# Boundary-Object Machines This is a note, not a theorem. It sketches a semantics in which a nonterminating computation is assigned a point on a Gromov boundary rather than a single bottom element `⊥`. ## 0. The primitive Let `M` be a deterministic Turing machine and `x` an input. Write ``` γ(M,x) = (C_0, C_1, C_2, ...) ``` for the configuration sequence, extended constantly if `M(x)` halts (so a halting run is an infinite dwell). Assume the configuration space `K` is compact metric, and fix a refining tower of finite partitions `𝒫 = (𝒫_j)_{j≥0}` with `mesh(𝒫_j) → 0`. For a traversal `γ`, let `D_j(γ)` be the finite word over `𝒫_j` listing cells in the order they are first encountered, later repeats deleted. Compatibility of successive resolutions (parent-and-reduce) gives ``` C^𝒫_nov(γ) = (D_0(γ), D_1(γ), D_2(γ), ...) ``` in the inverse limit ``` N_𝒫(K) = lim← (X_j, q_{j+1,j}). ``` Define the **boundary object** of the computation by ``` B_𝒫(M,x) := C^𝒫_nov(γ(M,x)) ∈ N_𝒫(K). ``` ## 1. Existence is not the problem **Lemma (total existence).** If `K` is compact metric and `𝒫` is a refining tower of finite partitions, then `B_𝒫(M,x)` exists for every machine `M` and input `x`. *Reason.* Each `D_j(γ)` is finite because `𝒫_j` is finite, and the compatibility equations `q_{j+1,j}(D_{j+1}) = D_j` hold by the coarse-graining lemma of the companion article. No convergence of `γ` is required; the coordinates exist level by level. So the object is always defined. The difficult content is **not** whether a nonterminating computation has a limit, but whether that limit is well-posed, realizable, canonic, and computable. ## 2. The upgrade over `⊥` Domain theory collapses nontermination to `⊥`. This construction does not. Two nonterminating runs can map to different points of `N_𝒫(K)` whenever their discovery structures differ at some resolution: ``` D_j(γ) ≠ D_j(γ') ⟹ B_𝒫(γ) ≠ B_𝒫(γ'). ``` Thus the candidate semantics is finer than "it diverges." The remaining question is whether that extra structure is the *right* extra structure. ## 3. The four sharpened questions ### Q1. Well-posedness of `(K, 𝒫)` What is the intended configuration space? - `K = Σ^ω × Q` and `K = Σ^ℤ × Q` are compact. - If head position is added as a coordinate in `ℕ`, then `K` is noncompact and the total-existence lemma must be restated; the map may enter the escape regime of the companion article. Which towers `𝒫_j` count as "finite observational distinctions"? Natural choices fix a finite tape window, the control state, and the head position at bounded resolution. The question is whether every such tower is refining with `mesh(𝒫_j) → 0` in the chosen metric. ### Q2. Realizability For a class `𝒯` of runs (all runs of a fixed machine family, or all computable runs), set ``` R_𝒯 = C^𝒫_nov(𝒯) ⊆ N_𝒫(K). ``` When is `closure(R_𝒯) = N_𝒫(K)`? This is the machine version of the finite-profile density criterion. The interesting new object is a coherent profile that is approached by computable runs but not realized by any: ``` finite computable approximants → ideal boundary object. ``` Those are the "computations" the completion adds. ### Q3. Canonicity Given two towers `𝒫` and `𝒬`, when does there exist a homeomorphism ``` Φ : N_𝒫(K) → N_𝒬(K) ``` intertwining the completion maps, i.e. `Φ ∘ C^𝒫_nov = C^𝒬_nov` on the run class? This is the companion article's canonicity problem with the run class restricted to machines. In general it should be expected to fail; the useful question is characterizing completion-equivalence through common refinement or an explicit re-coding condition on the bonding maps. ### Q4. Computability `D_j(γ)` is finite, but from a finite time prefix `γ[0..t]` one can only compute the *current* ordered discovery prefix. The decision "no new cell at level `j` appears after time `t`" is a nontermination/undiscovery question, and it is not decidable in general. Equivalently, the finite approximation `D_j^t(γ)` converges to `D_j(γ)`, but the modulus of convergence is not uniformly computable. So: ``` B_𝒫(M,x) exists — automatic for compact K j ↦ D_j is enumerable — as a monotone limit from the run j ↦ D_j is computable — not in general ``` For each fixed `j` and fixed run, `D_j^t(γ)` does eventually stabilize, because only finitely many cells can be newly discovered at level `j`. The issue is not the existence of a stabilization time `T_j(γ)`, but the absence of an effective certificate or uniformly computable modulus `T(j)` saying when stabilization is safe to declare. There is also a topological obstruction in the unrestricted setting. If a run has not yet visited some level-`j` cell, arbitrarily long temporal prefixes may still be extended by a previously unseen cell, changing the eventual discovery word. Thus `C^𝒫_nov` need not be continuous in the raw temporal-prefix topology. This depends on the chosen run class: restricting to runs that eventually visit every cell at each fixed resolution can restore local determination of those coordinates. Uniform effective computability remains a separate question. A TTE treatment therefore has to state effectivity relative to the representation of the run—and should compare the ordinary temporal representation with a discovery-adapted one. ## 4. A possible arithmetic bridge: Bost–Connes A natural comparison point is the [Bost–Connes system](https://doi.org/10.1007/BF01589495), introduced by Jean-Benoît Bost and Alain Connes as a quantum statistical dynamical system related to the distribution of prime numbers. It combines an algebra of observables with a time evolution and studies equilibrium (KMS) states and phase transitions. The connection here is structural, not an identification. `B_𝒫(M,x)` is an inverse-limit discovery object and, after a hyperbolic realization, a boundary point; it is not presently a C*-algebra, a time evolution, or a KMS state. A future construction could ask whether cylinder observables on discovery profiles carry a natural semigroup action or time evolution whose partition function or KMS states reflect the boundary geometry. No such action or phase-transition theorem is claimed in this note. ## 5. Status | Item | Status | | --- | --- | | Total existence of `B_𝒫` | Immediate, for compact `K` | | Separation of nonterminating runs | Immediate once `𝒫` separates configurations | | Well-posedness of `(K, 𝒫)` | Partially settled by compactness; modeling choice remains | | Realizability for machine classes | Open; finite-profile density restricted to computable runs | | Canonicity across towers | Open; expected false in general | | Type-2 computability of `B_𝒫` | Open; prefix-discontinuity is the first obstruction | ## 6. What is and is not claimed This note claims only that the ordered-novelty construction supplies **one** candidate boundary semantics for nontermination, and that the interesting questions are well-posed. It does **not** claim that every boundary point is the "correct" denotation of a computation, nor that the construction is canonic or computable. Those are the open content. The expected contribution, if the open questions resolve favorably, is: ``` automata semantics ↔ inverse limits ↔ hyperbolic boundaries ``` where the minimal predictive quotient of finite history and the geometric boundary of endless refinement meet.