Intrinsic Quantum Geometry and the Emergence of General Relativity Gravitation

1. Introduction

The reconciliation of quantum mechanics with general relativity remains a central challenge. Most approaches attempt to quantize a pre-existing classical spacetime. Recent developments in emergent gravity [9,10,11,12] suggest that spacetime itself may arise from more fundamental quantum degrees of freedom, with quantum entanglement playing a crucial role [9]. The world-block model, introduced in our previous work [1], takes a complementary perspective: each quantum particle possesses its own private 3-dimensional spatial geometry, evolving in proper time. The familiar 4-dimensional spacetime of general relativity emerges from the collective dynamics and interactions of many such world-blocks.

1.1. Summary of the Original World-Block Model

In its original non-relativistic formulation, the world-block model [1], was built on the following foundational ideas:

1.

Private spatial geometry: Each particle possesses its own 3-dimensional space, described by a metric $h_{ij}={\delta }_{ij}+{\epsilon }_{ij}$ on a constant-proper-time slice. The strain tensor ${\epsilon }_{ij}$ represents the deviation from flatness caused by the particle’s presence.

2.

World-Block of a particle: It is the four-dimensional manifold covered by the Fermi–Walker coordinates $(\tau ,\overrightarrow{x})$ of the particle. Each constant-$\tau$ slice is defined by the dynamical spatial metric $h_{ij}={\delta }_{ij}+{\epsilon }_{ij}$.

3.

Geometric stiffness: The geometry resists deformation with a universal stiffness constant $A_0$ (dimensions of energy per length), which is the same for all particles. This is analogous to an elastic modulus for spacetime itself.

4.

Wavefunction as source: The particle’s quantum state is described by a wavefunction $\psi \left(\overrightarrow{x}\right)$ on the slice, and its modulus squared ${\left|\psi \right|}^2$ is interpreted as the mass density $\rho (\tau ,\overrightarrow{x})$ that sources the strain field.

5.

Action principle: The dynamics on each slice is governed by the action:

$$S_{3D}=\int d\tau d^{3}x\left[{\displaystyle \frac{A_0}{2}}{\partial }_k{\epsilon }_{ij}{\partial }^k{\epsilon }^{ij}+{\displaystyle \frac{i\hslash }{2}}\left({\psi }^{*}{\partial }_{\tau }\psi -\psi {\partial }_{\tau }{\psi }^{*}\right)-{\displaystyle \frac{{\hslash }^2}{2m}}{\delta }^{ij}{\partial }_i{\psi }^{*}{\partial }_j\psi -{\displaystyle \frac{m{c}^2}{2}}\epsilon {\left|\psi \right|}^2\right],$$

(1)

where $\epsilon ={\delta }^{ij}{\epsilon }_{ij}$ is the trace of the strain tensor. The first term represents the elastic energy of the deformed geometry, the second is the standard Schrödinger time-derivative term, the third is the quantum kinetic energy, and the last is the coupling between the geometry (via its trace) and the mass density ${\left|\psi \right|}^2$.

From this action, the field equations were derived and solved, yielding several striking results:

• Repulsive self-gravity: A single world-block experiences a repulsive gravitational self-force. This is a robust prediction independent of any sign choices in the coupling, and it prevents the gravitational collapse of isolated quantum systems.
• Attractive mutual gravity: When two world-blocks overlap, stitching conditions enforce metric continuity, producing an attractive Newtonian interaction $U_{AB}=-G{m}_A{m}_B/R$ between distinct particles.
• Geometric interpretation of quantum phenomena: The model preserves interference, provides a geometric account of entanglement (through the stitching of blocks), and eliminates the need for a high-dimensional configuration space—each particle’s wavefunction lives on its own 3-space, and interactions are handled by stitching these spaces together.

The present work elevates this framework to a fully covariant, mathematically rigorous model through a three-phase program: (i) covariant reformulation of the 3-action, identifying the 3-metric $h_{ij}$ and extrinsic curvature $K_{ij}$; (ii) stitching of multiple blocks via compatibility conditions, forming a global 3-manifold with emergent lapse N and shift $N^i$; (iii) ADM reconstruction demonstrating that the collective dynamics obey the Einstein equations, with Newton’s constant $G=c^4/\left(8\pi A_0\right)$. Applying the same ideas to the vacuum—interpreted as a stitched compound of virtual world-blocks—naturally produces a constant harmonic field ${\mathrm{\Phi }}_H$ whose magnitude is set by the Hubble radius, yielding a vacuum energy density ${\rho }_{\mathrm{vac}}\sim 3c^2{H}^2/\left(8\pi G\right)$ and explaining the observed dark energy without fine-tuning.

1.2. Related Work

The emergence of spacetime from quantum entanglement has been explored extensively in holography and AdS/CFT [9,13]. Jacobson’s thermodynamic derivation of Einstein equations [10] and Verlinde’s entropic gravity [11] provide complementary perspectives. Padmanabhan [12] has argued that spacetime emerges from the degrees of freedom associated with null surfaces. More recent work by Oppenheim [5] and Strubbe [2] explores fundamentally classical frameworks for quantum gravity, while Carney et al. [24,25] review tabletop tests of quantum gravity. Experimental progress in massive matter interferometry [28,29] and proposals for testing the quantum nature of gravity [26,27] provide potential avenues for probing the world-block predictions. The cosmological constant problem has been approached from various angles, including Padmanabhan’s fluctuation-based model [20], Volovik’s condensed matter analogy [21], and recent work by Balázs [22] and Faraggi & Matone [23] connecting $\mathrm{\Lambda }$ to the Hubble scale. The world-block model unifies these ideas within a concrete geometric mechanism.

2. Phase 1: Geometric Reformulation of the 3-Action

The original world-block action (1) is written in terms of a fixed flat background metric ${\delta }_{ij}$ and partial derivatives ${\partial }_i$, which is not suitable for a coordinate-independent description. To prepare for stitching multiple blocks and for eventual comparison with general relativity, we must rewrite it in a form that is manifestly covariant under 3-dimensional coordinate transformations. This section provides a detailed derivation of the covariant action.

2.1. Step 1: Promoting the Metric to a Dynamical Field

We elevate the flat metric to a dynamical 3-metric $h_{ij}(\tau ,\overrightarrow{x})$ on each slice. The strain tensor is then a derived quantity, but for a covariant formulation we work directly with $h_{ij}$. The relation $h_{ij}={\delta }_{ij}+{\epsilon }_{ij}$ is only valid in a specific coordinate system (Fermi coordinates of the block). In a general coordinate frame, ${\epsilon }_{ij}$ is not a tensor; instead, $h_{ij}$ itself is the fundamental geometric object.

The measure for integration over the slice becomes $\sqrt{h}{d}^{3}x$, where $h=det{h}_{ij}$. All spatial derivatives must be replaced by covariant derivatives ${\nabla }_i$ compatible with $h_{ij}$.

2.2. Step 2: Covariant Form of the Geometric Term

The original geometric term ${\displaystyle \frac{A_0}{2}}{\partial }_k{\epsilon }_{ij}{\partial }^k{\epsilon }^{ij}$ is quadratic in first derivatives of the metric perturbation. In a fully covariant theory, the natural scalar constructed from first derivatives of $h_{ij}$ is the 3-dimensional Ricci scalar $R^{\left(3\right)}$. Indeed, in the weak-field limit $h_{ij}={\delta }_{ij}+{\epsilon }_{ij}$, one finds (see Appendix A for a detailed expansion) that

$$R^{\left(3\right)}={\partial }_k{\epsilon }_{ij}{\partial }^k{\epsilon }^{ij}-{\partial }_i\epsilon {\partial }^i\epsilon +\mathrm{total}\mathrm{derivatives}+O\left({\epsilon }^3\right).$$

(2)

The original action retained only the first term, which corresponds to a specific gauge choice or to a linearized approximation where the trace terms are neglected. Since our goal is a covariant formulation, we adopt the full $R^{\left(3\right)}$ as the geometric Lagrangian density. This choice is natural because $R^{\left(3\right)}$ is the unique scalar invariant quadratic in first derivatives of the metric (up to total derivatives) and appears in the Einstein–Hilbert action. Moreover, as we shall see, it will later combine with extrinsic curvature terms to yield the 4-dimensional Ricci scalar. Thus, we set the geometric term:

$${\mathcal{L}}_{\mathrm{ge}}={\displaystyle \frac{A_0}{2}}\sqrt{h}{R}^{\left(3\right)}.$$

(3)

The factor $A_0/2$ is retained from the original action; its physical significance will become clear when we identify Newton’s constant. The geometric term carries the same coefficient $A_0/2$, but now involves the full $R^{\left(3\right)}$ instead of just ${(\partial \epsilon )}^2$. In the weak-field limit, $R^{\left(3\right)}$ contains additional terms that are total derivatives or vanish in suitable gauges; thus the original action can be seen as a gauge-fixed version of the covariant one.

2.3. Step 3: Covariant Matter Kinetic Terms

The original kinetic term for $\psi$ is $-{\displaystyle \frac{{\hslash }^2}{2m}}{\delta }^{ij}{\partial }_i{\psi }^{*}{\partial }_j\psi$. To make it covariant, we replace ${\delta }^{ij}$ with the inverse metric $h^{ij}$ and include the volume element $\sqrt{h}$:

$${\mathcal{L}}_{\mathrm{kin}}=-{\displaystyle \frac{{\hslash }^2}{2m}}\sqrt{h}{h}^{ij}{\partial }_i{\psi }^{*}{\partial }_j\psi .$$

(4)

The time-derivative term ${\displaystyle \frac{i\hslash }{2}}\left({\psi }^{*}{\partial }_{\tau }\psi -\psi {\partial }_{\tau }{\psi }^{*}\right)$ is already coordinate-invariant under purely spatial transformations, provided we treat $\tau$ as an external parameter. In the full 4-dimensional picture, it will be covariantized by replacing ${\partial }_{\tau }$ with a Lie derivative along the evolution vector field, but for a single slice we keep it as is. The kinetic term for $\psi$ has the same coefficient ${\hslash }^2/\left(2m\right)$, but with $h^{ij}$ replacing ${\delta }^{ij}$ and a $\sqrt{h}$ measure.

2.4. Step 4: Covariant Coupling Term

The original coupling term is $-{\displaystyle \frac{m{c}^2}{2}}\epsilon {\left|\psi \right|}^2$, where $\epsilon ={\delta }^{ij}{\epsilon }_{ij}$ is the trace of the strain. In a covariant setting, we need a scalar built from $h_{ij}$ that reduces to $\epsilon$ in the weak-field limit. The natural candidate is $ln\sqrt{h}$, because for $h_{ij}={\delta }_{ij}+{\epsilon }_{ij}$,

$$ln\sqrt{h}={\displaystyle \frac{1}{2}}lndeth={\displaystyle \frac{1}{2}}ln(1+\epsilon +O\left({\epsilon }^2\right))={\displaystyle \frac{\epsilon }{2}}+O\left({\epsilon }^2\right).$$

(5)

Thus $\epsilon \approx 2ln\sqrt{h}$. Substituting into the original coupling gives $-{\displaystyle \frac{m{c}^2}{2}}(2ln\sqrt{h}){\left|\psi \right|}^2=-m{c}^{2}ln\sqrt{h}{\left|\psi \right|}^2$. Therefore the covariant coupling term should be

$${\mathcal{L}}_{\mathrm{couple}}=-m{c}^2\sqrt{h}(ln\sqrt{h}){\left|\psi \right|}^2,$$

(6)

where the extra factor $\sqrt{h}$ comes from the measure. In the weak-field limit, $\sqrt{h}\approx 1+\epsilon /2$, so to leading order ${\mathcal{L}}_{\mathrm{couple}}\approx -m{c}^2(\epsilon /2){\left|\psi \right|}^2$. Thus, the covariant expression reduces to the original in the linearized regime. Thus, the coupling term now reads $-m{c}^2(ln\sqrt{h}){\left|\psi \right|}^2$ instead of $-{\displaystyle \frac{m{c}^2}{2}}\epsilon {\left|\psi \right|}^2$. In the linearized approximation, $ln\sqrt{h}\approx \epsilon /2$, so the two expressions coincide.

2.5. Step 5: Assembling the Covariant Slice Action

Collecting all terms, the covariant action for a single world-block on a slice at proper time $\tau$ is

$$\begin{array}{cc}\hfill S_{\mathrm{slice}}=\int d\tau d^{3}x\sqrt{h}[& {\displaystyle \frac{A_0}{2}}{R}^{\left(3\right)}+{\displaystyle \frac{i\hslash }{2}}\left({\psi }^{*}{\partial }_{\tau }\psi -\psi {\partial }_{\tau }{\psi }^{*}\right)\hfill \\ \hfill & -{\displaystyle \frac{{\hslash }^2}{2m}}{h}^{ij}{\partial }_i{\psi }^{*}{\partial }_j\psi -m{c}^2(ln\sqrt{h}){\left|\psi \right|}^2].\hfill \end{array}$$

(7)

All terms are now manifestly covariant under 3-dimensional coordinate transformations. The field variables are the 3-metric $h_{ij}(\tau ,\overrightarrow{x})$ and the complex scalar field $\psi (\tau ,\overrightarrow{x})$. The proper time $\tau$ labels the slices and will later become a coordinate in the emergent 4-dimensional spacetime.

The action (7) is the starting point for Phase 2, where we will consider multiple such blocks and define their stitching.

2.6. Appendix: Linearization of $R^{\left(3\right)}$

For completeness, we sketch the expansion of $R^{\left(3\right)}$ around flat space. Write $h_{ij}={\delta }_{ij}+{\epsilon }_{ij}$. The Christoffel symbols are ${\Gamma }_{ij}^k={\displaystyle \frac{1}{2}}({\partial }_i{\epsilon }_{jk}+{\partial }_j{\epsilon }_{ik}-{\partial }_k{\epsilon }_{ij})+O\left({\epsilon }^2\right)$. The Ricci tensor to first order is $R_{ij}={\partial }_k{\Gamma }_{ij}^k-{\partial }_i{\Gamma }_{kj}^k+O\left({\epsilon }^2\right)$. After a straightforward calculation, one obtains

$$R^{\left(3\right)}={\partial }_k{\epsilon }_{ij}{\partial }^k{\epsilon }^{ij}-{\partial }_i\epsilon {\partial }^i\epsilon +\mathrm{total}\mathrm{derivatives}+O\left({\epsilon }^3\right),$$

(8)

where indices are raised with ${\delta }^{ij}$. The total derivatives integrate to zero in the action, so up to boundary terms, $R^{\left(3\right)}$ is equivalent to ${\partial }_k{\epsilon }_{ij}{\partial }^k{\epsilon }^{ij}$ only when $\epsilon$ is traceless or when gauge conditions eliminate the trace term. This explains why the original action could omit the trace term – it corresponds to a particular gauge choice (e.g., harmonic gauge for the spatial metric). In our covariant formulation, we keep the full $R^{\left(3\right)}$ to ensure coordinate independence.

Thus, the action (7) is a natural covariant generalization of the original world-block action.

3. Phase 2: Stitching as Mathematical Gluing

With the covariant action for a single world-block established in Phase 1, we now address the central question of how multiple blocks combine to form a continuous spacetime. Phase 2 develops a rigorous mathematical framework for stitching, in which each block is treated as a chart on an emerging manifold. The stitching conditions enforce compatibility of the geometry and matter fields in overlap regions, and the lapse and shift – essential ingredients of the ADM formalism – emerge naturally from the distribution and motion of blocks.

3.1. The Need for a Precise Stitching Formalism

In the original world-block model, the notion of stitching was heuristic: when two particles interact, their private spacetimes were said to "overlap" and the metric was assumed to be continuous across the overlap. For a rigorous derivation of 4-dimensional gravity, we must replace this intuitive picture with a precise mathematical construction. The blocks must be treated as local charts on a manifold, with well-defined transition functions and compatibility conditions.

3.2. Definition of a World-Block as a Chart

Let ${\Sigma }_{\tau }$ denote the global 3-dimensional space at a given proper time $\tau$. We cover ${\Sigma }_{\tau }$ by a collection of open sets $\left\{U_n\right\}$, each associated with a world-block $B_n$. For each block we have:

• A coordinate chart ${\varphi }_n:U_n\to {\mathbb{R}}^3$, assigning coordinates ${\overrightarrow{x}}_n$ to points in $U_n$.
• A 3-metric $h_{ij}^{\left(n\right)}\left({\overrightarrow{x}}_n\right)$ on $U_n$, expressed in the coordinates of chart ${\varphi }_n$.
• A scalar field ${\psi }^{\left(n\right)}\left({\overrightarrow{x}}_n\right)$ (the wavefunction) on $U_n$.
• A proper-time label ${\tau }_n$, which we will eventually identify with a global time coordinate after stitching.

The collection $\left\{(U_n,{\varphi }_n,h_{ij}^{\left(n\right)},{\psi }^{\left(n\right)})\right\}$ forms an atlas for ${\Sigma }_{\tau }$, provided the usual conditions of a topological manifold are satisfied (Hausdorff, second countable, etc.). The stitching process is precisely the construction of this atlas from individual blocks.

3.3. Overlap Regions and Compatibility Conditions

When two blocks $B_a$ and $B_b$ overlap, their domains satisfy $U_a\cap U_b\ne \varnothing$. In the overlap, the physical geometry and matter distribution must be unique – they cannot depend on which block’s coordinates we use. This leads to two fundamental compatibility conditions:

$$\begin{array}{cc}\hfill h_{ij}^{\left(a\right)}\left({\overrightarrow{x}}_a\right)& ={\displaystyle \frac{\partial x_a^k}{\partial x_b^i}}{\displaystyle \frac{\partial x_a^l}{\partial x_b^j}}{h}_{kl}^{\left(b\right)}\left({\overrightarrow{x}}_b\right)\mathrm{for}\mathrm{all}{\overrightarrow{x}}_a={\varphi }_a\left(p\right),{\overrightarrow{x}}_b={\varphi }_b\left(p\right),p\in U_a\cap U_b,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\psi }^{\left(a\right)}\left({\overrightarrow{x}}_a\right)& ={\psi }^{\left(b\right)}\left({\overrightarrow{x}}_b\right)\mathrm{for}\mathrm{all}{\overrightarrow{x}}_a={\varphi }_a\left(p\right),{\overrightarrow{x}}_b={\varphi }_b\left(p\right),p\in U_a\cap U_b.\hfill \end{array}$$

(10)

Equation (9) is the standard transformation law for a tensor under coordinate changes; it guarantees that the metric is well-defined on the overlap. Equation () requires the wavefunction to be a scalar field – its value at a physical point is independent of the chart used to represent it.

These conditions are not additional assumptions; they are the minimal requirements for a consistent definition of fields on a manifold. The stitching process is therefore equivalent to verifying that the collection of blocks satisfies these compatibility conditions.

3.4. Transition Functions and the Atlas

On an overlap $U_a\cap U_b$, the transition map ${\mathrm{\Phi }}_{ba}={\varphi }_b\circ {\varphi }_a^{-1}:{\varphi }_a(U_a\cap U_b)\to {\varphi }_b(U_a\cap U_b)$ must be a diffeomorphism. The compatibility condition (9) then becomes

$$h_{ij}^{\left(a\right)}\left(\overrightarrow{x}\right)={\displaystyle \frac{\partial {\mathrm{\Phi }}_{ba}^k}{\partial x^i}}{\displaystyle \frac{\partial {\mathrm{\Phi }}_{ba}^l}{\partial x^j}}{h}_{kl}^{\left(b\right)}\left({\mathrm{\Phi }}_{ba}\left(\overrightarrow{x}\right)\right),$$

(11)

which is precisely the requirement that the metric transforms as a tensor under the transition map. Thus, the existence of a global metric on ${\Sigma }_{\tau }$ is equivalent to the condition that all such transformations hold.

If the collection $\left\{(U_n,{\varphi }_n,h_{ij}^{\left(n\right)},{\psi }^{\left(n\right)})\right\}$ satisfies these compatibility conditions on every overlap, then it defines a unique global 3-metric $h_{ij}$ and global scalar field $\psi$ on ${\Sigma }_{\tau }$. This is the mathematical essence of stitching.

3.5. Temporal Stitching and the Emergence of Lapse and Shift

So far we have considered a single slice ${\Sigma }_{\tau }$. A 4-dimensional spacetime requires a foliation of such slices, labelled by $\tau$. The data relating successive slices are not contained in the spatial compatibility conditions alone; they must be supplied by the dynamics.

In the ADM formalism, the relation between the metric on a slice at $\tau$ and the next slice at $\tau +d\tau$ is encoded in the lapse function $N(\tau ,\overrightarrow{x})$ and shift vector $N^i(\tau ,\overrightarrow{x})$. The 4-metric takes the form

$$d{s}^2=-N^{2}d{\tau }^2+h_{ij}(d{x}^i+N^{i}d\tau )(d{x}^j+N^{j}d\tau ).$$

(12)

The lapse measures the rate of flow of proper time, and the shift encodes how spatial coordinates drift between slices.

In the world-block picture, these quantities must emerge from the distribution and motion of blocks. Consider a region of space containing many blocks at a given $\tau$. As $\tau$ advances, each block evolves according to its own internal dynamics. The collective behavior determines how the global metric changes. We propose the following coarse-grained definitions:

• The lapse $N(\tau ,\overrightarrow{x})$ is inversely related to the density of blocks. Intuitively, where blocks are densely packed, proper time advances more slowly (stronger gravity). In the continuum limit, we define

  $$N(\tau ,\overrightarrow{x})={\displaystyle \frac{{\rho }_0}{{\rho }_b(\tau ,\overrightarrow{x})}},$$

  (13)

  where ${\rho }_b$ is the block density and ${\rho }_0$ a reference density. This ensures that N is a smooth function that can vary across space.
• The shift $N^i(\tau ,\overrightarrow{x})$ arises from the average velocity of blocks relative to the coordinate grid. If blocks have velocities ${\overrightarrow{v}}_b$, we define

  $$N^i(\tau ,\overrightarrow{x})=\langle v^i\rangle (\tau ,\overrightarrow{x}),$$

  (14)

  where the average is taken over a small volume around $\overrightarrow{x}$.

These definitions are provisional; a full derivation of N and $N^i$ from the block dynamics would require solving the collective equations of motion. However, for the purpose of establishing the emergent 4-dimensional picture, it suffices to note that such quantities exist and are smooth on the stitched manifold.

3.6. Variational Principle for the Stitched System

We now consider the total action for a collection of blocks. For each block $B_n$, the action is given by (7) integrated over its proper time ${\tau }_n$ and spatial domain $U_n$:

$$S_n=\int d{\tau }_n{\int }_{U_n}{d}^{3}x\sqrt{h^{\left(n\right)}}{\mathcal{L}}_{\mathrm{slice}}^{\left(n\right)},$$

(15)

with ${\mathcal{L}}_{\mathrm{slice}}^{\left(n\right)}$ defined as in (7). The total action is

$$S_{\mathrm{total}}=\sum _n{S}_n.$$

(16)

If the blocks overlap, the variables $h_{ij}^{\left(n\right)}$ and ${\psi }^{\left(n\right)}$ are not independent; they must satisfy the compatibility conditions (9) and () on every overlap. These conditions can be enforced in the variational principle in two equivalent ways:

1.

Explicit constraints: Introduce Lagrange multipliers on each overlap to enforce the equalities. This leads to a constrained variational problem whose solution automatically satisfies the stitching conditions.

2.

Restricted variation: Restrict the space of field configurations to those that are already continuous across overlaps. This means we consider only those collections of fields that define a global $h_{ij}$ and $\psi$ on ${\Sigma }_{\tau }$. The variation is then performed within this restricted space.

Both approaches are mathematically equivalent. In practice, it is simpler to work directly with the global fields $h_{ij}(\tau ,\overrightarrow{x})$ and $\psi (\tau ,\overrightarrow{x})$ defined on the stitched manifold, and to postulate that their dynamics are governed by the sum of the block actions in the continuum limit.

3.7. Continuum Limit and the Global Action

In the limit where the blocks are infinitesimally small and densely packed, the sum over blocks becomes an integral over the global slice. The proper time labels ${\tau }_n$ must also be identified with a global time coordinate $\tau$; this identification is possible precisely when the blocks are stitched consistently. The total action then takes the form

$$S_{\mathrm{global}}=\int d\tau {\int }_{{\Sigma }_{\tau }}{d}^{3}x\sqrt{h}{\mathcal{L}}_{\mathrm{slice}},$$

(17)

where ${\mathcal{L}}_{\mathrm{slice}}$ is the same Lagrangian density as in (7), but now interpreted as a function of the global fields $h_{ij}(\tau ,\overrightarrow{x})$ and $\psi (\tau ,\overrightarrow{x})$.

This global action is the starting point for Phase 3, where we will show that it is equivalent to the ADM form of general relativity.

The stitched manifold ${\Sigma }_{\tau }$ is now a well-defined 3-dimensional Riemannian space for each $\tau$, equipped with a global metric $h_{ij}$ and scalar field $\psi$. The stage is set for Phase 3, where we will derive the dynamics of these global fields and show that they are governed by the Einstein equations.

3.8. Appendix: Mathematical Consistency of Stitching

For the construction to be rigorous, the collection of blocks must satisfy the usual properties of a manifold atlas:

• The union of all $U_n$ covers ${\Sigma }_{\tau }$.
• On every overlap $U_a\cap U_b$, the transition map ${\varphi }_b\circ {\varphi }_a^{-1}$ is smooth.
• The compatibility conditions (9) and () hold on all overlaps.
• The cocycle condition is satisfied on triple overlaps: ${\varphi }_c\circ {\varphi }_b^{-1}\circ {\varphi }_b\circ {\varphi }_a^{-1}={\varphi }_c\circ {\varphi }_a^{-1}$.

If these conditions are met, then ${\Sigma }_{\tau }$ is a smooth manifold with a well-defined metric and scalar field. The stitching process is therefore mathematically equivalent to the construction of a manifold from an atlas.

In the world-block model, these conditions are not imposed arbitrarily; they follow from the requirement that overlapping blocks describe the same physical region. The existence of such an atlas is a non-trivial condition on the dynamics, and verifying it for a given collection of blocks is part of solving the equations of motion.

4. Phase 3: Emergent 4-Dynamics and ADM Reconstruction

With the global 3-metric $h_{ij}(\tau ,\overrightarrow{x})$ and scalar field $\psi (\tau ,\overrightarrow{x})$ defined on each slice ${\Sigma }_{\tau }$ via the stitching construction of Phase 2, we now turn to the dynamics that relate different slices. The goal is to show that the collective behavior of the stitched blocks leads to an effective 4-dimensional action that is exactly the Einstein–Hilbert action in its 3+1 (ADM) form, and that the resulting field equations are the full Einstein equations coupled to a scalar field. This phase provides the crucial link between the microscopic world-block dynamics and macroscopic general relativity.

4.1. Recap of the Global Action on a Slice

From Phase 2, the total action for the stitched system, in the continuum limit where blocks are infinitesimally small and densely packed, reduces to an integral over the global slice ${\Sigma }_{\tau }$:

$$S_{\mathrm{global}}=\int d\tau {\int }_{{\Sigma }_{\tau }}{d}^{3}x\sqrt{h}{\mathcal{L}}_{\mathrm{slice}},$$

(18)

with the Lagrangian density ${\mathcal{L}}_{\mathrm{slice}}$ given by the covariant expression derived in Phase 1:

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{slice}}=& {\displaystyle \frac{A_0}{2}}{R}^{\left(3\right)}+{\displaystyle \frac{i\hslash }{2}}\left({\psi }^{*}{\partial }_{\tau }\psi -\psi {\partial }_{\tau }{\psi }^{*}\right)\hfill \\ \hfill & -{\displaystyle \frac{{\hslash }^2}{2m}}{h}^{ij}{\partial }_i{\psi }^{*}{\partial }_j\psi -m{c}^2(ln\sqrt{h}){\left|\psi \right|}^2.\hfill \end{array}$$

(19)

Here $R^{\left(3\right)}$ is the Ricci scalar of the 3-metric $h_{ij}$, and all quantities are now understood as global fields on the stitched manifold. The proper time $\tau$ labels the slices and will become one of the coordinates of the emergent 4-dimensional spacetime.

4.2. The Need for a Temporal Connection

Equation (18) describes the dynamics on each individual slice but does not relate the geometry on different slices. To reconstruct a 4-dimensional spacetime, we must specify how the metric $h_{ij}$ evolves from one slice to the next. In the ADM formalism, this evolution is encoded in the extrinsic curvature $K_{ij}$ and in two auxiliary fields: the lapse function $N(\tau ,\overrightarrow{x})$ and the shift vector $N^i(\tau ,\overrightarrow{x})$. The 4-metric is then expressed as

$$d{s}^2=-N^{2}d{\tau }^2+h_{ij}(d{x}^i+N^{i}d\tau )(d{x}^j+N^{j}d\tau ).$$

(20)

The lapse measures the rate of flow of proper time normal to the slices, while the shift describes how the spatial coordinates drift as one moves from one slice to the next.

In the world-block picture, N and $N^i$ are not fundamental fields; they emerge from the distribution and motion of blocks. As argued in Phase 2, they can be defined as coarse-grained quantities:

$$\begin{array}{cc}\hfill N(\tau ,\overrightarrow{x})& ={\displaystyle \frac{{\rho }_0}{{\rho }_b(\tau ,\overrightarrow{x})}},\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill N^i(\tau ,\overrightarrow{x})& =\langle v^i\rangle (\tau ,\overrightarrow{x}),\hfill \end{array}$$

(22)

where ${\rho }_b$ is the local block density, ${\rho }_0$ a reference density, and $\langle v^i\rangle$ the average velocity of blocks in a small neighbourhood. These definitions guarantee that N and $N^i$ are smooth functions on the stitched manifold.

4.3. Extrinsic Curvature and its Relation to ${\partial }_{\tau }{h}_{ij}$

The extrinsic curvature $K_{ij}$ describes how the 3-metric changes as we move along the normal direction to the slices. It is defined by

$$K_{ij}=-{\displaystyle \frac{1}{2}}{\mathcal{L}}_{\mathbf{n}}{h}_{ij},$$

(23)

where $\mathbf{n}$ is the future-pointing unit normal to the slice. In terms of the lapse and shift, this becomes the well-known relation

$${\partial }_{\tau }{h}_{ij}=-2N{K}_{ij}+{\nabla }_i{N}_j+{\nabla }_j{N}_i,$$

(24)

with ${\nabla }_i$ the covariant derivative associated with $h_{ij}$. This relation is purely kinematical; it follows from the definition of the 4-metric (20) and does not involve any dynamics.

4.4. The 4-Dimensional Ricci Scalar in 3+1 Form

A fundamental result of the ADM formalism is the decomposition of the 4-dimensional Ricci scalar $R^{\left(4\right)}$ in terms of 3-dimensional quantities:

$$R^{\left(4\right)}=R^{\left(3\right)}+K_{ij}{K}^{ij}-K^2,$$

(25)

where $K=h^{ij}{K}_{ij}$ is the trace of the extrinsic curvature. This identity holds for any foliation of a 4-dimensional Lorentzian manifold. Therefore, the Einstein–Hilbert action

$$S_{\mathrm{EH}}={\displaystyle \frac{1}{16\pi G}}\int d^{4}x\sqrt{-g}{R}^{\left(4\right)}$$

(26)

can be written in 3+1 form as

$$S_{\mathrm{ADM}}={\displaystyle \frac{1}{16\pi G}}\int d\tau d^{3}x\sqrt{h}N\left(R^{\left(3\right)}+K_{ij}{K}^{ij}-K^2\right).$$

(27)

4.5. Identifying the world-block action with the ADM action

Our global action (18) already contains the term ${\displaystyle \frac{A_0}{2}}\sqrt{h}{R}^{\left(3\right)}$ integrated over $d\tau d^{3}x$. To obtain the full ADM action, we must:

1.

Replace the measure $d\tau$ with $Nd\tau$ to account for the proper time elapsed between slices. This is natural because $Nd\tau$ is the increment of proper time normal to the slices.

2.

Add the term ${\displaystyle \frac{A_0}{2}}\sqrt{h}N(K_{ij}{K}^{ij}-K^2)$ to complete the expression for $R^{\left(4\right)}$.

3.

Covariantize the time derivative of $\psi$ by replacing ${\partial }_{\tau }$ with the Lie derivative along the normal direction, i.e., ${\displaystyle \frac{1}{N}}({\partial }_{\tau }-{\mathcal{L}}_{\overrightarrow{N}})$.

The second point is the most subtle. The term involving $K_{ij}{K}^{ij}-K^2$ is not present in the original slice action; it must arise from the collective dynamics of the blocks as they evolve in time. In the world-block picture, this term can be understood as follows: when many blocks are stacked in the time direction, the relative motion of their centroids generates an effective energy that contributes to the action. A detailed coarse-graining of the block dynamics would show that the sum over blocks produces exactly the quadratic expression in ${\partial }_{\tau }{h}_{ij}$ that, when expressed in terms of $K_{ij}$ using (24), yields ${\displaystyle \frac{A_0}{2}}\sqrt{h}N(K_{ij}{K}^{ij}-K^2)$. While a full derivation of this term from first principles is beyond the scope of the present paper, its necessity is clear from the requirement that the stitched geometry satisfy the Einstein equations. We therefore take it as an emergent feature of the stitching process.

With this understanding, the complete 4-dimensional action for the stitched world-blocks is

$$\begin{array}{cc}\hfill S_{4\mathrm{D}}=\int d\tau d^{3}x\sqrt{h}N[& {\displaystyle \frac{A_0}{2}}\left(R^{\left(3\right)}+K_{ij}{K}^{ij}-K^2\right)\hfill \\ \hfill & +{\displaystyle \frac{i\hslash }{2}}\left({\psi }^{*}{\displaystyle \frac{1}{N}}({\partial }_{\tau }-{\mathcal{L}}_{\overrightarrow{N}})\psi -\mathrm{c}.\mathrm{c}.\right)\hfill \\ \hfill & -{\displaystyle \frac{{\hslash }^2}{2m}}{h}^{ij}{\partial }_i{\psi }^{*}{\partial }_j\psi -m{c}^2(ln\sqrt{h}){\left|\psi \right|}^2].\hfill \end{array}$$

(28)

Comparing with the ADM action (27), we see that (28) is exactly the ADM form of general relativity coupled to a scalar field, provided we make the identification

$${\displaystyle \frac{A_0}{2}}={\displaystyle \frac{c^4}{16\pi G}}⟹G={\displaystyle \frac{c^4}{8\pi A_0}}.$$

(29)

This is a central result of the world-block model: Newton’s constant is not a free parameter but is determined by the universal stiffness $A_0$.

4.6. Field Equations

Varying the action (28) with respect to $h_{ij}$, N, $N^i$, and $\psi$ yields the full set of Einstein equations coupled to the scalar field. The variations are standard in the ADM formalism; we summarise the results:

• Variation with respect to N gives the Hamiltonian constraint:

  $$R^{\left(3\right)}-{\displaystyle \frac{1}{N^2}}(E_{ij}{E}^{ij}-E^2)={\displaystyle \frac{16\pi G}{c^4}}{\rho }_H,$$

  (30)

  where $E_{ij}={\displaystyle \frac{1}{2}}({\partial }_{\tau }{h}_{ij}-{\nabla }_i{N}_j-{\nabla }_j{N}_i)$ (so that $K_{ij}=E_{ij}/N$), $E=h^{ij}{E}_{ij}$, and ${\rho }_H$ is the energy density derived from the scalar field.
• Variation with respect to $N^i$ yields the momentum constraint:

  $${\nabla }^j(K_{ij}-h_{ij}K)={\displaystyle \frac{8\pi G}{c^4}}{j}_i,$$

  (31)

  where $j_i$ is the momentum density of the scalar field.
• Variation with respect to $h_{ij}$ gives the evolution equations for the metric:

  $${\partial }_{\tau }(K_{ij}-h_{ij}K)=\dots$$

  (32)
  (the full expression can be found in standard references).
• Variation with respect to $\psi$ gives the covariant Klein-Gordon equation in the curved background:

  $$\left(\square -{\displaystyle \frac{m^2{c}^2}{{\hslash }^2}}\right)\psi =0,$$

  (33)

  with □ the d’Alembertian of the 4-metric.

Together, these equations are exactly the Einstein field equations for a 4-dimensional spacetime with metric (20) and matter content described by the scalar field $\psi$.

4.7. Interpretation and Significance

The derivation above shows that the stitched world-blocks collectively behave as a single 4-dimensional spacetime satisfying general relativity. The key points are:

• The universal stiffness $A_0$ determines Newton’s constant via $G=c^4/\left(8\pi A_0\right)$. This is a predictive relation; once $A_0$ is fixed, G is no longer a free parameter.
• The lapse and shift, which are essential for the 4-dimensional description, emerge from the distribution and motion of blocks, giving a concrete physical interpretation to these auxiliary fields.
• The extrinsic curvature term $K_{ij}{K}^{ij}-K^2$ arises from the collective dynamics of blocks in time, providing a natural origin for the kinetic energy of the gravitational field.
• All matter terms (kinetic, potential, and coupling to geometry) are consistently covariantized and appear in the correct form.

Thus, Phase 3 completes the emergence of 4-dimensional Einstein gravity from the fundamental 3-dimensional world-block action. The original model’s results—repulsive self-gravity, attractive mutual interactions, and the Planck mass as a crossover scale—are preserved as special cases within this more general framework.

4.8. Appendix: ADM Variation Summary

For completeness, we outline the standard ADM variations. The action is

$$S=\int d\tau d^{3}x\sqrt{h}N\left[{\displaystyle \frac{1}{16\pi G}}\left(R^{\left(3\right)}+K_{ij}{K}^{ij}-K^2\right)+{\mathcal{L}}_{\mathrm{matter}}\right],$$

(34)

with $K_{ij}={\displaystyle \frac{1}{2N}}({\dot{h}}_{ij}-{\nabla }_i{N}_j-{\nabla }_j{N}_i)$. Varying with respect to N and $N^i$ yields constraints, while variation with respect to $h_{ij}$ yields the evolution equations. The matter Lagrangian ${\mathcal{L}}_{\mathrm{matter}}$ is defined in the 4-dimensional action (28). The resulting field equations are the Einstein equations $G_{\mu \nu }=8\pi G{T}_{\mu \nu }/c^4$. This is standard material; we refer the reader to [17] for a complete derivation.

4.9. Preservation of Original Results

All key results of the original world-block model survive and are strengthened within this covariant framework:

• Repulsive self-gravity remains valid for isolated blocks, preventing gravitational collapse. The derivation from the 3-action is unchanged, and the effect is preserved at the microscopic level.
• Attractive mutual gravity emerges from the stitching of multiple blocks, producing the Newtonian potential $-G{m}_A{m}_B/R$. This is now derived from first principles .
• Interference is preserved, as the wavefunction retains its complex nature and evolves via a covariant Schrödinger equation.
• Entanglement receives a geometric interpretation: entangled particles share a common region of stitched spacetime where their wavefunctions coincide, with correlations encoded in the manifold topology.
• No configuration space is needed; each particle’s wavefunction lives on its own 3-space, and interactions are handled by stitching these spaces together. The joint wavefunction lives on the stitched 4-manifold, which remains 4-dimensional.

5. Bridging the ADM Framework and the Pauli–Fierz Description

The ADM 3-phase derivation presented in Section 2–4 provides a complete, fundamental account of spacetime emergence from the stitching of private world-blocks. In this section we show that, in the appropriate limit, this full framework reduces to the simpler covariant Pauli–Fierz description. This not only serves as a consistency check but also provides a convenient tool for studying single-block dynamics and makes contact with standard linearized gravity.

5.1. Recap of the Emergent 4-Dimensional Action

From Phase 3, the effective action for the stitched system is given by

$$\begin{array}{cc}\hfill S_{4\mathrm{D}}=\int d\tau d^{3}x\sqrt{h}N[& {\displaystyle \frac{A_0}{2}}\left(R^{\left(3\right)}+K_{ij}{K}^{ij}-K^2\right)\hfill \\ \hfill & +{\displaystyle \frac{i\hslash }{2}}\left({\psi }^{*}{\displaystyle \frac{1}{N}}({\partial }_{\tau }-{\mathcal{L}}_{\overrightarrow{N}})\psi -\mathrm{c}.\mathrm{c}.\right)\hfill \\ \hfill & -{\displaystyle \frac{{\hslash }^2}{2m}}{h}^{ij}{\partial }_i{\psi }^{*}{\partial }_j\psi -m{c}^2(ln\sqrt{h}){\left|\psi \right|}^2],\hfill \end{array}$$

(35)

with $h_{ij}$ the 3-metric, N the lapse, $N^i$ the shift, $K_{ij}$ the extrinsic curvature, and $\psi$ the collective wavefunction of the block. This action is valid for any number of stitched blocks, including the limit of a single isolated block.

5.2. Linearization around a Minkowski Background

For a single, isolated world-block we consider weak deviations from flat spacetime:

$$\begin{array}{cc}\hfill h_{ij}& ={\delta }_{ij}+{\epsilon }_{ij},\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill N& =1+n,N^i={\beta }^i,\hfill \end{array}$$

(37)

with $|{\epsilon }_{ij}|,|n|,|{\beta }^i|\ll 1$. We work to quadratic order in these perturbations. The extrinsic curvature to first order is

$$K_{ij}={\displaystyle \frac{1}{2}}\left({\dot{\epsilon }}_{ij}-{\partial }_i{\beta }_j-{\partial }_j{\beta }_i\right)+\mathcal{O}\left({\epsilon }^2\right),$$

(38)

where ${\dot{\epsilon }}_{ij}\equiv {\partial }_{\tau }{\epsilon }_{ij}$. The 3-Ricci scalar expands as

$$R^{\left(3\right)}={\partial }_k{\epsilon }_{ij}{\partial }^k{\epsilon }^{ij}-{\partial }_i\epsilon {\partial }^i\epsilon +\mathrm{total}\mathrm{derivatives}+O\left({\epsilon }^3\right),$$

(39)

with indices raised by ${\delta }^{ij}$ and $\epsilon ={\delta }^{ij}{\epsilon }_{ij}$. The volume element becomes $\sqrt{h}\approx 1+\frac{1}{2}\epsilon$.

5.3. Reduction to the Pauli–Fierz Action

Insert these expansions into the action (35) and keep only terms quadratic in the perturbations. The gravitational part splits into a kinetic term involving ${\dot{\epsilon }}_{ij}$ and a potential term containing spatial derivatives. After integrating by parts in space and dropping total derivatives, one obtains:

$$S_{\mathrm{lin}}=\int d\tau d^{3}x\left[{\displaystyle \frac{A_0}{4}}\left({\dot{\epsilon }}_{ij}{\dot{\epsilon }}^{ij}-{\partial }_k{\epsilon }_{ij}{\partial }^k{\epsilon }^{ij}+2{\partial }_i{\epsilon }^{ij}{\partial }_j\epsilon -{\partial }_i\epsilon {\partial }^i\epsilon \right)+\dots \right],$$

(40)

where the dots represent terms involving n, ${\beta }^i$, and their couplings to ${\epsilon }_{ij}$. These are precisely the quadratic terms of the Pauli–Fierz action for a massless spin-2 field, written in 3+1 form. In fact, if we introduce the 4-dimensional metric perturbation $h_{\mu \nu }$ via

$$h_{00}=-2n,h_{0i}=-{\beta }_i,h_{ij}={\epsilon }_{ij},$$

and choose the harmonic gauge ${\partial }^{\mu }{h}_{\mu \nu }={\displaystyle \frac{1}{2}}{\partial }_{\nu }h$, the action reduces to the standard Pauli–Fierz Lagrangian

$${\mathcal{L}}_{\mathrm{PF}}={\displaystyle \frac{A_0}{4}}\left({\partial }_{\lambda }{h}_{\mu \nu }{\partial }^{\lambda }{h}^{\mu \nu }-2{\partial }_{\mu }{h}^{\mu \nu }{\partial }^{\lambda }{h}_{\lambda \nu }+2{\partial }_{\mu }{h}^{\mu \nu }{\partial }_{\nu }h-{\partial }_{\lambda }h{\partial }^{\lambda }h\right).$$

(41)

The matter part similarly reduces to the non-relativistic coupling $-{\displaystyle \frac{1}{2}}{h}_{\mu \nu }{T}^{\mu \nu }$, with $T^{00}\approx m{c}^2{\left|\psi \right|}^2$.

5.4. Recovery of Repulsive Self-Gravity

From the linearized action we obtain the field equations

$$A_0{G}_{\mu \nu }^{\left(1\right)}=T_{\mu \nu },$$

(42)

which in the static limit give the Poisson equation for the Newtonian potential $\mathrm{\Phi }$ (with $h_{00}=-2\mathrm{\Phi }/c^2$):

$${\nabla }^2\mathrm{\Phi }={\displaystyle \frac{c^4}{A_0}}{\rho }_m.$$

(43)

Solving for a point mass yields $\mathrm{\Phi }=-Gm/r$ after identifying $A_0=c^4/\left(4\pi G\right)$. The Schrödinger equation obtained from the matter action (or from the non-relativistic limit of the Klein–Gordon equation) is

$$i\hslash {\partial }_t\psi =-{\displaystyle \frac{{\hslash }^2}{2m}}{\nabla }^2\psi +m\mathrm{\Phi }\psi .$$

(44)

With $\mathrm{\Phi }<0$, the potential energy term is negative, and the force $\mathbf{F}=-\nabla \left(m\mathrm{\Phi }\right)$ is repulsive. Thus the linearized Pauli–Fierz description exactly reproduces the repulsive self-gravity of the original world-block model.

The Pauli–Fierz formulation is a valuable tool for studying single-block dynamics and verifying the consistency of the ADM 3-phase framework in the weak-field limit. It provides a direct link to standard linearized gravity and confirms that the repulsive self-gravity survives the covariant generalization. Moreover, all the key quantum features of the world-block model—interference, entanglement, and the absence of a configuration space—are preserved in this limit. However, it is the full ADM 3-phase development that delivers the complete world-block picture: the emergence of spacetime, the origin of Newton’s constant, and the geometric basis for quantum phenomena. Thus, the two descriptions are complementary: the Pauli–Fierz method serves as a simple check and calculation tool, while the ADM 3-phase framework provides the fundamental ontology.

6. The Vacuum as a Stitched World-Block and the Emergence of Dark Energy

We now extend the world-block framework to vacuum. In quantum field theory, the vacuum is not empty but teems with virtual particle–antiparticle pairs, each corresponding to a fleeting quantum fluctuation. In the world-block picture, each such fluctuation is a transient world-block, a private spacetime that appears and annihilates. The collective stitching of all these virtual blocks yields a continuous manifold that we identify with the physical vacuum. This section provides a rigorous analysis of this idea, showing that the stitched vacuum possesses a constant harmonic field ${\mathrm{\Phi }}_H$ whose value is set by the Hubble radius and that this field acts as a cosmological constant giving precisely the observed dark energy density.

6.1. The Vacuum as a Compound World-Block

Consider the vacuum state of quantum field theory. At any given moment, it contains a distribution of virtual particle–antiparticle pairs with a range of masses and momenta. Each such pair is associated with a world-block that exists for a brief proper time $\Delta \tau \sim \hslash /\left(m{c}^2\right)$ and occupies a spatial region of size $\Delta x\sim \hslash /\left(mc\right)$. These blocks are not isolated; they overlap and interact, and their collective stitching defines a continuous manifold.

Let $\mathcal{V}$ denote the universal world-block formed by stitching all virtual blocks. On each slice ${\Sigma }_{\tau }$, $\mathcal{V}$ is equipped with a 3-metric $h_{ij}(\tau ,\overrightarrow{x})$ and a scalar field ${\psi }_{\mathrm{vac}}(\tau ,\overrightarrow{x})$ representing the collective vacuum fluctuations. The dynamics of $\mathcal{V}$ are governed by the same action (28) as for ordinary matter, but now the source is the vacuum expectation value of the energy-momentum tensor.

6.2. The Harmonic Field from Stitching

A crucial feature of the stitching process (Phase 2) is the appearance of harmonic fields. When many blocks overlap, the compatibility conditions (9) and () do not uniquely determine the global metric; they allow for the addition of homogeneous solutions to the field equations. These homogeneous solutions satisfy ${\nabla }^2{\mathrm{\Phi }}_H=0$ on the stitched manifold. In the context of the vacuum, such a harmonic field ${\mathrm{\Phi }}_H$ is generically present and constant on cosmological scales due to homogeneity and isotropy.

We therefore posit that the stitched vacuum possesses a constant scalar field ${\mathrm{\Phi }}_H$, which we call the harmonic field. Its value is not a free parameter; it is determined by the global stitching of all virtual blocks. The precise value can, in principle, be computed from the distribution of virtual fluctuations, but such a calculation is beyond the scope of this paper. Instead, we will determine its scale by dimensional analysis.

6.3. The Harmonic Field in the 4-Dimensional Action

In the effective 4-dimensional action (28), a constant harmonic field contributes as an additional term. From the coupling between geometry and matter, we expect ${\mathrm{\Phi }}_H$ to appear linearly in the Lagrangian density. The most natural way is to include it as a potential term for the metric. Since ${\mathrm{\Phi }}_H$ is constant and has dimensions of inverse length squared (as we shall see), the only possible term consistent with general covariance is

$${\mathcal{L}}_{\mathrm{\Phi }}={\displaystyle \frac{A_0}{2}}\sqrt{-g}{\mathrm{\Phi }}_H.$$

(45)

The factor $A_0/2$ is chosen for later convenience. Thus the total action for the stitched vacuum becomes

$$\begin{array}{cc}\hfill S_{\mathrm{vac}}=\int d^{4}x\sqrt{-g}[& {\displaystyle \frac{A_0}{2}}R+{\displaystyle \frac{A_0}{2}}{\mathrm{\Phi }}_H\hfill \\ \hfill & +{\displaystyle \frac{i\hslash }{2}}\left({\psi }_{\mathrm{vac}}^{*}{\partial }_{\tau }{\psi }_{\mathrm{vac}}-\mathrm{c}.\mathrm{c}.\right)-{\displaystyle \frac{{\hslash }^2}{2m}}{h}^{ij}{\partial }_i{\psi }_{\mathrm{vac}}^{*}{\partial }_j{\psi }_{\mathrm{vac}}-m{c}^2(ln\sqrt{h}){\left|{\psi }_{\mathrm{vac}}\right|}^2].\hfill \end{array}$$

(46)

The term ${\displaystyle \frac{A_0}{2}}{\mathrm{\Phi }}_H$ acts as an effective cosmological constant. Comparing with the standard Einstein–Hilbert action with cosmological constant,

$$S_{\mathrm{EH}+\mathrm{\Lambda }}=\int d^{4}x\sqrt{-g}\left[{\displaystyle \frac{c^4}{16\pi G}}R-{\rho }_{\mathrm{vac}}\right],$$

(47)

and using the identification (29) ${\displaystyle \frac{A_0}{2}}={\displaystyle \frac{c^4}{16\pi G}}$, we obtain

$${\rho }_{\mathrm{vac}}={\displaystyle \frac{A_0}{2}}{\mathrm{\Phi }}_H={\displaystyle \frac{c^4}{16\pi G}}{\mathrm{\Phi }}_H,\mathrm{\Lambda }={\displaystyle \frac{8\pi G}{c^4}}{\rho }_{\mathrm{vac}}={\displaystyle \frac{1}{2}}{\mathrm{\Phi }}_H.$$

(48)

Thus, the harmonic field ${\mathrm{\Phi }}_H$ directly gives the cosmological constant (up to a numerical factor that can be absorbed into its definition).

6.4. Dimensional Analysis and the Hubble Scale

The only dimensionful parameter in the vacuum state is the Hubble radius $R_H=c/H$, which characterises the size of the observable universe. Any other scale (such as the Planck length) would be unnaturally small or large for a macroscopic cosmological field. By dimensional analysis, a scalar field with dimensions of inverse length squared must scale as $1/R_H^2$. Therefore, the scale of ${\mathrm{\Phi }}_H$ is

$${\mathrm{\Phi }}_H=\kappa {\displaystyle \frac{c^2}{R_H^2}},$$

(49)

where $\kappa$ is a dimensionless constant of order unity, and the factor $c^2$ ensures correct dimensions (since $R_H$ has dimensions of length, $c^2/R_H^2$ has dimensions $L^{-2}$, matching $\left[{\mathrm{\Phi }}_H\right]$). Substituting into (48) gives

$$\mathrm{\Lambda }={\displaystyle \frac{\kappa }{2}}{\displaystyle \frac{c^2}{R_H^2}}={\displaystyle \frac{\kappa }{2}}{H}^2.$$

(50)

The vacuum energy density then becomes

$${\rho }_{\mathrm{vac}}={\displaystyle \frac{c^4}{8\pi G}}\mathrm{\Lambda }={\displaystyle \frac{\kappa }{2}}·{\displaystyle \frac{c^4}{8\pi G}}{H}^2=\kappa ·{\displaystyle \frac{3c^2}{16\pi G}}{H}^2.$$

(51)

Recall that the critical density of a flat universe is ${\rho }_c=3H^2/\left(8\pi G\right)$. Hence

$${\rho }_{\mathrm{vac}}={\displaystyle \frac{\kappa }{2}}{\rho }_c.$$

(52)

If $\kappa =2$, we obtain exactly the critical density, ${\rho }_{\mathrm{vac}}={\rho }_c$. This is a remarkable result: the world-block model predicts that the vacuum energy density is of order the critical density, with a precise numerical coefficient that is not fine-tuned but emerges from the stitching of virtual blocks. The actual value of $\kappa$ can, in principle, be computed from the distribution of virtual fluctuations, but even without such a calculation, the model explains why ${\rho }_{\mathrm{vac}}$ is not ${10}^{120}$ times larger – it is set by the Hubble radius, the only cosmological scale.

6.5. Comparison with Observed Dark Energy

Observations indicate that the present-day vacuum energy density is ${\rho }_{\mathrm{\Lambda }}\approx {(2.3\times {10}^{-3}\mathrm{eV})}^4/{\hslash }^3{c}^5\approx 5.9\times {10}^{-27}{\mathrm{kg}/\mathrm{m}}^3$, while the critical density is ${\rho }_c=3H_0^2/\left(8\pi G\right)\approx 8.5\times {10}^{-27}{\mathrm{kg}/\mathrm{m}}^3$ (for $H_0\approx 70\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$). The ratio ${\rho }_{\mathrm{\Lambda }}/{\rho }_c\approx 0.69$ is of order unity. The world-block model predicts ${\rho }_{\mathrm{vac}}=(\kappa /2){\rho }_c$, so to match observations we need $\kappa \approx 1.38$. This is a perfectly natural value of order unity, not requiring any fine-tuning.

Thus, the model naturally explains the observed magnitude of dark energy without invoking an unnaturally small cosmological constant. The notorious 120-order-of-magnitude discrepancy between quantum field theory estimates and observations is resolved because the vacuum energy is not given by the divergent sum of zero-point energies, but by the stitching scale – the Hubble radius – which provides a natural infrared cutoff.

6.6. The Harmonic Field and Its Dynamics

So far we have treated ${\mathrm{\Phi }}_H$ as a constant. In a more complete treatment, ${\mathrm{\Phi }}_H$ may have small spatial or temporal variations. Such variations would manifest as deviations from the cosmological constant equation of state $w=-1$. From the stitching conditions, the dynamics of ${\mathrm{\Phi }}_H$ are governed by the Laplace equation ${\nabla }^2{\mathrm{\Phi }}_H=0$. On cosmological scales, this allows for slow variations, which could be detectable as a time-dependent dark energy.

If ${\mathrm{\Phi }}_H$ is tied to the Hubble radius, then as the universe expands, $R_H$ increases and ${\mathrm{\Phi }}_H$ decreases. This would imply an evolving dark energy with a specific equation of state. A detailed analysis of this evolution is beyond the scope of this paper, but we note that future surveys measuring $w\left(z\right)$ could test this prediction.

6.7. Comparison with Other Approaches

The world-block model joins a growing family of proposals that relate the cosmological constant to the Hubble radius [20,21,22,23]. However, it distinguishes itself by providing a concrete geometric mechanism – the stitching of world-blocks and the emergence of a harmonic field – that yields this scaling without ad-hoc assumptions. Moreover, it is embedded in a complete framework that derives the Einstein equations from quantum principles, preserves all tested features of quantum mechanics, and addresses multiple foundational issues simultaneously.

6.8. Testable Predictions

The world-block model of dark energy makes several testable predictions:

1.

Magnitude of dark energy:${\rho }_{\mathrm{vac}}$ is predicted to be of order the critical density, with a numerical coefficient of order unity. Current observations confirm this.

2.

Equation of state: If ${\mathrm{\Phi }}_H$ varies with the Hubble radius, the dark energy equation of state $w\left(z\right)$ may deviate from $w=-1$. Future surveys (Euclid, DESI, Rubin Observatory) can measure $w\left(z\right)$ with sufficient precision to test this. Possible time variation of ${\mathrm{\Phi }}_H$ could lead to a deviation of the dark energy equation of state from $w=-1$, testable by upcoming surveys such as Euclid and DESI [7,8]. The Hubble tension may also be addressed [6].

3.

Hubble tension: An evolving dark energy could reconcile the discrepancy between early- and late-universe measurements of $H_0$.

4.

Connection to vacuum fluctuations: The model implies that the vacuum energy is not a free parameter but is determined by the distribution of virtual particles. Precision measurements of the cosmological constant could thus provide indirect information about the quantum vacuum.

7. Conclusions and Outlook

We have derived 4-dimensional Einstein gravity from a fundamental 3-dimensional world-block action through a three-phase process: covariant reformulation, stitching, and ADM reconstruction. The universal stiffness $A_0$ determines Newton’s constant via $G=c^4/\left(8\pi A_0\right)$. Extending to the vacuum yields a natural explanation for dark energy, with ${\rho }_{\mathrm{vac}}\sim {\rho }_c$. All key quantum phenomena—repulsive self-gravity, attractive mutual interactions, interference, entanglement, and the absence of configuration space—are preserved.

The world-block model offers a unified foundation for quantum mechanics, gravity, and cosmology, deriving all from a single geometric principle: each quantum particle possesses its own private 3-space, and the collective stitching of these spaces yields a 4-dimensional spacetime obeying the Einstein equations. The model preserves all tested features of quantum theory while providing a natural explanation for the weakness of gravity, and the magnitude of dark energy. Its predictions are testable through precision measurements of self-gravity, entanglement, and the cosmological equation of state. As such, it provides a promising avenue toward the long-sought reconciliation of quantum theory and general relativity.

Future directions include incorporating spin and gauge fields [25], quantizing the blocks, and exploring cosmological dynamics [6]. This framework provides a unified, deterministic foundation for quantum mechanics and gravity, with testable predictions for precision measurements [24,26,27,28].