Persistent world simulation requires abandoning observation-buffer architectures in favor of a field-theoretic substrate. Instead of storing state as a finite history, the system maintains a continuous latent field:
X_t = (M_t, p_t, h_t)
- M_t — persistent latent field (world-state)
- p_t — agent position in the semantic manifold
- h_t — finite perceptual history
The field itself is structured as:
M_t(x) = (Φ_t, v_t, S_t, A_t, R_t, z_t)
Each component serves a distinct role:
- Φ_t — coherence potential (geometric admissibility)
- v_t — directed flow (semantic drift)
- S_t — entropy (uncertainty distribution)
- A_t — affordance structure (interaction algebra)
- R_t — causal residue (history encoding)
- z_t — latent appearance (resolved perception)
Once a region is observed, its appearance must be fixed:
z_{t+1}(x) ← z_t(x) + λ · ẑ(x)
This converts stochastic sampling into deterministic persistence.
Updates are decomposed into three stages to prevent global inconsistency.
The neural model generates a candidate update as a forcing term:
∂t Π = -δH/δM + F_ψ(M, p, a, o)
This stage introduces expressive local change without enforcing global validity.
The proposal is projected onto the admissible manifold:
Y' = Π_C(M_t + δ)
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This step enforces all structural constraints and removes invalid deformations.
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### Stage 3: Commit
The final state is selected via constrained optimization:
Y = argmin_{Y ∈ C} ||Y - (M_t + δ)||²
This ensures the committed state is both close to the proposal and globally valid.
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## 3. Field-Level Invariants
The admissible manifold C is defined by conserved quantities and structural constraints.
### Mass Consistency
∫ ρ_{t+1}(x) dx ≈ ∫ ρ_t(x) dx + sources - sinks
### Energy Budget
E_{t+1} ≤ E_t + W(a_t) - D(S_t)
### Topological Admissibility
Topology changes must preserve valid connectivity structure.
### Causal Locality
Updates propagate only within causal neighborhoods.
### Entropy Constraint
ΔS_t(x) ≥ 0 for x outside view(p_t)
### Identity Persistence
Entities must maintain consistent identity trajectories.
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### Hallucination As Cohomological Obstruction
Structural inconsistency is defined as:
H¹(𝒢∞) ≠ 0
This identifies failures of global coherence as topological defects.
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## 4. Operational Dynamics
The system operates on multiple temporal scales.
### Fast Dynamics
- z_t (appearance)
- v_t (local flow)
### Slow Dynamics
- Φ_t (structure)
- A_t (affordances)
- topology and identity
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### Read–Write Closure
Consistency is enforced through cycle closure:
M_t → O_θ → observation → Enc → M_{t+1}
The read operator must be locally Lipschitz to ensure stability under perturbation.
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## 5. Multi-Agent Synchronization
The system supports concurrent updates through causal merging.
### Commutative Updates
Independent edits combine without order dependence.
### Non-Commutative Updates
Conflicts are resolved via admissibility:
accept iff E constraint satisfied
Updates form a causal DAG, ensuring consistent shared evolution.
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## 6. Comparison With Degenerate Systems
Systems without a write operator exhibit:
- no persistent memory
- no causal residue
- repeated stochastic sampling
Formally, they implement only:
M_t → observation
and never complete the loop.
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## 7. Fixed Point Interpretation
A stable world satisfies:
M* = Update(M*)
At this point:
- updates preserve structure
- contradictions are eliminated
- the system becomes self-consistent
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## 8. Summary
The factored update mechanism guarantees persistence by combining:
- neural expressivity (proposal)
- structural enforcement (projection)
- constrained integration (commit)
The result is a system that evolves under invariant-preserving dynamics rather than unconstrained generation.
A world becomes stable when all updates remain within the admissible manifold defined by its own structure.