Causal Invariance
Run the rule in any order you like, take any branch you like — and the same causal graph comes out the other end. That stubborn agreement is what lets one shared reality exist at all.
In Multiway Systems we watched the rule branch every which way and then quietly reconverge — every path arriving at the same final state. That reconvergence wasn’t a lucky accident of one example. When it holds in general, it has a name: causal invariance. It’s arguably the single most load-bearing idea in the whole project.
Here’s the plain-English version. The rule can usually fire in many places, in many orders, and the multiway system can branch into genuinely different intermediate histories. Causal invariance says: none of that matters to the end result. Pick any order, follow any branch — you always get the same causal graph of what-depended-on-what. Diverging branches are guaranteed to merge back together.
That’s a strong promise, and a familiar one if you’ve ever rewritten an algebraic expression. There are many orders in which you can simplify , but they all land on . Causal invariance is that “doesn’t-matter-which-order” property, lifted to the rewriting of the universe.
Seeing branches re-merge
Same little string rule, BA → AB, starting from BABA. Watch it split into two
branches at the first step — and then watch those branches merge back into one
state. Every road leads to the same green final state, AABB:
BA → AB — wherever you see BA, you may replace it with AB. Green-outlined states are final (no move left).Try this: play it to the end and find the spot where two arrows point into the same node. That merge — two different histories collapsing onto one shared state — is causal invariance, made visible. The branching is real, but it’s never permanent.
If branches could split and never reconcile, different observers (or different “orders of events”) would end up in incompatible worlds, with no fact of the matter about what happened. Causal invariance is exactly the condition that rules this out.
Why the project leans on it so hard
Two big payoffs are claimed to ride on this one property. [setup]
- One consistent reality. Because all branches re-merge and all orderings give the same causal graph, there’s a single, observer-independent history that everyone agrees on — even though the underlying process was wildly nondeterministic. [setup]
- Relativity. In Causal Graphs we noted you can “slice” the causal graph into moments of time in many ways, each slicing a reference frame. For that to be consistent — for all frames to see the same physics — the causal graph had better not depend on which order events were processed in. Causal invariance guarantees precisely that, and the project treats it as the basis of relativistic invariance (“equivalent to Lorentz covariance”). [setup]
That second claim is the linchpin of the whole relativity story, so it’s worth being careful about how strong it is — see below.
The precise version: confluence and Church–Rosser
Formally, causal invariance is the property “whereby all possible evolution paths yield causal networks that are (eventually) isomorphic as directed acyclic graphs” — the same causal graph regardless of update order. [setup] It is the rewriting-system notion of confluence, also known as the Church–Rosser property: if a structure can be rewritten two different ways, those two results can always themselves be rewritten down to a common form. Diverging branches are guaranteed a common descendant.
This is why the multiway picture and the causal picture fit together. The multiway system shows branches that split and re-merge; causal invariance is the guarantee that the re-merging always happens, which is what lets a single causal graph (and so a single shared history) be recovered from all the branching. [setup]
Two honest caveats:
- Not every rule is causally invariant. It’s a special property a rule either has or lacks; the project’s interest is in rules that do have it (and in how near-invariance might still yield approximately consistent physics). [setup]
- “Causal invariance ⇒ Lorentz covariance / relativity” is a model-internal claim, not a theorem of accepted physics. Treat it as the framework’s argument, and lean on the formal paper rather than the blog essays when stating it precisely.
Grounded in the technical paper and Wolfram’s 2020 announcement.
Where the rigor lives: Gorard's relativity paper
The careful, peer-reviewed version of “causal invariance underlies relativistic invariance” is developed by Jonathan Gorard, not in the popular essays. He works out the continuum limit in which the discrete causal structure is argued to approach Lorentzian spacetime, and identifies causal invariance as the central condition under which relativistic invariance holds. [derived-in-model] When we write the relativity notes in detail, Gorard’s relativity paper is the source to trust.
This is the property that makes the model’s nondeterminism safe: it can branch freely without fracturing reality. The flip side is a different kind of limit on what we can know in advance — even with one guaranteed outcome, you may have no shortcut to it but to run the system. That’s Computational Irreducibility, next.
Sources for this page: Technical paper · 2020 announcement · Gorard — relativity