Lambda }={\displaystyle \frac{1}{T_{\Lambda }}}={\left({\displaystyle \frac{\Lambda c^7}{\hslash G}}\right)}^{1/4}.$$

(31)

Horizon thermodynamics selects a universal numerical factor so that UV saturation occurs at

$${\omega }_c\equiv {\omega }_{\Lambda }^{*}=2\pi {\nu }_{\Lambda }^{*},{\nu }_{\Lambda }^{*}={\displaystyle \frac{\zeta }{T_{\Lambda }}},$$

(32)

where $T_{\Lambda }=L_{\Lambda }/c$ and $\zeta$ is a pure number fixed by vacuum matching. Using

$$T_{\Lambda }={\left({\displaystyle \frac{\hslash G}{\Lambda c^7}}\right)}^{1/4},\zeta ={\left(16{\pi }^3\right)}^{-1/4},$$

one gets

$${\omega }_c=2\pi {\displaystyle \frac{\zeta }{T_{\Lambda }}}={\pi }^{1/4}{\left({\displaystyle \frac{\Lambda c^7}{\hslash G}}\right)}^{1/4}={\displaystyle \frac{{\pi }^{1/4}}{T_{\Lambda }}}.$$

(33)

The explicit integral with the matched upper limit

Inserting ${\omega }_c={\omega }_{\Lambda }^{*}$ into (29) (with $g\left(\omega \right)={\omega }^2/{\pi }^2{c}^3$) gives

$$u_{\Lambda }={\displaystyle \frac{\hslash }{8{\pi }^2{c}^3}}{\omega }_c^4={\displaystyle \frac{\hslash }{8{\pi }^2{c}^3}}{\left({\pi }^{1/4}\right)}^4\left({\displaystyle \frac{\Lambda c^7}{\hslash G}}\right)={\displaystyle \frac{\Lambda c^4}{8\pi G}}.$$

(34)

Thus the same quartic ZPE integral, evaluated at the $\Lambda$ cutoff ${\omega }_{\Lambda }^{*}$, reproduces exactly the GR vacuum density.

Massive fields decouple at the $\Lambda$ scale

For a species of mass m with angular frequency ${\omega }_k$,

$${\omega }_k=\sqrt{c^2{k}^2+{\left(m{c}^2/\hslash \right)}^2},$$

(35)

and the same $k_{max}$, the energy density becomes

$$u^{\left(m\right)}={\displaystyle \frac{s\hslash }{4{\pi }^2}}{\int }_0^{k_{max}}{k}^2\sqrt{c^2{k}^2+{\left(m{c}^2/\hslash \right)}^2}dk=\mathcal{O}\left({\displaystyle \frac{m{c}^2}{12{\pi }^2}}{k}_{max}^3\right)\ll u^{\mathrm{massless}},$$

(36)

with degeneracy s. Hence heavy fields are parametrically suppressed by powers of $k_{max}/(mc/\hslash )$. At the $\Lambda$ cutoff ($\hslash c/L_{\Lambda }\sim$ meV), only effectively massless modes (photons; plausibly gravitons) contribute appreciably.

• Interpretation.

The result above is not a fine–tuned cancellation but a horizon–imposed saturation of the mode count: a finite entropy/energy budget set by $\Lambda$. Radiative stability follows naturally—high–frequency loops do not shift $u_{\Lambda }$ [4,5,7,39].

Repeating the calculation with Planck’s ($T=0$) spectrum reproduces

$$u_{\mathrm{ZPE}}^{\left(\mathrm{Planck}\right)}={\displaystyle \frac{\pi \hslash }{c^3}}{\nu }_c^4,$$

(37)

with ${\nu }_c={\nu }_{\Lambda }^{*}$ (equivalently, $u_{\Lambda }=\hslash {\omega }_c^4/\left(8{\pi }^2{c}^3\right)$ for $\omega =2\pi \nu$), providing a useful consistency check; see Appendix C.

9. Law of Entropic Constraint (LoEC)

Across horizon thermodynamics, Casimir, quantum gases and EM flux, all ceilings track a single dimensionless constant:

$$\begin{array}{|c|}\hline {\alpha }_{\Lambda }\equiv {\displaystyle \frac{c^3}{G\hslash \Lambda }}\sim {10}^{123}\\ \hline\end{array}.$$

(38)

Fixed by horizon matching (Section 2.1, Appendix A), a direct corollary is the vacuum ceiling

$$u_{\Lambda }={\displaystyle \frac{\Lambda c^4}{8\pi G}},u\le u_{\Lambda }.$$

(39)

The cosmological constant and corresponding $u_{\Lambda }$ are small because the entropy of the universe is so large, bounded by Eq. (38). With Boltzmann’s relation (penned by Planck himself) [40,41],

$$S=k_{B}lnW,$$

(40)

we have

$$W=exp\left({\displaystyle \frac{S_{max}}{k_B}}\right)\approx exp\left({10}^{123}\right).$$

(41)

This is an immensely large entanglement entropy [5], meaning the number of accessible microstates available to the quantum vacuum is truly colossal. The smallness of $u_{\Lambda }$ is therefore a consequence of the largeness of the GFSC, ${\alpha }_{\Lambda }$. Furthermore, this finite number of microstates, according to the LoEC, cannot be created or destroyed arbitrarily: the total capacity is conserved [14]. A universe in which $\Lambda \to 0$ removes the bound, $S_{\mathrm{dS}}\to \infty$, and the $\Lambda$-scale would disappear.

Our preceding analysis has shown that seemingly independent domains converge on the same vacuum bound $u_{\Lambda }$. This unification points to a deeper principle: the vacuum encodes a maximum entropy, and all physical processes respect this limit.

Inertia emerges from entanglement across local Rindler horizons, while gravity reflects curvature-induced entropy bounds in de Sitter space.

If $\Lambda$ sets the universe’s entropy capacity, models that treat dark energy as a time-varying $\Lambda \left(t\right)$ are in tension with horizon thermodynamics, since $S_{\mathrm{dS}}\propto 1/\Lambda$ would drift unless compensated by entropy production. Likewise, varying c or ℏ would shift the same bound. The Hubble-tension [42,43] is therefore more plausibly traced to measurement systematics or new matter-sector physics than to variations in $\Lambda ,c,\hslash$. Gravity need not be fundamental: the $\Lambda$–framework suggests it is the macroscopic imprint of information loss across causal horizons.

Box 9.1: The Law of Entropic Constraint (LoEC): Summary

All physical processes are constrained by a maximum entropy determined by the vacuum’s causal structure, underpinned by the dimensionless GFSC. $$\begin{array}{|c|}\hline {\alpha }_{\Lambda }\equiv {\displaystyle \frac{c^3}{G\hslash \Lambda }}\sim {10}^{123}\\ \hline\end{array}\begin{array}{}\left(\mathrm{LoEC}\right)\end{array}$$ Its key consequences are: A finite vacuum energy density arises from the finite maximum entropy of a de Sitter causal patch. This provides a universal ceiling on coarse–grained energy densities accessible within a causal region: $u\le u_{\Lambda }\equiv \Lambda c^4/\left(8\pi G\right)$. Inertial, gravitational, and electromagnetic interactions respect the same horizon–thermodynamic bound. Consistency across sectors—bosons, fermions, electromagnetic flux, and the vacuum—means their extremal configurations saturate the same ceiling $u_{\Lambda }$; generic states satisfy $u<u_{\Lambda }$. In equilibrium, the LoEC favors a constant cosmological term $\Lambda$. Allowing $\Lambda \left(t\right)$ would require compensating entropy production (i.e., a non–equilibrium extension of the bookkeeping). Gravity can be interpreted as an emergent macroscopic response of geometry to entropy flow across causal horizons (à la Jacobson), rather than an independent fundamental force scale.

10. Discussion

Einstein’s aesthetic dilemma concerning the cosmological constant has haunted theoretical physics for more than a century. In extending his 1915 field equations [44],

$$G_{\mu \nu }={\displaystyle \frac{8\pi G}{c^4}}{T}_{\mu \nu },$$

(42)

with a cosmological constant [45],

$$G_{\mu \nu }+\Lambda g_{\mu \nu }={\displaystyle \frac{8\pi G}{c^4}}{T}_{\mu \nu }.$$

(43)

Einstein complained that he had “always had a bad conscience,” finding it “ugly indeed that the field law of gravitation should be composed of two logically independent terms” [46]. His discomfort reflected the sense that $\Lambda$ had been appended without justification. Yet historical precedent shows this unease was misplaced. Newton had already noted in the Principia that spherical symmetry admits not only the familiar inverse-square force law but also a linear term proportional to R [47]. See Figure 14.

What appeared to Einstein as a blemish was in fact the natural echo of a symmetry already recognised in classical gravity.

The thermodynamic interpretation of general relativity reveals why the cosmological term is inevitable. Jacobson demonstrated that the Einstein equations without $\Lambda$ are equivalent to the Clausius relation $\delta Q=TdS$ [5], expressing the balance of heat and entropy flow across local horizons.

Box 10.1: Thermodynamic Bookkeeping ↔ Einstein equation (with $\Lambda$)

$$\begin{array}{cc}\hfill \mathbf{First}\mathbf{law}:& dU=\delta Q-pdV(\delta Q=TdS)\hfill \\ \hfill \mathbf{Einstein}\mathbf{eq}.:& G_{\mu \nu }={\displaystyle \frac{8\pi G}{c^4}}{T}_{\mu \nu }-\Lambda g_{\mu \nu }\hfill \end{array}$$ Term-by-term correspondence: $$\begin{array}{ccc}\underset{\mathrm{internal}\mathrm{energy}}{\underbrace{{\displaystyle dU}}}& \leftrightarrow & \underset{\mathrm{internal}\mathrm{curvature}\left(\mathrm{geometry}\right)}{\underbrace{{\displaystyle G_{\mu \nu }}}}\\ \underset{\mathrm{heat}/\mathrm{energy}\mathrm{flux}}{\underbrace{{\displaystyle \delta Q(=TdS)}}}& \leftrightarrow & \underset{\mathrm{energy}--\mathrm{momentum}\mathrm{flux}}{\underbrace{{\displaystyle \frac{8\pi G}{c^4}{T}_{\mu \nu }}}}\\ \underset{\mathrm{work}}{\underbrace{{\displaystyle pdV}}}& \leftrightarrow & \underset{\mathrm{vacuum}\mathrm{term}}{\underbrace{{\displaystyle \Lambda g_{\mu \nu }}}}\end{array}$$ Interpretation. $G_{\mu \nu }$ plays the role of internal energy stored in curvature—the geometric response. ${\displaystyle \frac{8\pi G}{c^4}}{T}_{\mu \nu }$ maps to the heat/energy flux across a causal screen; via $\delta S=\delta Q/T$ it accounts for entropy flow (see Section 2.1). The $\Lambda$ term acts like a uniform vacuum pressure/tension. Writing $p_{\Lambda }=-{\displaystyle \frac{\Lambda c^4}{8\pi G}},$ the geometric piece $-\Lambda g_{\mu \nu }$ corresponds to the work term $-p_{\Lambda }dV$, matching the first law $dU=\delta Q-pdV$. The vacuum has equation of state $w=-1$ ($p=-\rho c^2$), so the geometric term $-\Lambda g_{\mu \nu }$ corresponds to the work piece with $p_{\Lambda }=-\Lambda c^4/\left(8\pi G\right)$.

Once the cosmological constant is restored, the equations assume the structure of the full first law of thermodynamics,

$$\Delta U=T\Delta S-p\Delta V,$$

(44)

with the $\Lambda g_{\mu \nu }$ contribution supplying the missing $pdV$ work term. In thermodynamic language, vacuum energy has the unique equation of state $p_{\Lambda }=-{\rho }_{\Lambda }{c}^2$, so the cosmological constant precisely mimics the work function of the vacuum. As Padmanabhan [7] emphasised, this endows spacetime with the ability to perform work and completes the thermodynamic analogy. Far from being an arbitrary appendage, $\Lambda$ is the key to upgrading Jacobson’s Clausius identity into the full first law, closing the circle of the thermodynamic derivation presented in Section 2.2.

Once $\Lambda$ is recognised as a fundamental constant, a new natural scale inevitably follows. The $\Lambda$-length,

$$L_{\Lambda }={\left({\displaystyle \frac{\hslash G}{\Lambda c^3}}\right)}^{1/4}=3^{-1/4}{\left(L_P{R}_{\mathrm{dS}}\right)}^{1/2},$$

(45)

Equation (45) shows that $L_{\Lambda }$ is simply the geometric mean scale derived from the Planck length $L_P$ and the de Sitter horizon radius $R_{\mathrm{dS}}$ (up to the fixed factor $3^{-1/4}$). This ties together quantum uncertainty, causality, and horizon thermodynamics in a single invariant.

A central point of comparison is with the traditional Planck scale. In Planck units, the vacuum energy density expected from zero-point fluctuations is catastrophically large, exceeding observation by some 120 orders of magnitude. This mismatch makes the observed value of

$$u_{\Lambda }={\displaystyle \frac{\Lambda c^4}{8\pi G}}$$

(46)

appear absurdly small [48,49,50]. Yet this impression arises only when Planck units are taken as the measure of naturalness [51]. Once $\Lambda$ is admitted as a fundamental constant, a new unit system emerges in which $u_{\Lambda }$ is not anomalously tiny but exactly the saturation value set by horizon thermodynamics. The $\Lambda$-length provides the correct cutoff, and the resulting vacuum energy is radiatively stable [4,52]. In this framework the Planck scale is revealed as inadequate for “weighing the vacuum,” while the $\Lambda$-scale resolves the paradox without fine-tuning. What appears as an inexplicable smallness is reinterpreted as the fundamental benchmark of information capacity. Thus the vacuum catastrophe is not a failure of physics but the signal that the Planck yardstick breaks down when applied to the quantum structure of empty space.

Several further consequences flow directly from treating $\Lambda$ as fundamental and recognising the $\Lambda$-scale as basic:

• Bound on entropy. $\Lambda$ imposes a universal entropy bound of spacetime. The Law of Entropic Constraint (LoEC) asserts that all physical processes are restricted by this bound, encoded in the de Sitter horizon entropy $S\propto 1/\Lambda$. The vacuum energy $u_{\Lambda }$ is a necessary corollary of the bound and the GFSC encodes it.
• Finite and stable vacuum energy. The LoEC forbids $\Lambda \left(t\right)$, since the de Sitter entropy bound would otherwise change in time. Time-varying models (quintessence/phantom) [53,54,55,56] are therefore not a true cosmological constant ($w\ne -1$) and proposals to solve the Hubble tension [57] with $\Lambda \left(t\right)$ contradict both LoEC and the horizon–thermodynamic derivation.
• Quantum derivation of vacuum energy. In $\Lambda$-units, the zero-point energy sum saturates at the $\Lambda$-scale reproducing the GR vacuum density in Eq.(46), and yielding the radiatively stable derivation that Zel’dovich and Sakharov first sought, unifying the quantum and cosmological vacua as a single entity [17,58].
• Cross-domain consistency. Independent systems all saturate the $\Lambda$-bound in the same way. Bosonic and fermionic gases at the $\Lambda$-temperature approach fixed fractions of $u_{\Lambda }$; Casimir pressures with separation $L_{\Lambda }$ reproduce the vacuum density of GR; electromagnetic fluctuations scale as ${\alpha }_E{u}_{\Lambda }$ via the Poynting analysis. These diverse phenomena confirm that the same entropy ceiling governs matter, radiation, and the vacuum alike [30,59,60].

These consequences show that the $\Lambda$-framework addresses simultaneously the vacuum catastrophe, the stability of the vacuum, and the constancy of couplings. In each case, the objections that once plagued the cosmological constant dissolve when it is treated as the natural thermodynamic completion of the first law and the basis of a new quantum scale.

In this light, Einstein’s dilemma disappears. The two-term structure that he found unsightly is the unavoidable expression of deeper principles. Just as Newton showed that two distinct force laws are permitted by symmetry, the thermodynamic derivation shows that two terms are demanded by consistency: the Einstein tensor encodes the response of spacetime geometry, while the $\Lambda$-term enforces its finite information capacity through the $\Lambda$-scale. Einstein’s infamous “blunder” is thereby reinterpreted as an inevitable consequence of symmetry and thermodynamics. Recognising $\Lambda$ as fundamental not only restores logical unity to the field equations but also reframes them as the complete first law of spacetime dynamics, with the $\Lambda$-scale as the bridge between quantum theory and cosmology.

Planck versus $\Lambda$ scales

The Planck system of units was constructed heuristically by equating a Schwarzschild radius with a Compton wavelength, and has long been assumed to define the natural scale of quantum gravity. Yet this construction is ad hoc and plagued by pathologies. It predicts a vacuum density overshooting observation by ${10}^{120}$ (the “vacuum catastrophe”) [4], a runaway instability at every instant [61,62], and a large conceptual gap (“Planck desert”) between $M_P$ and Standard Model scales [63,64]. Most significantly, as highlighted in Table 3, the Planck system is thermodynamically sterile: its base set $\{G,\hslash ,c\}$— is purely mechanical, with $k_B$ introduced only by convention—the $\Lambda$ base set $\{G,\hslash ,c,\Lambda \}$ by contrast, embeds a thermodynamic constant ab initio: $\Lambda$ fixes the de Sitter entropy bound, so information capacity is intrinsic rather than appended. Its base units follow not from heuristic dimensional analysis but from the Clausius relation $E=TS$, ensuring consistency with the observed vacuum energy density. In this sense, the $\Lambda$ framework subsumes and supersedes the Planck scale as a complete natural system, unifying quantum, thermodynamic, and gravitational perspectives.

11. Conclusions

The analysis presented suggests, $\Lambda$ is not a ’blunder’ but the missing piece of the cosmic–quantum jigsaw puzzle [45,46,65]. Its genuine inclusion at the foundations of physics reveals the vacuum not as a place of violent discontinuity, but as a regulated structure where spacetime remains smooth and continuous even at the smallest scales. Far from heralding the breakdown of the laws of physics, the $\Lambda$-scale completes them. The supposed realm of quantum gravity, long confined to the Planck scale, is displaced by some thirty orders of magnitude into a domain where thermodynamic and quantum principles converge. Taken together, our results recast $\Lambda$ from an expendable add-on to a fundamental constant that fixes the universe’s information capacity [14]. The $\Lambda$–framework turns the divergent zero-point sum into a finite non-zero value because the de Sitter horizon enforces geometric saturation: only modes that “fit” contribute, while heavy, short-wavelength modes are suppressed [66]. $\Lambda$ thereby emerges as a thermodynamic necessity, with the maximum entropy

$$S_{max}={\displaystyle \frac{A_{dS}}{4G\hslash }},$$

(47)

providing the global bound.

In this light, the Law of Entropic Constraint (LoEC), underpinned by the gravitational fine-structure constant

$${\alpha }_{\Lambda }={\displaystyle \frac{c^3}{G\hslash \Lambda }},$$

(50)

functions as a conservation-like principle: entropy can be redistributed locally through unitary dynamics, but the total number of accessible microstates is bounded globally. This explains why de Sitter spacetime lacks global energy conservation and unitarity, even while local conservation laws remain intact [22,67].

Historically, this framework fulfils what Planck originally sought. His aim was to resolve the UV catastrophe, via an exact law of entropy conservation [68,69]. Instead his journey initiated the quantum revolution. The $\Lambda$–framework, by contrast, subsumes and supersedes the Planck system, providing the entropy law he sought through the LOEC.

Most significantly, this perspective resolves Einstein’s misgivings about the “ugly” addition of two terms [46]. It provides the Zel’dovich–Sakharov program [17,58] with a successful, radiatively stable derivation of vacuum energy, and extends Jacobson’s thermodynamic derivation from local Rindler horizons to the full de Sitter horizon [5]. In doing so, the $\Lambda$–framework unifies quantum, gravitational, and thermodynamic perspectives.

In the gravitational context, thermodynamic constraints are not merely consequences of geometry; they give rise to it [5,6]. Spacetime’s structure may be the most elegant entropy-management architecture the universe has ever devised, with $\Lambda$ setting its intrinsic scale and, in the process, revealing gravity’s innate quantum nature.

Appendix A.1. Deriving and Fixing the Λ-Unit System

To formalize the inspection analysis above, we now write out the dimensional algebra explicitly. Using our four constants are defined in (A2), we consider a monomial

$$X=G^p{\hslash }^q{\Lambda }^r{c}^s,$$

(A10)

and demand $\left[X\right]=M^b{L}^a{T}^c$. Matching exponents of $M,L,T$ gives the linear system

$$\left(\begin{array}{cccc}-1& 1& 0& 0\\ 3& 2& -2& 1\\ -2& -1& 0& -1\end{array}\right)\left(\begin{array}{c}p\\ q\\ r\\ s\end{array}\right)=\left(\begin{array}{c}b\\ a\\ c\end{array}\right).$$

(A11)

This $3\times 4$ matrix has rank 3, hence a 1-dimensional nullspace. A basis vector is

$$\mathbf{n}={(1,1,1,-3)}^{\top },\left(\begin{array}{cccc}-1& 1& 0& 0\\ 3& 2& -2& 1\\ -2& -1& 0& -1\end{array}\right)\mathbf{n}=\mathbf{0}.$$

(A12)

Therefore any particular solution $(p_0,q_0,r_0,s_0)$ can be shifted by $\lambda \mathit{n}$ for arbitrary $\lambda$:

$$(p,q,r,s)=(p_0,q_0,r_0,s_0)+\lambda (1,1,1,-3).$$

(A13)

• Vacuum matching anchor and uniqueness.

Dimensional analysis leaves the affine family (A13). Physics fixes λ by the vacuum–matching postulate (QFT cutoff = GR vacuum):

$${\displaystyle \frac{\hslash c}{L^4}}={\displaystyle \frac{\Lambda c^4}{G}}.$$

(A14)

In exponent space (order $G,\hslash ,\Lambda ,c$) this requires

$$L^4\propto G^1{\hslash }^1{\Lambda }^{-1}{c}^{-3}⟹4(p,q,r,s)=(1,1,-1,-3),$$

so the unique length exponents are

$$(p,q,r,s)=\left(\frac{1}{4},\frac{1}{4},-\frac{1}{4},-\frac{3}{4}\right),L_{\Lambda }={\left({\displaystyle \frac{\hslash G}{\Lambda c^3}}\right)}^{1/4}.$$

(A15)

With the kinematic anchors $L/T=c$ and $M{L}^2/T=\hslash$ this yields

$$T_{\Lambda }={\displaystyle \frac{L_{\Lambda }}{c}},M_{\Lambda }={\displaystyle \frac{\hslash }{c{L}_{\Lambda }}}.$$

(A16)

Conclusion: with $(M,L,T)$ as bases and four constants $(G,\hslash ,c,\Lambda )$, every target dimension has a 1-parameter family of exponent quadruples. This is the mathematical origin of the “infinitely many $(p,q,r,s)$”.

Appendix A.2. Planck, Stoney, and Λ as “nullspace + anchors”

A unit system becomes unique once we add enough physical anchors to kill the nullspace freedom(s). With the vacuum matching (A14) selecting $\lambda =0$, the dimensionless values of the constants in $\Lambda$-units are summarized in Table A1.

Table A1. Dimensionless values in $\Lambda$-units. Anchors enforce $\tilde{c}=\tilde{\hslash }=1$. Vacuum matching selects the symmetric point, yielding $\tilde{G}\tilde{\Lambda }={\alpha }_{\Lambda }^{-1}$ and $\tilde{G}=\tilde{\Lambda }={\alpha }_{\Lambda }^{-1/2}$, where ${\alpha }_{\Lambda }\equiv c^3/(G\hslash \Lambda )$ is the GFSC.

Constant	Base dimensions	$\tilde{\mathit{X}}$ in $\mathbf{\Lambda }$-units	Note
c	$L{T}^{-1}$	1	anchor $L/T=c$
ℏ	$M{L}^2{T}^{-1}$	1	anchor $M{L}^2/T=\hslash$
G	$L^3{M}^{-1}{T}^{-2}$	${\alpha }_{\Lambda }^{-1/2}$	paired with $\tilde{\Lambda }$; product fixed
$\Lambda$	$L^{-2}$	${\alpha }_{\Lambda }^{-1/2}$	symmetric with $\tilde{G}$

Table A1. Dimensionless values in $\Lambda$-units. Anchors enforce $\tilde{c}=\tilde{\hslash }=1$. Vacuum matching selects the symmetric point, yielding $\tilde{G}\tilde{\Lambda }={\alpha }_{\Lambda }^{-1}$ and $\tilde{G}=\tilde{\Lambda }={\alpha }_{\Lambda }^{-1/2}$, where ${\alpha }_{\Lambda }\equiv c^3/(G\hslash \Lambda )$ is the GFSC.

Constant	Base dimensions	$\tilde{\mathit{X}}$ in $\mathbf{\Lambda }$-units	Note
c	$L{T}^{-1}$	1	anchor $L/T=c$
ℏ	$M{L}^2{T}^{-1}$	1	anchor $M{L}^2/T=\hslash$
G	$L^3{M}^{-1}{T}^{-2}$	${\alpha }_{\Lambda }^{-1/2}$	paired with $\tilde{\Lambda }$; product fixed
$\Lambda$	$L^{-2}$	${\alpha }_{\Lambda }^{-1/2}$	symmetric with $\tilde{G}$

• Planck units {G, ℏ, c} [16].

Here $N=3$ constants for 3 bases $(M,L,T)\Rightarrow$ no nullspace. Uniqueness comes for free once we impose the usual identifications $L/T=c$ and $McL=\hslash$.

• Stoney units {G, c, e} (+ EM convention) [71].

Relativity ($L/T=c$), gravity (G), and an electromagnetic normalization (choice of Coulomb constant / rationalization) fix the set. One recovers $c,G$ and a natural charge/action scale (e.g. $e^2/4\pi {\epsilon }_{0}c$ in SI) by construction.

• Λ-units {G, ℏ, c, Λ}.

Now $N=4$ for 3 bases ⇒one free parameter (the $\lambda$ in (A13)). Two standard anchors are kept:

$${\displaystyle \frac{L}{T}}=c,McL=\hslash .$$

(A17)

A single $\Lambda$-specific anchor removes the remaining freedom:

$$\mathit{Vacuum} \mathit{matching}(\mathit{thermo}/\mathit{GR}=\mathit{QFT} \mathit{cutoff}):{\displaystyle \frac{\hslash c}{L^4}}={\displaystyle \frac{\Lambda c^4}{G}}.$$

(A18)

This fixes the length scale uniquely:

$$L_{\Lambda }={\left({\displaystyle \frac{\hslash G}{\Lambda c^3}}\right)}^{1/4},T_{\Lambda }={\displaystyle \frac{L_{\Lambda }}{c}},M_{\Lambda }={\displaystyle \frac{\hslash }{c{L}_{\Lambda }}}={\left({\displaystyle \frac{{\hslash }^3\Lambda }{Gc}}\right)}^{1/4}.$$

(A19)

Thus the $\Lambda$-triple is unique once (A18) is adopted.

Equivalent algebraic symmetry. For length, write $L\propto G^p{\hslash }^q{\Lambda }^r{c}^s$. Solving (A11) for $\left[L\right]$ gives the family

$$p=q=-{\displaystyle \frac{s}{3}},r=p-{\displaystyle \frac{1}{2}}.$$

(A20)

Imposing the simple equal-weight symmetry $p=q=-r$ (“give quantum and gravity equal weight, oppose $\Lambda$ with equal magnitude”) yields

$$p=q=\frac{1}{4},r=-\frac{1}{4},s=-\frac{3}{4},$$

(A21)

i.e. exactly (A19). This symmetry condition is just the exponent form of the vacuum matching (A18).

Appendix A.3. Why most choices do not return c and ℏ

Because $(p,q,r,s)$ live on the affine line (A31), generic choices give

$${\displaystyle \frac{L}{T}}\ne c,M{\displaystyle \frac{L}{T}}L\ne \hslash .$$

The anchors (A17) pick out the subfamily consistent with relativity and quantum kinematics; the $\Lambda$-anchor (A18) then selects one point on that subfamily—giving (A19). This mirrors explicit counterexamples: other valid exponent choices produce a different velocity scale and a different action scale.

Appendix A.4. Temperature and k_B

Historically, Planck [16] introduced a fourth base dimension $\Theta$ and used $k_B$ to tie temperature to energy. Two consistent options:

• With $k_B$ (four bases $M,L,T,\Theta$). Add $k_B:\left[M{L}^2{T}^{-2}{\Theta }^{-1}\right]$. Then a natural $\Lambda$-temperature follows from horizon thermodynamics:

  $${\Theta }_{\Lambda }\sim {\displaystyle \frac{\hslash c}{k_B{L}_{\Lambda }}}={\displaystyle \frac{1}{k_B}}{\left({\displaystyle \frac{{\hslash }^3\Lambda c^7}{G}}\right)}^{1/4}.$$

  (A22)
  Up to order-unity factors, this is the de Sitter/Gibbons–Hawking scale.
• Without $k_B$ (modern HEP convention). Set $k_B=1$ so temperature is measured in energy; $\Theta$ is not an independent base dimension and (A19) suffices.

Appendix A.5. Practical “algorithm” (no trial & error)

• Write the dimensional matrix (A11); solve once to get a particular solution and the null vector $\mathbf{n}=(1,1,1,-3)$.
• Enforce the two standard anchors (A17) so that unit speed is c and unit action is ℏ.
• Impose the single $\Lambda$-anchor (A18) ⇒ fix the free parameter $\lambda$.
• Read off the exponents for $L_{\Lambda },T_{\Lambda },M_{\Lambda }$ as in (A19); if using $k_B$, define ${\Theta }_{\Lambda }$ via (A22).

• Remark on electromagnetism.

${\alpha }_E=e^2/4\pi {\epsilon }_0\hslash c$ is dimensionless and not set by units. One may choose EM conventions (e.g. Heaviside–Lorentz [27,72,73]) so that ${\epsilon }_0$ is absorbed, but the $\Lambda$-unit uniqueness rests entirely on the mechanical anchors plus the $\Lambda$-postulate, not on EM conventions.

Therefore: Admitting $\Lambda$ as a fundamental constant introduces one nullspace degree of freedom in the dimensional algebra; a single, physically motivated vacuum-matching postulate removes it and yields a unique $\Lambda$-scale. This both explains the infinity of formal solutions and why the physically relevant $\Lambda$-triple emerges uniquely.

Table A2. Comparison of natural unit systems. Enough anchors remove nullspace freedom.

System	Constants used	Base dims.	Anchors imposed	Unique outcome
Stoney (1881)	$\{G,c,e\}$ (with ${\epsilon }_0$)	$(M,L,T,Q)$	(i) $L/T=c$; (ii) G; (iii) EM normalisation (Coulomb const.)	Recovers $c,G$ and natural charge/action scale ($e^2/4\pi {\epsilon }_{0}c$ in SI)
Planck (1899)	$\{G,\hslash ,c\}$ (opt. $k_B$)	$(M,L,T)$ or $(M,L,T,\Theta )$	(i) $L/T=c$; (ii) $McL=\hslash$; (opt. iii) $k_B$ for $\Theta$	Unique $(L_P,T_P,M_P)$; with $k_B$, Planck temperature ${\Theta }_P$
$\Lambda$-scale (present)	$\{G,\hslash ,c,\Lambda \}$ (opt. $\Theta$ with $k_B$)	$(M,L,T)$	(i) $L/T=c$; (ii) $McL=\hslash$; (iii) $\hslash c/L^4=\Lambda c^4/G$	Unique $(L_{\Lambda },T_{\Lambda },M_{\Lambda })$; with $k_B$, ${\Theta }_{\Lambda }\sim \hslash c/\left(k_B{L}_{\Lambda }\right)$

Table A2. Comparison of natural unit systems. Enough anchors remove nullspace freedom.

System	Constants used	Base dims.	Anchors imposed	Unique outcome
Stoney (1881)	$\{G,c,e\}$ (with ${\epsilon }_0$)	$(M,L,T,Q)$	(i) $L/T=c$; (ii) G; (iii) EM normalisation (Coulomb const.)	Recovers $c,G$ and natural charge/action scale ($e^2/4\pi {\epsilon }_{0}c$ in SI)
Planck (1899)	$\{G,\hslash ,c\}$ (opt. $k_B$)	$(M,L,T)$ or $(M,L,T,\Theta )$	(i) $L/T=c$; (ii) $McL=\hslash$; (opt. iii) $k_B$ for $\Theta$	Unique $(L_P,T_P,M_P)$; with $k_B$, Planck temperature ${\Theta }_P$
$\Lambda$-scale (present)	$\{G,\hslash ,c,\Lambda \}$ (opt. $\Theta$ with $k_B$)	$(M,L,T)$	(i) $L/T=c$; (ii) $McL=\hslash$; (iii) $\hslash c/L^4=\Lambda c^4/G$	Unique $(L_{\Lambda },T_{\Lambda },M_{\Lambda })$; with $k_B$, ${\Theta }_{\Lambda }\sim \hslash c/\left(k_B{L}_{\Lambda }\right)$

Once $\{G,\hslash ,\Lambda ,c\}$ are admitted, dimensional analysis alone leaves a one-parameter family of monomials. A single, physically motivated postulate—matching the QFT cutoff to the GR vacuum density—selects a unique member: the $\Lambda$-scale. What looks like algebraic freedom is therefore resolved by thermodynamics, which also explains why the physically relevant $\Lambda$-triple emerges uniquely.

Appendix A.6. The Λ Scale as a Geometric Mean

An elegant property of the $\Lambda$ scale is that it lies midway, on a logarithmic scale, between the Planck length $L_P$ and the de Sitter horizon radius $R_{dS}$. This can be shown directly by inspection. Define the familiar quantities:

$$\begin{array}{cc}\hfill L_P& =\sqrt{{\displaystyle \frac{\hslash G}{c^3}}},R_{dS}=\sqrt{{\displaystyle \frac{3}{\Lambda }}},L_{\Lambda }={\left({\displaystyle \frac{\hslash G}{\Lambda c^3}}\right)}^{1/4}.\hfill \end{array}$$

(A23)

Taking the geometric mean of $L_P$ and $R_{dS}$ gives

$$\sqrt{L_P{R}_{dS}}={\left({\displaystyle \frac{\hslash G}{c^3}}·{\displaystyle \frac{3}{\Lambda }}\right)}^{1/4}=3^{1/4}{L}_{\Lambda }.$$

(A24)

Hence,

$$L_{\Lambda }={\displaystyle \frac{1}{3^{1/4}}}\sqrt{{\ell }_P{R}_{dS}}.$$

(A25)

If instead we adopt the common convention $R_{\Lambda }\equiv {\Lambda }^{-1/2}$ (dropping the factor $\sqrt{3}$), then $L_{\Lambda }$ is exactly the geometric mean:

Figure A1. Hierarchy of fundamental length scales. The Planck length $L_P$ anchors the UV, the de Sitter horizon $R_{\mathrm{dS}}$ anchors the IR, and the emergent $\Lambda$–scale $L_{\Lambda }$ sits at the geometric mean, $L_{\Lambda }=\sqrt{L_P{R}_{\mathrm{dS}}}$. Numerical values shown here are evaluated using CODATA values, with $\Lambda =1.1\times {10}^{-52}{m}^{-2}$ inferred from late–time cosmology.

Figure A1. Hierarchy of fundamental length scales. The Planck length $L_P$ anchors the UV, the de Sitter horizon $R_{\mathrm{dS}}$ anchors the IR, and the emergent $\Lambda$–scale $L_{\Lambda }$ sits at the geometric mean, $L_{\Lambda }=\sqrt{L_P{R}_{\mathrm{dS}}}$. Numerical values shown here are evaluated using CODATA values, with $\Lambda =1.1\times {10}^{-52}{m}^{-2}$ inferred from late–time cosmology.

Appendix A.7. Dimensionless G and Λ in Λ–units and the role of α_Λ

$$\tilde{X}\equiv {\displaystyle \frac{X}{{\left[X\right]}_{\Lambda }}}.$$

(A26)

The corresponding $\Lambda$–unit dimensions are

$$\begin{array}{cc}\hfill {\left[c\right]}_{\Lambda }& ={\displaystyle \frac{L}{T}},{[\hslash ]}_{\Lambda }=McL,\hfill \end{array}$$

(A27)

$$\begin{array}{cc}\hfill {\left[G\right]}_{\Lambda }& ={\displaystyle \frac{L^3}{M{T}^2}},{[\Lambda ]}_{\Lambda }=L^{-2}.\hfill \end{array}$$

(A28)

For gravity and the cosmological constant, define the dimensionless values $\tilde{G}\equiv G/(L^3/\left(M{T}^2\right))$ and $\tilde{\Lambda }\equiv \Lambda L^2$. Evaluated in the $\lambda =0$ (vacuum–matched) $\Lambda$–units,

$$\tilde{G}=G{\displaystyle \frac{M_{\Lambda }{T}_{\Lambda }^2}{L_{\Lambda }^3}}={\left({\displaystyle \frac{G\hslash \Lambda }{c^3}}\right)}^{1/2}={\alpha }_{\Lambda }^{-1/2},\tilde{\Lambda }=\Lambda L_{\Lambda }^2={\left({\displaystyle \frac{G\hslash \Lambda }{c^3}}\right)}^{1/2}={\alpha }_{\Lambda }^{-1/2},$$

(A29)

so that

$$\begin{array}{|c|}\hline \tilde{G}\tilde{\Lambda }={\alpha }_{\Lambda }^{-1}\\ \hline\end{array}.$$

(A30)

More generally, moving along the null direction by $\lambda$ (so $L\left(\lambda \right)=L_{\Lambda }{\alpha }_{\Lambda }^{-\lambda }$, $T\left(\lambda \right)=T_{\Lambda }{\alpha }_{\Lambda }^{-\lambda }$, $M\left(\lambda \right)=M_{\Lambda }{\alpha }_{\Lambda }^{\lambda }$) gives

$$\tilde{G}\left(\lambda \right)={\alpha }_{\Lambda }^{2\lambda -\frac{1}{2}},\tilde{\Lambda }\left(\lambda \right)={\alpha }_{\Lambda }^{-2\lambda -\frac{1}{2}},\tilde{G}\left(\lambda \right)\tilde{\Lambda }\left(\lambda \right)={\alpha }_{\Lambda }^{-1}.$$

(A31)

The vacuum–matching postulate (A14) fixes the remaining affine freedom by selecting the symmetric point $\lambda =0$. In the resulting vacuum–matched $\Lambda$ units the quarter–power base scales give

$$\tilde{c}=1,\tilde{\hslash }=1,\tilde{G}=\tilde{\Lambda }={\alpha }_{\Lambda }^{-1/2},$$

where ${\alpha }_{\Lambda }\equiv c^3/(G\hslash \Lambda )$ is a dimensionless invariant.Along the null rescaling direction one may shift weight between G and $\Lambda$ while keeping the product fixed:

$$\tilde{G}\left(\lambda \right)\tilde{\Lambda }\left(\lambda \right)={\alpha }_{\Lambda }^{-1}.$$

• Consequences.

• Vacuum matching enforces $\lambda =0$, so the chosen units are the symmetric quarter–power ones, with

  $$\tilde{G}=\tilde{\Lambda }={\alpha }_{\Lambda }^{-1/2},\tilde{c}=1,\tilde{\hslash }=1.$$

  (A32)
• One may re–scale along the null direction (choose $\lambda \ne 0$) to make either $\tilde{G}=1$ (take $\lambda =\frac{1}{4}$) or$\tilde{\Lambda }=1$ (take $\lambda =-\frac{1}{4}$), but not both simultaneously, since the product is invariant:

  $$\tilde{G}\left(\lambda \right)\tilde{\Lambda }\left(\lambda \right)={\alpha }_{\Lambda }^{-1}.$$

  (A33)

This parameter is the gravitational analogue of the electromagnetic fine–structure constant $\alpha$: it is dimensionless and invariant under unit choices. Here ${\alpha }_{\Lambda }$ encodes the simultaneous scaling of both G and $\Lambda$ alongside c and ℏ. In vacuum–matched $\Lambda$–units one has $\tilde{c}=\tilde{\hslash }=1$ and $\tilde{G}=\tilde{\Lambda }={\alpha }_{\Lambda }^{-1/2}$.

Closing summary. With four constants for three base dimensions, admitting $\Lambda$ leaves a one–parameter nullspace freedom in the dimensional algebra. The kinematic anchors $L/T=c$ and $M{L}^2/T=\hslash$ fix $\tilde{c}=\tilde{\hslash }=1$, and the vacuum–matching postulate

$${\displaystyle \frac{\hslash c}{L^4}}={\displaystyle \frac{\Lambda c^4}{G}}$$

(A32)

removes the remaining freedom, selecting the quarter-power length $L_{\Lambda }={\left(\hslash G/(\Lambda c^3)\right)}^{1/4}$ in (A32) and thereby determining the full $\Lambda$ unit set. In these units the dimensionless gravitational and cosmological constants share a single parameter,

$$\tilde{G}=\tilde{\Lambda }={\alpha }_{\Lambda }^{-1/2},{\alpha }_{\Lambda }\equiv {\displaystyle \frac{c^3}{G\hslash \Lambda }}$$

(A33)

so $\tilde{G}\tilde{\Lambda }={\alpha }_{\Lambda }^{-1}$ is the concise condition for G and $\Lambda$ to join c and ℏ in the $\Lambda$ system.

EM note. The electromagnetic fine-structure constant $\alpha =e^2/(4\pi {\epsilon }_0\hslash c)$ is dimensionless and invariant under unit choices; the uniqueness of the $\Lambda$ units arises from the mechanical anchors plus the vacuum-matching postulate (A32), independent of the EM sector.