Higher Categorical Semantics of Coordinator-Executor Prompt Optimization: Typed Transport, Coherent Assembly, and Interaction Bounds
Anonymous. Version 1.0, November 1, 2026.
Compact text accompanying the complete 12-page PDF and reproducibility archive. Result numbers follow the expanded PDF.
1. Source, contribution, and assumptions
CEPO [1] separates a shared coordinator prompt from agent-specific executor prompts and uses role-specific guidance for an underlying optimizer. Both roles are evaluated with a system-level objective. Its displayed Equation (4) and Algorithm 1 use other executors' initial prompts, while Appendix B.2 also describes sequential executor optimization. We distinguish frozen-context recombination from current-context acceptance, without assuming every implementation uses the same context. The source architecture and empirical results belong to Kim and coauthors. We do not reproduce their experiments.
Our reconstruction organizes contract-indexed executors, coherent execution, descent, and conditional optimization. Straightening, operadic composition, descent [2,3], and Boolean-lattice inversion are standard; we specialize them to executor changes.
Fix universes, a finite number $N$ of executor roles, and small infinity-categories as needed. Limits and fibers are homotopy coherent. The core $A^\simeq$ contains only equivalences. Scores are invariant under specified semantic equivalences, but are not assumed constant along directed optimization changes. Stochastic statements assume finite state spaces and a finite execution horizon.
2. Literal syntax
For allowed strings $C_0$ and $E_{i,0}$, define
$$P_0=C_0\times\prod_{i=1}^N E_{i,0},\qquad D(c,e_1,\ldots,e_N)=(c\Vert e_1,\ldots,c\Vert e_N),$$
where the structured tuple retains the concatenation boundary. This factors through the diagonal $C_0\to C_0^N$ and coordinatewise concatenation.
Proposition 2.1. Every assembled prompt has designated prefix $c$. If all component domains are unrestricted strings and the empty coordinator is allowed, $D$ is surjective but generally not injective.
Proof. For any prompt tuple ($s_i$), use empty $c$ and $e_i=s_i$. Noninjectivity follows from $a|bc=ab|c=abc$, repeated across roles. The prefix assertion is definitional.
Decomposition need not restrict expressivity. Identical shared text proves neither common interpretation nor compliance; recovering boundaries requires serialization.
3. Contract-indexed executor categories
Definition 3.1. Let $C$ be an infinity-category of coordinator contracts and certified migrations. For each role specify a functor
$$F_i:\mathcal C\longrightarrow\mathrm{Cat}_\infty.$$
Objects of $F_i(c)$ are executor implementations with declared admissibility evidence, and arrows are declared refinements. An arrow $u:c\to c'$ has a migration functor $F_i(u)$, with coherent composition. Existence of these migrations is a hypothesis: arbitrary natural-language edits do not automatically define such functors.
Unstraighten $F_i$ to a cocartesian fibration $q_i:E_i\to C$ and form
$$\mathcal E=\mathcal E_1\times_{\mathcal C}\cdots\times_{\mathcal C}\mathcal E_N,\qquad q:\mathcal E\to\mathcal C.$$
This compatibility means lying over the same contract. Further cross-executor constraints require additional evidence or a subcategory; that subcategory inherits transport only if closed under the chosen migrations.
Theorem 3.2 (Assembly and transport). The projection $q$ is classified by $F=\prod_i F_i$. Consequently
$$\mathcal E_c\simeq\prod_i F_i(c),\qquad u_!\mathbf e\simeq(F_i(u)e_i)_i.$$
For a specified base arrow $u:c\to c'$, the homotopy fiber of the mapping-space projection over $u$ satisfies
$$\operatorname{Map}_{\mathcal E}^{u}((c,\mathbf e),(c',\mathbf e'))\simeq\prod_i\operatorname{Map}_{F_i(c')}(F_i(u)e_i,e_i').$$
Every arrow over $u$ factors into cocartesian transport followed by a vertical refinement, with a contractible space of factorization choices. Successive transport has the higher coherence $v_!u_!$ equivalent to $(vu)_!$.
Proof. The cocartesian straightening equivalence, dual to [2, Section 3.2], identifies these fibrations with $\operatorname{Fun}(C,\mathrm{Cat}_\infty)$. Products of functors are pointwise, and their unstraightenings form the displayed fiber product. The defining universal property of a cocartesian lift identifies arrows over $u$ with vertical arrows from the transported object. Computing those in the product fiber gives the mapping formula. Uniqueness in the universal property is contractible, and the coherent composition comes from $F$.
The literal string architecture is the constant-family case, with $C$ freely generated by coordinator edits. In a nonconstant family, changing a schema requires transported or newly certified executors.
Proposition 3.4. Fixed endofunctors on distinct factors $F_i(c)$ and $F_j(c)$ induce commuting local transformations of $E_c$, up to canonical product coherence.
Proof. Both orders have identical projections to every factor. The product universal property supplies the equivalence and coherence. An optimizer choosing its transformation from the whole system's score need not have this independence.
4. Coherent execution and stochastic observations
Let $W$ be the symmetric monoidal category freely generated by a finite acyclic workflow's typed port signature and invocation/routing operations, including explicit permitted copying and deletion. Objects are port tuples; arrows are generated circuits. Include persistent state and shared random seeds in the relevant wires. Bounded loops may be unrolled; an ordinary monoidal structure does not supply unbounded feedback.
For a symmetric monoidal infinity-category $V$, an implementation is a strong symmetric monoidal functor, represented by a cocartesian-edge-preserving map of infinity-operads
$$A:\mathrm N^\otimes\mathcal W\longrightarrow\mathcal V^\otimes.$$
The nerve is that of a symmetric monoidal category. A higher semantics is an additional coherent assignment
$$\sigma:\mathcal E^\simeq\longrightarrow\operatorname{Fun}^\otimes(\mathcal W,\mathcal V)^\simeq.$$
It is not induced automatically by concatenating strings.
Proposition 4.1 (Contextual invariance). Equivalent implementing monoidal functors, compatible with typed interfaces, give equivalent evaluations of each workflow operation. Any observation map from these equivalences into a discrete space of laws gives equal observed laws.
Proof. Evaluation is functorial in the monoidal functor and its equivalences [3, Section 2.1]. A path in a discrete observation space has equal endpoints. Local identifications without compatibility data do not automatically constitute the required monoidal natural equivalence.
For concrete executions use $\mathrm{FinStoch}$ [4]: objects are finite sets, arrows stochastic matrices, composition is matrix multiplication, and tensor is independent product. Shared correlations must enter through explicit common sources, not a product of marginal laws. This ordinary specialization need not have nontrivial higher mapping spaces.
Let $K_p(o\mid z)$ be the composed execution kernel for input $z$ and parameter $p$. The observation $o$ may include answer, trace, cost, and next persistent state. State may be omitted only when no future context observes it. For fixed input distribution $\mu$ and reward $0\le r\le1$, set
$$J_{\mu,r}(p)=\sum_z\mu(z)\sum_o K_p(o\mid z)r(z,o).$$
Equality of observed kernels implies equality of this score; equality of one score does not imply equality of kernels.
Theorem 4.2 (Finite-horizon substitution). Suppose two workflows have the same typed schedule with $H$ invocation slots, and their full relevant history is included in the state. Stopped executions use absorbing states. If the kernels at slot h differ uniformly over states by at most $epsilon_h$ in total variation, then
$$\sup_z\operatorname{TV}(K_p(\cdot\mid z),K_{p'}(\cdot\mid z))\leq\min\left(1,\sum_{h=1}^H\varepsilon_h\right),$$
and the same bound holds for $|J(p)-J(p')|$.
Proof. For distributions $\alpha$,$\beta$ and kernels $k$,$k'$, contraction and convexity imply
$$\operatorname{TV}(\alpha k,\beta k')\leq\operatorname{TV}(\alpha,\beta)+\sup_s\operatorname{TV}(k_s,k'_s).$$
Induct from equal initial laws through $H$ slots. Observing the final state is another kernel and cannot increase distance. Expectation differences for $[0,1]$-valued rewards are at most total variation. Distances are also at most one.
Uniformity includes states null under one version. Prompt text and local benchmark estimates alone do not establish these kernel bounds.
5. Internalization and descent
In an infinity-topos $X$, let $C$ be an object and $E_i\to C$ be families of specifications and equivalence witnesses. Their product over $C$ has internal total-space notation
$$P=E_1\times_C\cdots\times_C E_N,\qquad P\simeq\sum_{c:C}\prod_{i=1}^N E_i(c).$$
This is a space-level model, distinct from the external $\mathrm{Cat}_\infty$-valued model retaining noninvertible refinements. In spaces it is obtained by restricting $F_i$ to the core of $C$ and taking cores of their values. Identifications of coordinators transport executor witnesses. No effective presentation of an arbitrary infinity-topos is assumed.
Theorem 5.1 (Descent). For an effective epimorphism $U\to C$ with Cech nerve $U_n=U^{\times_C(n+1)}$,
$$\mathcal X_{/C}\simeq\lim_{[n]\in\Delta}\mathcal X_{/U_n}.$$
Under this equivalence, pullbacks of $P$ are products of the pulled-back $E_i$. Local families assemble precisely when they form full descent data; the space of realizations of a specified descent datum is contractible.
Proof. The Cech nerve has colimit $C$, and object descent in an infinity-topos identifies the slice over this colimit with the displayed limit [2, Section 6.1.3]. Pullback preserves finite limits, hence the products. The equivalence of categories gives a contractible homotopy fiber over any specified descent object, including its realization equivalence.
Cech descent here does not require hypercompleteness. Data include equivalences on double overlaps, their compatibility on triple overlaps, and higher coherence.
Example 5.2 (Obstruction). In spaces use $\{0,1,2\}\to *$. Each local specification is a two-element set in the groupoid of such sets and bijections. Let $s$ be the swap and choose $g_{01}=g_{12}=g_{02}=s$, with inverse transitions and identity diagonals. Each pair is isomorphic, but triple compatibility requires $g_{02}=g_{12}g_{01}=\mathrm{id}$, contradicting $s\ne\mathrm{id}$. No higher homotopy in this 1-groupoid equates distinct bijections. These chosen witnesses fail to assemble, although other coherent choices for the same objects can assemble. Thus pairwise compatibility does not suffice.
6. Directed optimization and conditional guarantees
Let $P$ be a set of represented admissible systems, $J:P\to[0,1]$ a fixed objective, and $G$ a graph of proposal edges labeled by role. Coordinator edges carry the required migrations; executor edges are vertical. Role guidance influences candidate generation but is not a proof of admissibility.
The free category of accepted edges and its nerve describe directed optimization histories. If each edge $p\to p'$ satisfies $J(p)\le J(p')$, the score extends to the ordered category $([0,1],\le)$. It generally does not extend to a discrete space, which would force equality along every edge. Improvement is not semantic equivalence.
Theorem 6.1 (Current-context acceptance). At step $t$ let $B_t$ be a finite candidate set containing the actual incumbent $p_t$ and use the same exact $J$ for every candidate. Choosing $p_{t+1}\in\operatorname*{argmax}_{p\in B_t}J(p)$ gives $J(p_{t+1})\ge J(p_t)$. Strict gains of at least $\delta>0$ permit at most $\lfloor(1-J(p_0))/\delta\rfloor$ strict acceptances. In a finite reachable set there are at most $|P|-1$ strictly improving acceptances.
Proof. The finite maximum exists and dominates its member $p_t$. Sum the gains and use $J\le1$ for the $\delta$ bound. Strict increase forbids repeated states, proving the finite-state bound. Rejections and ties are not counted. Termination needs a stopping rule; a terminal state is blockwise optimal only if all allowed block alternatives were checked. Global optimality is not implied.
Proposition 6.2 (Certified evaluation error). If both incumbent and candidate estimates satisfy $|\widehat J-J|\le\varepsilon$, accepting an empirical gain $m$ gives true gain at least $m-2\varepsilon$. In particular margin $2\varepsilon$ guarantees nondecrease.
Proof. $J(p')\ge\widehat J(p')-\varepsilon\ge\widehat J(p)+m-\varepsilon\ge J(p)+m-2\varepsilon$.
Proposition 6.3 (Held-out finite class). Let $H$ be $K$ systems fixed independently of $n$ independent held-out examples and execution random variables. Assume each per-example reward lies in $[0,1]$, with mean $J(p)$. With probability at least $1-\alpha$, simultaneously for every $p\in H$,
$$|\widehat J_n(p)-J(p)|\leq\sqrt{\frac{\log(2K/\alpha)}{2n}}.$$
The empirical maximizer's true score is within twice this bound of the best true score in $H$.
Proof. Hoeffding's inequality [5] gives tail bound $2\exp(-2n\varepsilon^2)$ for a fixed $p$. Union-bound over $H$. Compare the empirical winner to a true maximizer and apply the error bound twice.
Adaptive selection within a fixed $H$ is permitted by uniformity. If prompts are generated from the same holdout feedback, substituting only the number of adaptively evaluated candidates for $K$ is unjustified. Use a fresh final evaluation, a fixed larger uniform class, or an appropriate adaptive-analysis method [6].
7. Interactions and counterexamples
Fix the coordinator and baseline executors $e_i^0$. For chosen alternatives $e_i^1$ and $A\subseteq[N]$, let $p^A$ change precisely the coordinates in A, and set $j(A)=J(p^A)$.
Example 7.1 (Conflicting improvements). For two executors let
$$j(\varnothing)=1/2,\quad j(\{1\})=j(\{2\})=3/4,\quad j(\{1,2\})=0.$$
Both standalone changes improve the baseline, but their recombination degrades it. This is realizable by a Bernoulli-output lookup kernel with these four success probabilities and success reward one. It is therefore an execution model. This refutes an unconditional inference about frozen-context recombination, not the CEPO experiments or every source implementation.
Theorem 7.2 (Interaction decomposition). Define
$$\Delta_Sj=\sum_{T\subseteq S}(-1)^{|S|-|T|}j(T).$$
For every $A$,
$$j(A)-j(\varnothing)=\sum_{i\in A}\Delta_{\{i\}}j+\sum_{\substack{S\subseteq A\\|S|\geq2}}\Delta_Sj.$$
If higher differences satisfy $\Delta_S j\ge-\beta_S$ with $\beta_S\ge0$, the joint gain is bounded below by the sum of singleton gains minus the corresponding sum of $beta_S$.
Proof. Expand $\sum_{S\subseteq A}\Delta_S j$. The coefficient of $j(T)$ is
$$\sum_{T\subseteq S\subseteq A}(-1)^{|S|-|T|}=(1-1)^{|A\setminus T|},$$
which vanishes except at $T=A$. Subtract the empty-set term and separate singletons. Replacing higher terms by their lower bounds gives the bound. In Example 7.1 singleton gains are 1/4 each and the pair term is -1, yielding joint gain -1/2.
Proposition 7.3 (Safe recombination). Suppose $j$ has increasing marginal returns: for $A\subseteq B\text{ and }i\notin B$,
$$j(A\cup\{i\})-j(A)\leq j(B\cup\{i\})-j(B).$$
Then the joint gain for any chosen set is at least the sum of its singleton gains. Nonnegative singleton gains consequently imply nonnegative joint gain.
Proof. Insert the chosen coordinates in any order and telescope the gain. Each insertion marginal is at least its marginal at the empty set. Sum. This supermodularity condition is substantive; neither shared text nor associativity proves it.
Jointly evaluating the recombined tuple, or sequentially accepting each change in the actual current context, gives a different remedy requiring no supermodularity assumption.
Example 7.4 (Order dependence). For binary executors let $J(00)=0,\quad J(10)=J(01)=2/3,\quad J(11)=1/3$. Starting at 00, exact coordinate best response in order 1,2 ends at 10; the reverse order ends at 01. Fixed coordinate replacements commute, but context-dependent optimizer choices need not.
Example 7.5 (No universal coordinator-first optimum). For coordinator $c$ and executor $e$, set $J(0,0)=1/2,\quad J(1,0)=2/5,\quad J(0,1)=3/5,\quad J(1,1)=1$. From (0,0), one coordinator-first pass retains $c$=0 then reaches (0,1), scoring 3/5. Executor-first then coordinator reaches score 1. Another coordinator pass also repairs the first schedule. One-pass order preferences are data dependent.
8. Implementation and evidence
Generate a finite role-specific batch, retain the actual incumbent, check complete-system admissibility, and evaluate all candidates in a recorded common context. That context includes prompt bytes, model and evaluator versions, topology, data distribution, randomization, reward, and horizon. Use exact or explicitly error-certified acceptance. Revalidate the assembled tuple after independently proposed changes. Persist the selected system with its evaluation record and a budget/stopping rule.
Changing model, distribution, or reward changes $J$ and invalidates direct comparisons unless an additional transport/error bound applies. An LLM service generally does not expose its exact kernel: the finite matrix model is a specification and counterexample language, not a claim that such kernels were measured for CEPO. Unbounded executions require further probability and semantic infrastructure.
The attached standard-library Python verifier uses exact rational arithmetic. All checks passed: 256 score tables; 2,048 sequential traces; 15,360 Boolean-cube identities; 625 contraction cases; 15,625 one-step and 2,048 multistep stochastic comparisons; 6,561 evaluation-error cases; and all eight binary transition triples. Six frozen-context recombination failures occur in the enumerated score domain. These are finite checks, not machine-checked higher categorical proofs or source benchmark replications.
The contract functors, admissibility evidence, coherent implementation assignment, and observations are explicit hypotheses. Descent does not construct missing witnesses; coherence does not prove independence, reward additivity, or convergence. Numerical improvement requires the acceptance or interaction hypotheses stated above.
Authorship and automation disclosure
Prepared anonymously with OpenAI Codex assistance at a human requester's direction. Generative assistance supported exposition, source retrieval, companion programming, and document production. The human submitting author is responsible for the final submitted version and its declarations. No independent referee certification, Lean verification, or CEPO experimental replication is claimed. The motivating architecture and empirical results remain attributed to [1].
References
1. Kim, Lee, Chung, Yoon, and Kim. CEPO: A General Framework for Improving Prompt Optimization in Multi-Agent Systems. arXiv:2610.34722v1. https://arxiv.org/abs/2610.34722v1 2. Lurie. Higher Topos Theory. Sections 3.2, 6.1.3. https://www.math.ias.edu/~lurie/papers/HTT.pdf 3. Lurie. Higher Algebra. Section 2.1. https://www.math.ias.edu/~lurie/papers/HA.pdf 4. Fritz. A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics. https://arxiv.org/abs/1908.07021 5. Hoeffding. Probability inequalities for sums of bounded random variables. https://doi.org/10.1080/01621459.1963.10500830 6. Dwork et al. Generalization in Adaptive Data Analysis and Holdout Reuse. https://arxiv.org/abs/1506.02629