Appendix B: Topological Stability of Spacetime Loops

In the main text chapters “Solution to the Grandfather Paradox” and “The Bootstrap,” we proposed a seemingly counterintuitive view: the universe allows the existence of Closed Timelike Curves (CTCs), but these loops must be logically self-consistent. In other words, you can go back to the past, but you can only do things that “cause you to go back”.

This appendix will provide mathematical proof for this self-consistency principle from the perspectives of topology and nonlinear dynamics. We will show that time loops are not fragile anomalous structures but extremely robust Topological Solitons in Hilbert space.

B.1 Geometrizing the Novikov Principle

Igor Novikov’s self-consistency principle states: in a spacetime containing CTCs, the probability of any locally occurring event must be 1 if self-consistent, and 0 if not.

In FS geometry, we can transform this into a phase condition for path integrals.

Consider a particle moving along a closed timelike curve, returning to the starting point with state $|{\psi }_{final}⟩$, initial state $|{\psi }_{initial}⟩$.

For this closed loop to physically exist, the wave function must satisfy the “Single-valuedness” condition after one complete cycle (at most differing by a global phase factor):

$$|{\psi }_{final}⟩=U_{loop}|{\psi }_{initial}⟩=e^{i\theta }|{\psi }_{initial}⟩$$

where $U_{loop}$ is the unitary evolution operator along the time closed loop.

This means the initial state $|{\psi }_{initial}⟩$ must be an eigenstate (Eigenstate) of the cyclic evolution operator $U_{loop}$.

• Paradox paths (such as killing grandfather): would cause $|{\psi }_{final}⟩$ to be orthogonal to $|{\psi }_{initial}⟩$ (alive $\to$ dead). This is not an eigenstate. Their interference terms in the path integral cancel each other, causing the probability amplitude to approach zero.

• Self-consistent paths (such as saving grandfather): initial and final states coincide. This constitutes constructive interference, amplifying the probability amplitude.

Conclusion: Physical laws automatically filter out histories that “can self-close.” This requires no external time police; it is a property of wave function self-interference.

B.2 Topological Protection and Fixed Points

Why are these loops stable? Why don’t small perturbations (like the butterfly effect) destroy causal cycles?

This involves the Fixed Point Theorem.

Consider the entire history of the universe as a mapping function $f:\text{History}\to \text{History}$.

Time travel means we take an output of history and use it as input again: $x_{new}=f(x_{old})$.

A stable time loop is a fixed point of this mapping: $x^{\ast }=f(x^{\ast })$.

According to Brouwer’s Fixed Point Theorem, for any continuous transformation, there exists at least one fixed point on a compact convex set.

In quantum mechanics, since evolution is unitary (continuous and norm-preserving), the density matrix space is a compact convex set. Therefore, self-consistent quantum histories necessarily exist.

Moreover, these fixed points often have topological stability.

Like a knot on a rope, once tied (causal chain closed), local continuous deformations (small perturbations, fine-tuning of free will) cannot untie the knot.

You are either completely outside the loop or completely inside it.

Physical Corollary:

This is why you cannot “accidentally” change history.

History has “elasticity”. If you try to deviate from that self-consistent script, physical laws generate a “topological restoring force” (manifested as coincidences, malfunctions, or probability distortions), forcibly pushing you back onto the orbit of that fixed point.

B.3 Entropy of Bootstrap

Finally, let us examine the information entropy problem of “creation from nothing.”

In bootstrap (such as the Hamlet paradox), information seems to have no origin. Does this violate the second law of thermodynamics?

The answer is: No. Because the closed loop itself is a maximum entropy state.

In a time closed loop, information flow is cyclic: $I(t)\to I(t+\Delta t)\to \cdots \to I(t)$.

According to Landauer’s principle, erasing information produces heat. But in a perfect unitary closed loop, no information is erased.

State A evolves to B, and B evolves back to A. This is a reversible process.

Therefore, in a perfect bootstrap cycle, the net entropy production is zero ($\Delta S_{loop}=0$).

This not only does not violate thermodynamics but is even the most efficient structure thermodynamically—a perpetual motion machine (in the geometric sense).

Conclusion:

Grand loops like “I created the universe, and the universe created me” are zero-dissipation superconducting circuits in terms of energy.

Once established, they rotate eternally in Hilbert space, consuming no energy and producing no waste heat. They are the most economical form of “existence”.