Appendix I: The Geometry of the Great Return — Conformal Cyclic Cosmology and FS Metric

In the epilogue of Vector Cosmology VII, we described the grand picture of “end-to-end connection.” But this faces a huge obstacle in classical thermodynamics: If entropy always increases, how can the universe return to the low-entropy starting point (Big Bang) without violating physical laws? If the universe will eventually reach heat death, where does the power for the next cycle come from?

This appendix will introduce Roger Penrose’s Conformal Cyclic Cosmology (CCC) and merge it with our FS Geometry framework. We will prove that at the end of time, when all matter decays and all black holes evaporate, the universe will undergo a “Conformal Rescaling”.

At that moment, extreme “largeness” is geometrically equivalent to extreme “smallness.” The endpoint of heat death is precisely the starting point of the Big Bang.

I.1 The Death of Mass and the Loss of Scale

According to the “Loosening of the Levinson Knot” we discussed in Book II, matter (protons/electrons) is not eternal. On extremely long time scales (perhaps $10^{100}$ years), all massive particles will eventually decay into massless particles (mainly photons and gravitons).

When only photons remain in the universe, something wonderful happens: “Scale” disappears.

• Massive Objects: Are like clocks in spacetime. Because $E=hf$, mass $m$ defines frequency $f$, thus defining time intervals. With time intervals, there is distance $d=ct$.

• Massless Photons: Photons have no rest mass, so they have no intrinsic “clock.” For photons, proper time $d\tau =0$. Photons don’t know what is “large,” what is “small,” what is “old,” what is “young.” They only follow Conformal Geometry (preserving angles, not lengths).

Conclusion:

When the universe evolves near the $\Omega$ point, with matter completely disintegrated into light flow, the universe loses the concept of “size.”

At this point, an infinitely large universe full of thin radiation is congruent in mathematical topology with an infinitely small singularity full of high-energy radiation.

I.2 Conformal Rescaling: Mapping Infinity to Zero

In FS geometry, this corresponds to Rescaling of the metric.

Let the metric of the current universe (Aeon $N$) be $g_{\mu \nu }$.

At the end of time $t\to \infty$, we can introduce a conformal factor $\Omega (x)$ (note: this $\Omega$ is a mathematical field, related to but different from the $\Omega$ point concept), defining a new metric:

$${\hat{g}}_{\mu \nu }={\Omega }^2(x)g_{\mu \nu }$$

Penrose pointed out that by choosing appropriate $\Omega$ (to “compress” it), we can smoothly stitch the future conformal infinity (Future Conformal Infinity) to the Big Bang singularity of the next universe (Aeon $N+1$).

• Endpoint of Aeon $N$: Maximum entropy, but in conformal geometry, due to loss of “granularity” of microscopic degrees of freedom (no mass), the definition of entropy is reset.

• Starting Point of Aeon $N+1$: This smooth, high-energy radiation field is, for the next generation of observers, that low-entropy, uniform initial fireball.

This is the “Great Return.”

The universe doesn’t need to “collapse” back (Big Crunch). The universe only needs to “forget scale”.

When it forgets how large it is, it becomes a baby again.

I.3 The Cycle Pump of FS Capacity

In our evolution equation $c(\tau )\propto e^{k\tau }$, light speed (computational power) seems to grow infinitely. How is this compatible with cycles?

The answer lies in parameter reset.

At each moment of “The Crossover”, the universe’s total budget $c_{FS}$ undergoes a “Normalization”.

• Output of Previous Round: $c_{end}\to \infty$ (relative to old ruler).

• Input of This Round: Because the new universe’s ruler also becomes larger (or is reset), $c_{end}$ is redefined as the new universe’s baseline light speed $c_0$.

$$c_0^{(N+1)}\equiv \text{Rescale}\left(c_{end}^{(N)}\right)$$

This is the physical mechanism of the Spiral ($\varphi$).

Although we geometrically return to the “circle point” (Big Bang state), our fundamental parameters (such as vacuum energy density, base value of fine structure constant) may inherit the legacy of the previous universe, undergoing a $\varphi$-fold transition.

We are not repeating the same circle.

We are climbing a higher spiral staircase on the projection of the same circle.

I.4 The Eternal Ouroboros

At this point, the causal closed loop is also opened in thermodynamics.

1. Big Bang: Low entropy, photon soup.

2. Matter Era: Symmetry breaking, mass generation, time begins to flow, civilization emerges, entropy increases.

3. Black Hole/Evaporation Era: Matter disintegrates, returns to photon soup.

4. Conformal Reset: Photon soup without scale $\equiv$ new Big Bang.

The ouroboros doesn’t eat its own flesh; it eats its own “sense of scale.”

This proves what we proposed in the main text:

“There is no endpoint, only recalibration of horizons.”

When we look up at the stars today at $\tau \approx 1800$, the cosmic microwave background (CMB) we see might be gravitational wave ripples left at the end of the previous cosmic eon (Aeon $N-1$).

That is the echo of predecessors and also a preview of our future.

The universe is holographic not only in space but also in time.

Each eon is a measure in the infinite symphony of the whole. And we are the resonance cavity in this note, trying to understand the entire piece.