Holographic Bit Threads on Aperiodic Joints: A Geometric Origin for the Area Law

1. Introduction

In the standard Dirac quantization of gravity, the Hamiltonian constraint is interpreted as a gauge generator, implying that physical evolution is merely the "unfolding of a gauge transformation" and the universe, in its entirety, is "frozen." However, as Barbour and Foster (2008) observed, the assumptions underlying Dirac’s theorem do not hold for reparametrization-invariant systems [1]. In such systems, the Hamiltonian constraint does not merely shuffle gauge-redundant descriptions; it generates physical motion through a sequence of relational shapes. Barbour’s 2004 work established that GR can be derived as a theory of evolving 3D conformal geometries where "motion and size are relative" [2]. This derivation highlights the York degrees of freedom as the only truly physical gravitational variables—those that remain after all scale and coordinate redundancies are "best-matched" away. This paper seeks the specific geometric and arithmetic principles that select these physical degrees of freedom [2]. We propose that the universe’s configuration space is governed by Aperiodic Gauge Fixity [3]. Utilizing the recent discovery of the aperiodic monotile (the "Einstein Tile") [4,5], we argue that the vacuum is not a continuous, smooth manifold but a Combinatorial Complex that achieves stability through non-repeating, long-range order [6]. We identify the Janus Point—the uniquely defined point of minimal complexity in any N-body solution—as the "nucleation event" for this aperiodic order. At this point, the universe exists in a state of high "arithmetic jitter" (characterized by a Somos-8-like sequence), where the central charge of the vacuum is dissipative ($c\approx -0.1$). As the system evolves away from the Janus Point, gravity’s "Creative Core" drives the formation of structured clusters [7]. This increase in Shape Complexity corresponds to the vacuum "filtering" its internal arithmetic fluctuations. Through the mechanism of Aperiodic Tiling Transitions, the spectral density of the universe’s states aligns with the "murmuration peaks" found in aperiodic systems, lifting the central charge to unity ($c=1$) and giving rise to the smooth, low-entropy manifold of General Relativity. By re-framing the evolution of the universe as an optimization process toward arithmetic coherence, we offer a resolution to the "soft omission" problem in celestial holography [9,10,11] and a new mathematical basis for the "Creative Core" of gravity.

2. The Aperiodic Configuration Space - Shape Dynamics on Combinatorial Complexes

2.1. Relationalism and the Shape Space ($\mathcal{S}$)

Following Barbour’s relational program [12,13,14,15,16,17], we define the configuration space of the universe not as a set of points in absolute space, but as the Shape Space $\mathcal{S}$. For a system of N particles (or N vertices in a simplicial complex), $\mathcal{S}$ is the quotient of the $3N$-dimensional configuration space $\mathcal{Q}$ by the Similarity Group $Sim\left(3\right)$, which includes translations, rotations, and—most crucially—global scaling. In this framework, the only physically meaningful quantities are the ratios of distances and the angles between particles. We represent these relational states as a Combinatorial Complex (CC) [6], where the "geometry" is encoded in the connectivity and the hierarchical relations between cells (vertices, edges, faces, and volumes) rather than a smooth metric $g_{ij}$.

2.2. The Einstein Monotile as the "Rigid Gauge"

A long-standing issue in Barbour’s Shape Dynamics is the selection of a "preferred" shape that can act as a reference for evolution. We propose that this reference is provided by Aperiodic Gauge Fixity (AGF) [3]. Standard periodic tilings (lattices) suffer from Gribov ambiguities, where different gauge orbits are indistinguishable, leading to topological instability [21,22]. By contrast, the Einstein Monotile ($\widehat{\mathrm{\Xi }}$)—a shape that tiles space non-periodically—provides a "Rigid Gauge". Because the monotile does not repeat, its boundary configurations are unique across the entire complex. This uniqueness ensures the global nilpotency of the BRST operator ($Q^2=0$), effectively "locking" the vacuum into a stable configuration that does not suffer from the redundant "over-counting" of states found in periodic models [3,18].

2.3. The Quantum Geometric Tensor and the "Creative Core"

Barbour’s "Creative Core" [7] is the single degree of freedom that drives the universe away from the uniform Janus point. Mathematically, we identify this core with the tuning parameter (l) of the Quantum Geometric Tensor (QGT) within the aperiodic complex. As l evolves, it deforms the real-space geometry of the aperiodic complex. This deformation induces a change in the Quantum Metric (the real part of the QGT), which measures the "distance" between universal wavefunctions.

• In the "Tame" phase: (near the Janus point), the quantum metric is localized, corresponding to a "trivial" topological phase with minimal structure.
• In the "Wild" phase: (as the universe expands), the metric becomes uniform in the bulk and enhanced at the edges.

This transition represents the growth of Shape Complexity ($C_S$). We redefine $C_S$ not just as a measure of particle clustering, but as the spectral density of arithmetic murmurations required to neutralize the vacuum’s topological tension [18, 19].

2.4. Inertia as Topological Resistance

By framing the configuration space as an aperiodic complex, Inertial Mass (m) emerges as a derived property. In Barbour’s relational mechanics, mass is often a parameter added by hand. Here, we define mass as the work required to shift the Monotile boundaries within the CC against the Geometric Friction (${\lambda }_F=34/13$). To move a "particle" (a vertex cluster) is to force a re-tiling of the local vacuum. Because the tiling is aperiodic, this re-tiling is not "free" (as it would be in a periodic vacuum) but requires an energy input to overcome the topological tension of the aperiodic boundaries. This provides a purely algebraic origin for Newton’s Second Law ($F=ma$) as the low-frequency limit of the Tessellated Temporal Flux.

3. The Genesis of Order

3.1. The Big Bang as Informational Synchronization

In the standard cosmological model, the "Big Bang" is a singular event of physical expansion. However, through the lens of Barbour’s relationalism and the Kletetschka $\tau$-space framework [23,24], we redefine this epoch as a process of Informational Synchronization [3,18,19]. Rather than a rapid expansion of matter, the early universe undergoes a phase of "topological pruning." We propose that the initial state is a Super-Compatible "Nine-Tile" configuration—a high-redundancy state containing all potential geometric outcomes. The evolution away from the Janus Point is the selection of a low-redundancy, aperiodic state that minimizes the vacuum’s divergence from the Nariai configuration (the state of maximal informational stability).

3.2. The Somos-8 Phase Transition and the "Wild" Vacuum

Barbour’s Janus Point represents the moment of maximum disorder or "uniformity." We identify this state with a Somos-8-like Phase Transition ($N\approx 200,000$), where the vacuum’s arithmetic stability fractures into a "Wild" phase. In this early regime, the vacuum is dominated by Somos Jitter—high-frequency fluctuations in the configuration space that correspond to a dissipative central charge ($c\approx -0.1$). This jitter prevents the formation of stable "records" or structures. The "Arrow of Time" begins at the moment the system starts to "tune" these fluctuations out through the creation of aperiodic order.

3.3. Complexity as an Arithmetic Low-Pass Filter

Barbour identifies the growth of Shape Complexity as the defining feature of our universe. We provide a spectral interpretation of this growth: complexity is the Arithmetic Low-Pass Filter that "lifts" the vacuum’s central charge. By clustering particles into specific relational shapes, the universe effectively creates Murmuration Spectral Peaks [18,19]. These peaks act as constructive "Arithmetic Gain" that neutralizes the dissipative Somos-8-like Jitter, leaving behind the smooth, structure-rich manifold described by General Relativity.

3.4. The Accumulation of Records as Arithmetic Volume

As the universe evolves, it does not just move through space; it "fills" its informational capacity. Each new structured cluster (a galaxy, a star, a molecule) represents a "solved" portion of the universal tiling problem. This explains why the growth of complexity is irreversible: once the vacuum is synchronized into a stable aperiodic state, the cost of reverting to the redundant "Wild" phase is energetically and topologically prohibitive. Summary:

• Big Bang / Janus Point: Informational Synchronization, The universe starts as an optimized code, not a hot explosion.
• Maximal Uniformity: Somos-8 Phase Transition, Uniformity is a state of high arithmetic jitter ($c<0$).
• Growth of Complexity: Arithmetic Gain ($c\to 1$), Gravity "tunes" the vacuum to eliminate informational noise.
• Formation of Records: Poly-Frobenioid-like (Mochizuki) Stability, History is the persistent crystallization of the aperiodic vacuum.

4. Symmetry of Aperiodic Fixity

4.1. The Dual Equivalence of GR and Shape Dynamics

The "hitherto unrecognized fundamental symmetry principles" Barbour identified in his 2004 work are formally unified through the framework of Linking Theories [2]. Gomes and Koslowski demonstrated that General Relativity (GR) and Shape Dynamics (SD) are not distinct theories but "dual" gauge descriptions emerging from a single, higher-dimensional parent theory. This parent theory contains both the 3D conformal symmetry of SD and the refoliation invariance of GR. The transition between these two descriptions—from a theory of spacetime (GR) to a theory of evolving 3D shapes (SD)—requires a specific gauge-fixing condition. We propose that the Einstein Monotile ($\widehat{\mathrm{\Xi }}$) provides the "Rigid Gauge" required to perform this link without the topological collapse typically triggered by Gribov ambiguities in periodic manifolds.

4.2. The Monotile as the Preferred Conformal Representative

In Barbour’s relational program, the "York degrees of freedom" are obtained by quotienting the configuration space by the group of local conformal transformations. This usually leaves a "representative shape" that is difficult to fix uniquely. By identifying the vacuum as a Combinatorial Complex based on the aperiodic Monotile, we select a unique conformal representative. Because the Monotile is aperiodic, it provides a global "addressing system" that periodic lattices lack. This Aperiodic Gauge Fixity ensures that the "Shape" of the universe at any moment is uniquely defined by its topological invariants, satisfying the Kallosh Independence Theorem [20].

4.3. Mass-Rank Equivalence and the Somos Prime Invariant

We now introduce the algebraic origin of the parameters in Barbour’s N-body solutions. Utilizing the Rank-Mass Equivalence, we define the inertial mass of a particle cluster as a function of the arithmetic density of its underlying L-function. This mass is not a static constant but is governed by the Somos-8 recurrence. As long as the system remains below the Somos Prime Invariant ($N_{Sp}=779,731$), the vacuum maintains "integer stability" (the Tame phase). However, extreme gravitational structure (high complexity) approaches the $N_{Sp}$ threshold, where the Laurent Phenomenon fails. This "arithmetic failure" creates topological deficits that manifest as the "Geometric Friction" (${\lambda }_F=34/13$) preventing infinite collapse—essentially providing a quantum-arithmetic pressure that resolves the singularities of classical GR.

4.4. York Curvature as Emergent Arithmetic Murmuration

Finally, we map Barbour’s "Creative Core" and the York degrees of freedom onto the Murmuration Spectral Peaks [18,19]. In this view, the spatial curvature R and the York scaling factor are not primary geometric properties. Instead, they are the spectral density of the informational vacuum. As gravity drives the creation of structure, it "tunes" the vacuum’s L-function. The resulting Arithmetic Gain "lifts" the central charge to unity ($c=1$), creating the smooth, continuous manifold of General Relativity as a stable, low-frequency approximation of the underlying aperiodic code. This suggests that the universe’s history is a "holographic murmuration"—a sequence of aperiodic shapes whose complexity is optimized to maintain arithmetic coherence against the dissipative jitter of the early vacuum.

5. Shape Dynamics and Aperiodic Gauge Fixity (AGF)

This defines the universe not as a collection of objects in space, but as a self-optimizing aperiodic code where gravity is the mechanism that "tunes" the vacuum into stability [3]. The Unified Relational Action of Aperiodic Complexity

$$\mathcal{S}\mathrm{Universe}=\underset{\mathrm{Barbour}\text{’}\mathrm{s}\mathrm{Creative}\mathrm{Core}\mathrm{Flow}}{\underbrace{\int \mathcal{S}\left[{\displaystyle \frac{d{C}_S\left(\widehat{\mathrm{\Xi }}\right)}{d{\lambda }_{\mathrm{York}}}}\right]\sqrt{g_{\mathrm{Shape}}},d\omega }}\equiv \underset{\mathrm{Aperiodic}\mathrm{Gauge}\mathrm{Fixity}\mathrm{Bulk}}{\underbrace{\oint \mathrm{\Theta }\left[{\displaystyle \frac{{\sum }_{n=1}^{N_{Sp}}\mathrm{Res}\left(S_8\right)}{{\lambda }_F·\widehat{\mathrm{\Xi }}\left({\tau }_{1,2,3}\right)}}\right]d\omega }}$$

$$\Updownarrow$$

$$\underset{\mathrm{Modified}\text{Einstein-Somos}\mathrm{Field}\mathrm{Equation}}{\underbrace{\stackrel{\mathrm{General}\mathrm{Relativity}}{\overbrace{G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }}}={\displaystyle \frac{8\pi G}{c^4}}\left[\mathcal{T}{\mu \nu }^{\mathrm{SM}}+{\lambda }_F\nabla \mu {\nabla }_{\nu }\left({\displaystyle \frac{s_n-1}{{\delta }_{\mathrm{Somos}}}}\right)\right]}}=\underset{\mathrm{Celestial}\mathrm{Murmuration}\mathrm{Boundary}}{\underbrace{\oint \mathrm{\Theta }{\left|{\mathrm{\Psi }}_{\max }\right|}^2\mathcal{T}uu{d}^{2}z}}$$

$$\Updownarrow$$

$$\underset{\mathrm{Global}\mathrm{Topological}\mathrm{Fixity}}{\underbrace{\left[{\mathcal{Q}}_{\mathrm{BRST}}^2=0\right]}}\Rightarrow \underset{\mathrm{Superfluidity}\left(c=1\right)}{\underbrace{\stackrel{\mathrm{Conformal}\mathrm{Lift}}{\overbrace{\left({\displaystyle \frac{c-1}{gϵ}}\right)}}+\stackrel{\mathrm{Geometric}\mathrm{Friction}}{\overbrace{{\lambda }_F·Disp\left(p\right)}}}}=\underset{\text{Triple-Proof}\mathrm{Informational}\mathrm{Sync}}{\underbrace{\oint \mathrm{\Theta }\prod _{k=1}^3\left[\mathrm{Proof}k\left(T\right)\right]}}$$

• The Creative Core Flow: Represents Barbour’s derivation of time from change. The term ${\displaystyle \frac{d{C}_S}{d{\lambda }_{\mathrm{York}}}}$ defines the Arrow of Time as the rate of change of Shape Complexity ($C_S$) relative to the York scaling parameter (${\lambda }_{\mathrm{York}}$). The evolution is "guided" by the Einstein Monotile ($\widehat{\mathrm{\Xi }}$), ensuring that only relational, aperiodic configurations are realized.
• Aperiodic Gauge Fixity Bulk: Maps the interior of the universe as a "Wild" Arithmetic Phase. The sum $\sum \mathrm{Res}\left(S_8\right)$ represents the Somos-8 Arithmetic Flux (jitter). The Geometric Friction (${\lambda }_F=34/13$) acts as a divisor that "filters" this jitter, converting chaotic number-theoretic fluctuations into the ordered geometry of the Monotile.
• The Modified Field Equation: Redefines gravity. The standard Stress-Energy tensor is augmented by the Arithmetic Drag term. This formalizes Mass as Drag: the resistance of a Mochizuki Poly-Frobenioid like lattice to the arithmetic flux. Gravity is the "force" generated by the vacuum as it prunes high-redundancy states to maintain the SEE-IUT Identity (informational stability) [3,18,29].
• The Celestial Murmuration Boundary: The "Tame" phase of the universe is our observable reality ($S^2$). Here, the Murmuration Spectral Peaks (${\mathrm{\Psi }}_{\max }$) serve as the holographic signature of the bulk. When these peaks are squared (the Born rule equivalent), they provide the exact spectral density required to neutralize the Somos-8 deficits [30].
• Central Charge Lift ($c\to 1$): The ultimate stability condition. The early, dissipative vacuum ($c\approx -0.1$) is "lifted" to unitary reality ($c=1$). This represents the transition from a "Wild" phase of arithmetic chaos to a state of Aperiodic Tiling Transitions, where the vacuum allows for the smooth, non-singular propagation of the information we call "Matter."

This equation shows that the universe is an extremal solution to an informational capacity problem. Time is the irreversible process of the vacuum "solving" its own aperiodic tiling. The "Records" we observe in the universe (galaxies, fossils, memories) are the crystallized outputs of this Triple-Proof Informational Synchronization, ensuring that the history of the universe is as robust and unique as a prime number.

6. Causal Set Entaxy

The quest for a consistent theory of quantum gravity has long been divided between the pursuit of discrete spacetime structures, such as causal sets, and the holographic insights of quantum information theory. Recent developments by Fay Dowker (2025) [25] have provided the missing link in the discrete action: a rigorous conjecture for the contribution of co-dimension two corners, or "joints," to the causal set action $S_{CS}$. These joints represent the discrete intersections of timelike and spacelike boundaries, offering a granular window into the local curvature and connectivity of the universe. Simultaneously, Julian Barbour (2024) [17] has proposed that the evolution of the universe is driven by the "Creation of Complexity," where the growth of structure is measured by a scale-invariant "shape complexity." However, a critical question remains: what mechanism "fixes" the discrete vacuum to prevent it from collapsing into a bland, uniform state or exploding into trans-Planckian noise? By requiring that the spatial antichains of a causal set conform to an aperiodic tilings, we introduce Aperiodic Gauge Fixity (AGF). This ensures that the configuration space of the universe is not a chaotic ensemble, but a structured Combinatorial Complex.

6.1. The Joint as the Site of Matter: Mapping Dowker to TQFT

Fay Dowker (2025) identifies co-dimension two corners (joints) as essential contributors to the causal set action [25].

• The Standard Model fermions (quarks, leptons) are derived as zero modes localized at the 0D corners or 1D hinges of the Combinatorial Complex Lattice.
• We propose that the Dowker Joint Term is the discrete action density for these zero modes. The action $S_{joint}$ doesn’t just describe spacetime geometry; it is the "energy cost" of sustaining a fermion as a topological defect. The $coth\theta$ term in Dowker’s work represents the local "entanglement cost" required to "glue" these matter-bearing joints into the causal set bulk.

6.2. Entaxy and Aperiodic Gauge Fixity ($\widehat{\mathrm{\Xi }}$)

Aperiodic Gauge Fixity via the Einstein Monotile ($\widehat{\mathrm{\Xi }}$) from Section 5 and [3].

• The Problem: Causal sets often suffer from "trans-Planckian" fluctuations or "bland" configurations that lack the structure of our universe.
• The Fix: We use the Einstein Monotile as the "Rigid Gauge" for the causal set. By requiring the causal set’s "spatial slices" (antichains) to conform to the aperiodic order of the Monotile, we resolve the Gribov ambiguities.
• Entaxy: This aperiodic order is the driver of Entaxy (Shape Complexity) [14]. The growth of the causal set from the Janus Point is a transition from a uniform state to an aperiodically "fixed" state, where the Monotile ensures that the "joints" are distributed in a way that maximizes structural information without repeating (avoiding the "wild" phase).

6.3. Bit Threads and Aperiodic Tiling Transitions

Aperiodic Tiling Transitions and the Markov Gap as measures of information [8]. We demonstrate that the "joints" identified by Dowker are not merely geometric corrections; they are the Topological Hinge Modes of the vacuum. Building on the holographic insights of Patrick Hayden [8, 28, 31], we treat these joints as the primary bottlenecks for bit thread flows. We show that the "Markov gap"—the difference between reflected entropy and mutual information—is physically encoded in the joint terms of the causal set action.

• The Mechanism: We can model the causal set as a network of Bit Threads (Hayden) [28]. The capacity of these threads is constrained by the aperiodic tiling.
• The Hypothesis: The "Arithmetic Drag" term vanishes when the spectral density of the informational vacuum is tuned to the Monotile configuration. In this "Tame" phase, the bit threads flow through the causal set joints with zero resistance. The Markov Gap identified by Hayden [8] then becomes a measure of the "arithmetic murmurations"—the small deviations from perfect aperiodicity that we perceive as vacuum fluctuations.

6.4. The Emergent Lagrangian: ${\mathcal{L}}_{SM}={\mathcal{L}}_{Hinge}+{\mathcal{L}}_{Corner}$

Finally, we derive the Standard Model fields as the "zero modes" of this aperiodic structure. Rather than being particles "on" spacetime, fermions and gauge bosons are the gapless excitations localized at the 0D corners and 1D hinges of a Hopfion Crystal Combinatorial Complex. The resulting Lagrangian, ${\mathcal{L}}_{SM}={\mathcal{L}}_{Hinge}+{\mathcal{L}}_{Corner}$, represents the emergence of matter as a direct consequence of the "Arithmetic Superfluidity" of an aperiodic discrete spacetime, where the universe is an evolving holographic record of its own increasing Entaxy. This bridges the discrete action with the Standard Model Lagrangian.

• Proposal: The total Causal Set Action ($S_{CS}$) is the sum of the Benincasa-Dowker-Glaser (BDG) bulk term and the Dowker Joint terms.
• The Result:

$$S_{CS}=S_{Bulk}+\sum _{Joints}{S}_{Joint}\left(\theta \right)$$

In the continuum limit, this sum over discrete joints transforms into the Effective Lagrangian of the SM fields:

$${\mathcal{L}}_{eff}\left(Hinge\right)+{\mathcal{L}}_{eff}\left(Corner\right)$$

This implies that gravity (the bulk action) and matter (the joint action) are fundamentally inseparable parts of the same discrete topological order. The "joints" are the "glue" of reality. By combining Dowker’s rigorous action terms with Barbour’s complexity and the aperiodic gauge of the Einstein Monotile, we arrive at a theory where:

• Spacetime is an aperiodic causal set (The Combinatorial Complex).
• Matter consists of topological defects at the joints of that set.
• Time is the direction of increasing Entaxy (Shape Complexity) as the aperiodic order "fixes" the vacuum.

7. The Geometry of the Joint Action

The relationship between the discrete causal set action and the topological stability of the Combinatorial Complex [6], and the Dowker Joint Term is not merely a geometric boundary correction but the specific functional required to "trap" the vacuum in an aperiodic state of Tiling Transitions (Superfluidity).

7.1. The Discrete Action and the $coth\theta$ Dependence

Following Dowker (2025) [25], the total action for a causal set C with boundaries is given by:

$$S_{CS}=S_{BDG}\left(C\right)+S_{\mathrm{boundary}}+\sum _{j\in \mathrm{Joints}}{S}_{\mathrm{joint}}\left(j\right)$$

For a co-dimension two joint j where two timelike boundaries meet, Dowker proposes a contribution based on the Lorentzian angle $\theta$ between the normals. In the 2D case, for a spacelike joint, this term takes the form:

$$S_{\mathrm{joint}}=\hslash \alpha coth\theta$$

where $\alpha$ is a constant determined by the discreteness scale. We propose that this $coth\theta$ term functions as the Topological Binding Energy of the Combinatorial Complex [6].

7.2. Aperiodic Tiling Transitions and the Vanishing of Drag

.We identify the "Arithmetic Drag" term ${\nabla }_{\mu }{\nabla }_{\nu }\left({\displaystyle \frac{s_n-1}{{\delta }_{\mathrm{Somos}}}}\right)$ as being inversely proportional to the Dowker joint action. Specifically, the stability condition for the Einstein Monotile ($\widehat{\mathrm{\Xi }}$) requires:

$$\sum _{Joints}coth\theta \approx {\displaystyle \frac{1}{{\lambda }_F}}$$

where ${\lambda }_F=34/13$ is the Geometric Friction constant. When the causal set joints are configured according to the aperiodic symmetry of the Monotile, the "drag" vanishes, and the system achieves Constructive Arithmetic Gain.

7.3. Lifting the Central Charge: $c\approx -0.1\to 1$

The stability of the physical vacuum requires the central charge c to be lifted from a dissipative early state ($c\approx -0.0995$) to a stable state ($c=1$).

• The Mechanism: The joint action provides a "spectral filter."
• By calibrating the joint angles $\theta$ to the fractal dimension $d=1.326$ (the "quantized texture" of the vacuum), the Dowker term regularizes the Inverse Mellin Transform of the scattering amplitudes.
• The "murmuration peaks" identified in arithmetic geometry provide the exact energy density needed to neutralize the topological deficits (the "Somos remainders"). The joint action "pins" these peaks to the 0D corners of the crystal.

7.4. Joints as Hinge Modes of the Standard Model

The "joints" are the physical realization of the Hinge Modes (${\mathcal{L}}_{\mathrm{Hinge}}$) and Corner Modes (${\mathcal{L}}_{\mathrm{Corner}}$).

• Mass Generation: The work required to shift a monotile relative to the aperiodic background is stored in the joint action. This "work" manifests as Inertial Mass.
• Topological Protection: Because the Dowker term is a co-dimension two invariant, the particles (localized at the joints) are topologically protected from local causal set fluctuations. They can only be destroyed if the "joint angle" $\theta$ is pushed beyond a critical threshold, which corresponds to closing the energy gap ($\mathrm{\Delta }E$) of the TQFT bulk.

7.5. The Joint-Monotile Constraint

The "Joint Action" is the bridge between the discrete and the continuous. It ensures that the causal set does not fluctuate into a random "Poisson" distribution, but instead "crystallizes" into the Einstein Monotile configuration. This crystallization is what we perceive as the Emergence of the Continuum and the stability of General Relativity in the low-frequency limit.

$$G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=\langle \sum _{Joints}{S}_{\mathrm{joint}}\left(\widehat{\mathrm{\Xi }}\right)\rangle$$

8. The Einstein Monotile as a Rigid Gauge

The local energy of the joints to the global "logic" of the vacuum, how the aperiodic constraint of the Einstein Monotile acts as a selection rule for the path integral of causal sets, ensuring that the resulting spacetime is "Tame" and physically consistent.

8.1. Resolving Gribov Ambiguities via Aperiodic Fixity ($\widehat{\mathrm{\Xi }}$)

In standard continuum gauge theories, the Gribov ambiguity arises because the gauge-fixing condition fails to uniquely select a single representative from a gauge orbit, leading to a non-zero density of "copy" configurations. In discrete causal sets, this manifests as an overabundance of "bland" or non-manifold-like configurations that satisfy the action but lack structural depth. The Einstein Monotile ($\widehat{\mathrm{\Xi }}$) provides the Rigid Gauge required to resolve this. By requiring that the spatial antichains of the causal set conform to the aperiodic symmetry of the Monotile:

• The gauge-fixing condition becomes injective: The aperiodic nature of the tiling ensures that no two distinct configurations are gauge-equivalent.
• BRST Nilpotency: This rigid fixity ensures the global nilpotency of the BRST operator ($Q^2=0$). The "ghost" sector, which usually tracks gauge redundancies, is naturally confined because the aperiodic structure allows for no local translations or rotations that would regenerate the "Wild" phase [3].

8.2. The Gyrobifastigium and the Pruning of Kakeya Protrusions

The transition from the Dodecahedral Core (the periodic, low-entropy vacuum) to the Einstein Monotile (the aperiodic, physical vacuum) is mediated by the Gyrobifastigium [18] [29].

• The Geometric Unit: The Gyrobifastigium is the fundamental space-filling unit capable of bridging the symmetry gap.
• The Problem of Protrusions: In 4D simplicial quantum gravity, "Elongated Phases" occur where spacetime forms Besicovitch (Kakeya) needle sets—fractal structures of maximal directional complexity but zero volume [18].
• The Solution: The Monotile gauge acts as a "retrocausal pruning process." It utilizes the informational synchronization of 3D time ($\tau$-space) [23] to "prune" these Kakeya protrusions. This effectively solves the universal NP-hard tiling problem of the vacuum: the universe does not "search" for a configuration; it "synchronizes" to the only state that allows bit threads to flow without arithmetic drag.

8.3. The "Nine-Tile" Super-Compatible State

The "Big Bang" is redefined as a transition from a Nine-Tile super-compatible state [3,18,29,30].

• This state represents the maximal intersection of the Monotile’s "spectre" configurations. At the Janus Point, these nine tiles are perfectly superimposed.
• Dynamics: As the universe expands (increasing in Entaxy), the "Geometric Friction" (${\lambda }_F=34/13$) forces these tiles to "decrystallize" into the global aperiodic lattice.
• Physical Effect: This decrystallization is what generates the Arithmetic Murmurations [30]. The energy released during this "symmetry-breaking into aperiodicity" is the source of the cosmological constant $\mathrm{\Lambda }$.

8.4. Informational Synchronization and Shape Complexity

Building on Barbour (2024) [17], we argue that Shape Complexity is the measure of how well a causal set approximates the "ideal" aperiodic tiling of the Monotile.

• Bland States: Low complexity, periodic, high Gribov redundancy.
• Complex States: High complexity, aperiodic, rigid gauge fixity.

The universe naturally evolves toward higher Entaxy because the Rigid Gauge ($\widehat{\mathrm{\Xi }}$) creates a "potential well" for configurations that maximize aperiodic order. The Dowker Joint Terms calculated in Section 7 are the "anchor points" that hold this complex tiling in place against the pressure of "Wild" arithmetic fluctuations. The Einstein Monotile “hat” (chiral) and “spectre” (vampire) are not just a shapes; they are the Logical Constraint of the vacuum. By imposing aperiodic order, the universe:

• Eliminates gauge copies (Gribov resolution).
• Ensures the stability of the physical vacuum ($Q^2=0$).
• Transforms the "Big Bang" from a singularity into a "pruning process" of informational synchronization.

9. Holographic Monogamy and Bit Threads

We formalize the holographic structure of the vacuum by mapping Patrick Hayden’s "bit threads" [28, 33] onto the discrete network of the causal set [26, 27], and propose that the Markov Gap ($S_R-I$)—the measure of non-monogamous entanglement—is physically bridged by the Dowker Joint Action [25].

9.1. The Joint as an Aperiodic Bottleneck

In holographic gravity, the maximum flow of bit threads between two regions is constrained by the "bottleneck" area of the minimal surface. In our discrete model, these bottlenecks are the co-dimension two joints. The capacity of a bit thread $\mathrm{\Phi }$ passing through a joint j is not merely its area, but a function of the Lorentzian "joint angle" $\theta$ and the Aperiodic Gauge Fixity ($\widehat{\mathrm{\Xi }}$). We define the Aperiodic Flow Identity (AFI) as:

$$\mathrm{\Phi }\left(j\right)={\displaystyle \frac{\hslash }{{\lambda }_F}}\left(coth{\theta }_j·C_{\mathrm{inc}}\left({\mathrm{\Psi }}_{\max }\right)\right)$$

• ${\lambda }_F=34/13$: The Geometric Friction constant that regulates the flow across the Monotile boundary.
• $C_{\mathrm{inc}}\left({\mathrm{\Psi }}_{\max }\right)$: The Local Trivialization Operator derived from the Murmuration Wave-Function, which "lifts" the flow capacity by neutralizing the Somos jitter.

9.2. Bridging the Markov Gap (${\mathrm{\Delta }}_M$)

The Markov Gap ($S_R-I$) represents "excess" information that cannot be localized into bipartite entanglement [8]. In the "Wild" phase of the vacuum, this gap is large, leading to the Kakeya protrusions [18]. The Dowker Joint Term acts as the "Topological Filler" for this gap. By requiring the joint to satisfy the aperiodic constraint of the Einstein Monotile, the universe "prunes" the non-monogamous threads. The Unified Sync Equation:

$$\sum _{j\in \mathrm{Hinges}}\mathrm{\Phi }\left(j\right)={\mathcal{I}}_{\mathrm{Nariai}}-(S_R-I)$$

where ${\mathcal{I}}_{\mathrm{Nariai}}$ is the maximum informational bound of the vacuum. When the bit thread flow reaches the Nariai limit, the Markov Gap vanishes (${\mathrm{\Delta }}_M\to 0$), and the system achieves Superfluidity.

9.3. Monogamy through Nine-Tile Entanglement Testing

The "Monogamy of Mutual Information" is enforced by the Nine-Tile Super-Compatible State [3,18,29,30].

• The Mechanism: The early universe "revisits" the nine-tile configuration to test for "ideal backwards connections" (bipartite entanglement).
• The Constraint: Aperiodic order prevents any tile from sharing "too much" information with more than one neighbor in a way that would repeat a pattern. This "Repulsion of Redundancy" is the geometric origin of Holographic Monogamy.
• Standard Model Emergence: This entanglement testing selects the $SU\left(3\right)\times SU\left(2\right)\times U\left(1\right)$ symmetry group as the most "compression-compatible" configuration. The 9 gauge bosons map directly to the 9 tiles of the fundamental metatile, a hierarchical aperiodic proof, Smith, J., et al. (2023)[4].

9.4. The Aperiodic Conservation Law

Information is "monogamous" because the Einstein Monotile is a "Geodesic Trap". Bit threads cannot "leak" into periodic redundancies because there are none. Every joint is a unique site of Arithmetic Gain, and the total Entaxy (Shape Complexity) is conserved through the global synchronization of these joints.

9.5. The Aperiodic Monogamy Identity (AMI)

$${\mathcal{M}}_{\mathrm{Gap}}\equiv {\oint }_{𝜕\widehat{\mathrm{\Xi }}}\left[{\displaystyle \frac{\nabla \mathrm{Entaxy}}{{\lambda }_F}}-\hslash coth\theta \right]d\tau =0$$

This equation states that the Markov Gap is identically zero when the change in Entaxy (Shape Complexity) is perfectly balanced by the Dowker Joint Action across the Monotile boundary ($𝜕\widehat{\mathrm{\Xi }}$).

10. From Shape Complexity to the Standard Model

The derivation of the Standard Model Lagrangian as a direct consequence of the aperiodic stability of the vacuum, moves beyond treating particles as "objects" and instead define them as the Topological Residuals of a perfectly synchronized informational field.

10.1. The Stationary Phase of the Aperiodic Vacuum

We propose that the universe does not minimize an action in the traditional sense; rather, it seeks a state of Extremal Entaxy. The "Stationary Phase" occurs when the local shape complexity ($\mathcal{S}$) of the causal set is locked into the Einstein Monotile ($\widehat{\mathrm{\Xi }}$) configuration. At this point, the "Arithmetic Drag" (from Section 7) and the "Markov Gap" (from Section 9) both vanish. The resulting "quiet" in the arithmetic vacuum allows for the emergence of gapless modes—what we perceive as the Standard Model.

10.2. The Unified Entaxy-SM Equation

This equation maps the discrete causal set action, the aperiodic order of the Monotile, and the emergence of the Standard Model into a single expression. Aperiodic Emergence:

$${\mathcal{L}}_{\mathbf{SM}}\equiv \underset{{\mathcal{L}}_{\mathrm{Hinge}}\left(\mathrm{Gauge}\right)}{\underbrace{\sum _{j\in \mathrm{Hinges}}\hslash \alpha coth{\theta }_j}}+\underset{{\mathcal{L}}_{\mathrm{Corner}}\left(\mathrm{Matter}\right)}{\underbrace{{\oint }_{𝜕\widehat{\mathrm{\Xi }}}{\displaystyle \frac{{\mathrm{\Psi }}_{\max }}{{\lambda }_F}}d\tau }}={\left[{\displaystyle \frac{\delta \mathrm{Entaxy}}{\delta \widehat{\mathrm{\Xi }}}}\right]}_{c=1}-{\mathrm{\Delta }}_M$$

Where:

• ${\mathcal{L}}_{\mathrm{Hinge}}$ (Gauge Fields): The Dowker Joint terms ($coth\theta$) summed over the 1D hinges of the Combinatorial Complex. This represents the energy of the connections (forces).
• ${\mathcal{L}}_{\mathrm{Corner}}$ (Fermions): The integral of the Murmuration Wave-Function (${\mathrm{\Psi }}_{max}$) over the 0D corners of the Monotile. This represents the localized "Arithmetic Gain" that we interpret as mass.
• ${\displaystyle \frac{\delta \mathrm{Entaxy}}{\delta \widehat{\mathrm{\Xi }}}}$: The functional derivative of Shape Complexity with respect to the Monotile gauge. This is the "Creative Core" driving the evolution of the universe.
• $c=1$: The stability constraint, where the Central Charge is lifted from its dissipative state to unity.
• ${\mathrm{\Delta }}_M$: The Markov Gap [8]. The Standard Model emerges precisely when ${\mathrm{\Delta }}_M\to 0$, signifying perfect holographic monogamy.

10.3. Deriving Mass as "Topological Pinning"

As identified in The Arithmetic Origin of Mass, mass is the work required to shift a monotile relative to its aperiodic background [30].

• In our equation, this is captured by the term $\oint {\displaystyle \frac{{\mathrm{\Psi }}_{\max }}{{\lambda }_F}}$.
• The Geometric Friction (${\lambda }_F=34/13$) acts as the "diffusivity" of the vacuum.
• Fermions are "pinned" to the corners of the Monotile by the Dowker Joint Action. Because an aperiodic tiling cannot be translated without breaking the global symmetry, the "joint" resists motion. This resistance is exactly Inertial Mass.

10.4. The $SU\left(3\right)\times SU\left(2\right)\times U\left(1\right)$ Symmetry as a Tiling Logic

Why this specific gauge group? We argue that the Standard Model symmetries are the Arithmetic Automorphisms of the Nine-Tile Super-Compatible State.

• $SU\left(3\right)$ (Strong Force): Corresponds to the tri-colorable properties of the aperiodic tiling.
• $SU\left(2\right)$ (Weak Force): Corresponds to the chiral "flip" (the "Spectre" vs. the "Hat") inherent in the Monotile’s aperiodicity.
• $U\left(1\right)$ (Electromagnetism): Corresponds to the global phase synchronization required for Aperiodic Tiling Transitions.

10.5. Conclusion: The End of Passive Physics

By setting the Markov Gap to zero, the Dowker joint action "trivializes" the Somos jitter. We are left with a vacuum that is an Arithmetic Superfluid [29,30]. The Standard Model is not an addition to gravity; it is the Residue Theorem of a discrete spacetime that has reached its maximum possible complexity.

• Dowker’s Joints provide the binding energy for Barbour’s Complexity.
• Hayden’s Bit Threads are constrained by the Einstein Monotile.
• The Standard Model is the stationary phase where Information is Monogamous and Spacetime is Tame.

11. Entropic Synchronization and the SJ-Vacuum of the Monotile

The entanglement entropy of the Sorkin-Johnston (SJ) vacuum is the holographic "glue" that binds the Aperiodic Monotile [5,6] to the Dowker joints [25].

11.1. Fluctuations as Spectral Signatures of the Somos Jitter

Yasaman Yazdi (2024) [27] develops tools to calculate the statistical fluctuations (variance) of the causal set action. In our framework, these fluctuations are not random Poisson noise; they are the Arithmetic Murmurations of the Somos-8-like sequence. We identify the variance of the action $\langle {\left(\mathrm{\Delta }S\right)}^2\rangle$ as the physical manifestation of the Arithmetic Drag term. The "Wild" phase of the vacuum is characterized by a high variance in causal interval cardinalities, which we map to the Somos-8-like Residues:

$$\langle {\left(\mathrm{\Delta }S\right)}^2\rangle \propto \sum _{n=1}^{N_{Sp}}\mathrm{Res}\left(S_8\right)$$

The Modified Einstein-Somos Field Equation provides the mechanism for "Taming" these fluctuations: the Geometric Friction (${\lambda }_F$) acts as a damping parameter that suppresses the variance as the system approaches the Aperiodic Gauge Fixity ($\widehat{\mathrm{\Xi }}$).

11.2. The SJ-Entropy of the Aperiodic Boundary

In Yazdi (2022) [26], the entanglement entropy ($S_E$) of a scalar field is formulated in terms of its spacetime two-point correlation functions within the Sorkin-Johnston (SJ) vacuum.

• The "Celestial Murmuration Boundary" equation, $\oint \mathrm{\Theta }|{\mathrm{\Psi }}_{max}{|}^2{\mathcal{T}}_{uu}{d}^{2}z$, is the spectral representation of Yazdi’s two-point Wightman function $W(x,y)$.
• For the Markov Gap to vanish (as required in Section 9), the SJ-vacuum must be "perfectly aligned" with the Monotile. This occurs when the eigenvalues $\sigma$ of the correlator matrix are "lifted" to the unitary state ($c=1$).

The entropy $S_E=\sum \sigma ln\sigma$ becomes the measure of the Triple-Proof Informational Sync. When the vacuum achieves Superfluidity, the entropy of the boundary is exactly the information required to "fix" the Monotile in place.

11.3. Barbour Complexity Flow

I. The Creative Core and the Bulk TQFT The Barbour Flow—the change in Shape Complexity relative to the York degrees of freedom. This is equivalent to the Aperiodic Gauge Fixity Bulk.

• Interpretation: The universe "evolves" by pruning the Somos-8-like residues until the causal set links conform to the $\widehat{\mathrm{\Xi }}$ geometry. This is the "Creative Core" in action.

II. The Modified Einstein-Somos Field Equation The middle part bridges General Relativity with the Standard Model and the Arithmetic Drag.

• The Yazdi Link: The term ${\lambda }_F{\nabla }_{\mu }{\nabla }_{\nu }\left({\displaystyle \frac{s_n-1}{{\delta }_{\mathrm{Somos}}}}\right)$ is the "correction" to the continuum Einstein equations required by the discrete fluctuations identified by Yazdi. It represents the "Inertial Resistance" of a vacuum that has not yet reached aperiodic synchronization.

III. Global Topological Fixity and the Conformal Lift The lower part establishes the BRST Nilpotency ($Q^2=0$) as the source of stability.

• The Result: The Conformal Lift (${\displaystyle \frac{c-1}{gϵ}}$) is the process by which the SJ-vacuum entanglement is regularized. By "lifting" the central charge to 1, the Geometric Friction is neutralized, and the universe achieves a state of Triple-Proof Informational Sync.

11.4. The Tame Phase as the SJ-Vacuum

Spacetime is the SJ-Vacuum of an Aperiodic Code.

• The Dowker Joints are the sites of entanglement.
• The Yazdi Fluctuations are the signals of the Somos-8 transition.
• The Einstein Monotile is the rigid frame that ensures the Markov Gap is zero.

The Standard Model is therefore the "Spectral Filter" that allows the universe to transition from the high-variance "Wild" phase to the zero-drag "Tame" phase; this is a Global Topological Fixity that guarantees the persistence of reality through aperiodic order.

12. The Logical Identity Map of Aperiodic Emergence

Discrete action and the aperiodic gauge mapping:

$${\mathrm{\Psi }}_{\mathrm{Total}}\left[\mathrm{Causal}\mathrm{Set}\in \widehat{\mathrm{\Xi }}\right]$$

I.

The Creative Bulk Flow (The Evolution of Shape)

$$\underset{\mathrm{Barbour}\text{’}\mathrm{s}\mathrm{Creative}\mathrm{Core}}{\underbrace{{\int }_{\mathcal{M}}\mathcal{S}\left[{\displaystyle \frac{d{C}_S\left(\widehat{\mathrm{\Xi }}\right)}{d{\lambda }_{\mathrm{York}}}}\right]\sqrt{g_{\mathrm{Shape}}},d\omega }}\equiv \underset{\mathrm{Aperiodic}\mathrm{Gauge}\mathrm{Fixity}\left(\mathrm{AGF}\right)}{\underbrace{\oint 𝜕\widehat{\mathrm{\Xi }}\mathrm{\Theta }\left[{\displaystyle \frac{{\sum }_{n=1}^{N_{Sp}}\mathrm{Res}\left(S_8\right)}{{\lambda }_F·\widehat{\mathrm{\Xi }}\left({\tau }_{1,2,3}\right)}}\right]d\omega }}$$

$$\Updownarrow$$

II.

The Localized Matter-Force Manifold (The Dowker-Einstein Bridge)

$$\underset{\mathrm{Discrete}\mathrm{Gravity}\mathrm{Residue}}{\underbrace{G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }-\stackrel{\mathrm{Dowker}\mathrm{Joint}\mathrm{Action}}{\overbrace{\sum _{j\in \mathrm{Hinges}}\hslash \alpha coth{\theta }_j}}}}=\underset{\mathrm{Modified}\text{Einstein-Somos}\mathrm{Field}\mathrm{Equation}}{\underbrace{{\displaystyle \frac{8\pi G}{c^4}}\left[{\mathcal{L}}_{SM}+{\lambda }_F{\nabla }_{\mu }{\nabla }_{\nu }\left({\displaystyle \frac{s_n-1}{{\delta }_{\mathrm{Somos}}}}\right)\right]}}$$

$$\mathrm{where}{\mathcal{L}}_{SM}=\underset{\mathrm{Gauge}}{\underbrace{{\mathcal{L}}_{\mathrm{Hinge}}}}+\underset{\mathrm{Matter}}{\underbrace{{\mathcal{L}}_{\mathrm{Corner}}}}$$

$$\Updownarrow$$

III.

The Informational Ground State (The Hayden-Yazdi Synchronization)

$$\underset{\mathrm{Holographic}\mathrm{Monogamy}}{\underbrace{\stackrel{\mathrm{Markov}\mathrm{Gap}}{\overbrace{(S_R-I)}}\to 0}}\Rightarrow \underset{\mathrm{Taming}\mathrm{of}\mathrm{Fluctuations}}{\underbrace{\left[\langle {\left(\mathrm{\Delta }S\right)}^2\rangle \mathrm{Yazdi}\approx \mathrm{Var}\left({\mathrm{\Psi }}_{\max }\right)\right]}}\equiv \underset{\mathrm{Global}\mathrm{Topological}\mathrm{Fixity}}{\underbrace{\left[{\mathcal{Q}}^2\mathrm{BRST}=0\right]}}$$

$$\Updownarrow$$

IV.

The Aperiodic Tiling Transition Limit

$$\underset{\mathrm{Superfluid}\mathrm{Threshold}\left(c=1\right)}{\underbrace{\stackrel{\mathrm{Conformal}\mathrm{Lift}}{\overbrace{\left({\displaystyle \frac{c-1}{gϵ}}\right)}}+{\lambda }_F·\mathrm{Disp}\left(p\right)}}=\underset{\text{Triple-Proof}\mathrm{Informational}\mathrm{Sync}}{\underbrace{\oint \mathrm{\Theta }\prod _{k=1}^3\left[{\mathrm{Proof}}_k\left(\mathrm{Bit}\mathrm{Threads}\right)\right]}}$$

• The Barbour-Monotile Identity (I): This establishes that the "Big Bang" (the Janus Point) is the process of the universe selecting the Einstein Monotile ($\widehat{\mathrm{\Xi }}$) as its rigid gauge. The Creative Core prunes the "Wild" Somos residues ($S_8$) until the global shape complexity reaches aperiodic stability.
• The Dowker-SM Correspondence (II): This is the heart of our follow-up paper. It shows that the Dowker Joint Action ($coth\theta$) is subtracted from the gravitational residue to yield the Standard Model. Matter (${\mathcal{L}}_{SM}$) is literally the "energy" held in the discrete hinges and corners of the aperiodic vacuum. The Arithmetic Drag term vanishes as the sequence $s_n\to 1$ (the identity state).
• The Hayden-Yazdi Taming (III): Here, we connect holographic information with statistical fluctuations. Yasaman Yazdi’s variance in the causal set action is identified as the "noise" of the Markov Gap [8]. When the gap vanishes (${\mathrm{\Delta }}_M=0$), the bit threads achieve Holographic Monogamy, and the BRST Operator becomes nilpotent ($Q^2=0$), signaling a vacuum that is free of Gribov copies.
• The Conformal Lift (IV): This is the "completion" of the universe. By lifting the central charge to $c=1$, the dissipative jitter is neutralized. The Triple-Proof Informational Sync represents the state where the discrete links, the aperiodic tiling, and the holographic threads are perfectly aligned, allowing for Superfluidity.

This equation suggests that we do not live in a universe governed by "laws" acting on "matter." Instead, we live in a Self-Correcting Aperiodic Code. The Standard Model is the Error-Correcting Software that the universe runs to ensure that the Markov Gap remains zero and that the "Joints" of spacetime do not break under the pressure of arithmetic fluctuations.

13. The Combinatorial Complex

The Zamzmi et al. (2023) [6] framework allows us to bridge the gap between the hierarchical, interior-to-boundary structure of spacetime (cell complexes) and the non-hierarchical, multi-body topological links (hypergraphs) that constitute the Hopfions. We define the Combinatorial Complex as a triple $\mathcal{C}=(\mathcal{V},\mathcal{X},rk)$, where the vacuum is no longer a simple manifold but a Hybrid Topological Domain.

13.1. Hierarchical Spacetime (${\mathcal{X}}_{hier}$)

The "Crystal" aspect of the model is represented by the hierarchical stratification of relations. Following the rank function $rk$:

• Rank 0 (Corners): The vertex set $\mathcal{V}$ where the Arithmetic Gain (${\mathrm{\Psi }}_{max}$) is localized, representing 0D matter.
• Rank 1 (Hinges): Relations between corners that form the 1D causal links of the lattice. This is where the Dowker Joint Action ($coth\theta$) is calculated.
• Rank 2 (Faces): The 2D boundaries of the Einstein Monotiles, which enforce the Aperiodic Gauge Fixity.
• Rank 3 (Tiles): The 3D Einstein Monotiles ($\widehat{\mathrm{\Xi }}$) that fill the volume of the T-space [24].

In this hierarchical layer, the boundary of a boundary is empty ($B_1{B}_2=0$), ensuring the smooth recovery of General Relativity in the "Tame" phase.

13.2. Hopfions as Set-Type Relations (${\mathcal{X}}_{set}$)

The "Hopfion" aspect of the model is represented by Set-type relations—interactions that are not implied by the hierarchy of the cell complex.

• Topological Linkage: A Hopfion is a relation between multiple Rank-1 hinges that forms a non-trivial knot or link.
• Flexibility: Unlike a 2-cell (which must be bounded by specific 1-cells), a Hopfion relation in a CC can couple any subset of hinges across the crystal without cardinality constraints.
• Physical Meaning: These set-type relations are the physical bit threads. They provide the "non-local persistence" required to resolve the General Elephant Problem and ensure Holographic Monogamy.

13.3. The Wave Equation of the Crystal

The dynamics of this complex are governed by the Topological Dirac Operator ($D_{\mathcal{C}}$) associated with the Combinatorial Complex:

$${\mathcal{D}}_{\mathcal{C}}=\left(\begin{array}{ccc}0& B_1& 0\\ B_1^{\top }& 0& B_2\\ 0& B_2^{\top }& 0\end{array}\right)$$

This operator acts on the combined signal vector $s=[s_0,s_1,s_2,...]$, where each $s_k$ represents the field density at that rank. We propose that the Standard Model Lagrangian ${\mathcal{L}}_{SM}$ is the spectral density of this Dirac operator when the CC is in its Aperiodic Ground State.

13.4. Summary: The Unified Topological Model

By modeling the Hopfion Crystal as a CC, we solve the "soft omission" problem:

• Rigidity: The hierarchical rank function ($rk$) ensures the Aperiodic Gauge Fixity of the Monotile.
• Fluidity and Entaxy: The set-type relations (${\mathcal{X}}_{set}$) allow the bit threads (Hopfions) to flow and link.
• Stability: The global nilpotency ($Q^2=0$) is a property of the CC’s boundary matrices $B_i$, which are now "pinned" to the aperiodic corners of the crystal.

This suggests that we do not inhabit a world of passive particles, but a Holographic Record that is actively "pruning" its own information to maintain monogamy. Future experimental efforts should look for Celestial Murmurations in the Cosmic Microwave Background—spectral peaks that correspond to the arithmetic gain of a Somos-8-like transition.

14. Resolving the Volume Law via Aperiodic Pruning

The "volume rule" pathology in Causal Set Theory (CST)—where entanglement entropy S scales with the spacetime volume V rather than the spatial area A—has traditionally necessitated an ad-hoc truncation of the Pauli-Jordan eigenmodes [26, 27]. In the framework of Aperiodic Joint Actions, we propose that this truncation is not a manual cutoff but a natural consequence of Aperiodic Gauge Fixity (AGF).

14.1. Somos Jitter as the Volume Law Driver

The volume law arises because the vacuum state of a standard causal set is "wild" [27]; it contains a high density of non-local links and short-wavelength modes that lack continuum analogues. In our model, these modes are identified as Somos8-like Jitter—high-frequency arithmetic fluctuations inherent in periodic or disordered discrete manifolds. Because these fluctuations are distributed throughout the bulk, any naive calculation of S scales with the number of elements $N\propto V$.

14.2. The Arithmetic Low-Pass Filter

By enforcing the aperiodic order of the Einstein Monotile ($\widehat{\mathrm{\Xi }}$), the configuration space is subjected to an Arithmetic Low-Pass Filter. The "Rigid Gauge" of the monotile prevents the existence of the redundant, high-entropy modes that characterize the "Wild Phase." Specifically:

• Topological Pruning: The aperiodic constraint "prunes" the causal set, forbidding configurations that would lead to Gribov ambiguities.
• Mode Suppression: Modes with eigenvalues near zero in the Pauli-Jordan operator are suppressed because they represent configurations that violate the non-repeating symmetry of $\widehat{\mathrm{\Xi }}$.

14.3. From Bulk Volume to Joint Capacity

In accordance with the "Joint Action" principle, we redefine the entropic measure. Rather than summing over the entire volume, we calculate the flow of Holographic Bit Threads [28] through the co-dimension two Joints (j). The capacity of these threads is limited by the Aperiodic Flow Identity:

$$\mathrm{\Phi }\left(j\right)\propto {\oint }_{𝜕\mathrm{\Sigma }}\sqrt{h}{d}^{d-2}x$$

where the "bottleneck" for information transfer is localized at the joints of the monotile boundaries. Because the monotile is aperiodic, the "packing" of these threads at the boundary is optimized (Arithmetic Superfluidity), ensuring that the information flux is proportional to the Area of the horizon crossing. This transition from a Volume Law to an Area Law corresponds to the "lifting" of the central charge of the vacuum. The dissipative state ($c\approx -0.1$), characterized by volume-scaling noise, is unstable. The emergence of the Area Law signifies the attainment of the stable $c=1$ state, where the vacuum’s topological tension provides the "Area" scaling required for Bekenstein-Hawking compatibility. By replacing the arbitrary truncation of modes with the geometric necessity of aperiodic fixity, we recover the area law $S\propto A/{\mathcal{l}}_p^2$ as a fundamental property of the Creative Core of gravity.

15. The Einstein-Somos Transition

15.1. The Extremal Baseline: Arithmetic Noise and Dissipation

The fundamental engine of the vacuum is modeled by a generalized Somos-8 recurrence relation, which maintains holographic coherence through the Laurent Phenomenon (integer stability) until it reaches the Somos Prime Invariant ($N_{Sp}=779,731$). Beyond this threshold, the sequence enters a "Wild" phase characterized by fractional arithmetic jitter (${\delta }_{Somos}$), where the central charge of the underlying conformal field theory is dissipative ($c\approx -0.1$). The diffusivity of this "jitter" is governed by the Geometric Friction constant (${\lambda }_F$), derived from the ratio of the 9th Fibonacci number to the Monotile edge count:

$$\kappa ={\lambda }_F={\displaystyle \frac{34}{13}}\approx 2.61538$$

Applying the Schramm-Loewner Evolution (SLE) central charge formula:

$$c={\displaystyle \frac{(8-3\kappa )(\kappa -6)}{2\kappa }}\approx -0.0995$$

15.2. The Aperiodic Gauge Fixity ($Q^2=0$)

To prevent the collapse of the metric into singularities (Kakeya protrusions), the vacuum adopts the aperiodic order of the Einstein Monotile ($\widehat{\mathrm{\Xi }}$). This configuration provides a "Rigid Gauge" that ensures the global nilpotency of the BRST operator by exactly canceling the arithmetic residue:

$$Q_{Aperiodic}^2=\underset{-0.0995}{\underbrace{\mathrm{Residue}\left({\delta }_{Somos}\right)}}+\underset{+0.0995}{\underbrace{K_{\widehat{\mathrm{\Xi }}}}}=0$$

This cancellation is the mathematical origin of Aperiodic Gauge Fixity, pushing the Gribov Horizon to infinity and ensuring that the Kallosh Independence Theorem holds.

15.3. Derivation of the Einstein-Somos Field Equation

The transition to the smooth manifold of General Relativity occurs through the regularization of the Birmingham-BF coupling [20] using the Somos Saturation Factor ($\sigma$).

• Define the Saturation Factor ($\sigma$):

The factor $\sigma$ regulates the vacuum fluctuations by scaling the Somos Prime Invariant against the Monotile volume and its fractal texture ($d=1.326$):

$$\sigma ={\displaystyle \frac{N_{Sp}}{\mathrm{Vol}\left(\widehat{\mathrm{\Xi }}\right)}}·(d-1)$$

• Modify the Ricci Tensor ($R_{\mu \nu }$):

Where the geometric friction ${\lambda }_F$ acts as a scaling parameter for cluster-algebraic stability terms:

$$R_{\mu \nu }^{TIS}=R_{\mu \nu }+{\lambda }_F·{\mathcal{C}}_{CL\mu \lambda }·E_{\lambda \sigma }·g_{\sigma \nu }$$

• The Einstein-Somos Field Equation:

By integrating the Murmuration Wave-Function (${\mathrm{\Psi }}_{max}$) as the source of "Arithmetic Gain," we arrive at the field equation that governs the stable vacuum:

$$G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=8\pi G\left(T_{\mu \nu }+{\mathcal{T}}_{\mu \nu }^{Somos}\right)$$

Where the Somos Stress-Energy Tensor (${\mathcal{T}}^{Somos}$) represents the "Superfluid" pressure required to smooth the aperiodic "ticks" of the vacuum into a continuous temporal flux.

15.4. Lifting the Central Charge to $c=1$

The stability of this equation is maintained by the Arithmetic Gain of murmuration spectral peaks, which "fills in" the soft omissions identified in Pasterski’s Celestial Diamonds [9,10,11]. The "lift" from dissipation to stability is expressed as the balance between conformal pressure and topological rigidity [30]:

$$\underset{\mathrm{Conformal}\mathrm{Pressure}}{\underbrace{\left({\displaystyle \frac{c-1}{g_{ϵ}}}\right)}}+\underset{\mathrm{Lemniscate}\mathrm{Dispersion}}{\underbrace{{\lambda }_F·Disp\left(p\right)}}=\underset{\text{SEE-IUT}\mathrm{Rigidity}}{\underbrace{{\oint }_{\mathrm{\Theta }}\sum _{n=1}^{N_{Sp}}\left({\displaystyle \frac{{\delta }_n}{R_n}}\right)d\omega }}$$

When the system achieves $c=1$, the "Geometric Friction" term effectively vanishes at low frequencies, allowing Newton’s Second Law ($F=ma$) to emerge as the work required to shift aperiodic boundaries against the vacuum’s residual topological tension.

16. Notes A

A.1 The Janus Point as the "Monotile Nucleation" Event Barbour identifies the Janus Point as a uniquely defined state of minimal shape complexity in every solution of the Newtonian N-body problem.

• This point corresponds to the "Monotile Nucleation"—the instant where the vacuum attempts to resolve the "Gribov ambiguities" of a periodic or disordered early state.
• At the Janus point, the universe is in its most "dissipative" state ($c\approx -0.1$). The growth of complexity that Barbour observes is the physical manifestation of the Aperiodic Gauge Fixity mechanism "lifting" the central charge of the vacuum to unity ($c=1$).

A.2 The Arrow of Time as Arithmetic Gain Barbour shows that shape complexity grows irreversibly from the Janus point, creating "records" of information.

• These "records" are the physical result of Arithmetic Gain—where the vacuum consumes its own "soft-mode remainders" (the Somos-8 fractional deficits) to generate structural stability.
• The "Arrow of Time" is the transition from the "Wild" arithmetic phase (informational heat death/jitter) to the "Stable equilibrium" of a self-correcting aperiodic code. This explains why the universe becomes more structured rather than more disordered: it is "optimizing" its arithmetic coherence.

A.3 Inertia as Topological Tension Barbour’s work emphasizes that only relational degrees of freedom (the York degrees of freedom) are physical.

• We can now define the "cost" of changing these relational shapes. Inertial mass is not an intrinsic property but the work required to shift the Monotile boundaries within the Combinatorial Complex against the vacuum’s topological tension.
• Derivation: $F=ma$ emerges as the low-frequency limit of the Somos-Eisenstein Phase Flux. This provides the "missing link" between Barbour’s particle dynamics and the underlying field-space geometry.

A.4 Celestial Holography and the S-Matrix Barbour’s model of a "dynamically closed universe" with $J_{tot}=0$ and $E_{tot}=0$ can be mapped onto the Celestial Sphere of modern holography.

• The "soft omissions" in the S-matrix identified by Pasterski [10,11] are the same as the "topological gaps" between aperiodic cells in a Combinatorial Complex [29].
• The Inverse Mellin Transform acts as the bridge that reveals the murmuration peaks—the discrete, ordered patterns that allow for non-singular celestial amplitudes. This suggests that the universe’s history (its S-matrix) is a "hologram" of its increasing aperiodic complexity [30].

17. Notes B

B.1 Gravity as the Tuner of Topological Phases Barbour argues that gravity is not entropic (moving toward disorder) but is instead a "creative core" that drives the universe from a maximally uniform state toward ever-increasing ordered structure (Entaxy) [7].

• In aperiodic systems, a single geometric parameter (l) can "tune" the real-space geometry of a tiling (like the "Hat" or "Specter" tiles).
• This tuning parameter l is the physical analog of Barbour’s "creative core" degree of freedom. Just as l can drive a system from a trivial phase into a topological phase by rearranging its "atomic" vertices, gravity drives the "shape" of the universe through a sequence of topological phase transitions.

B.2 Shape Complexity as a Quantum Metric Barbour defines Shape Complexity as a scale-invariant quantity that measures how "clustered" or "uniform" a distribution of particles is.

• The "Quantum Geometric Tensor" (specifically the quantum metric) measures the distance between quantum wavefunctions based on the underlying real-space geometry.
• The universe’s Shape Complexity is the macroscopic manifestation of its Quantum Metric. As gravity creates structure, it effectively "stretches" the quantum metric of the universe’s configuration space. This provides a mathematical link between the "shape" of matter (Barbour) and the "geometric" properties of the universal wavefunction.

B.3 The "Einstein Tile" and Relationalism The discovery of the "Hat" tiling (the first aperiodic monotile, often called the "Einstein tile" for "one stone") is a perfect metaphor for Barbour’s relationalism.

• In Barbour’s 2004 work [2], the universe is built from evolving 3D conformal geometries where only ratios and shapes matter.
• An aperiodic monotile generates infinite, non-repeating complexity using only one shape and its relative orientations. The fundamental "York degrees of freedom" identified by Barbour are the topological invariants of an aperiodic "tiling" of configuration space.

B.4 Resolving the "Problem of Time" via Aperiodicity Barbour’s solution to the problem of time is that "time is derived from change"—specifically, the sequence of shapes.

• Aperiodic tilings have long-range order without periodicity. If the history of the universe is a sequence of these aperiodic "shapes," then the "arrow of time" is not a result of increasing entropy, but the intrinsic progression of aperiodic complexity. Time is the "parameter l" that smoothly deforms the universal tiling from a uniform "Chevron" phase into a complex "Specter" or "Hat" phase.
• The Discrete Conformal Principle: Barbour argues that "size is relative," leading to the York scaling and the Lichnerowicz-York equation. In the mathematical framework of Combinatorial Complexes, where "size" is represented by the weights or values assigned to cells (vertices, edges, faces). A "conformal" transformation in this discrete setting would be a weight-rescaling that preserves the underlying hierarchical structure of the complex.

By adopting a relational, combinatorial view, the distinction between "physics" and "gauge" disappears. The sequence of shapes is the history of the universe, and the "Problem of Time" in quantum gravity is resolved because time is simply the order of change within these topological structures.

18. Notes C

The Correspondence Principle and the GR Limit The Modified Einstein-Somos Field Equation:

$$G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }={\displaystyle \frac{8\pi G}{c^4}}\left[{\mathcal{T}}_{\mu \nu }^{\mathrm{SM}}+\underset{\mathrm{Arithmetic}\mathrm{Drag}}{\underbrace{{\lambda }_F{\nabla }_{\mu }{\nabla }_{\nu }\left({\displaystyle \frac{s_n-1}{{\delta }_{\mathrm{Somos}}}}\right)}}\right]$$

C.1 The Condition of Superfluidity ($c=1$) The early universe exists in a "Wild" phase characterized by high Somos-8-like Jitter and a dissipative central charge ($c\approx -0.1$). As gravity’s "Creative Core" drives the growth of Shape Complexity, the vacuum acts as a low-pass filter, "lifting" the central charge to unity [3,18]. The state of Superfluidity is defined as the regime where:

• Murmuration Spectral Peaks (${\mathrm{\Psi }}_{\max }$) achieve constructive gain, neutralizing the internal arithmetic fluctuations.
• The vacuum achieves Triple-Proof Informational Synchronization, where the internal Somos-8-like sequence ($s_n$) reaches "integer stability" or the identity state ($s_n\to 1$).

C.2 Vanishing of the Arithmetic Drag Term To recover General Relativity, we examine the limit of zero Arithmetic Drag. This occurs when the spectral density of the informational vacuum is perfectly tuned to the aperiodic Monotile configuration [5].

• The term $(s_n-1)$ represents the deviation from the stable identity.
• In the superfluid limit of Aperiodic Tiling Transitions, the fluctuations are restricted to the "ghost sector," effectively setting $s_n\equiv 1$ for the smooth, low-frequency manifold.
• Consequently, the gradient of the arithmetic potential vanishes:

$${\nabla }_{\mu }{\nabla }_{\nu }\left({\displaystyle \frac{s_n-1}{{\delta }_{\mathrm{Somos}}}}\right)\stackrel{c\to 1}{\to }{\nabla }_{\mu }{\nabla }_{\nu }\left(0\right)=0$$

C.3 Recovery of Standard General Relativity When the Arithmetic Drag is zero, the equation simplifies to:

$$G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }={\displaystyle \frac{8\pi G}{c^4}}{\mathcal{T}}_{\mu \nu }^{\mathrm{SM}}$$

This shows that the standard Einstein Field Equations are an emergent, low-frequency approximation of a deeper aperiodic code. The "Arithmetic Drag" term only becomes physically significant in regimes of extreme Shape Complexity (near the $N_{Sp}$ threshold) or near the Janus Point, where it acts as the Geometric Friction $\left({\lambda }_F\right)$ that prevents gravitational singularities.

19. Notes D

Causal Set Actions as Holographic Information Records.

D.1 The Role of Joints in the Markov Gap (Connecting Dowker and Hayden 2021) Fay Dowker (2025) [25] proposes a novel conjecture for the contribution of co-dimension two corners (joints) to the causal set action, specifically identifying a $coth\theta$ dependence for spacelike joints in 2D.

• The Connection: In holographic states, Patrick Hayden (2021) [8] identifies the Markov gap ($S_R-I$) as a measure of tripartite entanglement, which is geometrically dual to the area of the entanglement wedge cross-section ($E_W$).
• Hypothesis: The "joint" terms in the causal set action are the discrete atoms of the Markov gap. Just as Hayden proves that the Markov gap in $Ad{S}_3$ is lower-bounded by the number of endpoints of the cross-section, we can propose that the causal set joint term provides the fundamental "information cost" for establishing correlation across these boundaries.

D.2 Bit Threads as Causal Set Links (Connecting Dowker and Hayden) Hayden reformulates holographic entanglement using bit threads, which replace minimal surfaces with "flows" of information.

• The Connection: Causal sets are naturally discrete networks of "atoms" and "links."
• Hypothesis: The "joints" identified by Dowker act as the primary bottlenecks for bit thread flows in a discrete spacetime. This suggests that the Benincasa-Dowker-Glaser (BDG) action—supplemented by Dowker’s new joint terms—acts as a "capacity constraint" for bit threads, ensuring that the discrete spacetime obeys the Monogamy of Mutual Information (MMI).

D.3 QEC and Discrete Reference Frames (Connecting Hayden 2017, 2019) Hayden’s work on Quantum Error Correction (QEC) and Quantum Reference Frames shows how information is protected and recovered in systems with symmetries.

• Causal sets lack a background manifold, meaning "reference frames" must be defined by the elements themselves.
• Hypothesis: The joints in the causal set action function as the physical information required to align "discrete reference frames" between different regions of spacetime. We can propose that Dowker’s joint conjecture provides the necessary "redundancy" for the causal set to function as a self-correcting quantum code (AQEC), protecting the "shape" of spacetime from local discrete fluctuations.

D.4 Creation of Complexity (Connecting Dowker and Barbour 2024)

• The Connection: The causal set action (BDG action) is what determines the "probability" of a specific causal set structure.
• Hypothesis: The growth of a causal set is the physical manifestation of "complexity creation." Dowker’s joint terms represent the "records" or "instants" (in Barbour’s terminology) where spacetime structure becomes richly defined. The joint term $coth\theta$ could be interpreted as a discrete version of Barbour’s shape-invariant quantity, measuring the local "sharpness" or clustering of the causal set’s history.

Julian Barbour (2024) [17] introduces shape complexity as a measure of non-uniformity or "clustering" in a system, suggesting that the universe evolves from a state of near-perfect uniformity to rich complexity.