Quantile Propagation Planning for Budgeted Tail-Risk Evaluation

# Quantile Propagation Planning for Budgeted Tail-Risk Evaluation
Source: [https://arxiv.org/html/2610.41351](https://arxiv.org/html/2610.41351)
\\coltauthor

1,2,1,3, and4 1University of Washington2University of Rochester3Independent Researcher4Georgia Institute of Technology

###### Abstract

Risk\-aware planning must identify rare, costly outcomes using only a limited number of model evaluations\. Monte Carlo rollouts spend this budget on noisy tail estimates, while planning from the mean can miss the tail entirely\. We introduce quantile propagation planning \(QPP\), which propagates a small, fixed set of disturbance quantiles through the dynamics and integrates the resulting cost curve to evaluate both mean and tail risk\. The method is exact at the propagated levels, converges as the quantile grid is refined, and admits a one\-sided conservatism guarantee under stated structural conditions\. In controlled overload and inventory tasks, QPP reduces realized tail cost and constraint violations relative to mean\-rollout planning\. On overload, it remains competitive with sampled evaluators that use substantially more trajectories\. These results position QPP as a transparent, budget\-controlled approach to tail\-risk evaluation\.

###### keywords

quantile propagation, risk\-aware planning, model predictive control, CVaR, cross\-entropy method, stochastic model\-based planning

## 1Introduction

Forecasts guide actions before their outcomes are known\. Retail demand forecasts drive purchasing, inventory, logistics, and pricing\([Fildes et al\., 2022](https://arxiv.org/html/2610.41351#bib.bib10)\); building and microgrid controllers schedule energy from uncertain load and weather\([Drgoňa et al\., 2020](https://arxiv.org/html/2610.41351#bib.bib9)\); and portfolio optimizers are sensitive to errors in estimated returns and covariances\([Kolm et al\., 2014](https://arxiv.org/html/2610.41351#bib.bib15)\)\. Learning\-based model predictive control \(MPC\) faces the same issue: stochastic learned models must be used without losing constraint reliability\([Hewing et al\., 2020](https://arxiv.org/html/2610.41351#bib.bib13)\)\. Optimization can therefore amplify a modest predictive error by selecting a different action and propagating its effect through repeated decisions\.

Rare errors also have asymmetric consequences\. A full\-scale vehicle study reports that a deterministic MPC baseline using one dynamics model may lose control in adverse conditions, whereas a risk\-averse controller accounts for uncertain friction and tire parameters\([Lew et al\., 2024](https://arxiv.org/html/2610.41351#bib.bib17)\)\. Worst\-case protection, however, can make a plan too conservative to use\. The practical goal is therefore to balance typical performance against costly tail outcomes, not merely to optimize either the mean or the worst case\.

This balance becomes a statistical bottleneck when a planner ranks many candidates under a fixed model\-call budget\. At tail massα\\alpha, empirical CVaR uses only the worstα\\alphafraction of sampled trajectories; withα=0\.1\\alpha=0\.1and five rollouts, the estimate is controlled by the single most costly rollout\. Concentration results formalize how the sample budget affects CVaR accuracy and selection error\([Prashanth et al\., 2020](https://arxiv.org/html/2610.41351#bib.bib26)\)\. Recent CVaR policy\-optimization work likewise traces poor sample efficiency to underused non\-tail trajectories\([Mead et al\., 2025](https://arxiv.org/html/2610.41351#bib.bib20);[Luo and Delage, 2026](https://arxiv.org/html/2610.41351#bib.bib19)\)\. Although those methods study policy learning, they expose the same scarcity of useful tail data\.

Stochastic MPC represents uncertainty through distributions, scenarios, or probabilistic constraints, but the general nonlinear problem remains computationally demanding\([Mesbah, 2016](https://arxiv.org/html/2610.41351#bib.bib21)\)\. Risk\-aware planners use, for example, stochastic search over nonlinear beliefs\([Wang et al\., 2021](https://arxiv.org/html/2610.41351#bib.bib34)\)or sample\-based CVaR constraints\([Lew et al\., 2024](https://arxiv.org/html/2610.41351#bib.bib17)\)\. A recent nonlinear SMPC framework still reports an accuracy–computation trade\-off among uncertainty\-propagation methods\([Landgraf et al\., 2026](https://arxiv.org/html/2610.41351#bib.bib16)\); analytic propagation avoids sampling variance but typically requires compatible distributional and value\-function structure\([Sharma and Precup, 2026](https://arxiv.org/html/2610.41351#bib.bib30)\)\. Mean\-only planning is cheaper but can hide skew and harmful tails\. The open question is how to rank many plans by mean and tail cost when each candidate can afford only a handful of model calls\.

We introduce*quantile propagation planning*\(QPP\), which evaluatesKKfixed disturbance levels and integrates their piecewise\-linear cost curve to obtain its mean and CVaR\. QPP holds the same disturbance rank across time and coordinates, so it evaluates a comonotonic surrogate rather than the independently disturbed execution process\. This shared\-rank coupling can be conservative and may change which plan is preferred\.

Our contributions are:

- •A deterministic level\-curve evaluator that usesKKmodel trajectories and does not require an ordering projection\.
- •Coupled\-process guarantees for exactness at the chosen levels, grid\-refinement convergence, and one\-sided conservatism under stated conditions\.
- •An evaluation across controlled overload and inventory tasks and a fitted microgrid, with budget\-matched sampling baselines and ablations that identify when QPP is effective\.

## 2Related Work

##### Sampling\-based planning\.

PETS\([Chua et al\., 2018](https://arxiv.org/html/2610.41351#bib.bib6)\)propagates uncertainty through an ensemble\. iCEM\([Pinneri et al\., 2021](https://arxiv.org/html/2610.41351#bib.bib25)\)improves CEM candidates, whereas the dsCEM\-style \(substituted point set\) construction\([Walker et al\., 2026](https://arxiv.org/html/2610.41351#bib.bib33)\)replaces random candidates by deterministic samples\. QPP instead changes candidate*evaluation*; the composition study therefore crosses QPP with both samplers and separates candidate\-distribution effects from loop mechanics\. On real\-load data, the fixed\-budget sampling comparison is a PETS\-style bootstrap ensemble \(Section[4](https://arxiv.org/html/2610.41351#S6.F4)\)\.

##### Risk\-aware objectives\.

The CVaR representation of[Rockafellar and Uryasev \(2000\)](https://arxiv.org/html/2610.41351#bib.bib29)is what makes the level\-curve tail analytically integrable\. Risk\-averse MPC supplies the dynamic\-control context\([Sopasakis et al\., 2019](https://arxiv.org/html/2610.41351#bib.bib31)\)\. Stochastic\-search CVaR control\([Wang et al\., 2021](https://arxiv.org/html/2610.41351#bib.bib34)\), sample\-based CVaR constraints\([Lew et al\., 2024](https://arxiv.org/html/2610.41351#bib.bib17)\), quantile\-gradient policy optimization\([Luo and Delage, 2026](https://arxiv.org/html/2610.41351#bib.bib19)\), distributional RL\([Dabney et al\., 2018b](https://arxiv.org/html/2610.41351#bib.bib8);[Dabney et al\., 2018a](https://arxiv.org/html/2610.41351#bib.bib7)\), and epistemic value distributions\([Luis et al\., 2023](https://arxiv.org/html/2610.41351#bib.bib18)\)all retain sampled or learned distributions\. QPP’s distinction is deterministic level\-curve evaluation inside planning\.

##### Deterministic uncertainty propagation\.

Non\-Gaussian stochastic MPC has used deterministic quadrature with stochastic Galerkin\([Chen and Zhang, 2019](https://arxiv.org/html/2610.41351#bib.bib3)\), polynomial\-chaos prediction and control\([Pan et al\., 2023](https://arxiv.org/html/2610.41351#bib.bib23)\), and monotonicity\-based reachable sets\([Heinlein et al\., 2025](https://arxiv.org/html/2610.41351#bib.bib12)\)\. QPP is narrower: it recursively propagates one scalar level through an evaluable one\-step model and integrates a piecewise\-linear mean/CVaR curve\. We claim neither general accuracy nor computational dominance over those methods\.

##### Learned distributional predictions\.

Local propagation of parametric laws\([Petersen et al\., 2024](https://arxiv.org/html/2610.41351#bib.bib24)\), deep multi\-step quantile MPC\([Chen et al\., 2025](https://arxiv.org/html/2610.41351#bib.bib4)\), and multi\-horizon quantile forecasting\([Wen et al\., 2017](https://arxiv.org/html/2610.41351#bib.bib35)\)differ from QPP’s recursive reuse of one latent level in a one\-step model\. Distribution\-guided diffusion planning\([Gong, 2026](https://arxiv.org/html/2610.41351#bib.bib11)\), quantile\-regularized representations\([Yu et al\., 2026](https://arxiv.org/html/2610.41351#bib.bib36)\), and point\-estimate CEM\([Rezazadeh et al\., 2025](https://arxiv.org/html/2610.41351#bib.bib27)\)are related hosts or alternatives, not evidence for nonlinear learned\-model performance here\.

## 3Method

##### Setting\.

A world model iszt\+1=f⁡\(zt,ut\)\+σ⁡\(zt,ut\)⊙εtz\_\{t\+1\}=f\(z\_\{t\},u\_\{t\}\)\+\\sigma\(z\_\{t\},u\_\{t\}\)\\odot\\varepsilon\_\{t\}with statezt∈ℝdz\_\{t\}\\in\\mathbb\{R\}^\{d\}, actionutu\_\{t\}, driftff, per\-coordinate loadingσ\\sigmaand a standardized scalar disturbanceεt\\varepsilon\_\{t\}with quantile functionF−1F^\{\-1\}; a per\-step costc⁡\(zt,ut\)c\(z\_\{t\},u\_\{t\}\)is charged on the pre\-step state\. The planner chooses an open\-loopu0:H−1u\_\{0:H\-1\}fromz0z\_\{0\}and executes it on the real, independently disturbed process;ff,σ\\sigma,ccneed only be evaluable in batch\.

##### Quantile\-vector state\.

Fix levels0<q1<⋯<qK<10<q\_\{1\}<\\dots<q\_\{K\}<1\(ℒ=\{0\.05,0\.25,0\.5,0\.75,0\.95\}\\mathcal\{L\}=\\\{0\.05,0\.25,0\.5,0\.75,0\.95\\\}here\)\. QPP replaces theSSparticles of a sampled rollout byKKdeterministic trajectories, one per level, all started atz0z\_\{0\}:

zt\+1\(k\)=f\(zt\(k\),ut\)\+σ\(zt\(k\),ut\)F−1\(qk\),k=1,…,K\.z^\{\(k\)\}\_\{t\+1\}=f\\big\(z^\{\(k\)\}\_\{t\},u\_\{t\}\\big\)\+\\sigma\\big\(z^\{\(k\)\}\_\{t\},u\_\{t\}\\big\)\\,F^\{\-1\}\(q\_\{k\}\),\\qquad k=1,\\dots,K\.\(1\)Trajectorykkreusesqkq\_\{k\}at every step and coordinate\. Thus theKKstates evaluate the comonotonic processZt​\(U\)Z\_\{t\}\(U\)\(one uniformUUshared across time and coordinates\) atU=qkU=q\_\{k\}\. SetCt\(k\)=c⁡\(zt\(k\),ut\)C\_\{t\}^\{\(k\)\}=c\(z\_\{t\}^\{\(k\)\},u\_\{t\}\); each step costsKKmodel evaluations per candidate\.

![Refer to caption](https://arxiv.org/html/2610.41351v1/fig0_method_overview.png)Figure 1:QPP in the closed\-loop planning pipeline\. Fixed disturbance levels generate coupled trajectories and an analytic mean–tail score; CEM ranks candidate plans, executes the first action, observes the next state, and replans\.
##### Level\-curve scoring\.

At each step, letCt​\(⋅\)C\_\{t\}\(\\cdot\)linearly interpolate the points\(qk,Ct\(k\)\)\(q\_\{k\},C\_\{t\}^\{\(k\)\}\)\. We extend it as a constant on\[0,q1\]\[0,q\_\{1\}\]and\[qK,1\]\[q\_\{K\},1\], and letU∼Unif⁡\(0,1\)U\\sim\\mathrm\{Unif\}\(0,1\)\. Its mean is a trapezoid rule whose end weights include the boundary masses; omitting them creates bias that does not vanish as the interior grid is refined\. We useα\\alphafor the*upper\-tail probability mass*, soCVaR0\.1\\operatorname\{CVaR\}\_\{0\.1\}is the mean of the worst10%10\\%\. We use the representation

CVaRα⁡\(X\)=minv⁡\{v\+α−1​𝔼​\[\(X−v\)\+\]\}\\operatorname\{CVaR\}\_\{\\alpha\}\(X\)=\\min\_\{v\}\\left\\\{v\+\\alpha^\{\-1\}\\mathbb\{E\}\\big\[\(X\-v\)^\{\+\}\\big\]\\right\\\}from[Rockafellar and Uryasev \(2000\)](https://arxiv.org/html/2610.41351#bib.bib29), because\(Ct−v\)\+\(C\_\{t\}\-v\)^\{\+\}has an analytic integral on every linear segment\. The implementation bracketsvvby the row minimum and maximum and uses6060bisection steps; Appendix[A\.2](https://arxiv.org/html/2610.41351#A1.SS2)gives the weights and cell integrals\. The result is exact to floating\-point precision for the constructed interpolant, not for the unobserved curve between or beyond the knots\. The cost row need not be sorted: a non\-monotone cost can make its largest value occur away from the largestqkq\_\{k\}, but integration still follows the level coordinateuu\.

##### Order restriction alongKK\.

Interpreting trajectorykkas a marginalqkq\_\{k\}quantile requires each coordinate ofZt​\(U\)Z\_\{t\}\(U\)to be non\-decreasing inUU\(Proposition[1](https://arxiv.org/html/2610.41351#Thmtheorem1)\(c\)\)\. Pool\-adjacent violators\([Barlow et al\., 1972](https://arxiv.org/html/2610.41351#bib.bib2);[Robertson et al\., 1988](https://arxiv.org/html/2610.41351#bib.bib28)\)or rearrangement\([Chernozhukov et al\., 2010](https://arxiv.org/html/2610.41351#bib.bib5)\)can impose this order \(Appendix[A\.5](https://arxiv.org/html/2610.41351#A1.SS5)\), but QPP uses neither: level\-curve scoring does not require ordered costs, and projecting a state would change the constructed trajectory whenever the condition fails \(Appendix[A\.4](https://arxiv.org/html/2610.41351#A1.SS4)\)\.

##### Objective\.

A candidatea=u0:H−1a=u\_\{0:H\-1\}is scored by

J⁡\(a\)=∑t=0H−1\[\(1−λ\)​𝔼​\[Ct​\(U\)\]\+λ​CVaRα​\[Ct​\(U\)\]\],λ∈\[0,1\],J\(a\)=\\sum\_\{t=0\}^\{H\-1\}\\Big\[\(1\-\\lambda\)\\,\\mathbb\{E\}\[C\_\{t\}\(U\)\]\+\\lambda\\,\\operatorname\{CVaR\}\_\{\\alpha\}\[C\_\{t\}\(U\)\]\\Big\],\\qquad\\lambda\\in\[0,1\],\(2\)lower is better: a sum of per\-step tail risks, a conservative surrogate for trajectory CVaR because subadditivity gives

∑tCVaRα⁡\[Ct\]≥CVaRα⁡\[∑tCt\]\.\\sum\_\{t\}\\operatorname\{CVaR\}\_\{\\alpha\}\[C\_\{t\}\]\\geq\\operatorname\{CVaR\}\_\{\\alpha\}\\\!\\left\[\\sum\_\{t\}C\_\{t\}\\right\]\.This surrogate uses the comonotonic coupling, not the independent process executed \(Sections[4](https://arxiv.org/html/2610.41351#S4),[6](https://arxiv.org/html/2610.41351#S6)\)\. Atλ=0\\lambda=0the score is risk\-neutral for the coupled process, which differs from a mean rollout whenccis nonlinear; atλ=1\\lambda=1it is the summed per\-step tail\. The level\-curve evaluator is the central component, whileλ\\lambdais an optional decision preference\. Appendix[A\.10](https://arxiv.org/html/2610.41351#A1.SS10)separately audits a single CVaR applied to the total trajectory cost\.

Algorithm[3](https://arxiv.org/html/2610.41351#S3.SS0.SSS0.Px5)collects the evaluator for one candidate; the planner evaluates candidates in batch\. The sameqkq\_\{k\}is retained throughout each trajectory, and the cost row is neither sorted nor projected\.\{algorithm\}\[t\]\\DontPrintSemicolonQPP level\-curve evaluation for one candidate action sequence\.\\KwInStatez0z\_\{0\}, actionsu0:H−1u\_\{0:H\-1\}, levelsℒ\\mathcal\{L\}, tail massα\\alpha, and weightλ\\lambda\\KwInEvaluableff,σ\\sigma, andcc, and disturbance quantile functionF−1F^\{\-1\}\\KwOutScoreJJand propagated states\{zt\(k\)\}\\\{z\_\{t\}^\{\(k\)\}\\\}J←0J\\leftarrow 0andz0\(k\)←z0z\_\{0\}^\{\(k\)\}\\leftarrow z\_\{0\}fork=1,…,Kk=1,\\ldots,K

\\For

t=0,…,H−1t=0,\\ldots,H\-1A\. Propagate the fixed levels

\\For

k=1,…,Kk=1,\\ldots,KCt\(k\)←c⁡\(zt\(k\),ut\)C\_\{t\}^\{\(k\)\}\\leftarrow c\(z\_\{t\}^\{\(k\)\},u\_\{t\}\)

zt\+1\(k\)←f⁡\(zt\(k\),ut\)\+σ⁡\(zt\(k\),ut\)​F−1​\(qk\)z\_\{t\+1\}^\{\(k\)\}\\leftarrow f\(z\_\{t\}^\{\(k\)\},u\_\{t\}\)\+\\sigma\(z\_\{t\}^\{\(k\)\},u\_\{t\}\)F^\{\-1\}\(q\_\{k\}\)

B\. Build the level curve

Form the constant\-extended piecewise\-linearCtC\_\{t\}through\(qk,Ct\(k\)\)k=1K\(q\_\{k\},C\_\{t\}^\{\(k\)\}\)\_\{k=1\}^\{K\}

C\. Evaluate and accumulate the score

mt←∫01Ct​\(u\)​𝑑um\_\{t\}\\leftarrow\\int\_\{0\}^\{1\}C\_\{t\}\(u\)\\,duandrt←CVaRα⁡\[Ct​\(U\)\]r\_\{t\}\\leftarrow\\operatorname\{CVaR\}\_\{\\alpha\}\[C\_\{t\}\(U\)\]by analytic integration

J←J\+\(1−λ\)​mt\+λ​rtJ\\leftarrow J\+\(1\-\\lambda\)m\_\{t\}\+\\lambda r\_\{t\}

returnJJand\{zt\(k\)\}\\\{z\_\{t\}^\{\(k\)\}\\\}

##### Planner, baselines, cost\.

The outer loop is the cross\-entropy method: drawPPcandidates from a diagonal Gaussian, clip, score, refit to the bestEE, and finally rescore the elite mean\. Both the sampler \(Gaussian, iCEM\-style\([Pinneri et al\., 2021](https://arxiv.org/html/2610.41351#bib.bib25)\), or dsCEM\-style\([Walker et al\., 2026](https://arxiv.org/html/2610.41351#bib.bib33)\)\) and evaluator are modular\. Monte Carlo usesSSindependent samples and averages the worst⌈α​S⌉\\lceil\\alpha S\\rceil; mean rollout propagatesffalone; moment matching uses a full central\-difference Jacobian with diagonal covariance \(Appendix[A\.1](https://arxiv.org/html/2610.41351#A1.SS1)\)\. QPP usesKKmodel calls per candidate\-step andO⁡\(K​B\)O\(KB\)scoring for fixedB=60B=60; the other costs areSScalls plus a sort, one call, and2​d\+12d\+1calls, respectively\. ThusS=KS=Kis the budget\-matched QPP–Monte Carlo comparison, withNevalN\_\{\\mathrm\{eval\}\}denoting the number of dynamics evaluations\.

## 4Theory

Zt​\(U\)Z\_\{t\}\(U\)denotes the*coupled process*of Section[3](https://arxiv.org/html/2610.41351#S3):

Zt\+1​\(U\)=f⁡\(Zt​\(U\),ut\)\+σ⁡\(Zt​\(U\),ut\)​F−1​\(U\),Z0=z0\.Z\_\{t\+1\}\(U\)=f\(Z\_\{t\}\(U\),u\_\{t\}\)\+\\sigma\(Z\_\{t\}\(U\),u\_\{t\}\)F^\{\-1\}\(U\),\\qquad Z\_\{0\}=z\_\{0\}\.One uniformUUis shared across steps and coordinates;Z~t\\tilde\{Z\}\_\{t\}is the*independent process*, a freshεt\\varepsilon\_\{t\}per step, on which plans are executed\. The guarantees below concern this scalar ordered disturbance construction; they do not cover arbitrary multivariate or non\-monotone learned dynamics\.

The next result separates three claims: exactness at the selected knots, convergence as the grid is refined, and the stronger conditions under which the propagated levels are marginal quantiles\. It applies to the states and level curves produced by Algorithm[3](https://arxiv.org/html/2610.41351#S3.SS0.SSS0.Px5)\.

###### Proposition 1\(Three guarantees for QPP\)\.

1. \(a\)Knot exactness\.For every action sequence,tt, andkk,zt\(k\)=Zt​\(qk\)z\_\{t\}^\{\(k\)\}=Z\_\{t\}\(q\_\{k\}\)\. Moreover,Ct​\(U\)C\_\{t\}\(U\)is the piecewise\-linear interpolation ofu↦c⁡\(Zt​\(u\),ut\)u\\mapsto c\(Z\_\{t\}\(u\),u\_\{t\}\)at the knotsℒ\\mathcal\{L\}, with no ordering assumption\.
2. \(b\)Grid\-refinement convergence\.Suppose this cost map is continuous on compact subintervals of\(0,1\)\(0,1\)and integrable\. If a sequence of grids has interior mesh tending to zero, endpoint knots tending to00and11, and uniformly integrable constant\-extended interpolants, then the level\-curve mean andCVaRα\\operatorname\{CVaR\}\_\{\\alpha\}converge to those ofc⁡\(Zt​\(U\),ut\)c\(Z\_\{t\}\(U\),u\_\{t\}\)\.
3. \(c\)Marginal\-quantile interpretation\.If, moreover,F−1F^\{\-1\}is locally absolutely continuous,ffandσ\\sigmaare continuously differentiable,σ≥0\\sigma\\geq 0, and, for everyu∈\(0,1\)u\\in\(0,1\)and every step,Jf\+F−1​\(u\)​JσJ\_\{f\}\+F^\{\-1\}\(u\)J\_\{\\sigma\}has non\-negative entries alongZt​\(u\)Z\_\{t\}\(u\), then each coordinate ofZt​\(U\)Z\_\{t\}\(U\)is non\-decreasing inUU\. Thuszt,i\(k\)z\_\{t,i\}^\{\(k\)\}is theqkq\_\{k\}quantile ofZt,i​\(U\)Z\_\{t,i\}\(U\), and the order restriction of Section[3](https://arxiv.org/html/2610.41351#S3)is a no\-op\.

The diagnostic in Appendix[A\.4](https://arxiv.org/html/2610.41351#A1.SS4)checks only the weaker discrete statement that theKKpropagated trajectories are ordered alongKKat every step\.

Figure 2:Planning holds one rank through the horizon \(left; colored endpoint labels give the five fixed quantile levels\), whereas execution draws a fresh rank at each step \(right; colors only distinguish sample paths\)\. Equal one\-step marginals need not imply equal temporal dependence or tail behavior\.The identities are exact at the selected knots\. Against a100,000100\{,\}000\-sample reference, the largest relative errors are0\.00860\.0086after one step and0\.00220\.0022after ten \(Appendix[A\.4](https://arxiv.org/html/2610.41351#A1.SS4)\)\. These values do not measure error between knots: on the monotone\-map diagnostic, relative level\-curve interpolation error is0\.11000\.1100at the deployedK=5K=5\(0\.13440\.1344/0\.09780\.0978atK=3K=3/99\)\. The sign condition holds for overload and the CityLearn surrogate, but fails for inventory and the rotation map\. The microgrid is order\-preserving in the shared level, including at charge clipping; because clipping is not continuously differentiable, it lies outside the proposition’s smooth sufficient condition \(Section[5](https://arxiv.org/html/2610.41351#S5); measured interpolation error at most0\.001170\.00117\)\.

The measured gap between propagated and independent\-process quantiles on the Gaussian monotone\-diagonal map is monotone inHHas \([3](https://arxiv.org/html/2610.41351#S4.E3)\) predicts and falls inside the band the closed form implies at every testedHHand persistence \(Appendix[A\.4](https://arxiv.org/html/2610.41351#A1.SS4)gives the values and the simulation check\)\. The closed form is for the constant\-loading scalar case: an analogy for overload \(action\-scaled loading\) and inapplicable to inventory’s stock \(a state\-dependent loading; Section[6\.1](https://arxiv.org/html/2610.41351#S6.SS1)reports the measured ratio\)\. At the microgrid’s fitted persistence andH=24H=24the ratio is within a percent of its saturating value, the analytic reason the coupled score there is a loose upper bound \(Section[6\.3](https://arxiv.org/html/2610.41351#S6.SS3)\)\.

###### Proposition 3\(One\-sided conservatism\)\.

Let the drift be linear with non\-negative coefficients and the loadings non\-negative constants, so that eachZt,iZ\_\{t,i\}is a non\-negative linear combination ofε1,…,εt\\varepsilon\_\{1\},\\dots,\\varepsilon\_\{t\}plus a deterministic term, and let every per\-step cost be a convex non\-decreasing function of a non\-negative linear combination of the state \(a ridge cost\), or a sum of such\. Assume also that the horizon\-summed costs and their stop\-loss transforms have finite expectations under both couplings\. Then the horizon\-summed cost is larger in the increasing convex order under the coupling than under independence; in particularCVaRα⁡\[∑tc⁡\(Zt​\(U\),ut\)\]≥CVaRα⁡\[∑tc⁡\(Z~t,ut\)\]\\operatorname\{CVaR\}\_\{\\alpha\}\[\\sum\_\{t\}c\(Z\_\{t\}\(U\),u\_\{t\}\)\]\\geq\\operatorname\{CVaR\}\_\{\\alpha\}\[\\sum\_\{t\}c\(\\tilde\{Z\}\_\{t\},u\_\{t\}\)\]for everyα\\alpha\.

The proof composes supermodularity of ridge costs and their increasing convex transforms with the extremal property of comonotonic coupling\([Kaas et al\., 2000](https://arxiv.org/html/2610.41351#bib.bib14)\); CVaR respects the resulting increasing convex order\. Together with subadditivity, this makes the exactλ=1\\lambda=1summed\-tail objective an upper bound under the stated hypotheses\. The finite interpolant has no general sign, however, so the implemented score is not a formal certificate without a certified envelope\. The hypotheses also matter: a concave rotation reverses the inequality \(−0\.855\-0\.855versus−0\.647\-0\.647\), and inventory receives no certificate\. Overload has a measured coupled\-minus\-executedCVaR0\.1\\operatorname\{CVaR\}\_\{0\.1\}of\+21\.4\+21\.4, while the microgrid gives302\.88302\.88versus8\.828\.82in the same conservative direction \(Appendix[A\.4](https://arxiv.org/html/2610.41351#A1.SS4)\)\. No rate is claimed for nonlinear drifts; monotone\-systems theory supplies a sign condition, not a rate\([Angeli and Sontag, 2003](https://arxiv.org/html/2610.41351#bib.bib1)\)\. Because the microgrid clips charge, its ordering remains empirical rather than a by\-construction proof\.

## 5Experimental Setup

##### Environments\.

Three environments with asymmetric tails carry the claims: two closed\-form and one built from real load data \(full definitions are in Appendix[A\.1](https://arxiv.org/html/2610.41351#A1.SS1)\)\.*Overload*has one state,H=8H=8, and an action\-scaled noise tail that trades throttle reward against squared\-hinge overload cost\.*Inventory*hasH=12H=12, persistent lognormal demand \(ρ=0\.8\\rho=0\.8\), and stockout cost ten times holding cost\. Synthetic maps provide calibration and negative controls; a fitted CityLearn surrogate is used only behind its validity gates\.

##### Microgrid\.

The third environment is battery dispatch against a real hourly load trace\. Its state is battery charge, an AR\(1\) standardized log\-load anomaly and hour;H=24H=24and the action is a charge/discharge fraction in\[−1,1\]\[\-1,1\]\. A pooled training\-client table supplies innovations; site\-specific quantities are fitted from the first70%70\\%of rows\. Twelve clients fit the pooled AR/noise model, while four retain local calibration\. Because the latter also enter the D1 design screen, they are not untouched: sitewise evidence is conditional \(Appendix[A\.3](https://arxiv.org/html/2610.41351#A1.SS3)\)\. The state is ordered in the shared level through charge clipping, although clipping lies outside Proposition[1](https://arxiv.org/html/2610.41351#Thmtheorem1)’s smooth condition\. The prospective endpointV​1V1is insufficient headroom for the next full\-power discharge, not actual import above capacity; that endpoint is a separately labelled post\-hoc replay\.

##### Planners and budget\.

Every planner uses CEM with1010iterations,1010elites and clipped candidates\. QPP \(K=5K=5,α=0\.2\\alpha=0\.2,λ∈\{0,0\.5,1\}\\lambda\\in\\\{0,0\.5,1\\\}\) and Monte Carlo \(S=5S=5\) useP=64P=64; mean rollout usesP​K=320PK=320; moment matching is within5%5\\%of QPP’s dynamics\-evaluation budget\.NevalN\_\{\\mathrm\{eval\}\}includes final elite\-mean scoring\. A control tests the different elite fractions \(Appendix[A\.4](https://arxiv.org/html/2610.41351#A1.SS4)\); composition studies otherwise vary only the sampler and loop refinements\. The microgrid also uses a PETS\-style TS\-∞\\inftybootstrap ensemble matched at2525calls per candidate\-step and evaluated by held\-out trace replay\.

Planning optimizes a20%20\\%per\-step tail surrogate, whereas evaluation reports the worst10%10\\%of realized episode costs; both choices were frozen, and are not asserted equivalent\. An exploratory audit separately testsα=0\.1\\alpha=0\.1, total\-trajectory CVaR and terminal alignment \(Appendix[A\.10](https://arxiv.org/html/2610.41351#A1.SS10)\)\.

##### Common random numbers and statistics\.

Repeatrrgives every planner the same noise streams, enabling paired contrasts\. Confirmatory tables usen=200n=200\(100100forλ\\lambdaand replanning sweeps\); the corrected inventory count audit usesn=400n=400\. EmpiricalCVaR0\.1\\operatorname\{CVaR\}\_\{0\.1\}averages the worst2020realized episode costs over optimizer and execution randomness, not fixed\-plan conditional risk\. Intervals use paired CRN bootstraps; mean, CVaR and violation rows use sign\-flip, stored bootstrap\-tail and exact McNemar procedures\. Each site table is one Holm family\. The four frozen sitewise bounds are unadjusted and descriptive; a labelled post\-hoc audit adds Holm tests and simultaneous Bonferroni bounds\. Prospective rules and test details are in Table[3](https://arxiv.org/html/2610.41351#A1.T3)and Appendix[A\.3](https://arxiv.org/html/2610.41351#A1.SS3); the budget is model calls, not runtime or memory\.

## 6Results

Conventions are those of Section[5](https://arxiv.org/html/2610.41351#S5);CVaR0\.1\\operatorname\{CVaR\}\_\{0\.1\}is the upper10%10\\%tail of the realized cost over repeats \(lower is better\)\.

Figure 3:Controlled\-environment results on overload\. \(a\) QPP moves along a mean–tail frontier rather than dominating every metric\. \(b\) Increasing trajectory count improves sampled evaluation, but the low\-budget QPP points remain competitive; bars show bootstrap intervals\. The appendix reports the full statistical comparisons and inventory counterexample\.### 6\.1Tail risk against noise\-blind planning

Table 1:Point estimates against mean rollout; lower is better andΔ\\Deltais QPP minus mean rollout\. Full results are in Appendix Tables[5](https://arxiv.org/html/2610.41351#A1.T5)and[6](https://arxiv.org/html/2610.41351#A1.T6)\.Table[1](https://arxiv.org/html/2610.41351#S6.T1)summarizes the main trade\-off\. On overload \(H=8H=8; Figure[3](https://arxiv.org/html/2610.41351#S6.F3)a\), QPP atλ=0\.5\\lambda=0\.5lowers the tail cost and reduces the overload rate from2\.5%2\.5\\%to0\.5%0\.5\\%, but its mean cost is worse\. The frozen B1 bound is met, so the result is a mean–tail frontier rather than overall domination\. A control with the same elite fraction as mean rollout changes the tail by less than one tenth of the QPP gaps on overload and inventory; full statistical audits are in Appendix[A\.4](https://arxiv.org/html/2610.41351#A1.SS4)\.

On inventory \(H=12H=12,ρ=0\.8\\rho=0\.8\), QPP records no stockout in200200episodes, whereas mean rollout has a rate of0\.6700\.670\. This tail improvement comes with a large mean\-cost penalty\. Moreover,S=5S=5Monte Carlo has a lower tail cost \(145\.252145\.252\), and QPP withλ=0\\lambda=0reaches144\.972144\.972\. Thus the nonzero risk weight is not the source of the inventory benefit\.

### 6\.2Against budget\-matched sampling

Table 2:RealizedCVaR0\.1\\operatorname\{CVaR\}\_\{0\.1\}in the trajectory\-count sweeps; lower is better\. Inventory reports the correctedn=400n=400audit; full statistical results and extended sweeps are in the appendix\.Table[2](https://arxiv.org/html/2610.41351#S6.T2)separates point estimates from interpretation\. At the matched evaluator budget \(S=K=5S=K=5\), the overload gap for QPP atλ=0\.5\\lambda=0\.5is−3\.210\-3\.210, but the adjusted test does not support a tail\-cost difference \(padj=0\.6183p\_\{\\mathrm\{adj\}\}=0\.6183\)\. The mean\-cost penalty is1\.8491\.849\(padj=0\.0015p\_\{\\mathrm\{adj\}\}=0\.0015\)\. The inventory matched\-budget tail comparison is also unresolved \(Table[16](https://arxiv.org/html/2610.41351#A1.T16)\), and QPP loses the mean comparison decisively\.

The trajectory\-count sweeps make the environment difference clear\. On inventory,S=20S=20Monte Carlo is already belowK=5K=5QPP, so B3 is not met\. On overload, noS≤50S\\leq 50arm comes within5%5\\%of QPP; the extension toS=200S=200is unresolved relative toS=50S=50, and a clipped coupled sampler first wins atS=100S=100\. The evidence therefore supports a low\-budget overload advantage, not a general advantage over sampling\.

On the fitted microgrid no IID arm comes within5%5\\%of QPP throughS=200S=200; IID tail cost remains99–15%15\\%worse and a clipped coupled sampler leaves a55–8%8\\%gap\. Population, elites and iterations are held fixed, so this is not a best\-allocation total\-budget comparison \(Appendix[A\.11](https://arxiv.org/html/2610.41351#A1.SS11)\)\.

### 6\.3A real\-data tail environment

##### Calibration and H1\.

Four locally calibrated clients also enter D1, so their evidence is conditional; pooled calibration passes only3/83/8split seeds, and two planned sites exceed the PIT tolerance\. Atn=200n=200andH=24H=24, all planners meet the budget gates\. Figure[4](https://arxiv.org/html/2610.41351#S6.F4)shows improvedV​1V1at all sites, B1 passing at242242,253253and279279, and B2 failing everywhere\. A post\-hoc replay gives actual\-import rates of28\.528\.5–37\.5%37\.5\\%for QPP versus43\.043\.0–62\.5%62\.5\\%for mean rollout\. This supports reduced incidence, not safety \(Appendix[A\.9](https://arxiv.org/html/2610.41351#A1.SS9)\)\. A Holm audit rejects the10%10\\%B1 null at three sites, but Bonferroni retains only242242and279279\. Atn=1,000n=1\{,\}000, neitherα=0\.1\\alpha=0\.1nor terminal alignment helps consistently; breaking rank sharing worsens tail cost and V1 \(Appendix[A\.10](https://arxiv.org/html/2610.41351#A1.SS10)\)\.

Figure 4:Sitewise one\-sided95%95\\%bounds for the fitted microgrid\. Dashed lines mark the gates: B1 passes at three sites, while V1 improves at all four\. Bounds are frozen and unadjusted; the multiplicity audit is in the appendix\.

## 7Limitations and Scope

##### Measurement\.

Model\-call matching omits runtime and memory; decision\-variance reduction is unsupported except through inventory saturation\. Other limits are loose coupling,11%11\\%interpolation error atK=5K=5, an unpriced terminal state, and 20 tail outcomes pern=200n=200record\.

##### Selection\.

D1 screens all sites; the construction fails D1 on a second dataset\([Miller et al\., 2020](https://arxiv.org/html/2610.41351#bib.bib22)\), and KPI calibration and CityLearn replanning fail\. Proposition[3](https://arxiv.org/html/2610.41351#Thmtheorem3)does not sign the finite interpolant, and a concave rotation reverses its ordering\. Evidence does not establish untouched\-site, learned\-model, epistemic, fixed\-plan or decision\-variance performance, nor an unconditional budget advantage\.

## 8Conclusion

QPP integrates deterministic comonotonic disturbance levels\. It is most useful when planning budgets are tight and shared\-rank tail structure is informative\. Its benefit can disappear when coupling or model fit is weak\.

### Generative AI Use

Generative AI tools were used to assist with writing and editing, figure refinement, and code review\. The authors reviewed and verified all AI\-assisted content and take responsibility for the final content\.

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## Appendix AAdditional proofs and experimental details

### A\.1Environment definitions

This section first defines the controlled environments and their baselines, then gives the fitted\-environment construction used in the application studies\. The final part defines the microgrid endpoints and ensemble evaluators\. This order separates the dynamics and costs from the data split and the later evaluation rules\.

#### A\.1\.1Controlled environments and baselines

The*overload*environment has one state and horizonH=8H=8\. Its dynamics arez′=a​z\+g​u\+\(s0\+s1​u\)​εz^\{\\prime\}=az\+gu\+\(s\_\{0\}\+s\_\{1\}u\)\\varepsilon, whereε\\varepsilonis standardized lognormal noise and the throttle satisfiesu∈\[0,1\]u\\in\[0,1\]\. The pre\-step cost is−r​u\+c​relu​\(z−θ\)2\-ru\+c\\,\\mathrm\{relu\}\(z\-\\theta\)^\{2\}\. The constants area=0\.7a=0\.7,g=0\.5g=0\.5,s0=0\.05s\_\{0\}=0\.05,s1=0\.5s\_\{1\}=0\.5,θ=2\.5\\theta=2\.5,c=8c=8,r=1r=1, andz0=0z\_\{0\}=0\. The deterministic path never crossesθ\\thetafor any constant throttle; the action\-scaled tail does\. The constants were tuned by a200,000200\{,\}000\-sample search for that separation; this is a disclosed design choice rather than an independently validated property\.

The*inventory*state is\[x,m\]\[x,m\], representing stock and a smoothed demand level, and the horizon isH=12H=12\. The dynamics arex′=x\+c​u−m−sD​m​εx^\{\\prime\}=x\+cu\-m\-s\_\{D\}m\\varepsilonandm′=ρ​m\+\(1−ρ\)​m¯\+sm​m​εm^\{\\prime\}=\\rho m\+\(1\-\\rho\)\\bar\{m\}\+s\_\{m\}m\\varepsilon, with one shared standardized lognormal shock per step and order fractionu∈\[0,1\]u\\in\[0,1\]\. The pre\-step cost ish​relu​\(x\)\+p​relu​\(−x\)\+co​uh\\,\\mathrm\{relu\}\(x\)\+p\\,\\mathrm\{relu\}\(\-x\)\+c\_\{o\}u\. The constants areh=1h=1,p=10p=10,co=0\.5c\_\{o\}=0\.5,ρ=0\.8\\rho=0\.8,m¯=1\\bar\{m\}=1,c=3c=3,sD=0\.5s\_\{D\}=0\.5,sm=0\.1s\_\{m\}=0\.1, andz0=\[1,1\]z\_\{0\}=\[1,1\]\. A step is a stockout when the post\-stepx<0x<0\. No constant was tuned against a planning outcome; the mean\- and tail\-optimal constant orders come from a200,000200\{,\}000\-sample grid search under the implemented cost timing \(0\.410\.41and0\.530\.53\)\.

The*synthetic*maps include monotone diagonal dynamics \(per\-coordinate persistenceai∼Unif⁡\(0\.5,0\.9\)a\_\{i\}\\sim\\mathrm\{Unif\}\(0\.5,0\.9\)\), densetanh\\tanh, and rotation dynamics with Gaussian, lognormal, or Student\-t3t\_\{3\}standardized noise\. The monotone\-diagonal family with a symmetric quadratic cost is the planning negative control\.

For moment matching, central differences estimate the full Jacobian\. The baseline propagates marginal variances while treating the covariance as diagonal\. With state meanμ\\mu, marginal variancevv, basis vectoreje\_\{j\}, and difference stepϵ\\epsilon, the update is

μ′=f⁡\(μ,u\),Ji​j=fi​\(μ\+ϵ​ej,u\)−fi​\(μ−ϵ​ej,u\)2​ϵ,vi′=∑j=1dJi​j2​vj\+σi​\(μ,u\)2\.\\mu^\{\\prime\}=f\(\\mu,u\),\\qquad J\_\{ij\}=\\frac\{f\_\{i\}\(\\mu\+\\epsilon e\_\{j\},u\)\-f\_\{i\}\(\\mu\-\\epsilon e\_\{j\},u\)\}\{2\\epsilon\},\\qquad v\_\{i\}^\{\\prime\}=\\sum\_\{j=1\}^\{d\}J\_\{ij\}^\{2\}v\_\{j\}\+\\sigma\_\{i\}\(\\mu,u\)^\{2\}\.It therefore uses2​d\+12d\+1dynamics evaluations per step: each perturbed call returns every output coordinate\. The method transmits marginal uncertainty across state coordinates but does not carry cross\-covariances between steps; it is a first\-order diagonal\-covariance approximation, not an exact Gaussian filter\.

#### A\.1\.2CityLearn surrogate

The CityLearn surrogate is a diagonal linear state\-space model whose dynamics are monotone\. Its noise table contains empirical residual quantiles\. The model is fitted to2,0002\{,\}000random transitions of the real simulator\. Planning uses a horizon ofH=24H=24hours on the surrogate, and execution uses the real environment\. The validity gates require costR2\>0R^\{2\}\>0and multi\-step quantile calibration error below0\.100\.10\.

#### A\.1\.3Microgrid dynamics and cost

The microgrid has state\[soc,ℓ,h\]\[\\mathrm\{soc\},\\ell,h\], actionu∈\[−1,1\]u\\in\[\-1,1\], and horizonH=24H=24\. The state contains the charge fraction, a standardized AR\(1\) log\-load anomaly, and an unbounded hour counter\. Positiveuucharges and negativeuudischarges\. Writingu\+=max⁡\(u,0\)u\_\{\+\}=\\max\(u,0\)andu−=max⁡\(−u,0\)u\_\{\-\}=\\max\(\-u,0\), the transition is

soc′=clip⁡\(soc\+ηc​u\+​pbat​Δ​tecap−u−​pbat​Δ​tηd​ecap,0,1\),ℓ′=ρℓ\+1−ρ2ε,h′=h\+1\.\\begin\{split\}\\mathrm\{soc\}^\{\\prime\}&=\\operatorname\{clip\}\\\!\\left\(\\mathrm\{soc\}\+\\frac\{\\eta\_\{c\}u\_\{\+\}p\_\{\\mathrm\{bat\}\}\\Delta t\}\{e\_\{\\mathrm\{cap\}\}\}\-\\frac\{u\_\{\-\}p\_\{\\mathrm\{bat\}\}\\Delta t\}\{\\eta\_\{d\}e\_\{\\mathrm\{cap\}\}\},\\,0,\\,1\\right\),\\\\ \\ell^\{\\prime\}&=\\rho\\ell\+\\sqrt\{1\-\\rho^\{2\}\}\\,\\varepsilon,\\qquad h^\{\\prime\}=h\+1\.\\end\{split\}Hereηc=ηd=0\.95\\eta\_\{c\}=\\eta\_\{d\}=0\.95,ecap=2\.0e\_\{\\mathrm\{cap\}\}=2\.0,pbat=0\.5p\_\{\\mathrm\{bat\}\}=0\.5, andΔ​t=1\.0\\Delta t=1\.0\. LetΔ​soc=soc′−soc\\Delta\\mathrm\{soc\}=\\mathrm\{soc\}^\{\\prime\}\-\\mathrm\{soc\}\. The bus\-side battery draw, positive when charging, is

B=\{Δ​soc​ecap/\(ηc​Δ​t\),Δ​soc≥0,Δ​soc​ecap​ηd/Δ​t,Δ​soc<0\.B=\\begin\{cases\}\\Delta\\mathrm\{soc\}\\,e\_\{\\mathrm\{cap\}\}/\(\\eta\_\{c\}\\Delta t\),&\\Delta\\mathrm\{soc\}\\geq 0,\\\\ \\Delta\\mathrm\{soc\}\\,e\_\{\\mathrm\{cap\}\}\\eta\_\{d\}/\\Delta t,&\\Delta\\mathrm\{soc\}<0\.\\end\{cases\}Normalized demand and net grid import are

Lpu=exp⁡\(profile⁡\(h\)\+sr​ℓ\),Gpu=Lpu\+B,L\_\{\\mathrm\{pu\}\}=\\exp\(\\mathrm\{profile\}\(h\)\+s\_\{r\}\\ell\),\\qquad G\_\{\\mathrm\{pu\}\}=L\_\{\\mathrm\{pu\}\}\+B,whereGpu<0G\_\{\\mathrm\{pu\}\}<0denotes export andprofile\\mathrm\{profile\}is a two\-harmonic Fourier curve\. Ifssites\_\{\\mathrm\{site\}\}is the reported site scale, thenLphysical=ssite​LpuL\_\{\\mathrm\{physical\}\}=s\_\{\\mathrm\{site\}\}L\_\{\\mathrm\{pu\}\}andGphysical=ssite​GpuG\_\{\\mathrm\{physical\}\}=s\_\{\\mathrm\{site\}\}G\_\{\\mathrm\{pu\}\}\. Thuspcapp\_\{\\mathrm\{cap\}\},pbatp\_\{\\mathrm\{bat\}\}andecape\_\{\\mathrm\{cap\}\}are in the normalized system’s per\-unit power and per\-unit\-hour coordinates; the tariff below prices per\-unit energy rather than claiming a physical currency bill\. With\[x\]\+=max⁡\(x,0\)\[x\]\_\{\+\}=\\max\(x,0\), the pre\-step cost is

c⁡\(z,u\)=π⁡\(h\)​\[Gpu\]\+−pexp​\[−Gpu\]\+\+wo​\[Gpu−pcap\]\+2\+wd​ueff2,c\(z,u\)=\\pi\(h\)\[G\_\{\\mathrm\{pu\}\}\]\_\{\+\}\-p\_\{\\mathrm\{exp\}\}\[\-G\_\{\\mathrm\{pu\}\}\]\_\{\+\}\+w\_\{o\}\[G\_\{\\mathrm\{pu\}\}\-p\_\{\\mathrm\{cap\}\}\]\_\{\+\}^\{2\}\+w\_\{d\}u\_\{\\mathrm\{eff\}\}^\{2\},whereueffu\_\{\\mathrm\{eff\}\}is the realized action after state\-of\-charge saturation\. The flat import tariff isπ=0\.20\\pi=0\.20, export creditpexp=0\.04p\_\{\\mathrm\{exp\}\}=0\.04, the overload\-penalty ratio is1010, and degradation is0\.100\.10per pu\-h\. Thusσ=\[0,1−ρ2,0\]\\sigma=\[0,\\sqrt\{1\-\\rho^\{2\}\},0\]is constant and non\-negative\.

#### A\.1\.4Microgrid data, split, and fitted quantities

The re\-dated ECL hourly mirror contains26,30426\{,\}304rows for321321clients from 2016\-07\-01 to 2019\-07\-02, matching the original release span\([Trindade, 2015](https://arxiv.org/html/2610.41351#bib.bib32)\)\. Of these clients,9292are strictly positive on every row\. The reproducible seed\-0 cohort contains clients 127, 130, 137, 189, 197, 202, 220, 225, 233, 240, 242, 253, 255, 257, 270, and 279\. Twelve form the pooled training set; sites 242, 253, 255, and 279 are excluded from pooled fitting and used for evaluation\.

Each site’s scale, profile,srs\_\{r\}, andpcapp\_\{\\mathrm\{cap\}\}are fitted from its first70%70\\%of rows; the final30%30\\%is reserved for evaluation\.pcapp\_\{\\mathrm\{cap\}\}is the site’s empirical0\.950\.95quantile of unshaved training load\. The standardized innovation table is fitted only from the twelve training clients and is cut at the0\.10\.1and99\.999\.9percentiles\. It contains220,932220\{,\}932values, has endpoints−4\.0116\-4\.0116and\+5\.1985\+5\.1985, raw mean0\.000090\.00009, and standard deviation0\.999960\.99996\. The fitted persistence isρ=0\.7525\\rho=0\.7525, with noise loading0\.65860\.6586\.

Figure 5:Site\-specific quantities fitted without pooling the four held\-out evaluation sites\. Labels give the exact scale, innovation scalesrs\_\{r\}, and capacity thresholdpcapp\_\{\\mathrm\{cap\}\}used by each site’s simulator\. The separate axes preserve the three quantities’ natural units rather than normalizing them to a common visual scale\.Two disclosed design revisions preceded freezing the D1 rule: a flat tariff replaced an originally planned two\-level tariff, and the innovation table was cut at its0\.1%0\.1\\%ends\. Both revisions make the tail smaller, working against a tail\-aware planner rather than in its favor\.

#### A\.1\.5Microgrid endpoints and ensemble evaluators

For the primaryV​1V1endpoint, define

Dmax​\(s\)=min⁡\(pbat,s​ecap​ηdΔ​t\),h⁡\(z\)=Lpu​\(z\)−Dmax​\(soc\)−pcap,D\_\{\\max\}\(s\)=\\min\\\!\\left\(p\_\{\\mathrm\{bat\}\},\\frac\{se\_\{\\mathrm\{cap\}\}\\eta\_\{d\}\}\{\\Delta t\}\\right\),\\qquad h\(z\)=L\_\{\\mathrm\{pu\}\}\(z\)\-D\_\{\\max\}\(\\mathrm\{soc\}\)\-p\_\{\\mathrm\{cap\}\},V1=\{max1≤t≤Hh\(zt\)\>0\}\.V1=\\mathbf\{1\}\\\!\\left\\\{\\max\_\{1\\leq t\\leq H\}h\(z\_\{t\}\)\>0\\right\\\}\.It is evaluated on every post\-transition state, includingzHz\_\{H\}\. Hence it marks loss of discharge headroom, not necessarily realized overload under the action already taken\. The separate exploratory event is𝟏​\{∃t∈\{0,…,H−1\}:Gpu​\(zt,ut\)\>pcap\}\\mathbf\{1\}\\\{\\exists t\\in\\\{0,\\ldots,H\-1\\\}:G\_\{\\mathrm\{pu\}\}\(z\_\{t\},u\_\{t\}\)\>p\_\{\\mathrm\{cap\}\}\\\}\. The exact\-stream replay uses the same execution seed sequence as the recorded cost andV​1V1observations\. Rates for every planner, paired bounds and the cross\-site multiplicity policy are reported with the H1 tables below\. Because this endpoint was added after the frozen protocol, it remains an exploratory operational diagnostic\.

##### Ensemble evaluators\.

Each bootstrap member resamples the training interval in168168\-hour blocks with replacement and refits the site models\. Its pooled AR coefficient and innovation table use a without\-replacement75%75\\%subsample of the twelve training clients;pcapp\_\{\\mathrm\{cap\}\}remains the reference site’s fixed per\-unit threshold\. For the trajectory ensemble withM=5M=5andK=5K=5, the same five levels are propagated separately through each of five members, giving anM×K=25M\\times K=25equal\-weight vertical mixture for mean and CVaR scoring\. The single\-model QPP arm withM=1M=1andK=25K=25propagates 25 levels through the reference model\. The PETS\-style ensemble withM=5M=5andS=5S=5draws five IID trajectories from each of five members and keeps each particle’s member fixed throughout the horizon \(TS\-∞\\infty\); its 25 realized trajectories are scored as ordinary equal\-weight Monte Carlo samples\.

### A\.2Level\-curve scoring details

ForK\>1K\>1, the constant\-extended trapezoidal mean is∑k=1Kwk​Ck\\sum\_\{k=1\}^\{K\}w\_\{k\}C\_\{k\}, where

w1=q1\+q2−q12,wk=qk\+1−qk−12​\(1<k<K\),wK=1−qK\+qK−qK−12\.w\_\{1\}=q\_\{1\}\+\\frac\{q\_\{2\}\-q\_\{1\}\}\{2\},\\qquad w\_\{k\}=\\frac\{q\_\{k\+1\}\-q\_\{k\-1\}\}\{2\}\\ \(1<k<K\),\\qquad w\_\{K\}=1\-q\_\{K\}\+\\frac\{q\_\{K\}\-q\_\{K\-1\}\}\{2\}\.The weights sum to one; forK=1K=1,w1=1w\_\{1\}=1\. For CVaR, the implementation brackets the thresholdvvby the row minimum and maximum and performs exactly6060bisection iterations\. On a sloped cell\[qk,qk\+1\]\[q\_\{k\},q\_\{k\+1\}\], it computes the crossing fractionr=clip⁡\(\(v−Ck\)/\(Ck\+1−Ck\),0,1\)r=\\operatorname\{clip\}\(\(v\-C\_\{k\}\)/\(C\_\{k\+1\}\-C\_\{k\}\),0,1\)and integrates the portion whereC⁡\(q\)\>vC\(q\)\>vby the exact trapezoid area\. Increasing and decreasing cells select opposite sides of the crossing; a flat cell is included wholly or not at all\. The two constant boundary cells are handled by the same rule\.

After bisection, letm\>m\_\{\>\}be the measure of\{q:C⁡\(q\)\>v⋆\}\\\{q:C\(q\)\>v\_\{\\star\}\\\}andI\>I\_\{\>\}the integral ofCCon that set\. The returned value is

I\>\+v⋆​\(α−m\>\)α,\\frac\{I\_\{\>\}\+v\_\{\\star\}\(\\alpha\-m\_\{\>\}\)\}\{\\alpha\},which supplies the required fraction of any atom atv⋆v\_\{\\star\}\. This is why unsorted level rows and thresholds crossing either direction are handled without ranking the values\.

### A\.3Gate statements and outcomes

This section is a registry of the paper’s decision rules\. It first reports the outcomes, then defines the controlled\-environment, composition, and microgrid gate families in that order\. A fraction such as3/43/4means that three of four site\- or row\-specific outcomes have the displayed status; it is not a rate computed from the underlying episodes\.

Table 3:Primary, calibration and application criteria\. “Sat\.” means satisfied, “Not sat\.” means not satisfied, “Inc\.” means inconclusive, “Negative” denotes a discharged negative result, and “N/A” denotes a criterion that was not evaluated\.†Satisfied by action\-bound saturation and not claimed \(Section[7](https://arxiv.org/html/2610.41351#S7)\)\. Microgrid planning gates are read per site;GE​2G\_\{E2\}andGE​3G\_\{E3\}are pooled cross\-client source gates\. A fraction reports the number of site\- or row\-specific outcomes with the displayed status\.Gate or diagnosticOver\.Inv\.Micro\.OtherC1 tail below MC under each sampler \(4\)Not sat\. 4/4Not sat\. 4/4C2 tail below mean rollout under each sampler \(4\)Sat\. 4/4Sat\. 4/4C3 tail gap within Gaussian\-row margin \(4\)Sat\. 2/4; Inc\. 2/4Not sat\. 2/4; Inc\. 2/4C4 candidate\-stream variance halved \(8\)Sat\. 3/8Not sat\. 8/8*Diagnostics without a prospectively specified bar*B2′both\-stream decision variance versus MCNot sat\.H1′tail and mean below mean rolloutNot sat\.E2/E3 epistemic decomposition/premiumNot sat\. 4/4; Not sat\. 4/4E4 shift amplifies premiumN/ATable 4:Composition gates and ungated diagnostics, using the status labels defined in Table[3](https://arxiv.org/html/2610.41351#A1.T3)\. C1–C4 were prospectively specified; the bottom block was not\.\\FloatBarrier

#### A\.3\.1Controlled\-environment gates

Figure 6:Overload B3 trajectory\-count follow\-up\. The curves show sampler\-minus\-QPP tail\-cost gaps\. IID intervals join the paired one\-sided95%95\\%lower and upper bounds used by the B3 branches; clipped\-coupled intervals are two\-sided95%95\\%intervals for the separate equivalence diagnostic\. The shaded region is the±0\.2411\\pm 0\.2411cost\-unit comparison band\.##### Accuracy and primary comparisons\.

Gates 1a–1c compare propagated quantiles with a coupled Monte Carlo reference\. B1 uses a one\-sided tail\-cost comparison: an absolute signed gap on overload and a relative gain on inventory\. H1 tests the overload gap against both noise\-blind baselines\. G2 checks surrogate skill and calibration, while B2 compares decision variance with budget\-matched Monte Carlo\.

##### Trajectory\-count comparison\.

B3 asks for the smallest Monte Carlo sample count within5%5\\%of QPP atK=5K=5, with the CEM search fixed\. That count must be at least5050, or ten times the per\-candidate evaluator cost\. The recorded rule labels the branch in which noS≤50S\\leq 50enters the band as*not measured*, not met, and the paper retains that verdict\.

The overload follow\-up extends the grid toS∈\{5,20,50,100,200\}S\\in\\\{5,20,50,100,200\\\}with the same candidate seeds; theS≤50S\\leq 50arms reproduce the closed record\. The absolute5%5\\%bar is0\.24110\.2411cost units\. No IID arm qualifies because its one\-sided upper bound stays above the bar\. AtS=200S=200, the point estimate0\.03580\.0358is below the bar, but the upper bound is1\.12861\.1286; the lower bound is−0\.7891\-0\.7891, so neither decisive B3 branch applies and the recorded verdict remains*not measured*\. This distinction between the estimate and the gate bound is visible in Figure[6](https://arxiv.org/html/2610.41351#A1.F6)\.

The clipped coupled sampler beats QPP atS=100S=100, with a difference of−0\.2545\-0\.2545and a95%95\\%interval from−0\.3507\-0\.3507to−0\.1472\-0\.1472\(Holm\-adjustedp=0\.001p=0\.001\)\. AtS=200S=200, its difference is−0\.1752\-0\.1752and the interval runs from−0\.3071\-0\.3071to0\.04660\.0466, so the apparent advantage is no longer conclusive\.

#### A\.3\.2Surrogate and composition gates

G2 requires a lower bound of at least0\.030\.03on the relativeCVaR0\.1\\operatorname\{CVaR\}\_\{0\.1\}gain over mean rollout on the CityLearn surrogate\. G2\-U3 uses a five\-way decision at the strongest disturbance: met, discharged negative, significant gain below the bar, inconclusive, or not measured\. The KPI gate checks simulator skill, PIT coverage, and support\.

For each of four gated sampler rows, C1 and C2 test QPP againstS=5S=5Monte Carlo and mean rollout, respectively, using the B1 tail\-cost rules\. C3 asks whether the row’s QPP\-minus\-Monte\-Carlo gap lies within the prespecified margin around the Gaussian row’s gap\. Because that margin came from an earlier exploratory gap rather than an application tolerance, C3 measures internal stability, not practical equivalence\. C4 requires the sampler’s candidate\-stream decision variance to be at most half the Gaussian sampler’s\. B2′and H1′are recorded diagnostics without a prospectively specified bar\.

#### A\.3\.3Microgrid gates

##### Design and calibration\.

D1 is the design\-time discrimination gate\. For every evaluation site and seed00–55, the constant\-reserve oracle uses200,000200\{,\}000CRN samples on a 41\-point reserve grid withα=0\.1\\alpha=0\.1,H=24H=24, a flat tariff, andpcapp\_\{\\mathrm\{cap\}\}atq0\.95q\_\{0\.95\}\. It computesrcvar−rmeanr\_\{\\mathrm\{cvar\}\}\-r\_\{\\mathrm\{mean\}\}and is met only if every seed’s median over evaluation sites is at least0\.150\.15\. One adjustment toq0\.90q\_\{0\.90\}was permitted before stopping\. The gate was met without that adjustment: the six per\-seed medians were0\.16250\.1625,0\.1750\.175,0\.1750\.175,0\.1750\.175,0\.16250\.1625, and0\.1750\.175, and the per\-seed minimum over sites was0\.150\.15\. The planning evidence is therefore conditional on this passed screen\.

GE​2G\_\{E2\}is a pooled cross\-client calibration gate\. It requires median one\-step log\-load skill over persistence above zero, every site’s skill above−0\.05\-0\.05, pooled PIT coverage within0\.050\.05of nominal at the five levels0\.050\.05,0\.250\.25,0\.50\.5,0\.750\.75, and0\.950\.95, and fewer than0\.010\.01of pooled held\-out innovations outside the fitted support\. Per\-site PIT deviations are diagnostics rather than gate components\.

##### Planning and ensemble comparisons\.

V1 tests the paired episode headroom\-loss rate against mean rollout with a one\-sided upper bound below zero\. B1 requires the one\-sided95%95\\%bootstrap lower bound on the relativeCVaR0\.1\\operatorname\{CVaR\}\_\{0\.1\}gain to be at least0\.100\.10\. B2 uses the first\-action decision\-variance ratio and the same1/31/3upper bar; B2eff applies that bar to planned\-reserve variance as an exploratory check\. Both budget\-match gates require the twoNevalN\_\{\\mathrm\{eval\}\}values to be within\(K−1\)​H\+10−6=96\.000001\(K\-1\)H\+10^\{\-6\}=96\.000001, and both hold at every site\.

The trajectory\-count gate varies onlySS: noS≤200S\\leq 200may enter the5%5\\%band, and the one\-sided lower bound atS=200S=200must exceed0\.050\.05\. The coupled\-sampler gate instead asks whether someS≤50S\\leq 50is equivalent within the same5%5\\%band\.GE​3G\_\{E3\}checks mixture PIT deviation and member\-level skill and support for the 25 bootstrap\-resampled models\. The replay\-scope design check applies the D1 rule to the execution\-scope choice\. E1 and E1v compare the quantile ensemble with PETS on tail cost and V1 using the B1 bound\. E2, E3, and E4 are recorded diagnostics for epistemic decomposition, a material premium with ratio at least0\.010\.01, and shift\-amplified premium\. The BDG2 check repeats D1 with the same bar, permitted adjustment, and stopping rule; it is not met at either cap \(Section[7](https://arxiv.org/html/2610.41351#S7)\)\.

#### A\.3\.4Baseline disclosures and claim scope

Theβ\\betasweep shares the headline’s repeats, andβ=0\\beta=0is the Gaussian row, so on overload every sweptβ\\betais weaker than Gaussian and the strongest recorded Monte Carlo baseline there is the Gaussian control with the iCEM loop\. Because selection and evaluation reuse repeats, inference involving the selectedβ\\betais descriptive rather than confirmatory\. The iCEM\-style row highlights three departures from the published iCEM algorithm: carried elites replace fresh draws rather than being appended \(budget neutrality\), the coloured noise is normalized to unit per\-position variance, and carried elites are re\-scored each iteration, which is information\-free for a deterministic evaluator\. The record documents eight baseline deviations across the complete iCEM\-style and dsCEM\-style sampler implementation\.

Taken together, the gates support the comparisons with noise\-blind planning that are stated in the main text\. They do not establish uniform superiority over budget\-matched Monte Carlo, a benefit from ensemble allocation, or transfer through the CityLearn surrogate\. The claims in the paper retain those limits\.

### A\.4Setup and measurement detail displaced from the main text

This section collects checks that support the experimental setup but are not needed to read the main results\. The material is grouped into search settings, record\-level robustness checks, theory diagnostics, and limitations\.

#### A\.4\.1Search settings

##### Elite fraction \(Section[5](https://arxiv.org/html/2610.41351#S5)\)\.

The elite fraction differs by evaluator:0\.1560\.156for QPP and Monte Carlo,0\.0310\.031for mean rollout,0\.0930\.093for moment matching\. At mean rollout’s fraction QPP’s tail changes by−0\.3824\-0\.3824on overload and−18\.9734\-18\.9734on inventory, under a tenth of either gap, so no B1 or C2 verdict depends on this confound\.

##### Sampler colour\.

β\\betais chosen per environment by a prospectively specified baseline\-favouring rule \(lowest realizedCVaR0\.1\\operatorname\{CVaR\}\_\{0\.1\}of that sampler’s Monte Carlo cell over\{1,2,3\}\\\{1,2,3\\\}\):11for both samplers on overload;22\(iCEM\-style\) and11\(dsCEM\-style\) on inventory\.

#### A\.4\.2Numerical validation

An earlier evaluator included a pool\-adjacent\-violators projection that could alter negatively loaded inventory trajectories\. All affected inventory results reported in this paper were recomputed without this projection\. For the controlled overload and synthetic studies, recomputation reproduced the reported outputs exactly\. In the remaining overload and surrogate studies, the measured projection displacement is zero on the evaluated trajectories, so removing the operation leaves the reported outputs unchanged\. Repeating Gate 1b without the projection gives the same error,0\.00200\.0020\.

The displayed overload table uses a10,00010\{,\}000\-replicate bootstrap interval and gives a B1 lower bound of−0\.8275\-0\.8275\. An additional sensitivity analysis with200,000200\{,\}000replicates gives a B1 overload bound of−0\.7739\-0\.7739\(SE0\.01060\.0106\), a B1 inventory bound of0\.22370\.2237\(SE0\.00050\.0005\), and G2 of0\.01240\.0124\(SE0\.00010\.0001\)\. No criterion outcome changes\. Because these analyses use different resampling counts, their numerical estimates are reported separately\.

#### A\.4\.3Theory diagnostics

##### The spread law, measured \(Section[4](https://arxiv.org/html/2610.41351#S4)\)\.

On the Gaussian monotone\-diagonal map the measured gap between propagated and independent\-process quantiles is0\.00480\.0048atH=1H=1\(where the couplings coincide and the value is the reference’s own error\), then0\.67370\.6737,1\.05971\.0597,1\.46941\.4694atH=3,5,10H=3,5,10; theH=10H=10value lies inside\[R⁡\(10,0\.5\)−1,R⁡\(10,0\.9\)−1\]=\[0\.7304,2\.0291\]\[R\(10,0\.5\)\-1,R\(10,0\.9\)\-1\]=\[0\.7304,2\.0291\], and simulation matches \([3](https://arxiv.org/html/2610.41351#S4.E3)\) within2%2\\%ona∈\{0\.5,0\.7,0\.9\}a\\in\\\{0\.5,0\.7,0\.9\\\},H≤10H\\leq 10\.

##### Level\-curve interpolation error \(Section[4](https://arxiv.org/html/2610.41351#S4)\)\.

Relative error0\.13440\.1344/0\.11000\.1100/0\.09780\.0978atK=3K=3/55/99on the monotone map\.

Figure 7:One\-step cost quantiles under standardized lognormal noise\. The deployed five\-level QPP approximation is shown together with a dense\-grid reference; both retain positive skew and upper\-tail curvature, whereas Gaussian moment matching is symmetric by construction\. This diagnostic concerns distributional shape, not calibration or planning performance\.\\FloatBarrier

#### A\.4\.4Limitations and transfer checks

##### KPI simulator \(Section[7](https://arxiv.org/html/2610.41351#S7)\)\.

Skill0\.01220\.0122; PIT deviation0\.13780\.1378;3\.39%3\.39\\%of residuals out of support\.

##### Measurement limitations \(Section[7](https://arxiv.org/html/2610.41351#S7)\)\.

The five\-level grid has only the0\.750\.75–0\.950\.95segment and a constant extension above0\.950\.95to describe the upper20%20\\%planning tail\. Costs are charged on the pre\-step state, while violations are measured after transition, so the terminal state is unpriced and the last action can affect a violation without incurring the corresponding state cost\. Realized tail metrics mix optimizer and execution randomness across repeats\. Runtime and memory were not matched\.

##### Second\-dataset diagnosis \(Section[7](https://arxiv.org/html/2610.41351#S7)\)\.

The unchanged microgrid construction has113113eligible buildings on BDG2\([Miller et al\., 2020](https://arxiv.org/html/2610.41351#bib.bib22)\); sixteen span four campuses and four are held out\. D1 fails at both permitted caps \(median gap0\.06250\.0625–0\.0750\.075against0\.150\.15\), stopping the study before calibration or planning\. Two buildings have zero gap for opposite reasons: one anomaly scale makes overload dominate every reserve, while the other is too mild for reserve to repay degradation\. ECL’s cohort spans anomaly scale0\.11660\.1166–0\.79620\.7962; persistence is0\.94410\.9441here against0\.75250\.7525on ECL, and the coupled\-to\- executed comparison is approximately1\.03×1091\.03\\times 10^\{9\}versus45\.745\.7, a factor of2\.2×1072\.2\\times 10^\{7\}against ECL’s factor of about thirty\-four\. This indicates severe surrogate looseness under the present construction\. The two datasets suggest that persistence and anomaly scale may govern comonotonic over\-dispersion, but they do not establish a cutoff or a general population scope\.

### A\.5Proofs

##### The two order\-restriction operators\.

Pool\-adjacent violators is the Euclidean projection onto the monotone cone: it keeps each entry’s position and replaces violating runs by their means, and it is non\-expansive because it is the metric projection onto a closed convex cone\. Rearrangement sorts the values: it keeps the multiset and permutes it, and it is non\-expansive by the Hardy–Littlewood–Pólya rearrangement inequality, not by convexity\. They disagree in general \(\(3,1,2\)↦\(2,2,2\)\(3,1,2\)\\mapsto\(2,2,2\)versus\(1,2,3\)\(1,2,3\)\), so a lemma about one does not transfer to the other\.

###### Proof\.

of Proposition[1](https://arxiv.org/html/2610.41351#Thmtheorem1), Part \(a\)\.Induction ontt:z0\(k\)=z0=Z0​\(qk\)z\_\{0\}^\{\(k\)\}=z\_\{0\}=Z\_\{0\}\(q\_\{k\}\), and ifzt\(k\)=Zt​\(qk\)z\_\{t\}^\{\(k\)\}=Z\_\{t\}\(q\_\{k\}\)then \([1](https://arxiv.org/html/2610.41351#S3.E1)\) is the definition ofZt\+1​\(qk\)Z\_\{t\+1\}\(q\_\{k\}\)withF−1​\(U\)F^\{\-1\}\(U\)evaluated atU=qkU=q\_\{k\}\. HenceCt\(k\)=c⁡\(Zt​\(qk\),ut\)C\_\{t\}^\{\(k\)\}=c\(Z\_\{t\}\(q\_\{k\}\),u\_\{t\}\), and the level curve interpolatesu↦c⁡\(Zt​\(u\),ut\)u\\mapsto c\(Z\_\{t\}\(u\),u\_\{t\}\)at the knots\.

Part \(b\)\.On every compact subinterval, continuity and vanishing interior mesh give uniform interpolation convergence\. Endpoint knots tending to00and11, together with uniform integrability, then giveL1L^\{1\}convergence on\(0,1\)\(0,1\)\. Hence the mean converges, and\|CVaRα⁡\(X\)−CVaRα⁡\(Y\)\|≤α−1​𝔼​\|X−Y\|\|\\operatorname\{CVaR\}\_\{\\alpha\}\(X\)\-\\operatorname\{CVaR\}\_\{\\alpha\}\(Y\)\|\\leq\\alpha^\{\-1\}\\mathbb\{E\}\|X\-Y\|gives CVaR convergence under the common\-UUcoupling\.

Part \(c\)\.For the marginal\-quantile statement, ifZt,i​\(u\)Z\_\{t,i\}\(u\)is non\-decreasing inuu, then

Pr\[Zt,i\(U\)<Zt,i\(q\)\]≤qandPr\[Zt,i\(U\)≤Zt,i\(q\)\]≥q\.\\Pr\[Z\_\{t,i\}\(U\)<Z\_\{t,i\}\(q\)\]\\leq q\\quad\\text\{and\}\\quad\\Pr\[Z\_\{t,i\}\(U\)\\leq Z\_\{t,i\}\(q\)\]\\geq q\.ThusZt,i​\(q\)Z\_\{t,i\}\(q\)is a generalizedqq\-quantile \(with equality in the second relation when the map is strictly increasing\)\. Non\-decreasing coordinates propagate: if every coordinate ofZt​\(u\)Z\_\{t\}\(u\)is non\-decreasing inuu, thendd​u​Zt\+1​\(u\)=\(Jf\+F−1​\(u\)​Jσ\)​dd​u​Zt​\(u\)\+σ​dd​u​F−1​\(u\)\\frac\{d\}\{du\}Z\_\{t\+1\}\(u\)=\\big\(J\_\{f\}\+F^\{\-1\}\(u\)J\_\{\\sigma\}\\big\)\\,\\frac\{d\}\{du\}Z\_\{t\}\(u\)\+\\sigma\\,\\frac\{d\}\{du\}F^\{\-1\}\(u\)has non\-negative entries whenever the bracketed matrix andσ\\sigmado, sinceF−1F^\{\-1\}is non\-decreasing\. The base derivative isd​Z0/d​u=0dZ\_\{0\}/du=0, andd​Z1/d​u=σ⁡\(z0,u0\)​\(F−1\)′​\(u\)≥0dZ\_\{1\}/du=\\sigma\(z\_\{0\},u\_\{0\}\)\(F^\{\-1\}\)^\{\\prime\}\(u\)\\geq 0almost everywhere; local absolute continuity then preserves monotonicity\. The condition is required for everyu∈\(0,1\)u\\in\(0,1\)\(non\-negativity at theKKpropagated levels alone says nothing about the order between knots\); the diagnostic in Appendix[A\.4](https://arxiv.org/html/2610.41351#A1.SS4)checks only that theKKpropagated trajectories are ordered alongKK, a necessary condition\. ∎

###### Proof\.

of Remark[2](https://arxiv.org/html/2610.41351#Thmtheorem2)\.Fromz0=0z\_\{0\}=0, the coupled process gives

ZH=s​ε​∑t<Hat=s​ε​1−aH1−a,sd⁡\(ZH\)=s​1−aH1−a\.Z\_\{H\}=s\\,\\varepsilon\\sum\_\{t<H\}a^\{t\}=s\\,\\varepsilon\\frac\{1\-a^\{H\}\}\{1\-a\},\\qquad\\mathrm\{sd\}\(Z\_\{H\}\)=s\\frac\{1\-a^\{H\}\}\{1\-a\}\.For the independent process,

Z~H=s​∑t<Hat​εH−1−t,sd⁡\(Z~H\)=s​1−a2​H1−a2\.\\widetilde\{Z\}\_\{H\}=s\\sum\_\{t<H\}a^\{t\}\\varepsilon\_\{H\-1\-t\},\\qquad\\mathrm\{sd\}\(\\widetilde\{Z\}\_\{H\}\)=s\\sqrt\{\\frac\{1\-a^\{2H\}\}\{1\-a^\{2\}\}\}\.Divide the two expressions and use1−a2​H=\(1−aH\)​\(1\+aH\)1\-a^\{2H\}=\(1\-a^\{H\}\)\(1\+a^\{H\}\)and1−a2=\(1−a\)​\(1\+a\)1\-a^\{2\}=\(1\-a\)\(1\+a\)\. ∎

###### Proof\.

of Proposition[3](https://arxiv.org/html/2610.41351#Thmtheorem3)\.The argument has four steps: write the total cost as a function of the shock sequence, establish its monotonicity and supermodularity, apply the comonotonic upper bound, and then use the stop\-loss representation of CVaR\. Under the hypotheses each per\-step cost isgt​\(vt⊤​Zt\+βt\)g\_\{t\}\(v\_\{t\}^\{\\top\}Z\_\{t\}\+\\beta\_\{t\}\)withvt≥0v\_\{t\}\\geq 0andgtg\_\{t\}convex non\-decreasing, andZtZ\_\{t\}is a non\-negative affine image of the shocks, so the summed cost isΦ\(ε1,…,εH\)=∑tgt\(wt⊤ε1:t\+bt\)\\Phi\(\\varepsilon\_\{1\},\\dots,\\varepsilon\_\{H\}\)=\\sum\_\{t\}g\_\{t\}\\big\(w\_\{t\}^\{\\top\}\\varepsilon\_\{1:t\}\+b\_\{t\}\\big\)withwt≥0w\_\{t\}\\geq 0\. \(A general convex coordinatewise non\-decreasing cost need not be supermodular, e\.g\.max⁡\(x1,x2\)\\max\(x\_\{1\},x\_\{2\}\); the ridge form is what the proof uses\.\) Each summand is non\-decreasing and supermodular inε\\varepsilon\(for twice\-differentiablegtg\_\{t\},∂2/∂εi∂εj=gt′′wt,iwt,j≥0\\partial^\{2\}/\\partial\\varepsilon\_\{i\}\\partial\\varepsilon\_\{j\}=g\_\{t\}^\{\\prime\\prime\}w\_\{t,i\}w\_\{t,j\}\\geq 0; the general case follows by approximation\), and sums preserve both properties\. For any non\-decreasing convexψ\\psi,ψ∘Φ\\psi\\circ\\Phiis again non\-decreasing and supermodular \(the cross derivative isψ′′​∂iΦ​∂jΦ\+ψ′​∂i​jΦ≥0\\psi^\{\\prime\\prime\}\\,\\partial\_\{i\}\\Phi\\,\\partial\_\{j\}\\Phi\+\\psi^\{\\prime\}\\,\\partial\_\{ij\}\\Phi\\geq 0\)\. The comonotonic coupling maximizes𝔼⁡\[h⁡\(ε\)\]\\mathbb\{E\}\[h\(\\varepsilon\)\]over all couplings with the same marginals for every supermodularhh\(the comonotonic upper bound,[Kaas et al\., 2000](https://arxiv.org/html/2610.41351#bib.bib14)\), so𝔼⁡\[ψ⁡\(Φcom\)\]≥𝔼⁡\[ψ⁡\(Φind\)\]\\mathbb\{E\}\[\\psi\(\\Phi^\{\\mathrm\{com\}\}\)\]\\geq\\mathbb\{E\}\[\\psi\(\\Phi^\{\\mathrm\{ind\}\}\)\]for every non\-decreasing convexψ\\psi, i\.e\.Φcom\\Phi^\{\\mathrm\{com\}\}dominatesΦind\\Phi^\{\\mathrm\{ind\}\}in the increasing convex order\. For everyvv, the stop\-loss functionx↦\(x−v\)\+x\\mapsto\(x\-v\)\_\{\+\}is increasing and convex, so

𝔼⁡\[\(Φind−v\)\+\]≤𝔼⁡\[\(Φcom−v\)\+\]\.\\mathbb\{E\}\[\(\\Phi^\{\\mathrm\{ind\}\}\-v\)\_\{\+\}\]\\leq\\mathbb\{E\}\[\(\\Phi^\{\\mathrm\{com\}\}\-v\)\_\{\+\}\]\.Applying this inequality inCVaRα⁡\(X\)=infv\{v\+α−1​𝔼​\[\(X−v\)\+\]\}\\operatorname\{CVaR\}\_\{\\alpha\}\(X\)=\\inf\_\{v\}\\\{v\+\\alpha^\{\-1\}\\mathbb\{E\}\[\(X\-v\)\_\{\+\}\]\\\}and then taking the infimum overvvgives the stated CVaR ordering\. Hereα\\alphais the upper\-tail mass, matching Section[3](https://arxiv.org/html/2610.41351#S3)\. The counterexample of Section[4](https://arxiv.org/html/2610.41351#S4)violates the hypotheses twice \(a rotation mixes signs inwtw\_\{t\}; a concave cost breaks supermodularity\) and reverses the inequality\. ∎

### A\.6Main\-text tables in full, and tables moved for space

The tables in this section expand results summarized in the main text\. They are ordered as comparisons with mean rollout, the budget\-matched Monte Carlo comparison, within\-sampler composition results, theλ\\lambdasweep, and the interaction contrasts\. Section[A\.7](https://arxiv.org/html/2610.41351#A1.SS7)begins only after this complete\-table block\. Tables write a two\-sided interval asm−a\+bm\_\{\-a\}^\{\+b\}, wheremmis the estimate anda,ba,bare the lower and upper95%95\\%offsets\. One\-sided intervals use∞\\inftyon the open side; a lone superscript gives an upper offset\.

Table 5:Overload environment, every row of the comparison against mean rollout; Section[6\.1](https://arxiv.org/html/2610.41351#S6.SS1)cites theλ=0\.5\\lambda=0\.5row\. One Holm family\.
Table 6:Inventory environment, every row of the comparison against mean rollout; Section[6\.1](https://arxiv.org/html/2610.41351#S6.SS1)cites theλ=0\.5\\lambda=0\.5row\. One Holm family\.
Table 7:Overload environment: every planner againstS=5S=5Monte Carlo at matchedNevalN\_\{\\mathrm\{eval\}\}, two\-sided\. One Holm family\.
Table 8:Composition study, overload: within each sampler row, QPP againstS=5S=5Monte Carlo and against mean rollout on absoluteCVaR0\.1\\operatorname\{CVaR\}\_\{0\.1\}\(one\-sided lower alternative\)\. Rows: Gaussian, iCEM\-style, dsCEM\-style, and the four Gaussian loop controls\. One Holm family\.
Table 9:Composition study, inventory: the same within\-row comparisons on the relativeCVaR0\.1\\operatorname\{CVaR\}\_\{0\.1\}gain \(one\-sided upper alternative\)\. One Holm family\.
Table 10:Synthetic environment: realized mean andCVaR0\.1\\operatorname\{CVaR\}\_\{0\.1\}of QPP atλ∈\{0\.25,0\.5,0\.75,1\}\\lambda\\in\\\{0\.25,0\.5,0\.75,1\\\}againstλ=0\\lambda=0, two\-sided,n=100n=100\. One Holm family\.
Table 11:Composition study, overload: each row’s QPP\-minus\-Monte\-CarloCVaR0\.1\\operatorname\{CVaR\}\_\{0\.1\}gap minus the Gaussian row’s gap \(two\-sided; zero means the sampler or loop leaves QPP’s advantage unchanged\)\. Main\-text rows of the prospectively specified family; QPP\-minus\-mean rows in the appendix\.
Table 12:Composition study, inventory: interaction contrasts as in Table[11](https://arxiv.org/html/2610.41351#A1.T11)\.
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### A\.7Composition study: appendix tables

Across the four prospectively gated sampler rows in each of two environments, QPP is below mean rollout on tail cost in8/88/8comparisons \(C2\) and is not below Monte Carlo in8/88/8\(C1\)\. C3 is satisfied or inconclusive on overload and not satisfied or inconclusive on inventory\. On both closed\-form environments,λ=0\\lambda=0has lower realized mean andCVaR0\.1\\operatorname\{CVaR\}\_\{0\.1\}point estimates thanλ=0\.5\\lambda=0\.5; because no direct paired family was specified, the evidence supports the evaluator but not a nonzero\-λ\\lambdabenefit\. On the symmetric control,λ\\lambdamoves the decision only above0\.50\.5and by little \(Table[10](https://arxiv.org/html/2610.41351#A1.T10)\)\.

##### Candidates versus loop\.

The sampler\-only contrasts \(each strong row minus its own loop control, so that only the candidate distribution differs; Tables[11](https://arxiv.org/html/2610.41351#A1.T11)and[12](https://arxiv.org/html/2610.41351#A1.T12)\) are null on inventory for both samplers \(9\.599\.59and3\.853\.85, intervals covering zero\) and on overload for dsCEM\-style; the coloured iCEM\-style candidates move the overload comparison in QPP’s favour \(−1\.02\-1\.02,padj=0\.0056p\_\{\\mathrm\{adj\}\}=0\.0056\)\. The iCEM loop moves the comparison in Monte Carlo’s favour on both environments \(Gaussian control with the iCEM loop:28\.428\.4on inventory,padj=0\.0048p\_\{\\mathrm\{adj\}\}=0\.0048;0\.7640\.764on overload\), the dsCEM\-style loop on neither\. Table[14](https://arxiv.org/html/2610.41351#A1.T14)attributes the loop effect to the decaying population schedule on both environments: the schedule\-only control reproduces the iCEM\-loop interaction \(28\.628\.6and1\.081\.08, both Holm\-significant; iCEM loop minus schedule alone null on both\), carryover alone \(in its budget\-neutral, re\-scored form\) does not \(null on both; iCEM loop minus carryover25\.925\.9and0\.4680\.468, both significant\), and the additivity residual is not rejected under the headline decision rule\. This is a measured attribution, not a mechanism proof: under the best\-of\-final\-elites rule the direct loop\-minus\-carryover contrasts lose significance and the overload additivity residual is rejected \(0\.7990\.799,padj=0\.0174p\_\{\\mathrm\{adj\}\}=0\.0174\); only the schedule\-only interaction rows keep theirs\. \(The recorded elite fraction under the schedule rises from0\.070\.07to0\.50\.5across iterations, the candidate explanation\.\)

##### Per environment\.

On overload the strong samplers reproduce the Gaussian tail gap within the equivalence margin \(C3 satisfied\); the C1 gates are not satisfied, the iCEM\-style one by a deciding bound of0\.00760\.0076against00with standard error0\.00710\.0071\(a boundary outcome read in neither direction\), and QPP pays1\.801\.80to1\.991\.99of realized mean against both baselines in every row \(Table[13](https://arxiv.org/html/2610.41351#A1.T13)\): the overload tail claim and this penalty are one statement\. On inventory there is no QPP\-versus\-Monte\-Carlo win to test \(C1 not satisfied in all four gated rows; mean penalty98\.498\.4to113113against Monte Carlo in every row\), the iCEM\-style loop widens the gap further \(interaction37\.9537\.95\), and C3 is not satisfied for the two iCEM\-loop rows and inconclusive for the two dsCEM\-style rows\.

Table 13:Composition study, overload: within each sampler row, QPP againstS=5S=5Monte Carlo and against mean rollout on the realized mean \(two\-sided\)\. QPP pays a mean penalty in every row; this table is the companion of Table[8](https://arxiv.org/html/2610.41351#A1.T8)\. One Holm family\.

Table 14:Loop decomposition on the QPP\-minus\-Monte\-Carlo interaction scale \(overload above, inventory below\): the iCEM loop control minus the carryover\-only control, minus the schedule\-only control, and the additivity residual \(carryover plus schedule minus the iCEM loop, each referenced to the Gaussian row\)\. Exploratory, one Holm family per environment\.
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### A\.8Exploratory two\-by\-two and CityLearn surrogate tables

This section collects diagnostics that are intentionally outside the frozen primary gate families\. The first two tables vary the candidate sampler and evaluator in a controlled2×22\\times 2design; the third records the simulator\- surrogate negative result\.

Table 15:Exploratory2×22\\times 2study on overload: quasi\-Monte\-Carlo against Gaussian candidates crossed with QPP againstS=5S=5Monte Carlo, interaction contrasts and comparisons against the Gaussian Monte Carlo cell\. The overload C3 margin of the composition study is half this table’s Gaussian QPP\-minus\-Monte\-CarloCVaR0\.1\\operatorname\{CVaR\}\_\{0\.1\}gap\.

Table 16:Exploratory2×22\\times 2study on inventory: the same layout as Table[15](https://arxiv.org/html/2610.41351#A1.T15)\. The inventory C3 margin of the composition study is half this table’s Gaussian QPP\-minus\-Monte\-CarloCVaR0\.1\\operatorname\{CVaR\}\_\{0\.1\}gap\.

Table 17:CityLearn surrogate \(plan on a fitted diagonal\-monotone linear quantile surrogate, execute on the real environment\): QPP atλ=0\.5\\lambda=0\.5, mean rollout andS=5S=5Monte Carlo at matchedNevalN\_\{\\mathrm\{eval\}\},n=200n=200\. One Holm family over both blocks\.
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### A\.9Microgrid: H1 in full, and versus budget\-matched sampling

This section keeps three comparisons separate: QPP against mean rollout, the exploratory operational and multiplicity audits, and QPP against budget\-matched Monte Carlo\. Each block is completed before the next begins\.

#### A\.9\.1Comparison with mean rollout

##### Against a constant\-reserve oracle\.

QPP over\-reserves against the CVaR\-optimal constant reserve by about a factor of two \(planned reserve0\.670\.67–0\.780\.78across the four sites against0\.2750\.275–0\.3750\.375\), the coupling’s conservatism atH=24H=24, yet its realized CVaR beats the best member of that one\-parameter family \(6\.8826\.882against7\.53617\.5361at site242242\) because the free planner shapes the whole2424\-step profile\. The oracle is a restricted\-family benchmark, not an upper bound, and its200,000200\{,\}000\-sample estimate is not the same estimator as a200200\-repeat realized CVaR\.

Table 18:Microgrid H1, every row of the comparison against mean rollout, evaluation site242242\(sites253253,255255,279279follow\): everyλ\\lambdavalue and moment matching, alongside theλ=0\.5\\lambda=0\.5row cited in Section[6\.3](https://arxiv.org/html/2610.41351#S6.SS3)\. One Holm family per site\.
Table 19:Microgrid H1, evaluation site253253\. One Holm family per site\.
Table 20:Microgrid H1, evaluation site255255\. One Holm family per site\.
Table 21:Microgrid H1, evaluation site279279\. One Holm family per site\.
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#### A\.9\.2Operational endpoint and multiplicity audit

##### Actual import on the recorded execution stream\.

The following summary reports episode\-level actual\-import exceedance percentages with exact binomial95%95\\%intervals for all six planners\. The final comparison reports QPP atλ=0\.5\\lambda=0\.5minus mean rollout in percentage points together with its one\-sided95%95\\%upper bound\. The correspondingpp\-values are adjusted by Holm’s procedure across the four sitewise contrasts\. The endpoint remains exploratory, but its replay now uses the identical execution seed sequence as the recorded cost andV​1V1observations\.

Table 22:Episode\-level actual\-import exceedance rates on the recorded execution stream\. Intervals are exact binomial95%95\\%intervals; the final columns compare QPP atλ=0\.5\\lambda=0\.5with mean rollout\. This endpoint is exploratory\.

##### Cross\-site B1 multiplicity audit\.

The frozen gate remains the four unadjusted sitewise lower bounds\. The table below is a post\-hoc robustness audit of the same paired arrays\. It adds a null\-centered one\-sided test against the actual0\.100\.10bar, applies Holm’s procedure across sites, and reports the conservative98\.75%98\.75\\%lower bound that gives Bonferroni simultaneous95%95\\%coverage\.

Table 23:Post\-hoc cross\-site audit of the B1 relativeCVaR0\.1\\operatorname\{CVaR\}\_\{0\.1\}gain\. The95%95\\%column reproduces the sitewise gate scale; the98\.75%98\.75\\%column and Holmppvalues address the four\-site family in two complementary ways\.\\FloatBarrier

#### A\.9\.3Comparison with budget\-matched Monte Carlo

The following four tables compare every planner withS=5S=5Monte Carlo at matched dynamics\-evaluation cost\. They use one Holm family per site and are kept in site order\.

Table 24:Microgrid H1, every planner againstS=5S=5Monte Carlo at matchedNevalN\_\{\\mathrm\{eval\}\}, two\-sided, evaluation site242242\(sites253253,255255,279279follow\)\. One Holm family per site\.
Table 25:Microgrid H1 versusS=5S=5Monte Carlo, evaluation site253253\. One Holm family per site\.
Table 26:Microgrid H1 versusS=5S=5Monte Carlo, evaluation site255255\. One Holm family per site\.
Table 27:Microgrid H1 versusS=5S=5Monte Carlo, evaluation site279279\. One Holm family per site\.
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### A\.10Exploratory objective and coupling sensitivity

##### Primary and exploratory analyses\.

The primary endpoints, evaluation sites, and decision thresholds were specified before conducting the sensitivity analyses reported in this section\. The exact\-stream evaluation, cross\-site multiplicity analysis, and objective and coupling sensitivity studies are therefore labeled exploratory\. These analyses do not alter the definitions or conclusions of the primary evaluations\.

#### A\.10\.1Objective sensitivity

The objective audit uses an independent seed block andn=1,000n=1\{,\}000CRN repeats per site, so each empiricalCVaR0\.1\\operatorname\{CVaR\}\_\{0\.1\}averages 100 tail outcomes\. All arms retainK=5K=5,P=64P=64,H=24H=24, ten CEM iterations and ten elites\. The objective audit changes one item at a time from QPP atα=0\.2\\alpha=0\.2: CVaR after summing the full trajectory, planning atα=0\.1\\alpha=0\.1, or an opt\-in terminal penalty equal to the running squared\-overload scale applied to terminal headroom shortfall\. These variants are evaluated separately from the primary objective\.

Table 28:Objective sensitivity relative to the original per\-step,α=0\.2\\alpha=0\.2QPP arm\. Entries are variant\-minus\-original paired differences with descriptive percentile\-bootstrap intervals; lower cost is better, and V1 differences are percentage points\. Total\-trajectory and per\-step scores select the same plans here because the shared\-rank cost paths remain ordered consistently across time\.

#### A\.10\.2Coupling and node sensitivity

The separaten=200n=200factorial arms use equal\-weight stratified scenarios and the same total\-trajectory objective\. Fixed versus randomized nodes isolates node randomization; shared versus time\-varying rank assignments isolates temporal coupling\. It is a mechanism diagnostic, not a claim that any arm is a better approximation of the independent execution process\.

Table 29:Matched2×22\\times 2coupling diagnostic relative to the original QPP arm\. Time\-varying ranks materially worsen the realized tail and V1 at every site; randomized shared ranks are also worse, while fixed shared ranks stay close\. This separates temporal coupling from determinism without changing the trajectory budget\.

### A\.11Microgrid: trajectory\-count sweep, IID and coupled

Figure[8](https://arxiv.org/html/2610.41351#A1.F8)gives the cross\-site pattern before the summary table\. With the search allocation fixed, all four IID comparisons meet the trajectory\-count rule because their one\-sided lower bounds remain above5%5\\%atS=200S=200\. The clipped coupled sampler does not meet its equivalence rule at any site\.

Figure 8:Microgrid trajectory\-count sweeps at all four held\-out sites\. Each panel compares IID Monte Carlo and the clipped coupled sampler withK=5K=5QPP asSSincreases\. Error bars are bootstrap95%95\\%intervals; the narrow blue region is the5%5\\%sampler\-denominated comparison band above QPP\. The CEM population, elite count and iteration count are fixed, so this is not a best\-allocation total\-budget frontier\.n=200n=200per point\.Table 30:Cross\-site decision summary atS=200S=200\. Positive gaps mean that the sampler has higher realizedCVaR0\.1\\operatorname\{CVaR\}\_\{0\.1\}than QPP\. IID entries pair the point estimate with the one\-sided95%95\\%lower bound used by B3; clipped\-coupled entries use the two\-sided interval required by the equivalence diagnostic\. All four B3 criteria are satisfied by the lower\-bound branch, while none of the four equivalence criteria is satisfied\.\\FloatBarrier

### A\.12Microgrid: quantile ensemble versus budget\-matched PETS

#### A\.12\.1Design and cross\-site summary

Replay windows come from each site’s final30%30\\%of rows\. A valid start is an hour\-of\-day\-zero row with at leastH\+1=25H\+1=25rows remaining, yielding328328eligible starts; each repeat draws one start uniformly with replacement and uses the same draw for every arm\. The cited200200repeats contain152152distinct starts\. Source admissibility is satisfied \(pooled PIT deviation0\.01130\.0113\)\. The design check that selects replay is also conditional on the eventual four sites \(median reserve gap0\.17500\.1750against0\.150\.15; site gaps0\.0250\.025/0\.2250\.225/0\.2000\.200/0\.1500\.150at242242/253253/255255/279279\)\. The quantile ensemble beats PETS on tail risk at three sites and onV​1V1at all four, but the single\-model2525\-level arm matches those margins, PETS is indistinguishable from plain Monte Carlo, and the epistemic\-premium gates are unsupported\. Accordingly the result supports deterministic quantile evaluation on replay, not ensemble allocation or epistemic\-risk claims\.

#### A\.12\.2Site\-level results

Table 31:Microgrid ensemble comparison at evaluation site242242\(sites253253,255255,279279follow\): five models with five quantile levels each, one reference model with2525levels, a2525\-model mean\-path ensemble and2525\-sample plain Monte Carlo, each against a budget\-matched PETS\-style ensemble\. Every arm uses2525evaluations per candidate and step and is executed by held\-out trace replay\. One Holm family per site\.
Table 32:Microgrid, quantile ensemble versus budget\-matched PETS, evaluation site253253\. One Holm family per site\.
Table 33:Microgrid, quantile ensemble versus budget\-matched PETS, evaluation site255255\. One Holm family per site\.
Table 34:Microgrid, quantile ensemble versus budget\-matched PETS, evaluation site279279\. One Holm family per site\.