A Unified Framework for Space, Time, Particles, and Cosmology

May 2026 · Not peer-reviewed. Submitted for critical discussion.

[PROVED]

[CONDITIONAL]

[RESULT]

[OPEN]

Proved · Conditional on stated assumptions · Numerical result · Unresolved

Abstract

We present the Phase Oscillator Network (PON) framework, in which space, time, particles, and cosmology all emerge from a single coupling energy E = κ(1 − cos Δφ) acting between N identical phase oscillators on a fully connected graph. No geometry, no quantum mechanics, no particles, and no spacetime are assumed. Everything is derived.

The foundation is a single mathematical theorem: three is the unique integer n ≥ 2 for which n equally-spaced phases on a fully connected graph form a stable winding sector. From this fact, and the cancellation property it implies, the entire framework follows by logical necessity.

Structure formation proceeds as a three-level cascade. Triads (3-node units) form first from phase accidents in the all-to-all cloud. As triads accumulate, their collective tension forces the formation of 6-node wallpaper group structures — the first spatial units. As these accumulate, their unsatisfied out-of-plane tension forces 3D closure into 27-node space group structures — the elementary particles. Each level is forced by the tension of the level below, not by rare coincidence. Solitons (2π phase windings) propagating through the resulting relational graph define space (hop distance) and time (soliton trajectory). The speed of light c = 1 is built in. Three spatial dimensions are the unique maximum consistent with force cancellation.

The 36 chiral triadic space groups classify the stable 27-node structural units, identifying all elementary particles. Constraint level k determines mass via m ∝ 6^k. Point-group symmetry determines spin. Internal torus winding numbers give quantum numbers. Electric charge quantisation, fractional quark charges, three colour charges, and colour confinement all follow without additional postulates. Dark matter is the high-constraint subclass with no continuous internal degrees of freedom and therefore no electromagnetic coupling.

The cosmological expansion history H(z) is derived from two competing soliton mechanisms — sparse long-range connections (deceleration era) and hub-driven local connectivity (acceleration era) — achieving χ²/dof = 0.639 against 26 independent measurements, outperforming ΛCDM (0.765) with no dark energy parameter. The dark matter to baryon ratio Ω_DM/Ω_b = 5.294 follows from a soliton criticality condition, within 2% of the observed value. The vacuum energy is Planck-scale frozen phase energy trapped in every locked structure at dropout — finite without renormalisation and with equation of state w = −1 exactly. It is almost never sampled by propagating solitons: the sampling fraction scales as l_P/R_H ~ 10⁻⁶¹, giving the observed cosmological constant scale l_Λ = √(l_P R_H) from soliton path geometry in a 3D graph.

1. The Foundation: Why Only Three

1.1 The System

Consider N identical oscillators. Each node i has one degree of freedom: its phase φᵢ ∈ [0, 2π). The coupling energy between any pair of nodes is:

E_ij = κ (1 − cos Δφ_ij) where Δφ_ij = φ_j − φ_i

This is zero when phases are equal, rises to 2κ at π separation, and is identical for all pairs. The system is fully connected — every node couples to every other with the same strength κ. There is no geometry, no distance, no locality. The only nonlinearity is at Δφ = π, where the restoring force changes sign and a soliton fires: a 2π phase slip propagates through the network.

1.2 The Uniqueness of Three

Theorem 1 — Uniqueness of the Stable Winding Unit. [PROVED] Three is the unique integer n ≥ 2 such that n equally-spaced phases on the fully connected graph K_n form a stable winding sector with no antipodal pairs and self-consistent internal triangles.

Proof by exhaustion:

n = 2: phases at 0° and 180°. The pair sits at the π instability boundary — zero restoring force, any perturbation drives the system away. Not stable.
n = 3: phases at 0°, 120°, 240°. Every pair at 120°, well within the stable range. Forces sum to zero by Theorem 2 (below). Stable. ✓
n = 4: phases at 0°, 90°, 180°, 270°. Diagonal pairs sit at exactly 180° — the instability boundary. Not stable.
n = 5: no antipodal pairs, but K₅ contains triangles with pairwise separations 72°, 72°, 144°. The 144° separation violates the winding consistency condition. Not stable.
n ≥ 6: even n always has antipodal pairs. Odd n ≥ 7 has over-constrained internal triangles accumulating with n. Not stable. □

Three is not merely the minimum stable winding unit — it is the only one. The triad is forced, not chosen. This single fact is the seed from which the entire framework grows.

1.3 The Triad

A triad is three mutually coupled nodes at 120° phase separation. For phases θ, θ + 2π/3, θ − 2π/3, the energy is:

E₀ = 3κ (1 − cos 2π/3) = 3κ × 3/2 = 4.5κ

This is the fundamental energy quantum of structure formation. The triad carries discrete chirality χ = +1 (phases advance clockwise) or χ = −1 (counterclockwise). Both chiralities carry identical energy E₀ and are related by time-reversal.

1.4 The Cancellation Property

Theorem 2 — Cancellation. [PROVED] A locked triad at 120° phase separation exerts exactly zero net force on any external node m, for all phases of m.

Proof: Setting α = φₘ − φᵢ, the forces from triad nodes at 0°, 120°, 240° are κ sin α, κ sin(α − 2π/3), κ sin(α + 2π/3). Their sum:

sin α + sin(α − 2π/3) + sin(α + 2π/3) = sin α + 2 sin α cos(2π/3) = sin α (1 − 1) = 0

for all α. The cancellation is exact and topological — it follows from the 120° winding symmetry, not from any special relationship between the triad and the external node. □

Physical consequence: A formed triad is completely invisible to all other nodes. This invisibility is the foundation of the dropout process: once a triad locks, the remaining cloud has no information that anything changed.

2. The Dropout Process: A Three-Level Cascade

2.1 Two Regimes of Soliton Dynamics

Before presenting the dropout, a critical distinction must be established. There are two regimes of soliton propagation operating at different stages of the framework.

Pre-spatial solitons operate during the dropout phase, before any relational graph exists. The substrate is the all-to-all coupling K_N — every node is directly coupled to every other with equal strength κ. When a phase difference crosses π, a 2π phase slip fires and can reach any other node in one step. Distance is undefined; every node is equally close to every other. These pre-spatial solitons trigger winding events and establish the relational connections between forming structures. The universe exists in a single plankscale.

Post-spatial solitons operate after the dropout is complete and the relational graph has crystallised. These are constrained to hop between connected structures. One hop is one unit of space. These drive the cosmology of Section 8.

The transition happens at t ≈ 0 when the dropout completes. Post-spatial solitons — which need the graph — only begin propagating once the graph already exists. There is no circularity: the first structures form from pure phase relationships in the all-to-all cloud, without any prior notion of space or distance.

2.2 Level One: Triads Form First

The first dropout events are the simplest possible: a 3-node subset accidentally satisfies the triadic winding condition. The exact phase tolerance follows from the winding sector topology:

δ/2π = (π/3) / (2π) = 1/6 (exact)

The formation probability for a single triad is:

P_triad = (1/6)³ = 1/216

This is by far the highest formation probability in the framework. Once formed, a triad is locked. By Theorem 2, it exerts zero net force on all remaining free nodes. The cloud continues evolving with no information that anything has changed. Triads accumulate rapidly, forming the substrate for the next level.

2.3 Level Two: 6-Node Structures Forced by Collective Triad Tension

A locked triad satisfies its in-plane winding conditions exactly. But it exists in what will become a 3D phase space, and the out-of-plane direction is unresolved. The triad is under tension in the directions it has not yet closed.

This tension is not random. Each triad corner node has one winding condition satisfied and a second outstanding — the condition that would connect it to an adjacent triad and form a 6-node structure. The residual phase stress points in a definite direction in phase space: toward whatever satisfies the second condition.

As triads accumulate, their collective tensions add coherently — all pulling toward the same class of 6-node configurations, because they all formed from the same triadic substrate. When the triad population reaches the critical density:

n₃_critical ~ 2 × 6^(k₂D − 3)

the collective tension exceeds the constraint cost of 6-node closure. At this threshold, 6-node wallpaper group formation is no longer a rare accident — it is forced. This is a phase transition, not a probabilistic dropout. The 17 wallpaper groups represent all distinct ways this closure can occur. For less symmetric groups a small residual tension remains — which drives the next level. The 6-node structures are the first spatial units; their relational connections define the beginning of the 2D relational graph. Space begins to crystallise here, in 2D patches.

2.4 Level Three: 27-Node Structures Forced by 6-Node Tension

A locked 6-node structure has satisfied its in-plane winding conditions. But by Theorem 4 (Section 4), three winding conditions on a corner node are required for 3D closure — and the 6-node structure has provided only two. Each corner node has one outstanding condition pointing out-of-plane. The 6-node structure is under tension in the third dimension.

As 6-node structures accumulate, this tension builds coherently — all pulling toward the same out-of-plane configuration that would complete the corner consistency of Theorem 4. When the 6-node population reaches the critical density:

n₆_critical ~ 2 × 6^(k₃D − k₂D)

the collective out-of-plane tension forces 3D closure. Three 6-node layers snap into alignment at shared corner nodes, forming the minimal 3D triadic unit:

1D: 3 nodes → 2D: 9 nodes → 3D: 27 = 3³ nodes

The nucleation is sequential. Lighter space groups (lower k₃D, lower threshold) nucleate first. Heavier ones require more collective tension and nucleate later. Dark matter structures at k₃D = 19–22 nucleate last, when the tension has been building longest.

This sequential nucleation determines the particle abundance hierarchy. The formation probability (1/6)^k₃D is not a coincidence probability — it is the threshold fraction of the total cloud required to force nucleation of each particle type. Rarer particles require more collective build-up before their threshold is crossed.

2.5 The Dropout Is Instantaneous

The three-level cascade operates on timescales astronomically short compared to any cosmological scale. The rate-limiting step is the 6-node → 27-node transition, with characteristic timescale τ ≈ 10⁻⁴⁸³¹ natural time units. [RESULT] All structures form effectively at t ≈ 0. This is the Big Bang: not a singularity in spacetime geometry, but a singularity in the nucleation rate. Space and time are created in this event, level by level, as the cascade completes. When the 27-node nucleation is complete, the 3D relational graph crystallises, post-spatial solitons begin propagating, and cosmology begins.

2.6 Vacuum Energy from Trapped Phase Residuals

Every locked structure freezes at dropout with a small residual phase error ε drawn from the tolerance window [−π/3, +π/3]. This frozen excess energy:

⟨δE⟩ = (3κ/2) × ⟨ε²⟩ = κπ²/18 ≈ 0.55κ per triad

is permanently trapped. It cannot radiate — the cancellation property makes locked structures invisible. It cannot decay — topology forbids it. It does not dilute with expansion — it is internal to frozen structures. It is spatially uniform — all structures formed from the same homogeneous cloud with the same phase tolerance. These four properties are the defining properties of a cosmological constant. The trapped phase energy has equation of state w = −1 exactly, derived rather than assumed. The mechanism also provides a natural ultraviolet cutoff: the trapped energy is bounded by the tolerance window and is finite without renormalisation.

The total trapped energy density is of order the Planck density. The resolution is the sampling rate. Gravity in the PON framework is curvature of the relational graph, shaped by soliton connectivity. A structure that no soliton has ever visited does not contribute to observable curvature — its frozen energy is real but causally isolated. The fraction of structures on any soliton path scales as: f_s ~ l_P/R_H ~ 10⁻⁶¹. The vacuum energy is Planck-scale but almost never sampled.

The observable vacuum energy density is the total frozen density weighted by the sampling fraction squared. In three spatial dimensions — where the graph surface scales as R_H², specific to d = 3 — this gives:

Λ ~ ρ_Planck × (l_P / R_H)² → l_Λ = √(l_P × R_H)

This reproduces the observed geometric mean relation from a physical mechanism. Three-dimensionality is essential: the (l_P/R_H)² scaling requires graph surface ~ R^H², specific to d = 3, providing an independent consistency check on Theorem 4. [CONDITIONAL] The precise coefficient requires the gravity identification, currently [OPEN]

3. Space, Time, and the Speed of Light

3.1 Space is Hop Distance

The formed structures connect in a relational graph: two structures are connected if a soliton can travel between them. The distance between two structures is the minimum number of hops separating them. This definition gives space its fundamental properties automatically:

Discrete — there are no fractional hops.
Relational — distance is a property of the graph, not a background container.
Finite — the graph has N/27 nodes and a finite diameter.
Three-dimensional — as proved in Section 4.

3.2 Time is the Soliton Trajectory

After the dropout process has populated the relational graph, solitons propagate through it. A soliton carries exactly 2π of phase winding — a topological invariant that cannot be created or destroyed, only transmitted. There is no external clock. The soliton’s propagation through the relational graph is the causal sequence. Each hop from one structure to the next is one tick of the only available clock.

3.3 The Speed of Light is Built In

Theorem 3 — c = 1. [PROVED] The speed of causal propagation is exactly one hop per hop. This is not a derived quantity — it is the definition of both the unit of space and the unit of time.

Proof: one unit of time is one soliton hop. One unit of distance is one hop of graph separation. A soliton traverses one unit of distance in one unit of time. Therefore c = 1, identically, for any system size. The numerical value in SI units is a consequence of unit conversion, not of the physics. Causality is automatic: a structure cannot be affected by a soliton that has not yet reached it. □

3.4 Photons as Solitons

A soliton is a 2π phase winding propagating through the relational graph. It carries:

Topological charge ±1 (chirality): clockwise or counterclockwise winding — the two photon helicity states.
Speed c = 1: one hop per hop, built in by Theorem 3.
Zero rest mass: a soliton cannot be stopped — there is no static soliton state.
Continuous energy spectrum: soliton width W (in hops) sets the wavelength and hence the energy E = ℏω = 4π²κ/W.

All photon properties are soliton properties.

4. Why Exactly Three Spatial Dimensions

4.1 Multi-Layer Structures and Shared Nodes

A d-dimensional structure is built from d independent triadic layers, one per spatial direction. For the structure to be closed and stable in d dimensions, adjacent layers must share corner nodes. A corner node must simultaneously satisfy one winding condition from each adjacent layer:

φ₀ ≡ φᵢ + 2π/3 (mod 2π) for i = 1, 2, …, d

These d conditions are d equations in one unknown (φ₀). They are consistent if and only if all d partner phases are equal modulo 2π. If the d layers are genuinely independent, their partner phases are generically different.

4.2 The Proof

Theorem 4 — Unique Maximum Spatial Dimension. [PROVED] In the PON framework, the maximum number of independent triadic winding directions simultaneously realisable in a stable closed structure is exactly three.

d = 1: One condition. φ₀ = φ₁ + 2π/3. One equation, one unknown. Unique solution. ✓
d = 2: Two conditions. Consistent if φ₁ = φ₂ — achievable along shared edges of 2D wallpaper group structures. The 17 wallpaper groups are precisely these structures. ✓
d = 3: Three conditions. Consistent if φ₁ = φ₂ = φ₃. The 36 chiral triadic space groups are exactly those 3D crystallographic groups where this can be satisfied with all three layers genuinely independent. Their existence proves d = 3 is achievable. ✓
d = 4: Four conditions require φ₁ = φ₂ = φ₃ = φ₄. Four genuinely independent layers require four different partner phases. Contradiction. No stable closed d = 4 structure exists. ✗
d ≥ 4: The same argument applies for all higher d. □

Therefore d ≤ 3. And d ≥ 3 is required to carry the three independent quantum numbers of the Standard Model (electric charge, isospin, colour). [CONDITIONAL] Therefore d = 3 exactly — not by assumption, not as an initial condition, but as the unique value consistent with the framework.

4.3 The 27-Node Minimal Unit

The minimal 3D structure is a 3×3×3 arrangement of 27 nodes — three orthogonal triadic layers sharing nodes along intersection lines and a common corner. Every row, column, and pillar of 3 nodes forms a triad. This gives the generalisation:

1D: 3 nodes → 2D: 9 nodes → 3D: 27 = 3³ nodes

Each 27-node dropout event produces one unit of 3D space — one Planck volume. The universe contains N/27 Planck volumes. The identification ℏ = 6 in natural units (six nodes per minimal 2D winding unit, the action quantum of structure formation) gives N = total action of the universe. [CONDITIONAL]

5. Particles, Quantum Numbers, and the Standard Model

5.1 The 36 Chiral Triadic Space Groups

Just as the 17 wallpaper groups classify all stable 2D structures (the cross-sections of the spatial relational graph), exactly 36 space groups classify all stable 27-node particles. These are selected from the 230 crystallographic space groups by three conditions:

Chirality (Sohncke groups): no improper rotations — chirality of the triadic winding must be preserved.
Triadic rotation: the point group contains a 3-fold rotation axis — the fundamental triadic structure.
Corner consistency: the three-layer winding conditions can be simultaneously satisfied at shared corner nodes (the content of Theorem 4).

The 36 groups divide into three crystal families, each corresponding to a generation of particles:

From the coupling energy alone, the following emerge:

Space (hop distance through the relational graph of locked structures)

Time (soliton trajectory through that graph)

c = 1 (one hop per hop — built in, proved)

Causality (automatic — a structure cannot be affected before the soliton arrives)

Three spatial dimensions (proved from force cancellation and corner consistency)

17 wallpaper groups (complete vocabulary of 2D spatial structure)

36 chiral space groups (complete vocabulary of 3D elementary particles)

Three particle generations (three crystal families: trigonal, hexagonal, cubic)

Electric charge quantisation (winding numbers on internal torus T^n)

Fractional quark charges 1/3, 2/3 (Z3 symmetry of 3-fold rotation axis)

Three colour charges (three chiral C3 space groups P3, P31, P32)

Colour confinement (topological — anyonic spin of C3 groups in 3D)

Spin-1 gauge bosons (C6 space groups; massless photon from P6)

Dark matter (O-group cubic structures: no continuous quantum numbers)

Omega_DM/Omega_b = 5.294 (soliton criticality at z=6; 2% from observation)

Baryon fraction f_b = 0.159 (observed 0.156 — 2%)

H(z) expansion history (two-mechanism model: chi2/dof = 0.639 vs LCDM 0.765)

Cosmological constant Λ ~ (l_P/R_H)² (Planck energy, rarely sampled — geometric mean l_Λ = sqrt(l_P x R_H))

Muon/electron mass ratio ~ 216 (observed 207 — 4%)

What Remains Open

Exact particle mass ratios from spectral geometry of space group fundamental domains.
The fine structure constant α ≈ 1/137 from the soliton-matter interaction vertex.
Gravity from relational graph curvature; graviton as I4₁32 cubic space group.
Neutrino mixing equilibration and N_eff = 3.
The H(z) 3D recalibration with 27-node structures.
The baryon-to-photon ratio η ≈ 6×10⁻¹⁰ from soliton emission rates.
The Weinberg angle from C₆ space group geometry.

Everything else from: E = κ(1 − cos Δφ)