Appendix B: Geometric Proof of Naimark’s Dilation

In the final chapter of Vector Cosmology II, we presented an ultimate vision full of Zen: the spiral is not the opposite of the circle; the spiral is merely a projection of a higher-dimensional great circle onto a lower-dimensional subspace. This view is not literary rhetoric in mathematics, but a direct physical application of Naimark’s Dilation Theorem in functional analysis.

This appendix will provide a rigorous geometric proof of this “great circle containing small circle” structure, starting from operator theory in Hilbert space. We will show that any seemingly open, dissipative, or non-unitary evolution trajectory must be a Contraction or Projection of some higher-dimensional closed system’s unitary evolution.

B.1 Non-Unitary Evolution of Open Systems

In the main text, we described two characteristics of the spiral universe:

1. Dissipation: Information flows to the environment (entropy increase).

2. Growth: The system absorbs negative entropy from the environment (life/civilization).

In quantum mechanics, this corresponds to an Open Quantum System. Its state evolution is no longer described by a unitary operator $U(\tau )$ (satisfying $U^{†}U=I$), but by a Contraction Semigroup or Completely Positive Trace-Preserving (CPTP) Map ${\mathcal{E}}_{\tau }$.

If we only focus on the system’s internal Hilbert space ${\mathcal{H}}_S$, we find that the total vector’s modulus (or coherence) is not conserved:

$$||{\mathcal{E}}_{\tau }|{\psi }_S⟩||\ne |||{\psi }_S⟩||$$

Geometrically, this means the trajectory no longer stays on the unit sphere of ${\mathcal{H}}_S$, but curls inward (spiral downward) or diverges outward when external pumping is introduced (spiral upward). This is precisely the “spiral” picture we saw in the second book.

B.2 Naimark’s Theorem: Finding the Larger Space

The core insight of Naimark’s theorem (and its subsequent generalizations, such as Stinespring dilation) is: Non-conservation is due to too narrow a view.

Theorem Statement:

Let ${\mathcal{H}}_S$ be a Hilbert space, and $\{V_{\tau }\}$ be a one-parameter contraction semigroup on it (i.e., the operator family describing spiral evolution).

Then, there necessarily exists a larger Hilbert space $\mathcal{K}$ such that ${\mathcal{H}}_S$ is a subspace of $\mathcal{K}$ (${\mathcal{H}}_S\subset \mathcal{K}$), and on $\mathcal{K}$ there exists a unitary group (i.e., the operator family describing great circle evolution) $\{U_{\tau }\}$ satisfying:

$$V_{\tau }=P_S{U}_{\tau }{P}_S$$

where $P_S$ is the orthogonal projection operator from the large space $\mathcal{K}$ back to the small space ${\mathcal{H}}_S$.

Physical Translation:

This mathematical formula is the most stunning physical metaphor of the entire book:

• $V_{\tau }$ (Spiral): The physical laws we observe in the macroscopic world, seemingly with birth and death.

• $U_{\tau }$ (Great Circle): The actually occurring, eternally conserved evolution in the global space ($\mathcal{K}={\mathcal{H}}_S\oplus {\mathcal{H}}_{env}$).

• $P_S$ (Projection): Our observational limitations. Because we are part of the system, we can only “see” the component projected onto ${\mathcal{H}}_S$.

This rigorously proves: Any spiral trajectory is the shadow of high-dimensional circular motion.

B.3 Geometric Reconstruction of the Environmental Term

The environment velocity $v_{env}$ we introduced in the first book finds its precise geometric definition in Naimark dilation.

In the global space $\mathcal{K}$, the unitary evolution $U_{\tau }$ preserves the total modulus, satisfying the global FS capacity identity:

$$||{\dot{\Psi }}_{total}|{|}_{FS}^2=C_{TOTAL}^2$$

When we project it back to the system space ${\mathcal{H}}_S$, the tangent vector is decomposed as:

$$|{\dot{\Psi }}_{total}⟩=|{\dot{\psi }}_{sys}⟩+|{\dot{\psi }}_{env}⟩$$

where:

• $|{\dot{\psi }}_{sys}⟩\in {\mathcal{H}}_S$: This is the system evolution velocity we observe (containing $v_{ext}$ and $v_{int}$).

• $|{\dot{\psi }}_{env}⟩\in {\mathcal{H}}_{env}$: This is the environmental evolution velocity orthogonal to the system.

According to the Pythagorean theorem (orthogonality in Hilbert space):

$$||{\dot{\psi }}_{sys}|{|}^2+||{\dot{\psi }}_{env}|{|}^2=||{\dot{\Psi }}_{total}|{|}^2$$

This directly leads to the extended capacity identity we used in the main text:

$$v_{sys}^2(\tau )+v_{env}^2(\tau )=C_{TOTAL}^2$$

Conclusion:

Naimark’s theorem mathematically guarantees from the bottom up that the “environment” is not an arbitrary trash can, but a necessary piece completing the geometric structure.

• When the system exhibits “dissipation,” it is actually the vector of $v_{sys}$ rotating toward the ${\mathcal{H}}_{env}$ direction.

• When the system exhibits “ascension” or “absorption of negative entropy,” it is actually the vector rotating back from the ${\mathcal{H}}_{env}$ direction to the ${\mathcal{H}}_S$ direction.

In that invisible $\mathcal{K}$ space, there is no dissipation, no growth, only rotation of angles. This is the geometric essence of “The Palm of the Buddha.”