Here is the the Coherence traveler movement visually explained
Note: this model is physically impossible as the movie is strictly physically speaking - well, impossible. But the movie still has some inner dynamics of which I tried to make sense of. Science Fiction is fun and this model should be treated as such
Abstract
I present an axiomatic model of a time travel mechanism called the twilight zone (TZ). TZ acts as a stochastic
timeline selector: every exit from TZ moves the traveler into a randomly selected, already existing timeline in
the multiverse rather than rewriting the origin timeline. The model enforces: (i) mandatory timeline switching
on each TZ exit, (ii) local observational similarity between origin and selected timeline up to the arrival time,
(iii) macroscopic coexistence of multiple instances of the same entity within one timeline, and (iv) trapping:
after exiting TZ, a traveler is stuck in that timeline unless they re enter TZ, which is only possible until a comet
passes. The model is designed for internal narrative coherence rather than physical realism.
1. Preliminaries
1.1 Time
Let objective time be discrete: t ∈ Z.
1.2 World states and timelines
Let S be the set of possible world states.
A timeline is a complete history function: h : Z → S.
Let Ω be the set of all possible tmielines.
2. Multitimeline structure
2.1 Timelines as a directed graph
Let the global multitimeline structure be a directed graph (V, R), where:
- each element v ∈ V corresponds to one timeline h_v ∈ Ω
- (v → u) ∈ R means “u is reachable from v via a twilight zone exit while the comet is active.
Let v0 ∈ V denote the traveler’s initial timeline.
We assume R has no self loops:
(v → v) ∉ R for all v ∈ V.
We do not assume R is transitive.
2.2 Measurable structure on the timeline set
Assume V is equipped with a sigma algebra Σ ⊆ P(V).
We interpret ∆(V, Σ) as the set of probability measures on (V, Σ).
Accordingly, for any admissible parameters, TZDist(v, t_d, e, x, t_a) is a probability measure on (V, Σ).
If V is countable, we take Σ = P(V).
3. Entities, internal states, and coexistence
3.1 Entities and internal states
Let E be the set of entity labels (abstract identities, for example “the traveler”).
Let X be the set of internal states (memories, body state, carried objects).
An instance is a pair (e, x) ∈ E × X.
3.2 World state decomposition
Each world state is a pair: h_v(t) = (W_v(t), M_v(t)), where:
- W_v(t) is the non agent environment component
- M_v(t) is a multiset of instances
A multiset is required rather than a set, so that multiple macroscopic copies of the same entity label can exist in one timeline.
Define an observation function:
O : S → Y,
where Y is the set of possible local observations available to the relevant agents. Define:
O_v(t) = O(h_v(t)).
Assume Y is equipped with a distance:
d_Y : Y × Y → R_{>= 0}.
3.3 Multiplicity of an entity label
Let µ_{v,t}(e) ∈ N be the multiplicity of entity label e in the multiset M_v(t).
Macroscopic coexistence of duplicates is expressed as:
µ_{v,t}(e) ≥ 2.
4. Baseline dynamics
Assume an abstract baseline evolution operator F : S → S.
In the absence of twilight zone events, the world evolves by:
h_v(t + 1) = F(h_v(t)).
This abstracts physics plus ordinary choices.
5. The comet window
Let t_c ∈ Z be the time at which the comet passes.
Define the predicate:
Active(t) ⇔ t < t_c.
The twilight zone mechanism is available only when Active(t) holds.
6. The twilight zone operator
6.1 Event signature
A twilight zone traversal event is a tuple:
ε = (v, t_d, e, x, t_a),
meaning:
- the event occurs in origin timeline v
- an instance (e, x) enters TZ at departure time t_d
- it exits at arrival time t_a.
We allow t_a ≤ t_d. For strictly backward travel, require t_a < t_d.
6.2 TZ as a stochastic selection operator
Define a conditional probability kernel over timelines:
TZDist : V × Z × E × X × Z → ∆(V, Σ),
where ∆(V, Σ) denotes the set of probability measures on (V, Σ).
If defined, it samples an already existing timeline node v′ ∈ V. Write:
v′∼ TZDist(v, t_d, e, x, t_a).
7. Axioms for twilight zone behavior
Coherence is cinematically constricted.
Axiom TZ1 (Comet restriction)
If the departure occurs after the comet, TZ cannot be used:
t_d ≥ t_c ⇒ TZDist(v, t_d, e, x, t_a) is undefined.
Axiom TZ2 (Stochastic timeline selection)
If the departure occurs while active, TZ selects a different reachable timeline:
Active(t_d) ⇒ ( v′ ∼ TZDist(v, t_d, e, x, t_a) ∧ (v → v′) ∈ R ∧ v′ ≠ v ).
Axiom TZ2b (Small but nonzero return probability)
Let p_k be the probability that the k-th TZ exit returns the traveler to the same timeline. We assume:
0 ≤ p_k ≤ δ,
where δ is a small constant.
In particular, for the first exit:
p_1 ≈ 0.
(This reflects that the space of possible local observation configurations is extremely large.)
Axiom TZ3 (Local observational similarity up to arrival)
The selected timeline is not required to be globally identical to the origin timeline. It must only be locally
observationally similar for the relevant agents
Let L ∈ N be a finite lookback window. Then:
d_Y( O_{v′}(t), O_v(t) ) ≤ ε for all t ∈ [t_a − L, t_a).
Here ε ≥ 0 is a small tolerance parameter.
Axiom TZ4 (Dispensing with coexistence)
At arrival time t_a, the selected timeline contains the arriving instance (e, x) and at least one other instance with
the same label e:
(e, x) ∈ M_{v′}(t_a) and µ_{v′,t_a}(e) ≥ 2.
This expresses coexistence without implying creation.
Axiom TZ4b (Selection bias toward counterpart existence)
TZDist assigns probability 1 only to timelines where a local counterpart exists at the arrival time:
TZDist(v, t_d, e, x, t_a)( { u ∈ V : µ_{u,t_a}(e) ≥ 1 } ) = 1.
Axiom TZ5 (Ordinary evolution after arrival)
For all t ≥ t_a, the selected timeline evolves under the same baseline operator F. All timelines share the same
dynamical laws, but because the selected timeline differs in state at t_a, the resulting trajectory differs:
h_{v′}(t_a) ≠ h_v(t_a) ⇒ ∃ t > t_a : h_{v′}(t) ≠ h_v(t).
Axiom TZ6 (Origin continuity)
The origin timeline is not retroactively changed by the act of time travel:
h_v continues to exist and evolves by F for all future times.
8. Traveler centered accessibility and trapping
8.1 Personal time and experienced position
Let personal time be τ ∈ Z.
Define the traveler’s experienced position as:
Γ(τ) = ( I(τ), T(τ), e(τ), x(τ) ),
where:
- I(τ) ∈ V is the currently experienced timeline
- T(τ) ∈ Z is the objective time within that timeline
- (e(τ), x(τ)) is the traveler instance
The experienced world state is:
s(τ) = h_{I(τ)}( T(τ) ).
We assume Γ(τ) is well defined for all τ ∈ Z.
A TZ exit event corresponds to a discontinuity in objective time T(τ) but not in personal time τ:
ExitTZ(τ) ⇒ T(τ) = t_d and T(τ + 1) = t_a.
8.2 The inside TZ predicate
Let InTZ(τ) be true exactly while the traveler is inside the twilight zone.
Let EnterTZ(τ) and ExitTZ(τ) be event predicates.
9. Axioms for trapping
Axiom S1 (No cross timeline motion outside TZ)
If the traveler is not in TZ, their timeline index cannot change:
¬InTZ(τ) ⇒ I(τ + 1) = I(τ).
Axiom S2 (Exit always changes timeline)
Exiting TZ forces a transition to a different reachable timeline:
ExitTZ(τ) ⇒ ∃v′ ∈ V : ( (I(τ) → v′) ∈ R ∧ I(τ + 1) = v′ ∧ v′ ≠ I(τ) ).
Axiom S3 (Comet disables entry)
If the traveler’s local objective time is at or after the comet time, entry is impossible:
T(τ) ≥ t_c ⇒ ¬EnterTZ(τ).
Thus, after the comet, trapping becomes permanent.
10. Derived results
Proposition 1 (Duplication theorem)
Assume the traveler exits TZ into timeline v′ at arrival time t_a. Then:
µ_{v′,t_a}(e) ≥ 2.
This follows directly from Axiom TZ4.
Proposition 2 (Each exit changes the traveler’s timeline index)
ExitTZ(τ) ⇒ ( I(τ) → I(τ + 1) ) ∈ R.
Proposition 3 (Stuckness until the comet)
Suppose the traveler exits TZ at τ0 into timeline node v1 and never enters TZ again. Then:
I(τ) = v1 for all τ ≥ τ0.
Proof: by Axiom S1, without being in TZ the timeline index cannot change. Moreover, if there exists τ_c such
that T(τ_c) ≥ t_c, then by Axiom S3 entry becomes impossible, so trapping is enforced for all later τ.
11. Conclusion
This is an axiomatic multitimeline model for the Coherence twilight zone time travel mechanism. Each TZ exit moves the traveler into a randomly selected, already existing timeline that is locally observationally similar to the origin up to the arrival time and contains an additional traveler instance at arrival. After exiting TZ, the traveler is trapped in that timeline unless they re enter TZ, and re entry becomes impossible after a comet passes. The resulting structure is a set of timelines with a reachability relation induced by TZ exits, with controlled accessibility and guaranteed duplication when arriving at times where an earlier self exists