C. Causal Isometry & Fractal Dragon Pearl: Geometric Notes on the Discrete–Continuous Handshake
This appendix provides a “hardcore version” explanation of Addendum A: we are not adding more metaphors, but supplementing a strict interface condition that allows “discrete integer updates” to seamlessly project onto “continuous spacetime geometry” without violating causality.
If you only need practical methods and training paths, you may read Addendum A first:
This appendix focuses on two objects:
- Causal Isometry: The mapping from discrete to continuous must preserve causal partial order.
- Fractal Dragon Pearl: How the universe archives losslessly in finite geometry when “historical length tends to infinity.”
C.1 The “Map Projection” Problem of the Discrete World
When projecting the discrete world onto the continuous world, the easiest mistake is to treat “distance” as the only geometric invariant. But in the context of relativity, the more fundamental invariant is not Euclidean distance, but causal structure.
Imagine you have an extremely fine grid paper (discrete integer space), and you try to draw a perfect world map (continuous spacetime) on it. When you wrap the planar grid onto a curved surface, distortion always occurs. Mathematically, this is just projection error; physically, it may mean more serious consequences: light cones are torn, causal partial order is reversed, and the system exhibits unrecoverable “temporal errors.”
We have previously argued that integer jumps at Planck scale can emerge as macroscopic smooth geometry. But there is still an interface condition that must be strictly stated: discrete updates, under continuous projection, must preserve the partial order relation of events.
Therefore, we introduce the definition:
Causal Isometry: In the “source code perspective,” if event is a necessary precondition for event (denoted ), then in the projection of physical spacetime, event must lie within the past light cone of event , i.e.,
This seems to merely write common sense as a formula, but its true meaning is: in the mapping from discrete to continuous, we allow length, area, and curvature to be rendered at different resolutions; but we do not allow causal partial order to break or invert in projection.
Why is this difficult? Because you are doing a dimensional reduction projection of high-dimensional information. It’s like projecting a knotted rope onto a wall: with a slight change in projection angle, the rope in the shadow may appear “broken” or “intersecting.” Such “rendering errors” are harmless in everyday vision, but in a universe-level temporal system, they become causal law collapse.
To stably preserve partial order, the universe cannot flatten high-dimensional causal chains into a line, but must adopt a folding strategy that can carry extreme complexity within finite regions.
C.2 Space-Filling Curves & Fractal Dragons
When we view “history” as an extremely long causal chain, the problem becomes concrete: how to embed a nearly one-dimensional “event sequence” into three-dimensional space and four-dimensional spacetime, while preserving causal partial order?
Mathematically, a powerful hint comes from space-filling curves: through continuous iteration and folding, a curve can approximate filling a region with arbitrary precision. Peano curves and Hilbert curves are classic examples.
In the symbolic system of this book, the most fitting for the “holographic–folding–interlocking” principle is the Heighway Dragon curve and its family. It is called “fractal dragon” not only because it looks like a dragon, but more because it carries two physically crucial properties:
- Self-similarity: Isomorphic folding laws appear at different scales, ensuring local recursive reuse.
- Interlocking: Folds at different scales can stably interlock, maintaining topological connectivity.
From this perspective, a “particle” need not be understood as a structureless point; it can be understood as a segment of highly coiled, invisible fractal trajectory at conventional resolution. Usually it is rendered as a point; under extreme curvature or extreme energy density, its folding levels are unfolded, connected, and interlocked, manifesting as larger-scale fractal structures.
From this angle, gravity need not be introduced as an additional “pull force.” When too many causal chains try to compress into the same region, space is forced to increase folding order to maintain partial order and connectivity; this “filling efficiency change” manifests macroscopically as spacetime curvature.
C.3 The Ultimate Memory: Fractal Dragon Pearl
Pushing the above mechanism to the extreme, you naturally arrive at black hole horizons or high-curvature regions of the early universe: there, information density and causal chain density are pushed to the limit, and the system must choose a geometric encoding that can carry extreme length.
Traditional holographic principle states: information inside a black hole is encoded on its surface area (Bekenstein–Hawking entropy). The intuitive puzzle is: how can a surface contain a volume?
In the language of this book, the answer is: the black hole core is not an “infinitely dense point.” That is more like a symbolic collapse of the continuous model at the interface. The answer closer to the discrete–continuous handshake is: the black hole core is a highly dense topological archive structure—the Fractal Dragon Pearl.
Imagine an extremely long causal chain (all history falling into the black hole); it is not flattened into “meaningless data,” but tightly wound layer upon layer according to fractal folding rules. The key to fractal structure is: it can accommodate massive “length” within finite “radius,” because the effective measure of length is determined by folding dimension (e.g., Hausdorff dimension).
Therefore, “the final patch” can be written as two implementation clauses:
- Lossless compression of information
- Information entering the horizon does not disappear; it is mapped onto the core fractal structure along paths satisfying causal isometry.
- Macroscopic distances are compressed, but causal order (the logical structure of information) is encoded in every folding turn.
- Pixelation of the horizon
- The black hole horizon seen from outside is a holographic projection of the internal “fractal dragon pearl.”
- Every Planck area on the horizon corresponds to a minimal unit segment of the internal folding structure: geometric discretization is the inevitable appearance of the encoding structure, not an approximation error.
Therefore, the densest celestial bodies can be understood as a universe-level “holographic hard drive”: macroscopically smooth, microscopically full of infinitely folded textures. That is not decoration, but frozen causality and time.
C.4 From Black Holes to a Glass of Water: Isomorphic Closed Loops
Addendum A emphasizes “micro-pearls”: drinking water, writing code, completing a decision. This statement receives a strict isomorphic explanation in this appendix:
- Black holes archive “causal chain compression” at extreme scales;
- Daily closed loops archive “causal chain compression” at human scales.
Their commonality is not scale, but structure: completing a verifiable causal handshake with minimal delay. When local closed loops continuously occur at the holographic level, they become composable components of larger-scale structures.
If you wish to further advance the geometric picture of “cosmic archiving” to the computational picture of “observer roles,” you may continue reading Appendix D: