Equivalence and Meaning

In the previous post I introduced the Semantic Filter as a refinement of exclusion. Exclusion filters prune Chaos by ruling out incoherent strings; semantic filters then assign lawful meaning to the survivors, interpreting them as trajectories. Here I want to formalize this idea: how semantic filters act as quotient maps that collapse strings into equivalence classes, each corresponding to a coherent world.

1. From Strings to Trajectories

Let Chaos be the set of all infinite binary strings. An exclusion filter picks out a subset F ⊆ Chaos: the strings that pass basic consistency constraints. A semantic filter is then a mapping:

S : F → T

where T is the set of lawful trajectories (quantum state evolutions, automaton runs, dynamical histories). Each string x ∈ F is interpreted as a trajectory τ = S(x).

2. Equivalence Classes of Strings

Different strings can map to the same trajectory:

S(x₁) = S(x₂) = τ

In this case, x₁ and x₂ belong to the same equivalence class under S. The preimage of a trajectory τ is the set of all strings that generate it:

S⁻¹(τ) = { x ∈ F : S(x) = τ }.

Thus, a semantic filter partitions the set F into equivalence classes, each class corresponding to one lawful world.

3. Examples

• Redundancy in Encoding: Suppose the semantic filter ignores every second bit. Then strings 0101… and 0001… both map to the same trajectory. They belong to the same equivalence class.

• Quantum Histories: Imagine a qubit measured repeatedly in the Z basis, but where some measurement outcomes are decohered away. Distinct bitstrings differing only in decohered positions map to the same physical trajectory.

4. Invertibility and Loss

• If S is one-to-one, every coherent string corresponds to a unique trajectory. The string is the trajectory.

• If S is many-to-one, strings collapse into equivalence classes. Some detail is lost; only the quotient structure remains.

• If S were one-to-many (rare), then a single string could correspond to multiple possible worlds; but in most formalizations we prefer determinism at the semantic level.

5. Why This Matters

• Exclusion filters define what survives.

• Semantic filters define how survivors are grouped into worlds.

• The equivalence class view makes clear that meaning emerges not from individual bitstrings, but from the partition structure induced by semantics.

In other words: coherence is not just survival; it is also identification — treating different raw sequences as the same world.

6. Place in the Arc

1. Chaos Reservoir — all random strings.

2. Exclusion Filters — prune incoherence.

3. Semantic Filters — partition survivors into equivalence classes, each a lawful world.

4. Constructors — stable patterns within worlds.

5. Life and Consciousness — self-maintaining and self-representing constructors.

Conclusion

Semantic filters are quotient maps: they collapse raw randomness into equivalence classes of coherent strings, each class corresponding to a world. This formalization makes precise the idea that semantics is not just interpretation but identification — the recognition that different chaotic traces can mean the same coherent trajectory.