Origin of the Universe

Section Status: Theorems + Philosophical Interpretation

This section contains a proven theorem (instability of the Source [T]), postulates (the Source itself), and philosophical interpretations (origin of "nothing"). Arguments about the origin of the Source are metaphysical in nature, but the instability of $\Gamma_{\odot}$ under the full UHM dynamics is a rigorous mathematical result.

The Problem of Beginning

Traditionally one asks: "What was before the Big Bang?"

In UHM the question transforms:

What is the structure of $\Gamma$ in the limit of minimal differentiation?

The Primordial State

The Source

Status: Postulate

The Source is postulated, not derived. It is the initial condition of the theory, not a consequence of it. The question "why this particular Source?" remains open.

Pure undifferentiated state [P] — superposition of all dimensions with equal amplitudes:

$$\Gamma_{\odot} = |\psi_{\odot}\rangle\langle\psi_{\odot}|, \quad |\psi_{\odot}\rangle = \frac{1}{\sqrt{7}} \sum_i |i\rangle$$

Properties:

• Purity: $P = \mathrm{Tr}(\Gamma_{\odot}^2) = 1$ (pure state)
• Maximal coherence: all $|\gamma_{ij}| = 1/7$
• Minimal differentiation: $D_{\text{diff}} = 1$

Why not a mixed state?

The maximally mixed state $\Gamma = I_7/7$ would have $P = 1/7$ — this is not a coherent state, but a classical ensemble without quantum correlations. UHM takes a pure superposition as the Source.

Open questions:

• Why equal amplitudes $1/\sqrt{7}$ specifically?
• Are alternative initial states possible?
• Connection to the Boltzmann Brain problem?

Spontaneous Symmetry Breaking

Instability of the Source

The Source is unstable [T] under the full dynamics of UHM. Any initial condition $\Gamma(0) = \Gamma_{\odot}$ with $\Delta F > 0$ inevitably evolves toward the structured attractor $\rho^*$.

Theorem (Source Instability)

Theorem. The state $\Gamma_{\odot}$ is unstable under the full UHM dynamics: $\frac{d\Gamma}{d\tau}\big|_{\Gamma_\odot} \neq 0$, the system drifts from $\Gamma_{\odot}$ at a finite rate, and $\kappa_0$ creates positive feedback that breaks $S_7$-symmetry.

Proof (three steps).

Step 1. $\Gamma_{\odot}$ is not a stationary state.

Compute $\frac{d\Gamma}{d\tau}\big|_{\Gamma_\odot}$ from the three terms of the evolution equation:

(a) Unitary term: $-i[H_{\text{eff}}, \Gamma_{\odot}]$. Since $H_{\text{eff}} = \sum_i \omega_i |i\rangle\langle i| + \ldots$, with unequal $\omega_i$ (guaranteed by the $G_2$-structure: $\lambda_E > \lambda_U > \lambda_L \geq \lambda_D \geq \lambda_S \geq \lambda_A \geq 0$ from A5), the commutator is non-zero — $\Gamma_{\odot}$ does not commute with $H_{\text{eff}}$.

(b) Dissipative term: $\mathcal{D}_\Omega[\Gamma_{\odot}]$. By theorem T6 (uniform contraction):

$$\mathcal{D}[\Gamma_{\odot}]_{ij} = \begin{cases} -\alpha \cdot \gamma_{ij} = -\alpha/7, & i \neq j \\ 0, & i = j \end{cases}$$

This is a non-zero operator: $\|\mathcal{D}[\Gamma_{\odot}]\|_F^2 = \alpha^2 \cdot 42/49 > 0$. Dissipation destroys coherences, reducing purity.

(c) Regenerative term: $\mathcal{R}[\Gamma_{\odot}, E] = \kappa(\Gamma_{\odot}) \cdot (\rho^* - \Gamma_{\odot}) \cdot g_V(P)$. For $P > P_{\mathrm{crit}}$ this term is non-zero, since $\rho^* \neq \Gamma_{\odot}$ (primitivity [T]: the unique stationary point $\rho^*$ is not a pure $S_7$-symmetric state).

Result: $\frac{d\Gamma}{d\tau}\big|_{\Gamma_\odot} \neq 0$ — $\Gamma_{\odot}$ is not a fixed point.

Step 2. Linearization around $\Gamma_{\odot}$.

Write $\Gamma = \Gamma_{\odot} + \delta\Gamma$, $\text{Tr}(\delta\Gamma) = 0$, $\delta\Gamma^\dagger = \delta\Gamma$. Linearized dynamics:

$$\frac{d\delta\Gamma}{d\tau} = \underbrace{-i[H_{\text{eff}}, \delta\Gamma]}_{\text{rotation}} + \underbrace{\mathcal{D}_{\text{lin}}[\delta\Gamma]}_{\text{contraction}} + \underbrace{F_0 + \mathcal{R}_{\text{lin}}[\delta\Gamma]}_{\text{shift + regeneration}}$$

• Unitary contribution: purely imaginary eigenvalues $\pm i(\omega_i - \omega_j)$ — rotations, do not change the distance from $\Gamma_{\odot}$.
• Dissipative contribution: $\mathcal{D}_{\text{lin}}[\delta\Gamma]_{ij} = -\alpha \cdot \delta\gamma_{ij}$ ($i \neq j$), $= 0$ ($i = j$). Eigenvalues: $-\alpha < 0$ for 42 off-diagonal components, $0$ for 6 diagonal ones.
• Constant shift: $F_0 = \kappa(\Gamma_{\odot})(\rho^* - \Gamma_{\odot}) \cdot g_V(P) \neq 0$ — a constant vector, independent of $\delta\Gamma$. This is a drift from $\Gamma_{\odot}$ in the direction of $\rho^*$.

Step 3. Mechanism of instability: drift + breaking of $S_7$-symmetry.

Even if linearized eigenvalues have $\text{Re}(\lambda) \leq 0$ (true for $\mathcal{D}$), instability arises from two mechanisms:

(I) Non-stationarity. $F_0 \neq 0$ means the system drifts from $\Gamma_{\odot}$ at a finite rate:

$$\|\Gamma(\delta\tau) - \Gamma_{\odot}\| \geq \|F_0\| \cdot \delta\tau - O(\delta\tau^2)$$

for small $\delta\tau > 0$. The drift is linear in time.

(II) Breaking of $S_7$-symmetry via $\kappa_0$. As soon as $\Gamma$ deviates from $\Gamma_{\odot}$, the formula $\kappa_0 = \omega_0 |\gamma_{OE}||\gamma_{OU}|/\gamma_{OO}$ (see categorical derivation) breaks $S_7$-symmetry: E and O are functionally distinguished. This creates positive feedback: deviation in the E-direction increases $\text{Coh}_E \to$ increases $\kappa \to$ increases regeneration in the E-direction.

Formally: the component $\delta\gamma_{EE}$ obeys the equation (to linear order):

$$\frac{d\delta\gamma_{EE}}{d\tau} = \kappa_0 \cdot (\rho^*_{EE} - 1/7) + \text{terms} \propto \delta\gamma_{EE}$$

The first term is a constant shift ($\rho^*_{EE} > 1/7$ for living systems). The second is feedback via $\partial\kappa/\partial\gamma_{EE} > 0$. Both increase $\delta\gamma_{EE}$.

(III) Result. The distance from $\Gamma_{\odot}$ grows monotonically:

From steps I–II we obtain $\|\Gamma(\tau) - \Gamma_{\odot}\|_F > 0$ for all $\tau > 0$. Since $d_B(\Gamma_1, \Gamma_2) > 0 \Leftrightarrow \Gamma_1 \neq \Gamma_2$ (the Bures metric is non-degenerate), from $\|\Gamma(\tau) - \Gamma_{\odot}\|_F > 0$ it follows:

$$d_B(\Gamma(\tau), \Gamma_{\odot}) > 0 \quad \forall \tau > 0$$

for any initial condition $\Gamma(0) = \Gamma_{\odot}$ with $\Delta F > 0$. The system inevitably leaves $\Gamma_{\odot}$ and converges to $\rho^*$. $\blacksquare$

Corollary: cosmogenesis as inevitability

The transition from the undifferentiated Source to structured configurations is not a random event, but a mathematical inevitability of UHM dynamics. From $\Gamma_{\odot}$ the system always evolves to $\rho^*$ (given $\Delta F > 0$).

Open question

The mechanism by which $\Delta F > 0$ arises in the primordial context is an open problem [P]. The instability theorem assumes $\Delta F > 0$; the question of why this condition holds lies beyond the scope of this result.

Self-Amplification

tip

Status: [T] (via $\kappa_0$)

Positive feedback is proved in step 3(II) of the instability theorem: the formula $\kappa_0 = \omega_0 |\gamma_{OE}||\gamma_{OU}|/\gamma_{OO}$ breaks $S_7$-symmetry and creates amplification in the E-direction.

Symmetry breaking self-amplifies [T] via positive feedback through $\kappa_0$:

Mechanism: $\kappa_0$ functionally distinguishes E and O among the seven dimensions (see categorical derivation of $\kappa_0$), directing evolution from the $S_7$-symmetric $\Gamma_{\odot}$ toward the structured $\rho^*$ with pronounced E-coherence.

Birth of Dimensions

From the primordial superposition the seven dimensions emerge:

$$|\psi_{\odot}\rangle = \frac{1}{\sqrt{7}}(|A\rangle + |S\rangle + |D\rangle + |L\rangle + |E\rangle + |O\rangle + |U\rangle)$$ $$\downarrow \text{decoherence via } \mathcal{D}[\Gamma]$$ $$\Gamma \to \sum_i p_i |i\rangle\langle i| + \sum_{i \neq j} \gamma_{ij} |i\rangle\langle j|$$

where $p_i = \gamma_{ii}$ are the dimension populations, $\gamma_{ij}$ are the coherences between them.

Evolution from the Source

Direction of Evolution

Warning: Non-Falsifiability

The statement $dD_{\text{diff}}/d\tau > 0$ is non-falsifiable: any observed decrease in differentiation can be interpreted as a local phenomenon within a global growth. This is a teleological assumption, not an empirical law.

Honest status: This is a philosophical position (directionality of evolution), not a formal UHM theorem.

The universe evolves in the direction of increasing differentiation while preserving integration:

$$\frac{dD_{\text{diff}}}{d\tau} > 0$$ $$\frac{d\Phi}{d\tau} \geq 0$$

where:

• $D_{\text{diff}} = \exp(S_{vN})$ — measure of differentiation (diversity of states)
• $\Phi$ — measure of integration (connectedness of dimensions)

Status: [H] Hypothesis. The connection to the second law of thermodynamics is conceptual, not formalized.

On notation

$D_{\text{diff}}$ is the measure of differentiation. Not to be confused with the Dynamics dimension $D$ (one of the seven Holon dimensions).

Phenomenology of Evolution

• Complexification of matter: from quarks to galaxies
• Evolution of life: from prokaryotes to minds
• Development of culture: from tribes to civilizations

Cosmogenesis Diagram

Diagram notation:

• $P = 1$ — purity (maximal coherence)
• $D_{\text{diff}} = 1$ — minimal differentiation (pure state)
• $\varphi(\Gamma) \approx \Gamma$ — self-modeling close to fixed point

Quantitative Estimates for the Cosmogenesis Epoch

Status: [C] Conditional estimates

The following estimates depend on the value of $\omega_0$ (Axiom A4 [P]) and on the model relating $\Delta F$ to physical scales. The orders of magnitude are approximate.

Differentiation Time

The characteristic instability scale is set by the spectral gap of the linearized dynamics:

$$\tau_{\text{diff}} \sim \frac{1}{\kappa_0} = \frac{7}{\omega_0} \cdot \frac{\gamma_{OO}}{|\gamma_{OE}||\gamma_{OU}|}$$

For the Source ($|\gamma_{OE}| = |\gamma_{OU}| = \gamma_{OO} = 1/7$):

$$\tau_{\text{diff}} \sim \frac{7}{\omega_0} \cdot \frac{1/7}{(1/7)^2} = \frac{7}{\omega_0}$$

At $\omega_0 \sim M_{\text{Planck}} = 1.22 \times 10^{19}$ GeV: $\tau_{\text{diff}} \sim 7 t_{\text{Planck}} \approx 3.8 \times 10^{-43}$ s.

Sequence of Events

Epoch	$\tau$	Event	Observable analogue
$0$	$0$	Source $\Gamma_\odot$	Planck singularity
$\sim 7 t_P$	$\sim 10^{-43}$ s	Breaking $S_7 \to G_2$	Inflation (?)
$\sim 10^2 t_P$	$\sim 10^{-42}$ s	Sector decomposition $7 = 1+3+3$	Spacetime formation
$\sim 10^{10} t_P$	$\sim 10^{-34}$ s	Electroweak scale	Higgs transition

Connection to inflation

The breaking of $S_7$-symmetry via $\kappa_0$ generates exponential growth of differentiation — this is the structural analogue of inflation. However in UHM, inflation is not a separate field (inflaton) but a consequence of autopoietic feedback. The detailed connection to observable parameters ($n_s$, $r$) is an open problem [P].

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Absence of "Before"

In UHM there is no "before the Big Bang":

• Time arises together with differentiation — via the Page–Wootters mechanism, which requires the O-dimension to be distinguished
• "Before" is a concept that requires time — in the Source all dimensions are equivalent, O is not distinguished
• The Source $\odot$ is outside of time (atemporal): $\tau$ is not defined for an $S_7$-symmetric state

Why Is There Something Rather Than Nothing?

Traditional question: "Why is there something rather than nothing?"

Status: Philosophical argument

This is not a formal theorem, but a philosophical position consistent with the UHM axiomatics. Formal proof is impossible — the question lies beyond any formal system.

UHM position: "Nothing" is unstable — it cannot be self-consistent, because self-consistency requires "something" that is consistent with itself.

$$\text{Nothing} \Rightarrow \text{inconsistency} \Rightarrow \text{impossibility}$$

$\Gamma$ exists because self-consistency requires existence [I].

Alternative positions:

• The question is meaningless (logical positivists)
• The answer lies beyond the rational (mysticism)
• Random without cause (some interpretations of QM)

UHM chooses the position of self-consistency as the most economical and explanatorily powerful.

What Is Formalized vs. Research Programme

Statement	Status	Comment
Source $\Gamma_{\odot}$ as initial condition	⚙️ Postulate	Not derived, accepted as an axiom
Instability of the Source	[T] Theorem	Proved: non-stationarity + drift $F_0 \neq 0$
Self-amplification of $S_7$-symmetry breaking	[T] Theorem	Positive feedback via $\kappa_0$ (step 3)
Condition $\Delta F > 0$	[P] Open question	Why is the free energy of the environment greater than the system's?
$dD_{\text{diff}}/d\tau > 0$	[I] Non-falsifiable	Teleological assumption
"Nothing" is unstable	[I] Philosophy	Metaphysical argument, not a theorem

Summary

This section contains theorems (instability of the Source [T], self-amplification via $\kappa_0$ [T]), postulates (the Source itself, the condition $\Delta F > 0$), and philosophical positions ("why is there something").

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Related documents:

• Spacetime — emergence of spacetime
• Coherence matrix — definition of $\Gamma$
• Evolution — dynamics of $\Gamma$
• Viability — purity measure $P$
• Self-observation — operator $\varphi$ and measure $D_{\text{diff}}$
• Unity dimension — integration measure $\Phi$
• Foundation (O) — connection to the Source
• Axiom Ω⁷ — ∞-topos $\mathrm{Sh}_\infty(\mathcal{C})$ as primitive