Topomatics: A Formal Framework for Normative Interaction Between Actors

We introduce topomatics, a formal framework for the analysis of normative interaction between actors. Each actor is defined by an axiomatic code Ω(A) = ⟨O(A), F(A)⟩: a finite weighted system of obligations and prohibitions interpreted over a finite set of possible worlds. The normative significance ρᴀ(Oᵢ) of each obligation is defined as the total weight of prohibitions logically violated by its non-fulfilment, inducing an endogenous ranking of the code from which the normative core Oᶜᵒʳᵉ(A, θ) and the target world W*(A, θ) are derived at each threshold θ ∈ (0, 1]. We prove four theorems on the normative core: (T1) W*(A, θ) is anti-monotone in θ; (T2) the maximum feasible threshold Θᴀ = sup{θ : W*(A, θ) ≠ ∅} is well-defined and attained; (T3) core coherence is equivalent to the existence of a world satisfying all core formulae simultaneously; (T4) core incoherence implies positive internal tension Sᵢⁿₜ(A) > 0, but not conversely. For two actors A and B we define the feasibility set Φ(A, B) = {(θᴀ, θᴮ) : θᴀ ≤ Θᴀ, θᴮ ≤ Θᴮ, W*(A, θᴀ) ∩ W*(B, θᴮ) ≠ ∅} and establish: (T5) a necessary and sufficient condition for Φ(A, B) ≠ ∅; (T6) a sufficient condition for Φ(A, B) = ∅ via the mirror core configuration; (T7) a threshold asymmetry criterion identifying which actor bears the greater burden of core reduction. Theorem T8 gives the three-way structural classification: mutual feasibility (θ₀, θ₀) ∈ Φ; deferred feasibility Φ ≠ ∅ but (θ₀, θ₀) ∉ Φ; and joint infeasibility Φ = ∅. The framework is related to STIT-logic and input/output logic; its distinguishing feature is that the normative ranking of the code is derived endogenously from the conflict structure rather than imposed externally.