Effective negotiation and higher agreements
Autonomous agents choose offers, issue counteroffers, and reserve resources under local information and bounded computation. Successful settlements can change the available fuel, storage, and verification rights for subsequent negotiations. The operational model distinguishes messages, consent, reservations, and atomic settlement. A guarded ledger proof establishes conservation, nonnegative balances, and at-most-once closing. Eventual settlement is conditional on persistent enabling and the stated scheduling hypothesis.
The higher semantics represents an open negotiation by a span of spaces. Its apex is the type of local agreement witnesses and its legs expose the interfaces. For a pair of components with a common interface, a joint witness consists of one witness from each component and a path between their interface values. Identity between two joint witnesses is described by two endpoint paths and a compatibility square in the mediator. Composition is by homotopy pullback. Units, associativity, substitution, and graph cuts preserve the corresponding witness types.
The effective presentation model uses Kan complexes with a finite description, a total complete list of simplices in each degree, and computable simplicial operations. The path-object construction computes the degreewise finite code of a homotopy pullback. Constant finite stages and uniform presented towers embed in the semantic tower calculus by interpretation. The ideal calculus permits arbitrary coherent diagrams of finite observation spaces, including those with no uniform presentation.
Structural comparison theorems
For a sequential tower the limit is the dependent sum of a sequence of points and a family of adjacent compatibility paths. A path in that limit consists of pointwise paths together with a family of squares. Applying this description to the stagewise match proves an equivalence between the limit of stagewise homotopy pullbacks and the homotopy pullback of the limits. The equivalence preserves the two outer projections and assembles the componentwise seam paths into one path in the limiting mediator.
To observe the composite of a cospan through homotopy degree $n$, its three inputs through degree $n+1$ suffice. For the mediator, the identity type of its $(n+1)$-truncation is the $n$-truncation of the original path type. The map on each seam fiber is therefore $n$-connected. Replacing the outside endpoints by their $(n+1)$-truncations is a base change of a connected comparison and has the same $n$-truncated result. The bound is sharp, using pointed singleton maps to $K(C_2,n+1)$. Their composite detects $K(C_2,n)$ even though the mediator has contractible $n$-truncation. For a finite probe of dimension $d$, the corresponding parametrized comparison costs $d$ further degrees. Levelwise gluing of canonical Postnikov towers and the true Postnikov tower of the composite define equivalent pro-objects, with a one-step shift witnessing the comparison.
Exact computability separations
For a bit sequence $b$, put $A_e$ equal to $C_2$ or zero according to $b(e)$. The split space is the product of $K(A_e,e+1)$, with Postnikov stages given by the finite products through depth $n$. A cocycle model gives each stage a finite description and finitely many simplices in each simplicial degree. The full product retains $b(e)$ in the order of its $(e+1)$-st homotopy group. In any degreewise finite Kan code, elementary homotopies in consecutive degrees allow this order to be computed uniformly. Conversely an oracle for $b$ gives the canonical code. Thus the least oracle degree for a presentation of the space is exactly the degree of $b$.
A second family has the same graded homotopy groups for every $b$. Each block is the fiber of either the zero map or the cup square $K(C_2,m) \to K(C_2,2m)$. The two choices have the same groups in degrees $m$ and $2m-1$. A finite skeleton of $K(C_2,m)$ detects the difference by asking for a lift that is nontrivial on the designated $m$-sphere. The cup square obstructs such a lift in the twisted case. The existence query is decidable from any degreewise finite Kan code, so the oracle lower bound persists when the hidden information consists entirely of Postnikov extension classes.
For the halting sequence, bounded evaluation generates uniform coherent towers of hypotheses at each search stage. At any fixed observation depth the hypotheses eventually agree with the exact finite stage. A computable choice of a sufficiently late stage would decide halting. The distinction persists for any effective resource translation of the search budget. In both families the existence of an eventually exact computable stream of hypotheses is equivalent to limit computability of the bits. For the extension-class block at degree $m$, the first depth at which its bit is determined by the homotopy type is $2m-1$. Pooled acquisition or resource growth can make search stages cofinal under the stated guards and scheduling assumptions, supplying eventual access at each fixed depth.
Applications and proof evidence
The manuscript treats escrow, delegation, common verification charges, compute allocation, the difference between feasible agreement and strategic stability, and nontrivial descent obstructions. It distinguishes provisional observations from certificates suitable for irreversible settlement. It also proves payoff and deviation-coverage bounds for transferring approximate equilibrium claims.
The attached Lean HoTT development uses proof-relevant identity, cubical squares, dependent products and sums, truncations, and Eilenberg–Mac Lane spaces. Its kernel-checked results include open composition and substitution, the sequential limit gluing theorem, the sharp cospan truncation bound, split tower reconstruction, and the equivalence between spaces of lifts and nullhomotopies for Postnikov extensions. The cup-square, simplicial, computability, strategic, and external topos arguments have written proofs in the full manuscript. The proof package includes a pinned upstream revision, build and verification commands, and the kernel dependency report. AI assistance was used in drafting the exposition, constructing the proof terms, and assembling the artifacts.