--- PAGE 1 --- [proposition] 1 --- PAGE 2 --- Salience as Contextual Budgeted Set Selection Under Uncertain Utility Abstract Many practical systems must choose a finite-cost subset of candidates when value depends on context, constraints, and interactions among the selected items. This paper formulates that problem as contextual budgeted set selection under uncertain utility. The true objective is latent expected downstream value, while the operational system optimizes a surrogate set utility that combines itemwise terms for novelty, goal-fit, long-run leverage, coherence, and temporal relevance with penalties for drag and redundancy and bonuses for complementarity. Hard safety, policy, validity, and compatibility constraints are represented as feasibility conditions rather than offsetting penalties. This yields a marginal-gain view of selection and clarifies the distinct roles of candidate generation, reranking, constrained optimization, control, and learning. Simpler salience-style formulas appear as restricted special cases. Stronger optimality claims require additional structure, such as modularity or submodularity, and should not be stated without those assumptions. 1 Introduction Finite-capacity selection recurs across retrieval, recommendation, memory management, routing, and attention support. A system faces more candidates than it can carry forward, each candidate consumes some resource, and the payoff of selecting a set depends both on context and on what else is selected. Many salience-style formulations compress this into a single scalar item score. That compression is often useful as a mnemonic, but it is too coarse as a general theory. The cleaner view is to treat salience as a constrained set-selection problem. A chooser operates under finite capacity, hard feasibility constraints, uncertain downstream utility, and interaction effects among selected items. In that setting, the right formal object is not a universal item score, but a contextual set objective together with a feasible family and a selection procedure. This reformulation makes four clarifications. First, it separates latent downstream value from the surrogate utility the system actually estimates and optimizes. Second, it places hard constraints in feasibility rather than in soft penalties. Third, it treats redundancy and complementarity as set properties rather than pretending they are intrinsic to isolated items. Fourth, it separates scoring from optimization, control, and learning. Those separations are structural, not cosmetic. 2 --- PAGE 3 --- 2 Formal Setup 2.1 Context, candidates, and feasibility Let x denote the exogenous context for a single decision episode. Context may include user state, task, session history, device, time horizon, risk posture, and any other state that changes what is valuable now. Unless stated otherwise,x is held fixed while the current selection problem is solved. If selection itself changes the relevant state, the model should be applied stagewise or lifted to a sequential decision framework. LetI(x)be the candidate universe available in contextx. Each candidatei∈I(x)has: •hard item-level eligibilityT i(x)∈ {0,1}, •costc i(x)>0when cost-normalized scoring is used. Some hard constraints are set-level rather than item-level. Let C(A, x)∈ {0,1} denote any additional hard feasibility condition on a setA, such as mutual exclusion, compatibility, quota, schema, or channel constraints. Define the eligible universe I +(x) ={i∈I(x) :T i(x) = 1} and the feasible family F(x) = ( A⊆I +(x) : X i∈A ci(x)≤B(x), C(A, x) = 1 ) . This formulation keeps item-level gating and set-level gating distinct while allowing both to be enforced before or during selection. 2.2 Latent downstream value Let V ∗(A, x) =E[Y(A, x)|x] denote the latent expected downstream value of selectingA in context x, where the expectation is over whatever uncertainty remains after conditioning on context. The ideal chooser solves A∗(x)∈arg max A∈F(x) V ∗(A, x). 3 --- PAGE 4 --- In practice V ∗ is not directly known. The system therefore estimates and optimizes a surrogate objective. 2.3 Remarks on zero-cost and sequential cases If genuinely zero-cost items exist, ratio scoring by1/ci(x)is not automatically meaningful. Those items should either be handled by raw marginal gain, by a separate convention, or by a richer cost model. If adding an item changes the context materially, write statesx0, x1, . . . and solve a sequence of stagewise selection problems or a fuller sequential decision problem. The present formulation addresses a single decision episode with context treated as fixed during selection. 3 Feature Vocabulary and Surrogate Utility 3.1 Stable feature vocabulary For each candidatei, define: •N i(x): novelty or information gain, •G i(x): goal-fit or immediate relevance to the current task, •L i(x): long-run leverage, such as reuse value, retention value, compounding value, or future usefulness, •H i(x): coherence with the current trajectory, working set, or dialogue state, •τ i(x): age, staleness, or lag, •D i(x): drag, meaning friction, interruption cost, cognitive burden, latency burden, or soft risk burden. For pairs of candidatesi, j, define: •K ij(x): complementarity, synergy, or unlock value, •Sim ij(x): redundancy, overlap, or substitutability. Pairwise terms are taken to be symmetric unless otherwise stated. They are an approximation chosen for tractability, not a claim that all real interaction effects are pairwise. 4 --- PAGE 5 --- 3.2 Base item utility Letθcollect the parameters of the surrogate model. Define the base item utility ui(x;θ) =w N(x)ϕ N(Ni(x);x) +w G(x)G i(x) +w L(x)L i(x) +w H(x)H i(x) +w F (x)ϕ F (τi(x);x). Here w∗(x)are contextual weights, ϕN is a learned or chosen novelty response, andϕF maps age or lag to temporal relevance. This is the value of candidatei before interactions and soft burdens are applied. The weights should depend on context. Research mode, emergency mode, cold start, and long-session expert use should not all share the same tradeoff profile. 3.3 Set utility Define the surrogate set utility ˆU(A, x;θ) = X i∈A ui(x;θ) + X i160. 6 --- PAGE 7 --- Proposition 3(Redundancy-only penalties imply submodularity).Suppose Kij(x; θ) = 0for all pairs andSim ij(x;θ)≥0. Define bi(x;θ) =u i(x;θ)−D i(x;θ) and f(A) = X i∈A bi(x;θ)−λ div(x) X i 0. Then the pairwise complementarity term is supermodular, and the full surrogate may fail to be submodular. Proof sketch.Consider f(A) = X i∈A bi +K ab 1[{a, b} ⊆A]. With A = ∅, B = {b}, adding a gives marginal gainsba versus ba + Kab. Since Kab > 0, marginal gain increases after selectingb, violating diminishing returns. Proposition 5(Finite penalties cannot uniformly encode hard exclusion).Suppose ineligibility is represented by subtracting a finite penaltyM >0instead of using a hard gate: ˜U(A, x) =U(A, x)−M X i∈A (1−T i(x)). Then for any fixedM, there exist instances in which an ineligible item is still selected. Proof sketch.Pick an ineligible itemq with Tq(x) = 0that fits within budget. If∆ q(∅, x) > M, then penalized marginal gain remains positive:∆q(∅, x)−M >0. 7 --- PAGE 8 --- 6 Algorithmic Implications The propositions above pin down what can and cannot be claimed. In the modular, equal-cost, cardinality-constrained regime, top-k ranking is exact. That is the cleanest setting in which a single itemwise score really does solve the problem. With modular utility and nonuniform costs, the problem becomes0–1knapsack. Even then, a density rule is only a heuristic, not a general optimum rule. With redundancy-only pairwise penalties and nonnegative similarities, the surrogate becomes submodular. In that regime, diminishing returns appears naturally, and submodular-style algorithms become mathematically natural. Once positive complementarity enters, the objective can become neither modular nor submodular. Then greedy procedures lose any blanket claim to optimality, and richer search or optimization procedures become appropriate. Depending on scale and latency budget, those may include beam search, local search, reranking loops, mixed-integer optimization, or learned approximate selectors. The safe statement is therefore narrow: Algorithmic guarantees depend on the structure of the surrogate objective and the feasible family. They do not follow from a salience score alone. 7 Architectural Decomposition A practical system should separate six layers. 7.1 Eligibility layer Apply hard feasibility checks. Item-level disqualifiers go intoTi(x). Set-level disqualifiers go into C(A, x). Examples include safety rules, policy rules, schema validity, source validity thresholds, compatibility checks, mutual exclusion, and quota rules. 7.2 Candidate generation Recall a broad, cheap pool from the candidate universe. This stage optimizes coverage and speed, not final precision. 8 --- PAGE 9 --- 7.3 Reranker Estimate surrogate utility terms and interactions. This stage computes itemwise value signals, drag estimates, redundancy estimates, and complementarity estimates. 7.4 Selector Choose the final feasible set underF(x). This stage is where greedy construction, beam search, local search, ILP, or other constrained optimization methods belong. 7.5 Controller Adjust exploration, aggressiveness, budget tightening, temperature, channel multipliers, or other policy knobs that govern behavior across contexts. Exploration belongs here, not inside a single salience score. 7.6 Learner Update θ, thresholds, transforms, interaction estimates, and controller policies from observed outcomes. Learning belongs here, not inside the same scalar equation used for online ranking. This decomposition is useful because it prevents hard constraints, soft utility, exploration, optimiza- tion, and learning from being crammed into one overworked formula. 8 Learning and Evaluation The separation betweenV ∗ and ˆUhas direct learning consequences. Observed feedback typically arrives only for the selected set, not for all counterfactual feasible sets. That means learning the surrogate is partly a counterfactual estimation problem. A system may estimate component models for itemwise terms and pairwise terms, fit a direct predictor of set-level outcomes, or combine both approaches. A generic learning target is to estimate parametersθ so that higher surrogate utility corresponds to better observed outcomes under deployment-relevant conditions. In symbolic form, one may fitθ by minimizing a loss of the form X t ℓ ψ(At, xt;θ), y t  + Ω(θ). where At is the selected set in episodet, yt is the observed outcome,ψ is the model output used for training, andΩis any regularizer or structural penalty. Two evaluation cautions follow. 9 --- PAGE 10 --- First, when interactions matter, itemwise ranking metrics can become a misleading proxy for decision quality. A system that ranks individual items well can still assemble poor sets. Second, evaluation should track the level at which the system acts. If the selector chooses sets, then at least part of the evaluation should be set-level or policy-level rather than purely item-level. This is one more reason the latent-versus-surrogate split is not just notation cleanup. It affects the learning target and the evaluation protocol. 9 Domain Instantiations 9.1 Retrieval and RAG In retrieval settings: •Gcaptures query relevance, •Lcaptures downstream usefulness for reasoning or future reuse, •Hcaptures coherence with the current dialogue or working set, •τcaptures source age or lag, •Simcaptures chunk overlap, •Kcaptures multi-hop complementarity across evidence pieces, •Dcaptures token cost, latency, or distraction burden, •TandCcapture trust, source validity, safety, and policy constraints. 9.2 Training data selection In training data selection: •Ncaptures undercoverage or surprise, •Gcaptures fit to the current training objective, •Lcaptures long-run capability gain, •Hcaptures curriculum coherence, •τcaptures distribution lag where relevant, •Simcaptures duplication or oversampling overlap, •Kcaptures complementary samples that unlock generalization, •Dcaptures labeling cost, noise burden, or instability risk, •TandCcapture license, validity, and policy constraints. 10 --- PAGE 11 --- 9.3 Product and engagement systems In product flows: •Gcaptures immediate task completion value, •Lcaptures retention or future utility, •Hcaptures journey coherence, •τcaptures timeliness, •Simcaptures recommendation redundancy, •Kcaptures next-step unlocks or useful bundles, •Dcaptures friction and interruption burden, •TandCcapture trust, safety, and experience constraints. 9.4 Human operator mode For human reasoning, a compact salience heuristic can still be useful. But it should be treated as an operator shorthand over this substrate, not as the substrate itself. 10 Scope of Claims The model supports strong structural claims and weaker universal claims. Keeping those separate prevents puffed-up nonsense. Safe claims: • Many practical selection systems can be modeled as contextual feasible-set selection under uncertain utility. •Itemwise salience formulas are restricted special cases of that broader problem. •Hard exclusions are more robustly modeled as feasibility conditions than as finite penalties. •Redundancy can produce diminishing returns, while complementarity can destroy them. Unsafe claims unless extra assumptions are stated: •One scalar item score explains all selection behavior. •Greedy selection is optimal in general. •Value-per-cost ranking solves the nonuniform-cost case. •Hard constraints can always be replaced by sufficiently large soft penalties. 11 --- PAGE 12 --- 11 Conclusion The useful general principle is not that one salience equation explains everything. The useful principle is that finite-capacity selection is a contextual feasible-set problem under uncertain utility. Candidates have costs. Some candidates are ineligible. Value depends on context. Redundancy and complementarity are properties of sets, not just of isolated items. The deployed chooser acts on an estimated surrogate through feasible marginal gains, often normalized by cost when that normalization is meaningful. That framing preserves what was good in the original salience idea while removing notation drift, layer confusion, and fake universality. A Compact Canonical Form Given contextx, define F(x) = ( A⊆I +(x) : X i∈A ci(x)≤B(x), C(A, x) = 1 ) . Latent downstream value: V ∗(A, x) =E[Y(A, x)|x]. Surrogate utility: ˆU(A, x;θ) = X i∈A ui(x;θ) + X i