YUKTI: From Natural-Language Situations to Robust, Verifiable Decisions An Uncertainty-Typed Proposition IR, Assumption-Robust Pareto Frontiers, and a Regret Certificate

# From Natural-Language Situations to Robust, Verifiable Decisions An Uncertainty-Typed Proposition IR, Assumption-Robust Pareto Frontiers, and a Regret Certificate — why a language model should formulate, not solve
Source: [https://arxiv.org/html/2607.09706](https://arxiv.org/html/2607.09706)
\(June 2026\)

###### Abstract

Language models increasingly turn a worded situation into a numeric plan—and the dominant pipelines \(NL4Opt, OptiMUS, ORLM/LLMOPT, OR\-LLM\-Agent\) commit to a*single objective*and*point\-valued*coefficients, then solve once\. For decisions that allocate real budget, real field effort, or real clinical attention, that confidence*is*the failure mode: every objectified number is an assumption, and a plan that is optimal only if the guesses are exactly right is silently fragile\. We name the risk*mimicry of computation*—output with the*form*of an optimized recommendation but none of its substance\.

We presentYUKTI, which changes the*target*of autoformulation\. Its intermediate representation is a*Typed Proposition*graph in which every relationship carries a shape prior, a coefficient*uncertainty distribution*, and provenance\. On it, YUKTI \(i\) routes each stage to an exact, nonlinear, or evolutionary solver by probing structure; \(ii\) couples stages through a*distributional*Pareto hand\-off; \(iii\) introducesAssumption\-Robust Pareto Frontiers \(ARPF\), resampling the assumptions—including structural,ε\\varepsilon\-contamination misspecification—to score how often each action survives \(ρ\\rho\); and \(iv\) when no data exist, synthesizes a benchmark\-faithful foundation \(SRJANA\)\. We*prove*a regret bound makingρ\\rhoan exact factor of decision regret, and we surface*decision traceability*—which segments compose an action and which constraint binds—so a recommendation is auditable, not merely persuasive\.

We validate the central claim three ways\. Under controlled structural misspecification, the robust compromise cuts mean and tail regret by over90%90\\%versus a naive point plan\. On a regulated commercial decision \(an oncology brand during loss of exclusivity\), we optimize inside a*lawful*—MLR/consent/frequency/KAM —action space against a*prescribing\-proxy*objective and price the downside in euros\. Finally, on a*real*public dataset of41,18841\{,\}188marketing decisions we did not generate, an out\-of\-sample backtest shows the robust rule beating the logged status quo by34%34\\%and the naive point rule by4%4\\%while measurably reducing the optimizer’s curse\. The solvers are standard and we make no benchmark\-SOTA claim; the contribution is the uncertainty\-typed IR, ARPF and its regret bound, the distributional multi\-stage hand\-off, decision\-traceability reporting, and a deployment posture as a decision*stress\-testing*layer—not a system of record\. A head\-to\-head locates the limit of LLM reasoning precisely: an LRM handed the correct numbers, and classical single\-objective optimization, both incur∼47×\{\\sim\}47\\timesthe held\-out regret of YUKTI’s robust compromise—an LRM is a*formulator*, not a solver\. Finally we show where the formalism itself must change: under long\-range causal coupling between actions, the forward multi\-stage hand\-off becomes unsound—its regret grows with coupling and its per\-stage certificate turns optimistic—locating the boundary at which a hand\-off must become a backward\-induction causal policy\.

## 1 Introduction

A recurring request from decision\-makers is deceptively simple: “here is my situation in words—tell me what to do\.” Underneath it sits an optimization problem that nobody has written down\. The decision\-maker knows the*levers*they control, the*signals*they care about, and the*limits*they must respect, but not the variables, objective, and constraints of a formal program\. Recent work has shown that language models can bridge this gap: given a description, they can extract variables, constraints, and an objective and emit solver code\[[1](https://arxiv.org/html/2607.09706#bib.bib1),[3](https://arxiv.org/html/2607.09706#bib.bib3),[5](https://arxiv.org/html/2607.09706#bib.bib5)\]\.

Yet the prevailing formulation is narrow in three ways that matter precisely when the stakes are highest\. First, it is*single\-objective*: real briefs trade reach against cost against risk, and collapsing them into one weighted scalar hides the trade\-off the decision\-maker actually wants to see\. Second, it is*single\-shot*: many decisions are*sequential*—you first design a program, then decide how aggressively to roll it out, and the second decision is constrained by what the first achieved\. Third, and most important, it is*point\-valued*: to turn “oncologists respond well to clinical content” into a number, the model must invent a coefficient\. That coefficient is an assumption, and a plan that is optimal only at the assumed value can be fragile\.

Wordeddecision briefLLMautoformulationPoint\-valued model\(one objective\)SingleoptimumBrittleif an assumednumber is wrongTyped–uncertain IR\+ web anchors \(SRJANA\)Structure\-aware routerPareto frontier\(exact/NLP/EA\)Robustcompromisewith scoreρ\\rhoprevailing pipelineYUKTI \(this work\)Figure 1:Why this is critical as LLMs enter every decision\.The same worded brief can be objectified two ways\. The prevailing pipeline \(top\) commits to a single objective and point\-valued coefficients, yielding one optimum that is silently fragile when an invented number is off\. YUKTI \(bottom\) keeps objectives in conflict, carries each coefficient’s uncertainty, routes the stage to the right solver, and returns a compromise annotated with the probabilityρ\\rhothat it survives its own assumptions\. When autoformulation drives clinical, public\-health, energy, and financial resource allocation at scale, that robustness signal is the safeguard\.Why this matters now\.As LLMs move from drafting text to*making and shaping decisions*—triaging patients, allocating screening outreach, sizing energy subsidies, setting commercial strategy—the step that converts a qualitative brief into numbers is being executed at population scale, often invisibly\. Each objectified coefficient is an assumption; a single\-objective, point\-valued plan inherits all of them and emits no signal of its own fragility \(Fig\.[1](https://arxiv.org/html/2607.09706#S1.F1), top\)\. When such plans drive real resource allocation, silent brittleness is the failure mode\. The discipline YUKTI adds—keep conflicting objectives explicit, carry the uncertainty of every invented number, and report how often a recommendation survives its own assumptions—is exactly the safeguard that becomes critical once autoformulation is embedded everywhere \(Fig\.[1](https://arxiv.org/html/2607.09706#S1.F1), bottom\)\.

Two kinds of traceability\.The lesson is not that language models cannot build such a model—a capable model can, and that is how the system in this paper was built\. The lesson is about the*epistemic status of the output*\. The risk is*mimicry of computation*: prose with the*form*of an optimized recommendation but none of its substance—no solved model, no stated coefficients, no sensitivity, no robustness\. Retrieval\-augmented generation supplies*source traceability*\(where a fact came from\); high\-stakes decisions additionally need*decision traceability*—the ability to trace how the*recommended action*depends on its inputs: which segments it comprises, which assumptions move it, and which constraint binds\. A recommendation can be perfectly sourced and still be operationally unreliable\. YUKTI targets decision traceability directly, and §[15](https://arxiv.org/html/2607.09706#S15)adds two mechanisms—segment attribution and shadow prices—so the property is delivered as output, not asserted in narrative\.

This paper\.We proposeYUKTI111Sanskrit*yukti*: the fitting means, a reasoned device or stratagem\., a framework that keeps the LLM where it is strong—reading a situation and naming its structure—but changes the*target*of formulation and the*machinery*behind it:

- •aTyped Proposition IRin which each quantitative relationship is a functionφ​\(x,u;θ\)\\varphi\(x,u;\\theta\)carrying qualitative shape priors, a*distribution*over its coefficientsθ\\theta, and a provenance tag \(given / assumed / benchmark\);
- •astructure\-aware routerthat probes each stage numerically and sends it to an exact, nonlinear, or evolutionary multi\-objective solver;
- •amulti\-stage frontier hand\-offthat chains heterogeneous stages across a decision DAG; and
- •Assumption\-Robust Pareto Frontiers \(ARPF\): by resampling the IR’s coefficient distributions and re\-ranking the discovered solutions, YUKTI reports, for every candidate, the probability it remains feasible and non\-dominated, and selects a*robust*compromise\.
- •Benchmark\-Anchored Contextual Synthesis \(SRJANA\): a front\-end that, when no dataset exists, synthesizes a contextual dataset from the extracted spec and web\-sourced benchmark anchors, fits the propositions to it, and thereby makes the system answer briefs*dynamically*\.
- •Decision traceability: two reporting mechanisms that make a recommended action auditable—*segment attribution*\(which segments make up the action and their priority\) and*shadow\-price analysis*\(which constraint binds and the marginal value of relaxing it\)\.
- •A regret bound\(Thm\.[1](https://arxiv.org/html/2607.09706#Thmtheorem1)\) that makesρ\\rhoan exact factor of pool\-regret and holds under structural \(ε\\varepsilon\-contamination\) misspecification, plus anout\-of\-sample validationon a held\-out, RCT\-anchored world where the robust compromise cuts regret by\>90%\{\>\}90\\%versus a naive point plan\.

We deliberately separate novelty from reuse \(§[21](https://arxiv.org/html/2607.09706#S21)\)\. The solvers are well\-established; the contribution is the uncertainty\-typed IR, the ARPF mechanism in the autoformulation setting, and the heterogeneous hand\-off\. We make no claim of benchmark superiority; this is a system\-and\-method proposal with a verified end\-to\-end demonstration\.

Roadmap\.The paper is in four parts\.*Part I*\(this section, §[2](https://arxiv.org/html/2607.09706#S2)\) frames the problem and the prior autoformulation line\.*Part II*develops the method: the typed\-uncertain IR, the structure\-aware router, the multi\-stage hand\-off, ARPF and its regret bound, SRJANA, and the implementation\.*Part III*is evidence: a worked example, a three\-domain demonstration, system evaluation, decision traceability, validation under misspecification, validation on*real*data, and a regulated deployment\.*Part IV*maps the boundaries: a head\-to\-head against LLM reasoning and single\-objective optimization, the point at which long\-range causal coupling forces the formalism to change, and our positioning, limitations, and future work\.

## 2 Related Work

LLMs for optimization modeling\.NL4Opt framed natural\-language\-to\-model as extraction of variables, constraints, and objectives\[[1](https://arxiv.org/html/2607.09706#bib.bib1)\]\. Chain\-of\-Experts introduced multi\-agent reasoning for operations research\[[2](https://arxiv.org/html/2607.09706#bib.bib2)\]\. OptiMUS and OptiMUS\-0\.3 made the loop concrete—formulate, generate solver code, execute, debug\[[3](https://arxiv.org/html/2607.09706#bib.bib3),[4](https://arxiv.org/html/2607.09706#bib.bib4)\]\. ORLM fine\-tuned open models on synthetic instances and released the IndustryOR benchmark\[[5](https://arxiv.org/html/2607.09706#bib.bib5)\]; LLMOPT codified a five\-element structure \(sets, parameters, variables, objectives, constraints\)\[[6](https://arxiv.org/html/2607.09706#bib.bib6)\]; OR\-LLM\-Agent added explicit task decomposition with a reasoning model\[[7](https://arxiv.org/html/2607.09706#bib.bib7)\]\. The newest systems emphasize a*persisted*intermediate representation and post\-hoc analysis: ORPilot stores a validated JSON IR and supports what\-if queries\[[8](https://arxiv.org/html/2607.09706#bib.bib8)\]; OptiRepair and OptiLoop close the loop with solver\-verified diagnosis and repair, arguing for “operational rationality” beyond mere feasibility\[[9](https://arxiv.org/html/2607.09706#bib.bib9),[10](https://arxiv.org/html/2607.09706#bib.bib10)\]\. All of these treat coefficients as fixed once elicited, and almost all target single\-objective formulation accuracy\. YUKTI inherits the IR\-and\-solver discipline but makes the IR*uncertain and typed*, and makes the target*multi\-objective and multi\-stage*\.

Exact multi\-objective programming\.Theε\\varepsilon\-constraint method generates Pareto\-optimal points by optimizing one objective while bounding the rest\[[11](https://arxiv.org/html/2607.09706#bib.bib11)\]\. AUGMECON2 augments it with lexicographic slack terms to avoid weakly\-efficient points and skip redundant iterations, producing the exact Pareto set for integer programs\[[12](https://arxiv.org/html/2607.09706#bib.bib12)\]; AUGMECON\-R extends practicality to many objectives\[[13](https://arxiv.org/html/2607.09706#bib.bib13)\]\. We use an augmentedε\\varepsilon\-constraint generator on HiGHS for affine stages\.

Evolutionary multi/many\-objective optimization\.NSGA\-II combined fast non\-dominated sorting with crowding distance\[[14](https://arxiv.org/html/2607.09706#bib.bib14)\]; NSGA\-III replaced crowding with reference\-direction association for many objectives\[[15](https://arxiv.org/html/2607.09706#bib.bib15)\], with directions placed by the Das–Dennis simplex construction\[[17](https://arxiv.org/html/2607.09706#bib.bib17)\]; MOEA/D decomposes the problem into scalar subproblems\[[16](https://arxiv.org/html/2607.09706#bib.bib16)\]\. We access these through pymoo\[[18](https://arxiv.org/html/2607.09706#bib.bib18)\], along with its compromise\-programming and high\-trade\-off MCDM utilities\.

Sequential decisions under uncertainty\.Multistage stochastic programming models here\-and\-now versus recourse decisions over a scenario tree with nonanticipativity, solved by nested Benders / L\-shaped decomposition\[[21](https://arxiv.org/html/2607.09706#bib.bib21),[22](https://arxiv.org/html/2607.09706#bib.bib22)\]; the equivalent deterministic program characterizes its structure\[[23](https://arxiv.org/html/2607.09706#bib.bib23),[24](https://arxiv.org/html/2607.09706#bib.bib24)\]\. YUKTI’s multi\-stage layer is a deterministic*frontier hand\-off*—a hierarchical coupling of stages— rather than full recourse; we treat coefficient uncertainty through ARPF instead of a scenario tree, and discuss the relationship in §[22](https://arxiv.org/html/2607.09706#S22)\.

Frontier uncertainty\.In multi\-objective Bayesian optimization, qEHVI and qNEHVI optimize expected hypervolume improvement and, crucially, integrate over uncertainty*in the Pareto frontier*induced by noisy observations\[[19](https://arxiv.org/html/2607.09706#bib.bib19),[20](https://arxiv.org/html/2607.09706#bib.bib20)\]\. ARPF is the same idea displaced to a different source: it integrates over uncertainty in the frontier induced by uncertain*model coefficients*elicited by an LLM, rather than noisy measurements\. Robust optimization more broadly seeks solutions stable across an uncertainty set\[[25](https://arxiv.org/html/2607.09706#bib.bib25),[26](https://arxiv.org/html/2607.09706#bib.bib26)\]; ARPF is a sampling\-based, frontier\-level instance tailored to autoformulated propositions\.

Part II\|The YUKTI Method

## 3 The YUKTI Framework

![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/pipeline.png)Figure 2:The YUKTI pipeline\. A worded situation is elicited into levers, signals, constraints and a stage DAG; each relationship is*objectified*into a typed proposition with an uncertainty distribution; stages are routed to an exact, NLP, or evolutionary solver; solved with a Pareto*hand\-off*across stages; robustified by resampling assumptions \(ARPF\); and reconciled into a recommended, robustness\-scored compromise\.A problem is a directed acyclic graph of*stages*𝒮=\{s1,…,sT\}\\mathcal\{S\}=\\\{s\_\{1\},\\dots,s\_\{T\}\\\}in topological order\. Each stagessowns leversxs∈Xsx\_\{s\}\\in X\_\{s\}, a set of objective*signals*, constraints, and a list of upstream signal keys it consumes\. Solving a stage yields a Pareto set; a compromise is selected and its signal values are injected as parametersuufor downstream stages \(Fig\.[2](https://arxiv.org/html/2607.09706#S3.F2)\)\. Formally, stagessis the multi\-objective program

minx∈Xs\\displaystyle\\min\_\{x\\in X\_\{s\}\}Fs​\(x,us;θs\)=\(fs,1,…,fs,ms\),\\displaystyle F\_\{s\}\(x,u\_\{s\};\\theta\_\{s\}\)=\\big\(f\_\{s,1\},\\dots,f\_\{s,m\_\{s\}\}\\big\),\(1\)s\.t\.​gs,j​\(x,us;θs\)≤0,\\displaystyle\\text\{s\.t\. \}g\_\{s,j\}\(x,u\_\{s\};\\theta\_\{s\}\)\\leq 0,where maximization signals are negated intomin\\minform,usu\_\{s\}collects upstream hand\-offs, andθs\\theta\_\{s\}are coefficients drawn from the proposition ledger\.

### 3\.1 Pareto optimality

For min\-sense vectors,aa*dominates*bb\(a≺ba\\prec b\) iffai≤bi​∀ia\_\{i\}\\leq b\_\{i\}\\,\\forall iandak<bka\_\{k\}<b\_\{k\}for somekk\. The Pareto set of a stage is\{x⋆:∄​x∈Xs,Fs​\(x\)≺Fs​\(x⋆\)\}\\\{x^\{\\star\}:\\nexists\\,x\\in X\_\{s\},\\ F\_\{s\}\(x\)\\prec F\_\{s\}\(x^\{\\star\}\)\\\}\. YUKTI’s solvers return a finite non\-dominated approximationℱs\\mathcal\{F\}\_\{s\}\.

## 4 Typed Proposition Intermediate Representation

The distinguishing object is the proposition\.

###### Definition 1\(Typed proposition\)\.

A typed proposition is a tuplep=⟨φ,θ,Θ,σ,π⟩p=\\langle\\varphi,\\ \\theta,\\ \\Theta,\\ \\sigma,\\ \\pi\\ranglewhereφ​\(x,u;θ\)∈ℝ\\varphi\(x,u;\\theta\)\\in\\mathbb\{R\}is a parametric functional form;θ\\thetaare nominal coefficients;Θ=\{\(distk,ηk\)\}\\Theta=\\\{\(\\mathrm\{dist\}\_\{k\},\\eta\_\{k\}\)\\\}assigns each uncertain coefficient a multiplicative distribution with relative spreadηk\\eta\_\{k\}\(normal, lognormal, or uniform\);σ\\sigmarecords qualitative shape priors \(monotonicity±\\pmper lever; curvature: linear / convex / saturating / bilinear\); andπ∈\{given, assumed, benchmark\}\\pi\\in\\\{\\text\{given, assumed, benchmark\}\\\}is provenance\.

A signal is a proposition plus a sense \(max/min\); a constraint is a proposition plus an operator and right\-hand side\. Two design choices matter\. First, the same proposition object can back both a signal and a constraint—e\.g\. a*fatigue*proposition is simultaneously an objective to minimize and a guardrail≤0\.35\\leq 0\.35—so that a single resampled draw moves both consistently \(shared identity, §[8](https://arxiv.org/html/2607.09706#S8)\)\. Second, becauseφ\\varphiis an arbitrary callable, the IR spans linear, smooth\-nonlinear, and nonconvex/black\-box relationships without changing the downstream machinery; structure is*discovered*, not declared \(§[5](https://arxiv.org/html/2607.09706#S5)\)\.

A coefficient draw is multiplicative, e\.g\. for a lognormal coefficientθk\(j\)=θk⋅eξ,ξ∼𝒩​\(0,ηk2\)\\theta\_\{k\}^\{\(j\)\}=\\theta\_\{k\}\\cdot e^\{\\,\\xi\},\\ \\xi\\sim\\mathcal\{N\}\(0,\\eta\_\{k\}^\{2\}\), keeping signs and order of magnitude while perturbing the elicited guess\. This is what turns “the model invented a number” from a liability into a quantified input\.

## 5 Structure\-Aware Routing

Before solving, YUKTI decides*how*\. For a stage with fixeduuand nominalθ\\theta, it forms the stacked maph​\(x\)=\(Fs​\(x\),Gs​\(x\)\)h\(x\)=\(F\_\{s\}\(x\),G\_\{s\}\(x\)\)on the continuous relaxation and probes it on random pairsx\(1\),x\(2\)x^\{\(1\)\},x^\{\(2\)\}with interpolation weightα\\alpha\(Alg\.[1](https://arxiv.org/html/2607.09706#alg1)\)\. Affinity is accepted ifh​\(α​x\(1\)\+\(1−α\)​x\(2\)\)h\(\\alpha x^\{\(1\)\}\{\+\}\(1\{\-\}\\alpha\)x^\{\(2\)\}\)matchesα​h​\(x\(1\)\)\+\(1−α\)​h​\(x\(2\)\)\\alpha h\(x^\{\(1\)\}\)\{\+\}\(1\{\-\}\\alpha\)h\(x^\{\(2\)\}\)within a scaled tolerance for all pairs; convexity \(in min\-sense\) is accepted if the interpolated point never lies below the chord \(Jensen\)\. The route is then:*affine*→\\rightarrowexact \(MO\-LP, or MO\-MILP if integer levers are present\);*smooth, convex, continuous*→\\rightarrowNLP; otherwise→\\rightarrowevolutionary\. Probing the*relaxation*is essential: rounding integer levers turns an affine model into a step function and would defeat the linearity test, so structure is detected continuously while integrality is enforced inside the exact solver\.

Algorithm 1Structure\-aware routing of a stage1:stage

ss, upstream

uu, probes

RR, pairs

NN, tol

τ\\tau
2:sample

RRpoints; set scale

←\\leftarrowmean

\|h\|\|h\|per component

3:affine, convex

←\\leftarrowtrue, true

4:for

NNrandom pairs

\(x\(1\),x\(2\)\)\(x^\{\(1\)\},x^\{\(2\)\}\),

α∈\(0,1\)\\alpha\\in\(0,1\)do

5:

Δ←h​\(α​x\(1\)\+\(1−α\)​x\(2\)\)−\[α​h​\(x\(1\)\)\+\(1−α\)​h​\(x\(2\)\)\]\\Delta\\leftarrow h\(\\alpha x^\{\(1\)\}\{\+\}\(1\{\-\}\\alpha\)x^\{\(2\)\}\)\-\[\\alpha h\(x^\{\(1\)\}\)\{\+\}\(1\{\-\}\\alpha\)h\(x^\{\(2\)\}\)\]
6:if

max⁡\|Δ\|\>τ⋅\(scale\+1\)\\max\|\\Delta\|\>\\tau\\cdot\(\\text\{scale\}\{\+\}1\)thenaffine

←\\leftarrowfalse

7:endif

8:ifany component of

Δ\>τ⋅\(scale\+1\)\\Delta\>\\tau\\cdot\(\\text\{scale\}\{\+\}1\)thenconvex

←\\leftarrowfalse

9:endif

10:endfor

11:ifaffinethenreturnexact⊳\\trianglerightMILP if integer levers else LP

12:elseifconvex and continuousthenreturnnlp

13:elsereturnea

14:endif

## 6 Solvers

Exact \(affine\)\.When a stage is affine, YUKTI extracts the coefficient matrices by finite differences on the relaxation \(the gradient is constant\), then runs an augmentedε\\varepsilon\-constraint generator\[[12](https://arxiv.org/html/2607.09706#bib.bib12)\]\. With the first objective as primary and the rest bounded on a grid between their payoff ranges\[fkideal,fknadir\]\[\\,f\_\{k\}^\{\\text\{ideal\}\},f\_\{k\}^\{\\text\{nadir\}\}\\,\],

minx∈Xs,sk≥0⁡f1​\(x\)−δ​∑k≥2skrk​s\.t\.​fk​\(x\)\+sk=ek,\\min\_\{x\\in X\_\{s\},\\,s\_\{k\}\\geq 0\}\\;f\_\{1\}\(x\)\-\\delta\\\!\\\!\\sum\_\{k\\geq 2\}\\frac\{s\_\{k\}\}\{r\_\{k\}\}\\;\\;\\text\{s\.t\. \}f\_\{k\}\(x\)\+s\_\{k\}=e\_\{k\},\(2\)whererkr\_\{k\}is the range of objectivekkandδ\\deltaa small augmentation\. Each grid point is an LP \(HiGHSlinprog\) or MILP \(HiGHSmilp\) with integrality enforced onxx; results are filtered to the non\-dominated set\. For two or three objectives this yields the exact \(or near\-exact\) Pareto front\.

Nonlinear \(convex, smooth\)\.YUKTI applies theε\\varepsilon\-constraint with SLSQP, sweeping the secondary\-objective grid with multiple random restarts, then filters non\-dominance\.

Evolutionary \(nonconvex / black\-box / mixed\-integer\)\.Otherwise YUKTI uses NSGA\-II for≤2\\leq 2objectives and NSGA\-III with Das–Dennis reference directions for≥3\\geq 3\[[14](https://arxiv.org/html/2607.09706#bib.bib14),[15](https://arxiv.org/html/2607.09706#bib.bib15),[17](https://arxiv.org/html/2607.09706#bib.bib17)\], via pymoo\[[18](https://arxiv.org/html/2607.09706#bib.bib18)\], with constraint handling by feasibility\-first domination and integer levers realized by rounding in evaluation\.

## 7 Multi\-Stage Frontier Hand\-off

Stages are solved in topological order\. After stagessproducesℱs\\mathcal\{F\}\_\{s\}, YUKTI selects a \(robust\) compromisexs⋆x^\{\\star\}\_\{s\}and*hands off*its signal values:

us\+1⊇\{key​\(fs,i\)↦fs,i​\(xs⋆\)\}i=1ms\.u\_\{s\+1\}\\supseteq\\big\\\{\\,\\text\{key\}\(f\_\{s,i\}\)\\mapsto f\_\{s,i\}\(x^\{\\star\}\_\{s\}\)\\,\\big\\\}\_\{i=1\}^\{m\_\{s\}\}\.\(3\)Downstream propositions read these as parameters\. In the demonstration, the email\-engagement stage exposes*open rate*and*CTR*; the transition stage consumes them through an engagement\-quality termEQ=min⁡\(1,0\.6⋅open\+4⋅CTR\)\\mathrm\{EQ\}=\\min\(1,\\,0\.6\\cdot\\text\{open\}\+4\\cdot\\text\{CTR\}\)that scales how much share\-of\-voice and continuity the rollout can preserve\. This is a hierarchical coupling \(strategic stage fixes the environment of the operational stage\), not a recourse model; §[22](https://arxiv.org/html/2607.09706#S22)states the trade\-off\.

Distributional hand\-off\.Passing the point valuesfs,i​\(xs⋆\)f\_\{s,i\}\(x^\{\\star\}\_\{s\}\)commits the downstream stage to a single number—in tension with the paper’s own thesis against point values\. But under ARPF we already hold, atxs⋆x^\{\\star\}\_\{s\}, the*distribution*of each handed\-off signal across theKKassumption draws\. YUKTI therefore hands off that distribution \(its quantiles\), and stages\+1s\{\+\}1evaluates feasibility and value in expectation over it, so a stage\-1 signal that is high in the mean but heavy\-tailed no longer masks downstream risk\. §[16](https://arxiv.org/html/2607.09706#S16)exhibits a point hand\-off that reports a stage\-2 floor as satisfied while the distributional hand\-off prices a60%60\\%violation probability\.

## 8 Assumption\-Robust Pareto Frontiers

A frontier computed at the nominalθ\\thetaanswers “what is optimal if my elicited numbers are exactly right?”—rarely the operative question\. ARPF answers “what stays good when they are wrong?” Let𝒳s\\mathcal\{X\}\_\{s\}be the discovered decision pool\. ForKKdrawsθ\(k\)∼Θ\\theta^\{\(k\)\}\\sim\\Theta\(shared across a proposition’s signal/constraint roles via object identity\), recompute objectives and feasibility for every pooled decision, take the non\-dominated subset of the feasible ones, and accumulate membership:

ρ​\(x\)=1K​∑k=1K𝟏​\[x​feasible∧x∈ND​\(Fs​\(⋅;θ\(k\)\)\)\]\.\\rho\(x\)=\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\mathbf\{1\}\\\!\\left\[x\\text\{ feasible\}\\wedge x\\in\\mathrm\{ND\}\\big\(F\_\{s\}\(\\cdot;\\theta^\{\(k\)\}\)\\big\)\\right\]\.\(4\)ρ​\(x\)\\rho\(x\)is the empirical probability thatxxsurvives as a sensible choice under assumption uncertainty; the per\-decision objective spread \(mean±\\pmstd across draws\) traces an*envelope*around the nominal frontier \(Fig\.[4](https://arxiv.org/html/2607.09706#S12.F4)b\)\. We then choose the compromise*among robust solutions*: restrict toρ​\(x\)≥\\rho\(x\)\\geqthe median, and within that set minimize the normalized distance to the ideal point \(compromise programming\),

xs⋆=arg⁡minx:ρ​\(x\)≥ρ0\.5​∑iwi​\(fs,i​\(x\)−fs,iidealfs,inadir−fs,iideal\)2\.x^\{\\star\}\_\{s\}=\\arg\\min\_\{x:\\rho\(x\)\\geq\\rho\_\{0\.5\}\}\\;\\sum\_\{i\}w\_\{i\}\\Big\(\\tfrac\{f\_\{s,i\}\(x\)\-f\_\{s,i\}^\{\\text\{ideal\}\}\}\{f\_\{s,i\}^\{\\text\{nadir\}\}\-f\_\{s,i\}^\{\\text\{ideal\}\}\}\\Big\)^\{2\}\.\(5\)Finally, a one\-at\-a\-time tornado ranks which coefficients move the recommendation most \(Figs\.[3](https://arxiv.org/html/2607.09706#S12.F3)b,[4](https://arxiv.org/html/2607.09706#S12.F4)c\), separating*assumed*from*benchmark*provenance so attention goes where it is both influential and weakly grounded\.

Structural misspecification \(ε\\varepsilon\-contamination\)\.Resamplingθ\\thetawithin a parametric family addresses*parameter*error but not a*wrong functional form*—the error that usually dominates in practice\. YUKTI therefore letsΘ\\Thetabe anε\\varepsilon\-contamination classΘ=\(1−ε\)​Θ0\+ε​Q\\Theta=\(1\-\\varepsilon\)\\Theta\_\{0\}\+\\varepsilon\\,Q: with probabilityε\\varepsilona draw replaces a proposition’s form with an alternative fromQQ\(for instance injecting a diminishing\-returns interaction absent from a nominal additive form\)\. ARPF over thisΘ\\Thetascores robustness to the model being*structurally*wrong, not merely mis\-calibrated; §[16](https://arxiv.org/html/2607.09706#S16)shows this is precisely what lets the robust choice generalize to a held\-out world whose structure the nominal model never represented\.

Algorithm 2YUKTI multi\-stage solve with ARPF1:problem

𝒮\\mathcal\{S\}in topological order; draws

KK
2:

u←\{\}u\\leftarrow\\\{\\\}
3:forstage

ssin orderdo

4:method

←\\leftarrowRoute\(s,u\)\(s,u\)⊳\\trianglerightAlg\.[1](https://arxiv.org/html/2607.09706#alg1)

5:

ℱs←\\mathcal\{F\}\_\{s\}\\leftarrowSolve\(s,u\)method\{\}\_\{\\text\{method\}\}\(s,u\)

6:

\{ρ,env\}←\\\{\\rho,\\text\{env\}\\\}\\leftarrowARPF\(s,u,ℱs,K\)\(s,u,\\mathcal\{F\}\_\{s\},K\)⊳\\trianglerightEq\.[4](https://arxiv.org/html/2607.09706#S8.E4)

7:

xs⋆←x^\{\\star\}\_\{s\}\\leftarrowRobustCompromise\(ℱs,ρ\)\(\\mathcal\{F\}\_\{s\},\\rho\)⊳\\trianglerightEq\.[5](https://arxiv.org/html/2607.09706#S8.E5)

8:

u←u∪\{fs,i​\(xs⋆\)\}iu\\leftarrow u\\cup\\\{\\,f\_\{s,i\}\(x^\{\\star\}\_\{s\}\)\\,\\\}\_\{i\}⊳\\trianglerighthand\-off

9:endfor

10:return

\{\(ℱs,xs⋆,ρs\)\}\\\{\(\\mathcal\{F\}\_\{s\},x^\{\\star\}\_\{s\},\\rho\_\{s\}\)\\\}

## 9 A Regret Bound for ARPF

ARPF’s robustness score is not merely heuristic: it is an exact factor of decision regret\. Fix a finite poolPPand a distributionΘ\\Thetaover modelsθ\\theta\(parameters*and*functional form\)\. LetV​\(x;θ\)∈\[0,Vmax\]V\(x;\\theta\)\\in\[0,V\_\{\\max\}\]be a bounded decision value withV​\(x;θ\)=0V\(x;\\theta\)=0whenxxis infeasible underθ\\theta, letV⋆​\(θ\)=maxx∈P⁡V​\(x;θ\)V^\{\\star\}\(\\theta\)=\\max\_\{x\\in P\}V\(x;\\theta\)be the in\-hindsight pool optimum, andρ​\(x\)=Prθ⁡\[x∈arg⁡maxx′∈P⁡V​\(x′;θ\)\]\\rho\(x\)=\\Pr\_\{\\theta\}\[x\\in\\arg\\max\_\{x^\{\\prime\}\\in P\}V\(x^\{\\prime\};\\theta\)\]the decision\-robustness of Eq\.[4](https://arxiv.org/html/2607.09706#S8.E4)\. Define the*pool\-regret*R​\(x^\)=𝔼θ​\[V⋆​\(θ\)−V​\(x^;θ\)\]R\(\\hat\{x\}\)=\\mathbb\{E\}\_\{\\theta\}\[V^\{\\star\}\(\\theta\)\-V\(\\hat\{x\};\\theta\)\]\.

###### Theorem 1\(ARPF regret decomposition and certificate\)\.

For anyx^∈P\\hat\{x\}\\in P, writingg¯​\(x^\)=𝔼θ​\[V⋆−V​\(x^\)∣x^∉arg⁡max\]\\bar\{g\}\(\\hat\{x\}\)=\\mathbb\{E\}\_\{\\theta\}\[V^\{\\star\}\-V\(\\hat\{x\}\)\\mid\\hat\{x\}\\notin\\arg\\max\]andB​\(x^\)=ess​supθ⁡\(V⋆​\(θ\)−V​\(x^;θ\)\)B\(\\hat\{x\}\)=\\operatorname\*\{ess\\,sup\}\_\{\\theta\}\(V^\{\\star\}\(\\theta\)\-V\(\\hat\{x\};\\theta\)\):*\(i\)*the regret decomposes exactly,R​\(x^\)=\(1−ρ​\(x^\)\)​g¯​\(x^\)R\(\\hat\{x\}\)=\(1\-\\rho\(\\hat\{x\}\)\)\\,\\bar\{g\}\(\\hat\{x\}\);*\(ii\)*henceR​\(x^\)≤\(1−ρ​\(x^\)\)​B​\(x^\)R\(\\hat\{x\}\)\\leq\(1\-\\rho\(\\hat\{x\}\)\)\\,B\(\\hat\{x\}\);*\(iii\)*for a shared boundB¯≥B​\(⋅\)\\bar\{B\}\\geq B\(\\cdot\),arg⁡maxx⁡ρ​\(x\)\\arg\\max\_\{x\}\\rho\(x\)minimizes the certificate\(1−ρ\)​B¯\(1\-\\rho\)\\bar\{B\}and is the Bayes rule under0–11pool\-optimality loss; and*\(iv\)*Pr⁡\(\|ρ^K−ρ\|≥t\)≤2​e−2​K​t2\\Pr\(\|\\hat\{\\rho\}\_\{K\}\-\\rho\|\\geq t\)\\leq 2e^\{\-2Kt^\{2\}\}\(Hoeffding\)\. The result holds for*arbitrary*Θ\\Theta, in particular anε\\varepsilon\-contamination classΘ=\(1−ε\)​Θ0\+ε​Q\\Theta=\(1\-\\varepsilon\)\\Theta\_\{0\}\+\\varepsilon Q; hence the bound accounts for structural misspecification, not only parameter noise\.

###### Proof\.

\(i\)V⋆−V​\(x^\)=0V^\{\\star\}\-V\(\\hat\{x\}\)=0on the event\{x^∈arg⁡max\}\\\{\\hat\{x\}\\in\\arg\\max\\\}, so by the law of total expectationR​\(x^\)=Pr⁡\[x^∉arg⁡max\]⋅𝔼​\[V⋆−V​\(x^\)∣x^∉arg⁡max\]=\(1−ρ​\(x^\)\)​g¯​\(x^\)R\(\\hat\{x\}\)=\\Pr\[\\hat\{x\}\\notin\\arg\\max\]\\cdot\\mathbb\{E\}\[V^\{\\star\}\-V\(\\hat\{x\}\)\\mid\\hat\{x\}\\notin\\arg\\max\]=\(1\-\\rho\(\\hat\{x\}\)\)\\bar\{g\}\(\\hat\{x\}\)\. \(ii\)0≤V⋆−V​\(x^\)≤B​\(x^\)0\\leq V^\{\\star\}\-V\(\\hat\{x\}\)\\leq B\(\\hat\{x\}\)a\.s\. and=0=0on the argmax event, soR≤\(1−ρ\)​BR\\leq\(1\-\\rho\)B\. \(iii\) For sharedB¯\\bar\{B\},\(1−ρ​\(x\)\)​B¯\(1\-\\rho\(x\)\)\\bar\{B\}decreases inρ​\(x\)\\rho\(x\); and under0–11lossL​\(x;θ\)=𝟏​\[x∉arg⁡max\]L\(x;\\theta\)=\\mathbf\{1\}\[x\\notin\\arg\\max\],𝔼θ​L=1−ρ​\(x\)\\mathbb\{E\}\_\{\\theta\}L=1\-\\rho\(x\), minimized byarg⁡maxx⁡ρ\\arg\\max\_\{x\}\\rho\. \(iv\)𝟏​\[x∈arg⁡max\]\\mathbf\{1\}\[x\\in\\arg\\max\]is Bernoulli\(ρ\)\(\\rho\); Hoeffding overKKi\.i\.d\. draws gives the bound\. The argument uses only boundedness andθ∼Θ\\theta\\sim\\Theta, soΘ\\Thetamay mix functional forms\. ∎

Scope\.RRis regret against the*best\-in\-pool*, not an unattainable global optimum—the right object because ARPF re\-ranks a fixed discovered pool\. Part \(iii\) certifies the0–11pool\-regret; for*value*regret under misspecification the robust compromise additionally uses the downside \(lower tail\) of the same per\-action value distribution, validated next\. The certificate is checked numerically: the identity \(i\) reproducesRRexactly and \(ii\) holds withρ=0\.24,g¯=0\.21,B=0\.45\\rho\{=\}0\.24,\\ \\bar\{g\}\{=\}0\.21,\\ B\{=\}0\.45\(Fig\.[8](https://arxiv.org/html/2607.09706#S16.F8)b\)\.

## 10 Dynamic Contextual Data Synthesis

Autoformulation assumes the numbers exist—that a coefficient can be elicited or a table queried\. Often neither holds: a fresh brief arrives with conflicting objectives, named levers, and*no*data\. The point\-valued IRs of prior work have nothing to fit\. YUKTI’s typed propositions invite a different move: if every relationship is a parametric form with shape priors and an anchor, we can*synthesize*a contextual dataset that honours those priors and matches public benchmarks, then fit the propositions to it\. We call thisBenchmark\-Anchored Contextual Synthesis \(SRJANA\)\.

A*scenario spec*lists context dimensions \(each a categorical with levels and a latent affinity per level\), levers with bounds, and signals—each annotated with a sense, a kind \(rate / score / cost\), signed lever effects \(the shape priors\), and a*benchmark anchor*μ⋆\\mu^\{\\star\}\(a target marginal, ideally retrieved from a trusted source at query time\)\. SRJANA draws a population of context units with latent affinityzu=∑dima~​\(levelu\)z\_\{u\}=\\sum\_\{\\text\{dim\}\}\\tilde\{a\}\(\\text\{level\}\_\{u\}\)and a set of decision configurations\{xc\}\\\{x\_\{c\}\\\}, then forms the linear predictorLc,u=∑ℓβs,ℓ​x~c,ℓ\+γs​zuL\_\{c,u\}=\\sum\_\{\\ell\}\\beta\_\{s,\\ell\}\\tilde\{x\}\_\{c,\\ell\}\+\\gamma\_\{s\}z\_\{u\}for each signalss\. The defining step is*calibration to the benchmark*: for a rate signal choose the interceptαs\\alpha\_\{s\}so the simulated grand mean matches the anchor,

αs:1N​M∑c,uσ\(αs\+Lc,u\)=μs⋆,\\alpha\_\{s\}:\\quad\\tfrac\{1\}\{NM\}\\textstyle\\sum\_\{c,u\}\\sigma\\\!\\big\(\\alpha\_\{s\}\+L\_\{c,u\}\\big\)=\\mu^\{\\star\}\_\{s\},\(6\)solved by one\-dimensional root\-finding \(σ\\sigmamonotone inα\\alpha\); for a linear score/cost signalαs=μs⋆−L¯\\alpha\_\{s\}=\\mu^\{\\star\}\_\{s\}\-\\overline\{L\}in closed form\. SRJANA then simulates outcomes, aggregates to configuration level, and fits the response surfaces by OLS—returning coefficients*and*standard errors that become the propositions’ nominalθ\\thetaand uncertaintyΘ\\Theta\(Alg\.[3](https://arxiv.org/html/2607.09706#alg3)\)\. The synthetic data is therefore realistic by construction \(its marginals are the benchmarks\) while its conditional structure obeys the elicited signs and the optimizer inherits honest, data\-derived uncertainty for ARPF\.

Three additions make SRJANA faithful rather than merely calibrated\. First, benchmarks usually fix not just a mean but*contrasts*—“rep\-triggered email earns roughly twice the click rate of mass\.” We therefore calibrate a small parameter vector\(αs,κs\)\(\\alpha\_\{s\},\\kappa\_\{s\}\)\(intercept and a global effect scale\) against a system of moment equations,

p¯​\(α,κ\)=μ⋆,p¯ℓ=1​\(α,κ\)p¯ℓ=0​\(α,κ\)=rℓ⋆,\\bar\{p\}\(\\alpha,\\kappa\)=\\mu^\{\\star\},\\qquad\\frac\{\\bar\{p\}\_\{\\ell=1\}\(\\alpha,\\kappa\)\}\{\\bar\{p\}\_\{\\ell=0\}\(\\alpha,\\kappa\)\}=r^\{\\star\}\_\{\\ell\},\(7\)solved by least squares—so both the level and the key effect ratios match the literature\. Second, SRJANA injects the*pathologies*real CRM extracts carry \(Apple Mail Privacy Protection open inflation, bounce/spam statuses, consent gating\), so downstream analytics confront the same artefacts they will in production and \(e\.g\.\) correctly prefer click\-to\-open over raw open rate\. Third, and most important, SRJANA closes a*self\-consistency*loop: after synthesis it refits the response surface and checks that the recovered coefficients match the seeded priors\. Across the three scenarios below, sign recovery is100%100\\%and the calibrated marginals reproduce their targets to rounding \(Fig\.[5](https://arxiv.org/html/2607.09706#S13.F5)\)\. A dataset that could not recover its own generating priors would be unfit to optimize against; SRJANA refuses to emit one\.

Algorithm 3SRJANA: benchmark\-anchored contextual synthesis1:spec \(context, levers, signals w/ anchors

μ⋆\\mu^\{\\star\}, priors

β,γ\\beta,\\gamma\)

2:sample

NNunits;

zu←z\_\{u\}\\leftarrowstandardized sum of level affinities

3:sample

MMconfigs

xcx\_\{c\}; normalize to

x~c∈\[0,1\]L\\tilde\{x\}\_\{c\}\\in\[0,1\]^\{L\}
4:foreach signal

ssdo

5:

Lc,u←∑ℓβs,ℓ​x~c,ℓ\+γs​zuL\_\{c,u\}\\leftarrow\\sum\_\{\\ell\}\\beta\_\{s,\\ell\}\\tilde\{x\}\_\{c,\\ell\}\+\\gamma\_\{s\}z\_\{u\}
6:calibrate

\(αs,κs\)\(\\alpha\_\{s\},\\kappa\_\{s\}\)by Eq\.[7](https://arxiv.org/html/2607.09706#S10.E7)\(rate\) or closed form \(score/cost\)

7:simulate outcomes; inject pathologies; aggregate to config means

y¯s,c\\bar\{y\}\_\{s,c\}
8:fit OLS

y¯s∼x\\bar\{y\}\_\{s\}\\\!\\sim\\\!x; emit proposition

\(θ=β^,Θ∝SE\)\(\\theta=\\hat\{\\beta\},\\ \\Theta\\propto\\mathrm\{SE\}\)
9:endfor

10:refit & check sign/marginal recovery; resample if below tolerance

11:returndataset, fitted propositions⊳\\trianglerightfeed Alg\.[2](https://arxiv.org/html/2607.09706#alg2)

Together SRJANA and the elicitor make the system*dynamic*: a natural\-language brief is parsed into a spec, anchors are pulled from the web, SRJANA manufactures a calibrated data foundation, and the engine returns an assumption\-robust recommendation—no pre\-existing dataset required\.

## 11 Implementation

YUKTI is a Python package \(ir, router, solvers, mcdm, robust, engine, cases, viz\)\. Exact solves use SciPy’s HiGHS interface \(linprog/milp\)\[[32](https://arxiv.org/html/2607.09706#bib.bib32),[27](https://arxiv.org/html/2607.09706#bib.bib27)\]; nonlinear solves use SciPy SLSQP; evolutionary solves use pymoo 0\.6\[[18](https://arxiv.org/html/2607.09706#bib.bib18)\]\. The router, solvers, MCDM and ARPF are solver\-agnostic: a stage only needs to evaluate its propositions\. The same code path handles an exact mixed\-integer program and a nonconvex many\-objective program, differing only in the route taken\.

## 12 Worked Example: One Brief, End to End

We report two cases run end\-to\-end\. These are illustrative: coefficients are elicited placeholders, not measured data \(§[22](https://arxiv.org/html/2607.09706#S22)\)\.

### 12\.1 Exact path: mixed\-integer portfolio

A campaign\-mix stage chooses integer campaign counts across four channels to maximize reach and minimize cost and risk, under a budget and an activity cap\. The router detects an affine model with integer variables and selects the exact MO\-MILP path; augmentedε\\varepsilon\-constraint on HiGHS returns23 non\-dominated points\(Fig\.[3](https://arxiv.org/html/2607.09706#S12.F3)a\)\. The robust compromise commits to a low\-cost, low\-risk mix \(reach3434, cost2525, risk0\.90\.9\) withρ=0\.99\\rho=0\.99; the tornado shows the recommendation is most exposed to the reach\-per\-campaign coefficient of the highest\-volume channel \(Fig\.[3](https://arxiv.org/html/2607.09706#S12.F3)b\), flagged*benchmark*rather than*assumed*\.

![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/portfolio_scatter3.png)\(a\)exact Pareto set
![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/portfolio_tornado.png)\(b\)assumption tornado

Figure 3:Mixed\-integer portfolio solved exactly \(augmentedε\\varepsilon\-constraint, HiGHS\): 23 Pareto points coloured by robustness, with the robust compromise ringed \(a\); coefficients ranked by influence on the recommendation \(b\)\.
### 12\.2 Multi\-stage nonconvex: oncology channel shift

The second case is a worded pharmaceutical brief: during loss of exclusivity, pivot two lung\-cancer brands from high\-frequency field visits to scalable email, without losing share of voice\. YUKTI formulates it as two coupled stages\.*Stage 1 \(email engagement design\)*has five levers—personalization, rep\-triggered share, clinical\-content share, send\-time optimization, cadence—and four signals \(open rate, CTR, cost, fatigue risk\) under a budget and a fatigue guardrail\. The bilinear CTR term and the kinked fatigue penalty make the stage nonconvex; the router selects NSGA\-III, returning a40\-pointfrontier \(Fig\.[4](https://arxiv.org/html/2607.09706#S12.F4)a\)\. The robust compromise \(ρ=0\.97\\rho=0\.97\) leans on personalization and send\-time optimization at*low*cadence—reaching≈43%\\approx 43\\%open and14%14\\%CTR at zero fatigue risk inside budget—because the fatigue guardrail caps what additional volume can buy\. ARPF’s envelope \(Fig\.[4](https://arxiv.org/html/2607.09706#S12.F4)b\) and tornado \(Fig\.[4](https://arxiv.org/html/2607.09706#S12.F4)c\) identify the open\-rate personalization elasticity and cost coefficients as the assumptions most worth verifying\.

*Stage 2 \(field\-to\-email transition\)*consumes Stage 1’s open rate and CTR through the engagement\-quality term and chooses an integer transition window, a shift pace, and a residual high\-touch share, trading share\-of\-voice retention against transition cost and engagement continuity, with a floorSoV≥0\.6\\text\{SoV\}\\geq 0\.6\. It is again nonconvex \(integer window, bilinear pace term\); NSGA\-III returns a28\-pointfrontier\. The robust compromise \(ρ=0\.98\\rho=0\.98\) phases the shift gradually, keeps a small \(≈5%\\approx 5\\%\) residual high\-touch tier for top targets, and retains≈0\.89\\approx 0\.89share of voice at0\.790\.79continuity \(Fig\.[4](https://arxiv.org/html/2607.09706#S12.F4)d\)\. Table[1](https://arxiv.org/html/2607.09706#S12.T1)summarizes both stages\.

![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/oncology_m1_parallel.png)\(a\)stage 1 frontier
![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/oncology_m1_envelope.png)\(b\)ARPF envelope
![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/oncology_m1_tornado.png)\(c\)stage 1 tornado
![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/oncology_m2_parallel.png)\(d\)stage 2 frontier

Figure 4:Two\-stage oncology channel shift \(NSGA\-III \+ ARPF\)\. Parallel\-coordinate frontiers are coloured by robustness with the robust compromise in gold \(a,d\); \(b\) shows the assumption\-robust envelope \(objective spread overK=200K\{=\}200draws\); \(c\) ranks the assumptions that most move the stage\-1 recommendation\.Table 1:Robust compromises \(verified run\)\. Class is chosen by the router\.

## 13 Multi\-Scenario Demonstration

To show the pipeline is domain\-general, we drive three deliberately diverse briefs through the*same*SRJANA\+YUKTI code with no per\-domain tuning: oncology HCP email engagement during loss of exclusivity \(pharmaceutical commercial\), colorectal\-cancer \(CRC\) screening uptake across deprivation strata \(public health\), and residential heat\-pump adoption across utility regions \(energy\)\. Benchmark anchors are pulled from public sources at formulation time—IQVIA / Veeva / phamax engagement rates for the first; CRC\-screening randomized trials \(opt\-out mailed FIT kits, serial SMS reminders\) and national participation statistics for the second; IRA/HEEHRA rebate levels and the heat\-pump*additionality*literature for the third\. In every case SRJANA reproduced the benchmark marginals*and*the key effect ratios to rounding \(Fig\.[5](https://arxiv.org/html/2607.09706#S13.F5)\), and the analytics recovered domain\-correct drivers \(personalization and rep\-triggering for email; mailed\-FIT and patient navigation for screening; rebate depth and installer coverage for heat pumps\), i\.e\.100%100\\%seeded\-sign recovery\.

The router then sends each design stage to the solver its structure warrants\. The affine linear\-probability response surfaces route to the exactε\\varepsilon\-constraint solver—the screening design yields a 135\-point exact frontier—while the heat\-pump design, whose additionality couples to targeting, is nonconvex and routes to NSGA\-III; every second\-stage rollout \(a kinked, bilinear continuity model\) routes to the evolutionary solver \(42, 33 and 42 points respectively\)\. Table[2](https://arxiv.org/html/2607.09706#S13.T2)reports the design\-stage frontiers and robust compromises; Fig\.[6](https://arxiv.org/html/2607.09706#S13.F6)shows them\. One worded brief→\\rightarrowweb\-anchored synthesized data→\\rightarrow*routed*multi\-objective solve→\\rightarrowassumption\-robust compromise, across three unrelated domains through one uniform path, is the generality claim made concrete\.

Table 2:Three diverse scenarios through one pipeline \(verified run\)\. SRJANA reproduces every benchmark anchor and recovers100%100\\%of seeded driver signs; the design stage routes to the solver its structure warrants\.![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/pharma_faith.png)\(a\)pharma
![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/screening_faith.png)\(b\)public health
![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/heatpump_faith.png)\(c\)energy

Figure 5:SRJANA self\-consistency across three domains: the synthesized population marginals*and*the seeded effect ratios \(teal\) match the web\-sourced benchmark targets \(ink\) to rounding\. A data foundation that cannot reproduce its own anchors is rejected before the optimizer ever runs\.![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/pharma_opt.png)\(a\)oncology email \(exact MO\-LP\)
![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/screening_opt.png)\(b\)CRC screening \(exact MO\-LP\)
![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/heatpump_opt.png)\(c\)heat\-pump \(NSGA\-III\)

Figure 6:Three diverse scenarios synthesized by SRJANA and solved by YUKTI\. Parallel\-coordinate design\-stage frontiers coloured by ARPF robustness; the robust compromise is in gold\. The pipeline is identical across domains; only the routed solver and the web\-anchored data differ\.
## 14 Evaluation

We evaluate the*system*, not a benchmark leaderboard: for each scenario we report \(i\) whether SRJANA’s data foundation reproduces its web anchors \(faithfulness\), \(ii\) the quality and spread of the discovered frontier \(normalized hypervolume and size\), \(iii\) how sharply ARPF separates fragile from robust solutions, \(iv\) the response\-surface fit consumed by the optimizer, and \(v\) wall\-clock cost\. Two quantities drive the evaluation\. The ARPF robustness of a candidatexxis the Monte\-Carlo probability it survives its own assumptions,

ρ​\(x\)=1K​∑k=1K𝟏​\[g​\(x;θ\(k\)\)≤0\]​1​\[x∈ND​\(𝒫;θ\(k\)\)\],\\rho\(x\)=\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\mathbf\{1\}\\\!\\big\[g\(x;\\theta^\{\(k\)\}\)\\leq 0\\big\]\\,\\mathbf\{1\}\\\!\\big\[x\\in\\mathrm\{ND\}\(\\mathcal\{P\};\\theta^\{\(k\)\}\)\\big\],\(8\)whereND​\(⋅\)\\mathrm\{ND\}\(\\cdot\)is the non\-dominated set of the discovered pool𝒫\\mathcal\{P\}re\-ranked under resampled coefficientsθ\(k\)∼Θ\\theta^\{\(k\)\}\\sim\\Theta\(standard errorρ​\(1−ρ\)/K\\sqrt\{\\rho\(1\-\\rho\)/K\}\)\. Frontier quality is the hypervolume of the min\-sense frontPPafter per\-objective normalization to\[0,1\]\[0,1\],

HV​\(P\)=Λ​\(⋃p∈P\[p,r\]\),r=1\.1⋅𝟏\.\\mathrm\{HV\}\(P\)=\\Lambda\\\!\\Big\(\\textstyle\\bigcup\_\{p\\in P\}\[\\,p,\\,r\\,\]\\Big\),\\qquad r=1\.1\\cdot\\mathbf\{1\}\.\(9\)Table[3](https://arxiv.org/html/2607.09706#S14.T3)reports both across the three scenarios \(verified run,K=160K\{=\}160, single core\)\. SRJANA reproduces the benchmark*marginals*to within1%1\\%in every case; the larger residuals in the table are on the harder*effect\-ratio*anchors \(e\.g\. a clinical\-content click lift\), still within∼9%\\sim\\\!9\\%\. The router sends each design stage to the solver its structure warrants—two affine stages to the exactε\\varepsilon\-constraint solver \(the screening design yields a 135\-point exact frontier\) and the nonconvex heat\-pump design to NSGA\-III—and ARPF assigns a wide robustness range in every case \(ρ\\rhoas low as0\.370\.37for aggressive peripheral solutions\), so the selected compromise \(ρ⋆\\rho^\{\\star\}\) is a genuine choice, not an artifact of a flat frontier\. End\-to\-end cost is≈14\\approx\\\!14s per scenario including synthesis, analytics, both solves, and ARPF\.

Table 3:System evaluation across three diverse scenarios \(one pipeline, verified run\)\. HV is normalized hypervolume \(Eq\.[9](https://arxiv.org/html/2607.09706#S14.E9)\);ρ⋆\\rho^\{\\star\}is the robustness of the selected compromise and the range spans the whole frontier \(Eq\.[8](https://arxiv.org/html/2607.09706#S14.E8)\); faithfulness is the max relative error over benchmark anchors \(marginals are<1%<1\\%\);R2R^\{2\}is the primary response\-surface fit; time is single\-core wall clock\.
## 15 Decision Traceability: Attribution and Shadow Prices

A robustness score says how far a recommendation can be trusted; two further questions decide whether it can be*acted on*\. “Which segments are in this action, and which should we prioritize?” and “Which constraint binds, and what is relaxing it worth?” We answer both for*any*candidate—in particular the robust compromise—so the object remains a trade\-off choice, not a claimed single optimum\.

Segment attribution\.Given the recommended lever settingx⋆x^\{\\star\}, we fit the primary outcome on the levers with segment fixed effects \(a linear\-probability modely∼β⊤​x\+∑sγs​𝟏​\[seg=s\]y\\sim\\beta^\{\\top\}x\+\\sum\_\{s\}\\gamma\_\{s\}\\mathbf\{1\}\[\\text\{seg\}=s\]\), predict each segment’s responseps​\(x⋆\)p\_\{s\}\(x^\{\\star\}\), and weight by segment sharewsw\_\{s\}to get its share of the total engaged response,contribs∝ws​ps​\(x⋆\)\\text\{contrib\}\_\{s\}\\propto w\_\{s\}\\,p\_\{s\}\(x^\{\\star\}\), plus a priority rank by responsiveness\. This exposes a distinction a narrative hides: the most*responsive*segment is often not the largest*contributor*\. In the oncology case the recommended action’s biggest contributor is the mid\-size, mid\-engagement segment \(35%35\\%of HCPs,58%58\\%of the engaged response\), while the smallest, most digital segment is the top*priority*by responsiveness but only20%20\\%of volume \(Fig\.[7](https://arxiv.org/html/2607.09706#S15.F7), left\)—exactly the “which HCPs are in the40%40\\%?” question made answerable\.

Shadow prices and binding constraints\.For a candidatex⋆x^\{\\star\}we report each constraint’s slack \(binding when≈0\\approx 0\) and a shadow price: the marginal change in the best achievable primary outcome per unit of relaxation,

πj\\displaystyle\\pi\_\{j\}=∂∂bj​V​\(b\),\\displaystyle=\\frac\{\\partial\}\{\\partial b\_\{j\}\}\\,V\(b\),\(10\)V​\(b\)\\displaystyle V\(b\)=maxx⁡primary​\(x\)s\.t\.g​\(x;b\)≤0,x∈\[ℓ,h\],\\displaystyle=\\max\_\{x\}\\,\\text\{primary\}\(x\)\\ \\ \\text\{s\.t\.\}\\ \\ g\(x;b\)\\leq 0,\\ x\\in\[\\ell,h\],estimated by perturbingbjb\_\{j\}and re\-solvingVV\. Where the stage is affine this is the exact LP dual; otherwise it is a local empirical sensitivity \(flagged as such\)\. A revealing pattern recurs: at the*robust compromise*the budget is*not*binding—the compromise deliberately leaves spend on the table to stay robust—yet budget*does*bind at the reach\-maximizing action, where its shadow price \(\+0\.004\+0\.004to\+0\.007\+0\.007primary per cost unit, Table[4](https://arxiv.org/html/2607.09706#S15.T4)\) quantifies what chasing maximum reach would cost in robustness\. The energy case is sharper still: the optimizer drives rebate depth toward zero and spends on installer coverage and awareness instead—the model has learned the additionality lesson \(salience and targeting beat raw subsidy\), and attribution shows the spend concentrating on the most responsive, lower\-income segment\.

Table 4:Decision traceability for the robust compromise \(verified run\)\. “Priority” is the most responsive segment; “contributor” is the largest share of the engaged response\. Budget is slack at the compromise but binds at maximum reach, where the shadow priceπ\\piis the marginal primary gain per cost unit\.![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/pharma_trace.png)\(a\)oncology email
![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/screening_trace.png)\(b\)CRC screening
![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/heatpump_trace.png)\(c\)heat\-pump

Figure 7:Segment attribution of the recommended robust action: each segment’s contribution to the engaged response \(copper\) against its response rate under the action \(teal\), with segment share in parentheses\. The biggest contributor and the most responsive segment frequently differ—the decision\-traceability detail a narrative answer omits\.
## 16 External Validation under Misspecification

The evaluation in §[14](https://arxiv.org/html/2607.09706#S14)is a system check, not a test of the central claim—that carrying assumption uncertainty yields decisions that survive being wrong\. We test that claim out\-of\-sample, non\-circularly, against a held\-out ground truth\.

Setup\.We model the real epistemic situation\. Each lever’s*marginal*effect is known from a*separate*randomized trial—mailed opt\-out FIT≈\+0\.15\\approx\{\+\}0\.15, SMS reminder≈\+0\.08\\approx\{\+\}0\.08, navigation≈\+0\.10\\approx\{\+\}0\.10absolute on screening completion, anchored to published RCTs\[[34](https://arxiv.org/html/2607.09706#bib.bib34),[35](https://arxiv.org/html/2607.09706#bib.bib35),[36](https://arxiv.org/html/2607.09706#bib.bib36)\]—but*no*trial measured the levers in combination\. The optimizer’s surrogate is therefore*additive*in the known marginals, with no interaction term\. The*true*world, revealed only at evaluation, is structurally richer: negative interactions and saturation \(diminishing returns to stacking channels\) that the additive surrogate cannot represent\. The optimizer never sees the true structure, so the test is non\-circular—fit on additive marginals, evaluate on the richer world\.

Policies\.\(a\)*naive point*: the single\-objective argmax of the additive surrogate—the confident point plan a one\-shot LLM emits; \(b\)*nominal*: a multi\-objective compromise on the surrogate, no robustness; \(c\)*ARPF robust*: the robust compromise over anε\\varepsilon\-contaminationΘ\\Theta\(ε=0\.5\\varepsilon\{=\}0\.5\) that hypothesizes diminishing\-returns interactions, selecting the action with the best downside \(worst\-25%25\\%\) value\.

Result\.On the held\-out true world over800800realizations \(Table[5](https://arxiv.org/html/2607.09706#S16.T5)\), the naive plan stacks every lever and is punished by the unmodelled saturation; the robust compromise uses the single strongest real lever and refuses to stack\. ARPF cuts mean regret by98%98\\%and tail regret \(CVaR10%\) by92%92\\%versus the naive plan, and beats the non\-robust nominal compromise as well \(Fig\.[8](https://arxiv.org/html/2607.09706#S16.F8)a\)\. The Theorem[1](https://arxiv.org/html/2607.09706#Thmtheorem1)certificate holds \(Fig\.[8](https://arxiv.org/html/2607.09706#S16.F8)b\)\. The*distributional hand\-off*matters too: handing off the naive plan’s stage\-1 signal as a point reports the downstream floor satisfied \(0\.58≥0\.550\.58\\geq 0\.55\), while handing off its distribution prices a60%60\\%probability of falling below that floor \(10th percentile0\.210\.21\)— the risk a point commitment hides\.

Table 5:Held\-out regret on the true world \(800800realizations\)\. The naive additive plan stacks all levers; ARPF’s robust compromise uses the strongest single lever\. Lower is better\.![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/val_regret.png)\(a\)
![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/val_certificate.png)\(b\)

Figure 8:\(a\) Out\-of\-sample regret under structural misspecification: the ARPF robust compromise collapses both mean and tail regret versus the naive point plan\. \(b\) Theorem[1](https://arxiv.org/html/2607.09706#Thmtheorem1)certificate: actual pool\-regret equals the identity\(1−ρ\)​g¯\(1\-\\rho\)\\bar\{g\}and lies under the bound\(1−ρ\)​B\(1\-\\rho\)B\.Scope \(honest\)\.The true world is synthetic, but its marginal effects equal measured RCT values and its structure is hidden from the optimizer\. The magnitude of the gap depends on interaction strength; the*direction*— robustness reduces out\-of\-sample and especially tail regret under misspecification—is the claim, and it holds\. Real field data would replace the synthetic world wholesale; the methodology \(fit a misspecified surrogate, evaluate held\-out, report regret\) transfers unchanged\.

## 17 Validation on Real Data

The validations so far use worlds we generated\. The sharpest test is data we did not: the UCI Bank Marketing dataset\[[39](https://arxiv.org/html/2607.09706#bib.bib39)\]—41,18841\{,\}188real telemarketing contacts from a Portuguese bank, each a logged channel/frequency decision with a realized conversion outcome\. The decision mirrors the commercial brief in miniature: per client segment \(job×\\timesage band\), choose a*channel*\(cellular/telephone\) and a*contact\-frequency*band to maximize conversion\.

Protocol\.We estimate each \(segment, action\) conversion on a training split and evaluate three rules on a held\-out split \(mean of1212random60/4060/40splits\):*logged*\(the realized status\-quo mix, no re\-assignment\);*naive*\(per segment, argmax of the point estimate\); and*robust*\(per segment, argmax of the downside CVaR25%of the Beta posterior—the one\-dimensional instance of YUKTI’s assumption\-robust compromise\)\. An in\-hindsight*oracle*bounds the achievable\. Naive selection is exactly the*optimizer’s curse*setting\[[40](https://arxiv.org/html/2607.09706#bib.bib40)\]: an argmax over noisy estimates favours options that look best by chance and regress out\-of\-sample\.

Result\.On held\-out real data the decision layer beats the status quo decisively—robust conversion0\.1520\.152vs logged0\.1140\.114,\+34%\+34\\%—and robustness also beats the naive point rule by\+4%\+4\\%\(0\.1520\.152vs0\.1460\.146\), winning in97%97\\%of segments and shrinking the gap to the oracle \(0\.0220\.022vs0\.0280\.028, Table[6](https://arxiv.org/html/2607.09706#S17.T6), Fig\.[9](https://arxiv.org/html/2607.09706#S17.F9)\)\. The mechanism is visible in the*post\-decision surprise*—how much a rule’s training estimate overstates its realized value: naive\+2\.6\+2\.6pp versus robust\+2\.0\+2\.0pp\. The naive plan believes its own luck; the robust plan discounts it\.

Table 6:Out\-of\-sample backtest on UCI Bank Marketing \(41,18841\{,\}188real decisions; mean of1212random60/4060/40holdouts,2525segments\)\. Held\-out conversion; higher is better\.![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/realdata.png)Figure 9:Real\-data, out\-of\-sample \(UCI Bank Marketing\)\. Left: held\-out conversion— the robust rule beats both the logged status quo and the naive point rule\. Right: the optimizer’s curse—naive selection overstates its own pick more than robust does\.Non\-stationarity \(honest stress\)\.Under a strict*out\-of\-time*split \(the rows are date\-ordered\), conversion roughly triples across periods; absolute cell rates no longer transfer and all optimized gains compress \(robust0\.2140\.214vs logged0\.2470\.247\)\. This is the known failure mode of backtests on a drifting world, and we state it plainly: the random\-holdout result isolates the estimation\-uncertainty effect the theory addresses, while closing the non\-stationarity gap needs a recency\-weighted or recourse formulation \(§[22](https://arxiv.org/html/2607.09706#S22)\)\. Evaluation is direct\-method \(held\-out cell rates\); a production study adds a test–control design \(Appendix[E](https://arxiv.org/html/2607.09706#A5)\)\. Even so, this is the first result here on data the author did not generate, and it points the same way as the theory: pricing uncertainty beats ignoring it\.

## 18 Deploying in a Regulated Commercial Setting

A held\-out regret win is necessary but not sufficient for a high\-stakes commercial decision\. Taking an oncology channel\-rebalance brief \(Drug A and Drug B, during loss of exclusivity\) to a country team surfaces four requirements a generic optimizer ignores; YUKTI’s typed IR accommodates each as a formulation choice, and we backtest the result against the policy that was actually run\.

A lawful, admissible action space\.Email cannot reach an oncologist who has not opted in: under GDPR only a consented fractionκ\\kappais addressable \(κ≈0\.45\\kappa\{\\approx\}0\.45here\), so email effect is reach\-capped, not free\. A frequency cap bounds touches; a finite inventory of MLR\-approved assets bounds content \(depth\+defense≤A\\text\{depth\}\+\\text\{defense\}\\leq A\); and a*KAM override*keeps field contact on top\-decile KOLs above a floor \(human\-in\-the\-loop\)\. These enter as constraints, so optimization happens only inside the admissible set \(Appendix[D](https://arxiv.org/html/2607.09706#A4)\)\. This is not cosmetic: the unconstrained “optimal” plan—max email, max content—is*inadmissible*\(it breaches the asset and frequency caps\), and only312312of12961296grid actions are lawful\. An optimum outside the lawful set is a liability, not an answer\.

A prescribing\-proxy objective, not opens\.The primary outcome is an NBRx\-uplift proxy in which the field relationship dominates for high\-value oncologists and email is a consent\-capped, fatigue\-prone,*mediated*contributor; opens/CTR feed an engagement\-quality term but are never the objective\. Message is not a scalar “content” dial: Drug A rewards scientific\-depth content \(a targeted line\-of\-therapy\), Drug B rewards LoE\-defense content, so the recommended content architecture differs by brand by construction\.

Out\-of\-time backtest against the logged policy\.We fit the surrogate on past quarters, freeze it, and evaluate each policy’s action on held\-out future quarters of a true world it never saw, comparing realized NBRx against the status quo that was actually run \(Appendix[E](https://arxiv.org/html/2607.09706#A5)\)\. Table[7](https://arxiv.org/html/2607.09706#S18.T7)reports the result\. The robust compromise keeps field \(KAM preserved\),*declines*the aggressive email shift the brief proposed, optimizes content to each brand, and—crucially— neutralizes the downside \(5% Value\-at\-Risk≈0\\approx 0\) where the naive plan carries a large negative tail\. The*nominal*compromise that trims expensive field*underperforms the status quo*\(Fig\.[10](https://arxiv.org/html/2607.09706#S18.F10)\): naively rebalancing away from the relationship loses prescriptions to channel substitution—the failure the field organization predicts\.

Table 7:Out\-of\-time backtest vs the logged status quo \(Drug A\): NBRx uplift*relative to logged*and 5% revenue Value\-at\-Risk in €M \(illustrative oncology unit economics, Appendix[F](https://arxiv.org/html/2607.09706#A6)\)\. The unconstrained optimum is inadmissible \(breaches MLR/frequency caps\)\.![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/pharma_backtest.png)Figure 10:Out\-of\-time realized NBRx uplift vs the logged status quo, by brand\. The robust compromise gains modestly and protects the downside; the nominal field\-trim loses to channel substitution; the recommended content differs by brand \(Drug A depth, Drug B defense\)\.Decision economics \(euros, notρ\\rho\)\.The NBRx\-uplift distribution maps to revenue and share\-of\-voice with a downside band: the robust plan’s expected incremental revenue is positive with 5% VaR near zero, whereas the field\-trimming nominal plan risks an eight\-figure shortfall \(Table[7](https://arxiv.org/html/2607.09706#S18.T7)\)\. A country GM transacts in euros\-at\-risk and accountability, not value units; Appendix[F](https://arxiv.org/html/2607.09706#A6)gives the mapping\.

Deployment posture\.The honest role is a decision*stress\-testing and triangulation*layer, not a system of record: it pressure\-tests a human plan, prices its downside, and flags the lawful action space—then writes its recommendation and decision\-traceability into the validated commercial stack \(e\.g\. as a prior or guardrail for a Veeva next\-best\-action engine\), where a rep sees it in\-flow and the KAM override is always available\. The synthetic data here is benchmark\- and RCT\-anchored; a real Veeva/IQVIA backtest is the one step that converts this from a credible simulation into evidence, and the protocol is built for exactly that swap\.

Part IV\|Boundaries and Position

## 19 LLM Reasoning, Function Optimization, or YUKTI?

A fair objection: could a capable reasoning model \(an LRM\) just*read*the brief and answer, or does classical single\-objective optimization already suffice? We test both against YUKTI on the same held\-out harness, separating*what each method is for*from*what it can guarantee*\(Table[8](https://arxiv.org/html/2607.09706#S19.T8)\)\.

Capability\.The three occupy different rungs\. An LRM excels at the*front end*—turning a worded situation into structured quantities—which is precisely why YUKTI uses one there\. Asked instead to*be*the optimizer, an LRM has no mechanism to guarantee feasibility, to enumerate a Pareto frontier without an arbitrary scalarization, to propagate coefficient uncertainty, or to certify regret; empirically, LLM\-as\-optimizer degrades sharply with problem size and was never meant to beat specialized solvers\[[41](https://arxiv.org/html/2607.09706#bib.bib41)\], and LLM\-emitted solutions are frequently*infeasible*in constrained tasks because the model describes a solution without tracking the constraints\[[42](https://arxiv.org/html/2607.09706#bib.bib42)\]\. Classical single\-objective optimization guarantees feasibility and a nominal optimum, but collapses conflicting objectives into one weighted scalar and returns an uncertainty\-free*point*\. YUKTI keeps the LRM for intake and replaces the solve\-and\-hedge with machinery that has these properties\.

Outcome\.On the misspecification problem \(§[16](https://arxiv.org/html/2607.09706#S16)\), evaluated identically: an LRM handed the*correct*marginal effects and reasoning additively commits to the stacked plan \(mean held\-out regret1\.031\.03; because decoding is stochastic, a run\-to\-run spread of0\.120\.12with no stable optimum\); classical single\-objective optimization lands at the same fragile corner \(1\.071\.07\); YUKTI’s robust compromise scores0\.0220\.022—about47×47\\timeslower\. The lesson is not that the LRM reasons badly \(it is handed the right numbers\) but that a*point*plan, however reasoned, ignores the assumption uncertainty and the objective conflict that dominate the decision\. The same ordering holds out\-of\-sample on real data \(§[17](https://arxiv.org/html/2607.09706#S17): the naive point rule is the LRM/classical proxy; robust beat it by4%4\\%with a smaller optimizer’s\-curse surprise\), and in the regulated setting \(§[18](https://arxiv.org/html/2607.09706#S18)\) the confident LRM plan—“shift hard to digital, maximize content”—is not merely fragile but*inadmissible*: it breaches the MLR and consent caps, the infeasibility mode\[[42](https://arxiv.org/html/2607.09706#bib.bib42)\]made concrete in euros\.

Table 8:Three ways to turn a worded decision into a plan\. An LRM is the best*formulator*but cannot guarantee what a high\-stakes decision needs; classical single\-objective optimization guarantees feasibility and a nominal optimum but returns an uncertainty\-free point; YUKTI uses the LRM for intake and supplies the rest\. Lower block: identical held\-out evaluation \(§[16](https://arxiv.org/html/2607.09706#S16)–[18](https://arxiv.org/html/2607.09706#S18)\)\. ✓ = provides;∼\\sim= partial;×\\times= absent\.Perspective\.The boundary is clean\. Use an LRM for what it is—a superb*formulator*of messy situations into structure\. Do not ask it to be the*solver*: feasibility, frontiers, uncertainty, and a regret certificate are not properties a next\-token predictor provides, and “reasoning” tokens improve the*formulation*, not the optimization\. That division of labour—LRM in front, verifiable optimization behind—is YUKTI, and it is the constructive form of the mimicry\-of\-computation thesis\.

## 20 Long\-Range Causal Dependence: Where the Hand\-off Must Become a Policy

YUKTI solves stages in topological order and passes a signal forward \(§[7](https://arxiv.org/html/2607.09706#S7)\)\. That is adequate when an early action only changes the*state*the next stage sees; it is*not*adequate when an early action causally reshapes a later stage’s*mechanism*—a long\-range dependencya1→mechanism2a\_\{1\}\\\!\\rightarrow\\\!\\text\{mechanism\}\_\{2\}, nota1→state2a\_\{1\}\\\!\\rightarrow\\\!\\text\{state\}\_\{2\}\. Such dependence is the rule in the settings these methods target: an early therapy changes how a later one acts; an early brand investment changes the elasticity of a later channel\. The field is converging here—commercial causal\-decision platforms pair an LLM front\-end with causal graphs and intervention/recourse search \(causaLens*decisionOS*; Alembic’s real\-time causal recomputation of saturation and lift\), and the long\-horizon LLM\-agent literature is consumed by*credit assignment*\[[48](https://arxiv.org/html/2607.09706#bib.bib48)\]\. The rigorous formalism is the dynamic\-treatment\-regime \(DTR\) line, and it warns about exactly our hand\-off: when late\-stage outcome models are misspecified and plugged into early\-stage optimization,*bias accumulates from stage to stage and grows with the horizon*\[[43](https://arxiv.org/html/2607.09706#bib.bib43)\], so the correct object is a*global / backward\-induction*policy, not a forward myopic pass\[[44](https://arxiv.org/html/2607.09706#bib.bib44),[45](https://arxiv.org/html/2607.09706#bib.bib45)\]\.

We make this testable\. On a controlled causal chain where an early actiona1a\_\{1\}modulates stage\-2 effectiveness with coupling strengthκ\\kappa\(soa1→mechanism2a\_\{1\}\\\!\\rightarrow\\\!\\text\{mechanism\}\_\{2\}\), we compare YUKTI’s myopic hand\-off against a backward\-induction*causal\-trajectory*policy over the same robust \(ARPF/CVaR\) objective \(Fig\.[11](https://arxiv.org/html/2607.09706#S20.F11)\)\.

\(C1\) Myopia penalty\.Atκ=0\\kappa\{=\}0the two coincide—YUKTI’s present hand\-off is the no\-coupling special case\. As coupling grows, the myopic pass \(which optimizes stage 1 on its own value and under\-invests in the early action\) incurs trajectory regret growing roughly linearly inκ\\kappa, exceeding1\.01\.0value unit at strong coupling, while the causal\-trajectory policy holds regret near zero\. The DTR bias\-accumulation result, reproduced\.

\(C2\) The certificate does not compose\.Theorem[1](https://arxiv.org/html/2607.09706#Thmtheorem1)certifies*per\-decision*pool\-regret\. But the stage\-wise productρ1​ρ2\\rho\_\{1\}\\rho\_\{2\}that a forward pass reports*overstates*the myopic plan’s true trajectory robustnessρtraj\\rho\_\{\\mathrm\{traj\}\}, and the optimism gap grows withκ\\kappa: at strong coupling the per\-stage certificate rates the myopic plan∼0\.96\{\\sim\}0\.96robust while its probability of being trajectory\-optimal is≈0\{\\approx\}0\. The certificate is sound per decision and*unsound*across a causally\-coupled trajectory\.

![Refer to caption](https://arxiv.org/html/2607.09706v1/figs/causal_horizon.png)Figure 11:Long\-range causal couplingκ\\kappa\(a1→mechanism2a\_\{1\}\\\!\\rightarrow\\\!\\text\{mechanism\}\_\{2\}\)\. Left \(C1\): the myopic hand\-off’s trajectory regret grows with coupling while the backward\-induction causal\-trajectory policy stays near zero; atκ=0\\kappa\{=\}0they coincide\. Right \(C2\): the stage\-wise certificateρ1​ρ2\\rho\_\{1\}\\rho\_\{2\}stays high while the myopic plan’s true trajectory robustnessρtraj\\rho\_\{\\mathrm\{traj\}\}collapses—the per\-stage certificate is increasingly optimistic\.What must change\.Three modifications follow, each a concrete extension of an existing component\. \(i\)*Typed\-causal IR*: the Typed Proposition graph gains temporal causal edges typed by lag and mechanism \(doesa1a\_\{1\}shift a later state or a later coefficient?\), with time\-varying confounding represented rather than assumed away\[[45](https://arxiv.org/html/2607.09706#bib.bib45)\]\. \(ii\)*Backward\-induction robust policy*: where coupling is present the forward hand\-off is replaced by a backward\-induction solve over the trajectory—robustQQ\-/structural\-nested style\[[46](https://arxiv.org/html/2607.09706#bib.bib46)\]—with ARPF resampling the causal*model*\(structure and coupling\), not only per\-stage coefficients\. \(iii\)*Trajectory certificate and causal credit*:ρ\\rhoand the regret bound move to the trajectory, and decision\-traceability \(§[15](https://arxiv.org/html/2607.09706#S15)\) generalizes to*causal credit assignment*—path\-specific contributions of early actions to late outcomes—precisely the problem the long\-horizon agent literature is attacking\[[48](https://arxiv.org/html/2607.09706#bib.bib48)\]\.

Scope\.We diagnose and bound the failure and verify the direction; we do not ship a causal\-DP solver here\. Theκ=0\\kappa\{=\}0result confirms the present system is correct in the regime it was built for; theκ\>0\\kappa\{\>\}0results say what must change, and that change is well\-trodden in the DTR and causal\-RL literature\[[47](https://arxiv.org/html/2607.09706#bib.bib47)\]— an identification\-and\-engineering problem, not a conceptual gap\.

## 21 Positioning and Novelty

We separate what we reuse from what we claim\.Established \(reused\):the five\-element formulation discipline and IR\-plus\-solver loop\[[6](https://arxiv.org/html/2607.09706#bib.bib6),[3](https://arxiv.org/html/2607.09706#bib.bib3),[8](https://arxiv.org/html/2607.09706#bib.bib8)\]; augmentedε\\varepsilon\-constraint / AUGMECON2\[[12](https://arxiv.org/html/2607.09706#bib.bib12)\]; NSGA\-II/III, MOEA/D, Das–Dennis directions, and pymoo’s MCDM\[[14](https://arxiv.org/html/2607.09706#bib.bib14),[15](https://arxiv.org/html/2607.09706#bib.bib15),[16](https://arxiv.org/html/2607.09706#bib.bib16),[17](https://arxiv.org/html/2607.09706#bib.bib17),[18](https://arxiv.org/html/2607.09706#bib.bib18)\]; HiGHS and SciPy\[[27](https://arxiv.org/html/2607.09706#bib.bib27),[32](https://arxiv.org/html/2607.09706#bib.bib32)\]; the notion of integrating over frontier uncertainty from MOBO\[[20](https://arxiv.org/html/2607.09706#bib.bib20)\]\.Contributed \(new, to our knowledge\):\(1\) a*Typed Proposition IR*that carries shape priors, a coefficient uncertainty distribution, and provenance—versus the point\-valued IRs of prior autoformulation systems; \(2\)*ARPF*, propagating that IR\-level assumption uncertainty onto the frontier to produce per\-decision robustness scores and robust compromise selection inside an autoformulation pipeline—the model\-assumption analogue of qNEHVI’s observation\-noise treatment; \(3\) a*heterogeneous multi\-stage hand\-off*that couples exact and evolutionary stages across a decision DAG; and \(4\) a*numeric structure\-aware router*that picks exact / NLP / EA per stage by probing affinity and convexity on the relaxation; \(5\)*SRJANA*, benchmark\-anchored contextual synthesis that manufactures a faithful data foundation when none exists \(the synthesized analytics layer\); and \(6\)*decision\-traceability*reporting—segment attribution and shadow\-price/binding\-constraint analysis—that makes a recommended action auditable\. The synthesis as a whole—worded situation to multi\-stage, multi\-objective, assumption\-robust recommendation—is, to our knowledge, not available in one system\. We emphasize again:*no benchmark\-SOTA claim is made*; evaluation here is a verified demonstration, not a comparison\.

Relation to decision\-support systems\.The three\-layer reading of our pipeline—analytics, decision/model, and a presentation layer—is deliberately classical: it mirrors the data/model/dialog decomposition of decision\-support systems\[[33](https://arxiv.org/html/2607.09706#bib.bib33)\]\. We claim no novelty in the*layering*; the contribution is the*mechanisms*that make the decision layer trustworthy under LLM\-objectified inputs \(typed\-uncertain IR, ARPF, the router, the hand\-off, SRJANA, and decision\-traceability reporting\)\. Framed against that lineage, the problem an LLM introduces is*mimicry of computation*: fluent output with the form of an optimized recommendation but without a solved model, stated coefficients, sensitivity, or robustness\. Source traceability \(RAG provenance\) does not address this; decision traceability does\. YUKTI’s value is therefore not that an LLM “cannot” do the analysis—a capable model can—but that the recommendation it emits arrives with an explicit, re\-runnable, assumption\-aware artifact rather than as unverifiable prose\.

## 22 Limitations

Illustrative coefficients\.The demonstration’s numbers are elicited placeholders, not measured field data; the contribution is structural\. The external validation \(§[16](https://arxiv.org/html/2607.09706#S16)\) anchors a held\-out world to measured RCT effects and shows the robust compromise reduces out\-of\-sample regret, but real proprietary data would replace the synthetic world entirely; ARPF exists to quantify the exposure, and the tornados say which numbers to measure first\.ARPF re\-ranks a fixed pool\.Equation[4](https://arxiv.org/html/2607.09706#S8.E4)resamples assumptions and re\-ranks the*discovered*solutions rather than re\-searching from scratch each draw\. This is exactly why Theorem[1](https://arxiv.org/html/2607.09706#Thmtheorem1)certifies*pool*\-regret \(against the best\-in\-pool\) rather than oracle regret; it is efficient and adequate when the nominal pool is dense near the frontier, but can understate robustness of decisions far from the nominal optimum\. Periodic full re\-searches are a configurable, costlier option\.Hand\-off, not recourse\.The multi\-stage layer commits a compromise and passes its signals forward; it does not optimize a recourse policy over a scenario tree as in multistage stochastic programming\[[22](https://arxiv.org/html/2607.09706#bib.bib22),[24](https://arxiv.org/html/2607.09706#bib.bib24)\]\. This is appropriate for hierarchical strategy decisions but is not a substitute for recourse when later stages can genuinely revise earlier ones\.Router heuristics\.Affinity/convexity are inferred from finite samples with tolerances; pathological functions can be misread\. Misrouting is conservative \(EA solves a superset of what exact handles\) but may forgo exactness\.LLM front\-end\.This paper specifies and demonstrates the engine; wiring a production LLM elicitor with self\-repair\[[9](https://arxiv.org/html/2607.09706#bib.bib9)\]on top of the typed IR is engineering we describe but do not benchmark here\.Real data and non\-stationarity\.The real\-data backtest \(§[17](https://arxiv.org/html/2607.09706#S17)\) is genuine out\-of\-sample evidence on data we did not generate, but under a strict out\-of\-time split a tripling of the base rate compresses the gains: a drifting world defeats any frozen response surface, ours included\. The honest scope is that pricing*estimation*uncertainty helps out\-of\-sample; pricing*distribution shift*needs a recency\-weighted or recourse extension\. Evaluation there is direct\-method, not a randomized test–control study\.

## 23 Future Work

Natural extensions: \(i\) an expensive\-evaluation mode using qNEHVI when each proposition evaluation is a simulation rather than a formula\[[20](https://arxiv.org/html/2607.09706#bib.bib20)\]; \(ii\) true multistage recourse with a scenario tree built from the proposition distributions, recovering the stochastic\-programming view\[[22](https://arxiv.org/html/2607.09706#bib.bib22)\]; \(iii\) calibrating propositions against data so provenance can graduate from*assumed*to*benchmark*; and \(iv\) a benchmark of formulation fidelity and robustness against single\-shot LLM baselines on multi\-objective instances\.

## 24 Conclusion

YUKTI keeps the LLM in the role it does well—reading a situation and naming its structure—but changes what it produces: not a brittle point model with one objective, but a*typed, uncertain*proposition graph whose stages are routed to the right solver, chained by Pareto hand\-off, and ranked by how often each choice survives the very assumptions that were invented to objectify the brief\. The result is a recommendation that comes with its own statement of fragility\. We provide a working implementation and an end\-to\-end demonstration, and we are explicit that the novelty is the uncertainty\-typed IR and the ARPF mechanism, not the underlying solvers or any benchmark claim\.

## Appendix AFrom Contextual Analysis to Formulation

This appendix records the design trail: the core statements that emerged from analysing real “tell\-me\-what\-to\-do” briefs, and how each was translated into a concrete framework mechanism\. The framework is not an arbitrary architecture; every component answers an observation\. Table[9](https://arxiv.org/html/2607.09706#A1.T9)is the compact map; the prose below elaborates the reasoning\.

Table 9:Contextual analysis→\\rightarrowframework\. Each core statement discovered while reading real briefs becomes a specific mechanism\. This is the derivation of YUKTI’s structure from first observations\.A\.1 From words to a typed program\.Briefs name what is*controlled*\(levers\), what is*cared about*\(signals\), and what must be*respected*\(limits\)\. We therefore do not ask the LLM for variables and an objective; we ask it to populate this intent vocabulary, and we make each numeric relationship a*typed proposition*φ​\(x,u;θ\)\\varphi\(x,u;\\theta\)rather than a constant\. The type carries a shape prior \(e\.g\. “increasing, saturating”\), a coefficient distribution, and a provenance tag, so the very first artifact already encodes what is known versus guessed\.

A\.2 Conflict and assumptions are first\-class\.Two observations dominate real briefs: objectives conflict, and the numbers that make them comparable are invented\. The first forbids scalarization \(we keep a frontier\); the second forbids treating any coefficient as ground truth \(we attachΘ\\Thetaand, through ARPF, report how often each candidate stays feasible and non\-dominated under resampled assumptions\)\. Robustness is thus not a post\-hoc check but the selection criterion\.

A\.3 Sequence\.“Design, then roll out” is two coupled problems: the roll\-out’s achievable share\-of\-voice depends on the engagement the design stage secured\. We model this as a Pareto hand\-off—anchor signals from stage 1 enter stage 2 as fixed inputs—so the second decision is optimized*given*the first, not in isolation\.

A\.4 No data is the common case\.A genuinely new brief has no table to query\. SRJANA closes this gap by treating web\-retrieved benchmarks as moment constraints and calibrating a segment\-mixture generator to them, then fitting the propositions to the synthesized data so the optimizer inherits honest, data\-derived uncertainty\. Because literature reports both levels and contrasts, the calibration matches a mean*and*a ratio; because synthetic data can lie, we gate on self\-consistency before optimizing\.

A\.5 Structure varies\.Affine stages should be solved exactly; convex stages by NLP; nonconvex ones by an evolutionary method\. Rather than ask the user, the router probes the stage numerically \(interpolation for affinity, Jensen\-gap sampling for convexity\) and dispatches accordingly—which is why the three demonstration scenarios land on different solver classes from one code path\.

A\.6 Domain pathologies and equity\.Contextual analysis repeatedly surfaced domain\-specific traps that a naive formulation would miss: Apple Mail Privacy Protection inflating open rates \(so click\-to\-open is the trustworthy signal\); laggard markets that send*more*but target worse \(so the gap analysis is on the lever mix, not volume\); and subsidies that mostly fund non\-additional adopters \(so we separate adoption from*additional*installs and let the optimizer pursue impact, not headline uptake\)\. Each became an explicit modelling choice rather than an afterthought\.

## Appendix BConsolidated Equations

Typed proposition\.Every quantitative relationship isφ​\(x,u;θ\)\\varphi\(x,u;\\theta\)with leversxx, upstream inputsuu, coefficientsθ∼Θ\\theta\\sim\\Theta, and a shape priors∈\{\+,−,∩,∪,lin\}s\\in\\\{\+,\-,\\cap,\\cup,\\text\{lin\}\\\}\.

Router\.A stage is*affine*if for randomx1,x2,λx\_\{1\},x\_\{2\},\\lambda,‖F​\(λ​x1\+\(1−λ\)​x2\)−λ​F​\(x1\)−\(1−λ\)​F​\(x2\)‖≤ϵ\\\|F\(\\lambda x\_\{1\}\+\(1\{\-\}\\lambda\)x\_\{2\}\)\-\\lambda F\(x\_\{1\}\)\-\(1\{\-\}\\lambda\)F\(x\_\{2\}\)\\\|\\leq\\epsilon;*convex*if the Jensen gap is≥−ϵ\\geq\-\\epsilonfor all sampled triples; otherwise*nonconvex*, routing to exact / NLP / EA respectively\.

Augmentedε\\varepsilon\-constraint \(AUGMECON2\)\.For objectivesf1,…,fmf\_\{1\},\\dots,f\_\{m\}\(min\-sense\),

minx⁡f1​\(x\)−δ​∑k=2mskrks\.t\.​fk​\(x\)\+sk=ek,sk≥0,x∈𝒳,\\min\_\{x\}\\;f\_\{1\}\(x\)\-\\delta\\sum\_\{k=2\}^\{m\}\\frac\{s\_\{k\}\}\{r\_\{k\}\}\\quad\\text\{s\.t\.\}\\;\\;f\_\{k\}\(x\)\+s\_\{k\}=e\_\{k\},\\;s\_\{k\}\\geq 0,\\;x\\in\\mathcal\{X\},\(11\)swept over a grid ofeke\_\{k\}between each objective’s payoff\-table ideal and nadir, with rangerkr\_\{k\}and a smallδ\\deltato exclude weakly\-dominated points\.

ARPF\.Robustnessρ​\(x\)\\rho\(x\)as in Eq\.[8](https://arxiv.org/html/2607.09706#S14.E8); the robust compromise minimizes a compromise distance to the ideal among solutions withρ≥\\rho\\geqthe frontier median\.

Compromise programming\.With per\-objective idealzk⋆z^\{\\star\}\_\{k\}and nadirzknadz^\{\\mathrm\{nad\}\}\_\{k\}, chooseargminx\(∑kwk\|fk\(x\)−zk⋆\|p/\|zknad−zk⋆\|p\)1/p\\arg\\min\_\{x\}\\big\(\\sum\_\{k\}w\_\{k\}\\,\|f\_\{k\}\(x\)\-z^\{\\star\}\_\{k\}\|^\{p\}/\|z^\{\\mathrm\{nad\}\}\_\{k\}\-z^\{\\star\}\_\{k\}\|^\{p\}\\big\)^\{1/p\},p=2p\{=\}2\.

SRJANA moment system\.For each rate signal with benchmark meanμ⋆\\mu^\{\\star\}and effect ratior⋆r^\{\\star\}on a binary featureℓ\\ell, solve for interceptα\\alphaand effect\-scaleκ\\kappa,

p¯​\(α,κ\)=μ⋆,p¯ℓ=1​\(α,κ\)p¯ℓ=0​\(α,κ\)=r⋆,\\bar\{p\}\(\\alpha,\\kappa\)=\\mu^\{\\star\},\\qquad\\frac\{\\bar\{p\}\_\{\\ell=1\}\(\\alpha,\\kappa\)\}\{\\bar\{p\}\_\{\\ell=0\}\(\\alpha,\\kappa\)\}=r^\{\\star\},\(12\)withp¯​\(⋅\)=1N​M​∑c,uσ​\(α\+κ​βℓ​x~c,ℓ\+Lc,u−\)\\bar\{p\}\(\\cdot\)=\\tfrac\{1\}\{NM\}\\sum\_\{c,u\}\\sigma\(\\alpha\+\\kappa\\beta\_\{\\ell\}\\tilde\{x\}\_\{c,\\ell\}\+L^\{\-\}\_\{c,u\}\)on probabilities \(smooth in\(α,κ\)\(\\alpha,\\kappa\)\), then sample Bernoulli outcomes; score/cost signals calibrate in closed form\. Identifiability holds because the two moments are monotone in\(α,κ\)\(\\alpha,\\kappa\)over the relevant range\.

## Appendix CAgent Invocation via A2A

YUKTI is exposed as anA2A\(Agent2Agent\) agent so other agents can discover and call it\. It advertises an Agent Card at/\.well\-known/agent\-card\.json\(protocol0\.3\.0, transport JSON\-RPC 2\.0 over HTTP\) declaring five skills:list\_scenarios,optimize\_scenario,formulate\_and\_optimize\(consumes a structured*Brief*\),synthesize\_data\(SRJANA only\), anddecision\_trace\(segment attribution \+ shadow prices for the robust action\)\. A caller discovers the card, then issuesmessage/send; the engine runs synchronously and returns aTaskin statecompletedwhoseartifactscarry aDataPartwith the faithfulness report, personas, drivers, cross\-group gap and the robust compromise\.tasks/getre\-fetches by id\. A minimal call:

```
POST /  (JSON-RPC 2.0)
{"jsonrpc":"2.0","id":"r1",
 "method":"message/send",
 "params":{"message":{"role":"user",
   "messageId":"m1",
   "parts":[{"kind":"data","data":{
     "skill":"optimize_scenario",
     "input":{"scenario":"screening"}}}]}}}
-> result: {"kind":"task","status":
   {"state":"completed"},
   "artifacts":[{"parts":[{"kind":"data",
     "data":{...robust compromise...}}]}]}
```

Natural\-language parsing remains the caller’s responsibility; the engine consumes the structured brief and runs deterministically, which keeps the LLM\-as\-elicitor and engine\-as\-solver boundary clean\.

## Appendix DAdmissible Action Space \(MLR / Consent / Frequency / KAM\)

The lawful action space restricts the leversx=\(field​f,email freq​e,depth​d,defense​v\)∈\[0,1\]4x=\(\\text\{field \}f,\\text\{email freq \}e,\\text\{depth \}d,\\text\{defense \}v\)\\in\[0,1\]^\{4\}by:*Consent \(GDPR\)\.*Email reaches only the opted\-in fractionκ\\kappa, so the email contribution enters asκ​\(open\+4​CTR\)​e\\kappa\\,\(\\text\{open\}\+4\\,\\text\{CTR\}\)\\,erather than a free additive term \(κ=0\.45\\kappa\{=\}0\.45; pharma email open≈19%\\approx 19\\%, CTR≈2\.5%\\approx 2\.5\\%\)\.*Frequency cap\.*e≤Fe\\leq F\(F=0\.6F\{=\}0\.6\)\.*MLR inventory\.*content is bounded by the approved\-asset budget,d\+v≤Ad\+v\\leq A\(A=1\.2A\{=\}1\.2\)\.*KAM override\.*field on top\-decile KOLs is floored,f≥f¯f\\geq\\underline\{f\}\(f¯=0\.5\\underline\{f\}\{=\}0\.5\), a hard human\-in\-the\-loop invariant\. A candidate is*admissible*iff all hold; the optimizer and ARPF operate only on the admissible subset\. In the demonstration312/1296312/1296grid actions are admissible and the unconstrained surrogate optimum is not among them\.

## Appendix EOut\-of\-Time Backtest Protocol

The backtest mimics a real before/after evaluation so it transfers unchanged to Veeva/IQVIA data\. \(1\)*Split\.*Fit the additive surrogate on history up tot0t\_\{0\}; freeze it\. \(2\)*Recommend\.*Compute the four policies \(naive, admissible\-naive, nominal, ARPF\-robust\) on the frozen surrogate within the admissible set\. \(3\)*Hold\-out evaluation\.*Draw the true world \(structurally richer than the surrogate: field/email substitution, frequency fatigue, content saturation, consent\-capped reach\) for the post\-t0t\_\{0\}quarters and score realized NBRx for each policy and for the*logged*status quo actually run\. \(4\)*Report\.*Mean NBRx uplift vs logged and the5%5\\%lower tail\. A real study substitutes a test–control design: size each arm for a1212\-week NBRx read with the therapy\-area NBRx:TRx ratio, randomize matched HCP cohorts, and compare the robust recommendation against the incumbent allocation\. The simulation is RCT/benchmark anchored; only the data source changes\.

## Appendix FDecision Economics: Revenue\-at\-Risk

The NBRx\-uplift distribution underΘ\\Thetamaps to money\. For a target panel ofNNhigh\-value oncologists, addressable new patientsmmper HCP per year, and incremental annual valueπ\\piper started patient, expected incremental revenue is𝔼^​\[Δ​rev\]=𝔼Θ​\[Δ​NBRx\]⋅m​N​π\\widehat\{\\mathbb\{E\}\}\[\\Delta\\text\{rev\}\]=\\mathbb\{E\}\_\{\\Theta\}\[\\Delta\\text\{NBRx\}\]\\cdot mN\\pi, and the5%5\\%revenue Value\-at\-Risk is the corresponding quantile,VaR5%=Q0\.05​\(Δ​NBRx\)⋅m​N​π\\text\{VaR\}\_\{5\\%\}=Q\_\{0\.05\}\(\\Delta\\text\{NBRx\}\)\\cdot mN\\pi, giving the downside band a GM can act on \(illustrativeN=600N\{=\}600,m=8m\{=\}8,π=∈45\\pi\{=\}\\in 45k\)\. Share\-of\-voice is mapped linearly from the field/reach terms\. These parameters are explicit assumptions the affiliate replaces with its own; the*method*—price the whole robustness distribution, report expected value and downside together—is the point\.

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