$$
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\newcommand{\BumplessTransferProgress}{\hat{t}}
\newcommand{\BumplessTransferStartInstant}{t_{\text{start}}}
\newcommand{\BurstArrivalCount}{N_{\text{burst}}}
\newcommand{\BurstSurvivalTime}{t_{\text{limit}}}
\newcommand{\BusBitRate}{R_{\text{bit}}}
\newcommand{\BusCapacitorEnergy}{E_C}
\newcommand{\BusCapacitorVoltage}{V_C}
\newcommand{\BusSerializationTime}{t_{\text{bus}}}
\newcommand{\ButterflyAllReduceTime}{T_{\text{butterfly}}}
\newcommand{\ByteAlphabet}{\Sigma}
\newcommand{\ByteString}{u}
\newcommand{\ByzantineClientCount}{f}
\newcommand{\ByzantineFaultCount}{f}
\newcommand{\CableFlexCycleCount}{n_{\text{cable}}}
\newcommand{\CableRatedFatigueLife}{L_{\text{cable}}}
\newcommand{\CacheMissStallFraction}{f}
\newcommand{\CachedInputTokenCount}{n_{\text{cache}}}
\newcommand{\CachedInputTokenPrice}{c_{\text{cache}}}
\newcommand{\CalibratedAcceptanceEstimate}{\hat{p}}
\newcommand{\CalibrationBinCount}{N_{\text{bins}}}
\newcommand{\CalibrationBinExampleSet}{B_m}
\newcommand{\CalibrationBinIndex}{m}
\newcommand{\CalibrationConstraintMatrix}{\mathbf{V}}
\newcommand{\CalibrationDataDistribution}{p_{\text{cal}}}
\newcommand{\CalibrationDataset}{\mathcal{C}_{\text{cal}}}
\newcommand{\CalibrationExampleCount}{n_{\text{cal}}}
\newcommand{\CalibrationModelPoint}{\mathbf{M}}
\newcommand{\CalibrationPoseIndex}{k}
\newcommand{\CalibrationRotationAxisCount}{N}
\newcommand{\CalibrationSceneIndex}{j}
\newcommand{\CalibrationTemperature}{T_{\text{cal}}}
\newcommand{\CalibrationViewCount}{N}
\newcommand{\CallLatency}{T_{\text{call}}}
\newcommand{\CallPostprocessingTime}{T_{\text{post}}}
\newcommand{\CallPreparationTime}{T_{\text{prep}}}
\newcommand{\CallTailProbability}{p_{\text{tail}}}
\newcommand{\CallUsefulWorkTime}{W}
\newcommand{\CameraCenterArcPosition}{x_{\text{rot}}}
\newcommand{\CameraExtrinsicRotation}{\mathbf{R}_{bc}}
\newcommand{\CameraExtrinsicTransform}{\mathbf{T}_{bc}}
\newcommand{\CameraExtrinsicTranslation}{\mathbf{p}_{bc}}
\newcommand{\CameraFrameDepth}{Z_c}
\newcommand{\CameraFrameHorizontalCoordinate}{X_c}
\newcommand{\CameraFramePeriod}{T_{\text{sample}}}
\newcommand{\CameraFramePoint}{\mathbf{p}_c}
\newcommand{\CameraFramePointCovariance}{\boldsymbol{\Sigma}_c}
\newcommand{\CameraFrameVerticalCoordinate}{Y_c}
\newcommand{\CameraIdentifierEmbedding}{\mathbf{e}_{\text{cam}}}
\newcommand{\CameraImage}{\mathbf{I}}
\newcommand{\CameraImuTimeOffset}{\Delta t_{\text{imu}}}
\newcommand{\CameraIntrinsicCalibrationMatrix}{\mathbf{K}}
\newcommand{\CameraIntrinsicInverseTranspose}{\mathbf{K}^{-\top}}
\newcommand{\CameraLateralStandardDeviation}{\sigma_{x_c}}
\newcommand{\CameraMountTiltAngle}{\theta_{\text{ext}}}
\newcommand{\CameraObjectTransform}{\mathbf{T}_{\text{camera}}^{\text{object}}}
\newcommand{\CameraOpticalCenter}{\mathbf{o}}
\newcommand{\CameraReferenceFrame}{\mathcal{F}_{\text{cam}}}
\newcommand{\CameraRotationColumn}{\mathbf{r}}
\newcommand{\CameraSampleTimestamp}{t_{\text{cam}}}
\newcommand{\CameraSkewCoefficient}{\gamma}
\newcommand{\CameraTargetTransform}{\mathbf{T}_{\text{cam}}^{\text{target}}}
\newcommand{\CameraToWorldRotation}{\mathbf{R}_{WC}}
\newcommand{\CameraToWorldTransform}{\mathbf{T}_{WC}}
\newcommand{\CameraToWorldTranslation}{\mathbf{t}_{WC}}
\newcommand{\CampaignActiveTime}{T_{\text{active}}}
\newcommand{\CanHighLineVoltage}{V_{\text{CANH}}}
\newcommand{\CanLowLineVoltage}{V_{\text{CANL}}}
\newcommand{\CanaryAllocationFraction}{\rho}
\newcommand{\CanaryInterventionRate}{\lambda_{\text{canary}}}
\newcommand{\CanaryLabelDelay}{t_{\text{label}}}
\newcommand{\CanarySpeedBound}{v_{\text{canary}}}
\newcommand{\CanaryStageDuration}{T_{\text{stage}}}
\newcommand{\CanaryTrafficFraction}{p_{\text{stage}}}
\newcommand{\CandidateAcceptanceEstimate}{\hat{p}_{\text{cand}}}
\newcommand{\CandidateActionVector}{\hat{\mathbf{a}}}
\newcommand{\CandidateArchitecture}{a}
\newcommand{\CandidateAvailabilityEstimate}{\widehat{q}}
\newcommand{\CandidateAvailabilityProbability}{q}
\newcommand{\CandidateCheckingCost}{v}
\newcommand{\CandidateClientCount}{K_{\text{candidates}}}
\newcommand{\CandidateControlCommand}{\mathbf{u}^{\text{cand}}}
\newcommand{\CandidateControlCommandDerivative}{\dot{\mathbf{u}}^{\text{cand}}}
\newcommand{\CandidateDefectRate}{\epsilon}
\newcommand{\CandidateGenerationCost}{g}
\newcommand{\CandidateIndex}{i}
\newcommand{\CandidateIntervalLowerLimit}{l_{\text{cand}}}
\newcommand{\CandidateIntervalUpperLimit}{u_{\text{cand}}}
\newcommand{\CandidateLowerLimitDistance}{\delta^-_{\text{cand}}}
\newcommand{\CandidateOnlyAcceptedTaskCount}{c}
\newcommand{\CandidatePolicyAcceptanceProbability}{p_{Y}}
\newcommand{\CandidatePolicyAction}{\mathbf{a}_t^{\text{cand}}}
\newcommand{\CandidatePolicyCost}{C_{Y}}
\newcommand{\CandidatePoolSize}{N}
\newcommand{\CandidateProposal}{p}
\newcommand{\CandidateRoutingPolicy}{Y}
\newcommand{\CandidateScoreVector}{\mathbf{c}}
\newcommand{\CandidateScoringCost}{c_{\text{infer}}}
\newcommand{\CandidateScoringError}{\epsilon}
\newcommand{\CandidateSkill}{a}
\newcommand{\CandidateSpeedCommand}{v_{\text{cand}}}
\newcommand{\CandidateTailCompletionTime}{\widehat{T}_{c,0.99}}
\newcommand{\CandidateToolCallDistribution}{p_{c}}
\newcommand{\CandidateTrackingErrorStandardDeviation}{\sigma_{\text{track}}}
\newcommand{\CandidateTrajectoryGoodput}{\mathcal{G}_{c}}
\newcommand{\CandidateUpperLimitDistance}{\delta^+_{\text{cand}}}
\newcommand{\CandidateUtility}{U}
\newcommand{\CandidateUtilityMean}{\mu}
\newcommand{\CandidateUtilityStandardDeviation}{\sigma_{U}}
\newcommand{\CandidateViolationCount}{V_{c}}
\newcommand{\CandidateWinCount}{j}
\newcommand{\CapabilityGrant}{\Gamma}
\newcommand{\CaptureAgeEmbedding}{\mathbf{e}_{\Delta t}}
\newcommand{\CaptureSampleIndex}{t_0}
\newcommand{\CaptureSampleKalmanGain}{\mathbf{G}_{t_0}}
\newcommand{\CaptureSampleMeasurementMatrix}{\mathbf{H}_{t_0}}
\newcommand{\CaptureSamplePredictedCovariance}{\mathbf{P}_{t_0|t_0-1}}
\newcommand{\CaptureSamplePredictedEstimate}{\hat{\mathbf{x}}_{t_0|t_0-1}}
\newcommand{\CaptureSampleUpdatedCovariance}{\mathbf{P}_{t_0|t_0}}
\newcommand{\CaptureSampleUpdatedEstimate}{\hat{\mathbf{x}}_{t_0|t_0}}
\newcommand{\CaptureTimestamp}{t_{\text{capture}}}
\newcommand{\CapturedSensorObservation}{\mathbf{z}_{t_0}}
\newcommand{\CartesianAcceleration}{\ddot{\mathbf{p}}}
\newcommand{\CartesianPosition}{\mathbf{p}}
\newcommand{\CartesianXCoordinate}{x}
\newcommand{\CartesianYCoordinate}{y}
\newcommand{\CartesianZCoordinate}{z}
\newcommand{\CascadeEscalationProbability}{p_{\text{escalate}}}
\newcommand{\CascadeGateCost}{g}
\newcommand{\CascadeGateDuration}{v}
\newcommand{\CascadeInferenceEnergy}{E_{\text{cascade}}}
\newcommand{\CascadeTaskCost}{C_{\text{cascade}}}
\newcommand{\CascadeTaskDuration}{T_{\text{cascade}}}
\newcommand{\CascadeTierCost}{c^{\text{tier}}}
\newcommand{\CascadeTierDuration}{t}
\newcommand{\CascadeTierIndex}{j}
\newcommand{\CascadedBarrierCertificate}{\psi}
\newcommand{\CascadedBarrierClassKFunction}{\alpha}
\newcommand{\CascadedBarrierDerivative}{\dot{\psi}}
\newcommand{\CaseAmbientThermalResistance}{R_{\text{th,case-amb}}}
\newcommand{\CaseTemperature}{T_{\text{case}}}
\newcommand{\CaseThermalCapacitance}{C_{\text{case}}}
\newcommand{\CaseThermalTimeConstant}{\tau_{\text{th,case}}}
\newcommand{\CatalogDefinitionTokens}{S_{\text{catalog}}}
\newcommand{\CatalogStagingCallCount}{N}
\newcommand{\CategoryRecordCount}{n}
\newcommand{\CausalInterventionMask}{\mathcal{M}}
\newcommand{\CausalMediator}{M}
\newcommand{\CausalTokenPosition}{s}
\newcommand{\CbsRuntimeBudget}{Q}
\newcommand{\CellOccupancyProbability}{p}
\newcommand{\CenterCount}{k}
\newcommand{\CentralizedTrainingEnergy}{E_{\text{centralized}}}
\newcommand{\CentroidalAngularMomentum}{\mathbf{h}_{\text{ang}}}
\newcommand{\CentroidalAngularMomentumDerivative}{\dot{\mathbf{h}}_{\text{ang}}}
\newcommand{\CentroidalLinearMomentum}{\mathbf{h}_{\text{lin}}}
\newcommand{\CentroidalLinearMomentumDerivative}{\dot{\mathbf{h}}_{\text{lin}}}
\newcommand{\CentroidalMomentum}{\mathbf{h}_G}
\newcommand{\CentroidalMomentumMatrix}{\mathbf{A}_G}
\newcommand{\CentroidalMomentumMatrixDerivative}{\dot{\mathbf{A}}_G}
\newcommand{\CentroidalMomentumRate}{\dot{\mathbf{h}}_G}
\newcommand{\CertifiedCommandWindow}{\mathbf{u}^*_{t:t+H-1}}
\newcommand{\CertifiedControlAction}{\mathbf{u}^*}
\newcommand{\CertifiedJointTorque}{u^*}
\newcommand{\CertifiedRadius}{R}
\newcommand{\ChainRuleInput}{x}
\newcommand{\ChainRuleIntermediate}{u}
\newcommand{\ChainRuleOutput}{v}
\newcommand{\ChainedOperationCount}{n_{\text{ops}}}
\newcommand{\ChanceAnnotatorAgreement}{p_e}
\newcommand{\ChannelAuthorityState}{S}
\newcommand{\ChannelCommand}{u}
\newcommand{\ChannelStoredEnergy}{E_{\text{channel}}}
\newcommand{\CharacteristicPolynomial}{p}
\newcommand{\CharacteristicPolynomialCoefficient}{c}
\newcommand{\ChassisThermalResistance}{R_{\text{th,chassis}}}
\newcommand{\CheapConfigurationAttemptCost}{c_{\text{small}}}
\newcommand{\CheapConfigurationPassProbability}{p_{\text{small}}}
\newcommand{\CheckIntervalStepCount}{m}
\newcommand{\CheckPassProbability}{p^{\text{chk}}}
\newcommand{\CheckSlotTime}{T}
\newcommand{\CheckWindowTurnCount}{w}
\newcommand{\CheckedSignal}{x}
\newcommand{\CheckedSignalLimit}{L_{\text{limit}}}
\newcommand{\CheckpointAttemptDuration}{T_{\text{att}}}
\newcommand{\CheckpointExtraPauseTime}{T_{\text{pause}}}
\newcommand{\CheckpointInterval}{T_{\text{ckpt}}}
\newcommand{\CheckpointLoadTime}{T_{\text{load}}}
\newcommand{\CheckpointLogPosition}{j_{\text{ckpt}}}
\newcommand{\CheckpointOrdinalIndex}{m}
\newcommand{\CheckpointOverheadFraction}{f_{\text{ckpt}}}
\newcommand{\CheckpointOverheadRatio}{w}
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\newcommand{\CheckpointRestoreTime}{T_{\text{restore}}}
\newcommand{\CheckpointStride}{k}
\newcommand{\CheckpointSuffixReadTime}{T_{\text{read}}}
\newcommand{\CheckpointWasteFraction}{W}
\newcommand{\CheckpointWriteTime}{T_{\text{write}}}
\newcommand{\ChildBatchSize}{n_{\text{batch}}}
\newcommand{\ChildBudgetAllocation}{C_{\max,C}}
\newcommand{\ChildCapabilityGrant}{\Gamma_{\text{child}}}
\newcommand{\ChildFreeBalance}{F_{C}}
\newcommand{\ChildPruningProbability}{p_{\text{prune}}}
\newcommand{\ChildReservedFunds}{R_{C}}
\newcommand{\ChildSettledCharges}{E_{C}}
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\newcommand{\ChirpDuration}{T_c}
\newcommand{\ChirpSlope}{S}
\newcommand{\ChopperClampVoltage}{V_{\text{clamp}}}
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\newcommand{\ChopperTurnOnVoltage}{V_{\text{clamp,on}}}
\newcommand{\ChunkActionDimension}{d_{\text{chunk}}}
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\newcommand{\ChunkImitationLoss}{\mathcal{L}_{\text{chunk}}}
\newcommand{\ChunkLease}{T_{\text{lease}}}
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\newcommand{\ChunkReactionLatency}{T_{\text{reaction}}}
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\newcommand{\ChunkRenewalRate}{f_{\text{replan}}}
\newcommand{\ChunkStepIndex}{i}
\newcommand{\ClampedControlCommand}{\mathbf{u}_{\text{clip}}}
\newcommand{\ClassCount}{K}
\newcommand{\ClassIndex}{j}
\newcommand{\ClassKFunction}{\kappa}
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\newcommand{\ClassProbabilityVector}{\mathbf{p}}
\newcommand{\ClassificationThreshold}{\gamma_{\text{cls}}}
\newcommand{\CleanLabelSampleCount}{D_{\text{clean}}}
\newcommand{\CleanTrainingDataset}{\mathcal{S}}
\newcommand{\CleanupGraphEdgeSet}{E}
\newcommand{\CleanupGraphVertexSet}{V}
\newcommand{\ClearDistance}{D_{\text{clear}}}
\newcommand{\ClearanceAcceleration}{\ddot{D}_{\text{clear}}}
\newcommand{\ClearanceAllowance}{C}
\newcommand{\ClearanceExhaustionTime}{t_{\text{impact}}}
\newcommand{\ClearanceInset}{C_{\text{inset}}}
\newcommand{\ClearanceQueryTime}{t_{\text{query}}}
\newcommand{\ClearanceRate}{\dot{D}_{\text{clear}}}
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\newcommand{\ClearanceTimeBudget}{T_{\text{total}}}
\newcommand{\ClfClassKFunction}{\alpha_V}
\newcommand{\ClfDescentControlSet}{\mathcal{U}_{\text{CLF}}}
\newcommand{\ClientGradientVector}{\mathbf{g}}
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\newcommand{\ClientLocalDataset}{\mathcal{D}}
\newcommand{\ClientLocalTrainingEnergy}{E_{\text{client}}}
\newcommand{\ClientMaskVector}{\mathbf{r}}
\newcommand{\ClientObjectiveWeight}{w}
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\newcommand{\ClippedSurrogateObjective}{\mathcal{M}}
\newcommand{\ClippingRangeLowerBound}{\alpha}
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\newcommand{\ClockConversionBound}{\epsilon_{\text{sync}}}
\newcommand{\ClockConversionIntercept}{b_{\text{clk}}}
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\newcommand{\ClockFrequency}{f}
\newcommand{\ClockOffset}{\theta_{\text{offset}}}
\newcommand{\ClockPeriod}{T_{\text{clk}}}
\newcommand{\ClockSynchronizationSkew}{\delta_{\text{skew}}}
\newcommand{\ClosedLoopTrajectoryDensity}{p_{\text{cl}}}
\newcommand{\ClosingSpeedBound}{v_{\text{close}}}
\newcommand{\CloudInferenceEnergy}{E_{\text{cloud}}}
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\newcommand{\CloudTrainingRunEnergy}{E_{\text{cloud,train}}}
\newcommand{\ClusterAvailability}{A_{\text{cluster}}}
\newcommand{\ClusterCount}{k}
\newcommand{\ClusterHeadroom}{N}
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\newcommand{\CoarseCellCount}{C}
\newcommand{\CodecDecodeTime}{T_{\text{decode}}}
\newcommand{\CodecEncodeTime}{T_{\text{encode}}}
\newcommand{\CodecOverheadTime}{T_{\text{overhead}}}
\newcommand{\CognitivePolicyAction}{\mathbf{a}}
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\newcommand{\ColdCopperLossPower}{P_0}
\newcommand{\ColdResetCycleTime}{T_{\text{cycle,cold}}}
\newcommand{\CollaborativeContactForce}{F_{\text{contact}}}
\newcommand{\CollectedPhotocharge}{Q}
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\newcommand{\CollectionStateVisitationDistribution}{p^{\pi_{\text{col}}}}
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\newcommand{\CollectiveInputTensor}{\mathbf{X}}
\newcommand{\CollectivePayloadSize}{M}
\newcommand{\CollectiveTensorSize}{M_{\text{coll}}}
\newcommand{\CollisionHullCount}{N_{\text{hulls}}}
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\newcommand{\ColumnModeComponentVector}{\mathbf{b}_s}
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\newcommand{\ColumnModeFactorSlice}{\mathbf{b}}
\newcommand{\ColumnParallelWeightMatrix}{\mathbf{A}}
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\newcommand{\CombinedPlanValidationAllowance}{t_{\text{validate}}}
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\newcommand{\CommandTransportTime}{T_{\text{bus}}}
\newcommand{\CommandedActuatorEffort}{u_{\text{cmd}}}
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\newcommand{\CommandedTrackedPosition}{r_{\text{cmd}}}
\newcommand{\CommandedTractiveForce}{F_{\text{cmd}}}
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\newcommand{\CommitmentInstant}{t_{\text{commit}}}
\newcommand{\CommitmentPathPosition}{q_{c}}
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\newcommand{\CommunicationComputeRatio}{\rho}
\newcommand{\CommunicationIntensity}{\text{CI}}
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\newcommand{\CommunicationTime}{T_{\text{comm}}}
\newcommand{\CommunicationTimeFraction}{f}
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\newcommand{\CompactionPaybackCallCount}{N_{\min}}
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\newcommand{\CompactionTriggerLength}{S_{\text{high}}}
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\newcommand{\ComparedPeriodAccuracy}{a}
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\newcommand{\CompilationBreakevenCount}{N_{\text{breakeven}}}
\newcommand{\CompilationTime}{T_{\text{compile}}}
\newcommand{\CompiledExecutionTime}{T_{\text{compiled}}}
\newcommand{\CompilerBackendCount}{N_{\text{backend}}}
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\newcommand{\CompliantElementStiffness}{k_{\text{spring}}}
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\newcommand{\ComponentReliability}{R_{\text{single}}}
\newcommand{\ComposedChainTransform}{\mathbf{T}_{AC}}
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\newcommand{\CompressedGradientUpdate}{\mathbf{v}}
\newcommand{\CompressedGradientUpdateElement}{v}
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\newcommand{\ComputationalDimensionCount}{N_{\text{comp}}}
\newcommand{\ComputationalGraph}{G}
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\newcommand{\ComputeConstraintMultiplier}{\lambda}
\newcommand{\ComputeDemandVolume}{Q}
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\newcommand{\ComputeOptimalTokenCount}{D_{\text{opt}}}
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\newcommand{\ConditionalDamageProbability}{p}
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\newcommand{\ConditionalFlowMap}{\psi}
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\newcommand{\ConfidenceCriticalQuantile}{z_{\alpha/2}}
\newcommand{\ConfidenceLevel}{C}
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\newcommand{\ConfirmationHitCount}{M}
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\newcommand{\ConfluentVandermondeMatrix}{\mathbf{M}}
\newcommand{\ConformalMiscoverageRate}{\epsilon}
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\newcommand{\ConformalScoreQuantile}{\widehat{q}_{1-\epsilon}}
\newcommand{\ConservativePenaltyWeight}{\alpha_{\text{CQL}}}
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\newcommand{\ConsolidationOverheadTime}{T_{\text{overhead}}}
\newcommand{\ConsolidationStride}{c}
\newcommand{\ConstantDivisorGrpoObjective}{\widehat{\mathcal{J}}_{\text{const}}}
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\newcommand{\ConstrainedParameterVector}{\mathbf{w}}
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\newcommand{\ConstraintLagrangeMultiplier}{\lambda}
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\newcommand{\ContactDirectionSpeed}{v_c}
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\newcommand{\ContactForceBarrierFunction}{h_{\text{force}}}
\newcommand{\ContactForceCap}{F_{\text{cap}}}
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\newcommand{\ContactGapVelocity}{\dot{\phi}}
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\newcommand{\ContactNormal}{\hat{\mathbf{n}}}
\newcommand{\ContactNormalDeflection}{x}
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\newcommand{\ContactNormalJacobianDerivative}{\dot{\mathbf{J}}_n}
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\newcommand{\ContactNormalVelocity}{v_n}
\newcommand{\ContactPatchRegion}{\mathcal{A}}
\newcommand{\ContactPointForce}{\mathbf{f}}
\newcommand{\ContactPointIndex}{i}
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\newcommand{\ContactPointJacobianDerivative}{\dot{\mathbf{J}}_{\text{c}}}
\newcommand{\ContactPointOffset}{\mathbf{r}}
\newcommand{\ContactPointPosition}{x}
\newcommand{\ContactPointTorque}{\boldsymbol{\tau}}
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\newcommand{\ContactResponseLatency}{t_{\text{resp}}}
\newcommand{\ContactStiffness}{k_{\text{contact}}}
\newcommand{\ContactTorqueCeiling}{\tau_{\max}}
\newcommand{\ContactTripwireForce}{F_{\text{trip}}}
\newcommand{\ContactWrench}{\mathbf{w}}
\newcommand{\ContactWrenchCeiling}{\mathbf{w}_{\max}}
\newcommand{\ContactWrenchEnvelope}{\mathbf{w}_{\text{contact}}}
\newcommand{\ContactWrenchTransform}{\mathbf{X}_k^*}
\newcommand{\ContactYieldDeadline}{t_{\text{yield}}}
\newcommand{\ContainerInnerRadius}{R}
\newcommand{\ContainmentForceCeiling}{F_{\max}}
\newcommand{\ContendingApplicationCount}{N}
\newcommand{\ContentionCoefficient}{\sigma_{\text{cont}}}
\newcommand{\ContextChunkTokenCount}{n}
\newcommand{\ContextHiddenRepresentation}{\mathbf{h}_{\text{context}}}
\newcommand{\ContextParallelDegree}{c}
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\newcommand{\ContextSaveTime}{t_{\text{context\_save}}}
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\newcommand{\ContinuationReturnStandardDeviation}{\sigma}
\newcommand{\ContinuityCorrectedCrossProduct}{A}
\newcommand{\ContinuousActionComponent}{a}
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\newcommand{\ContinuousActionVector}{\mathbf{a}}
\newcommand{\ContinuousCurrent}{I_{\text{cont}}}
\newcommand{\ContinuousFactorialFunction}{\Gamma}
\newcommand{\ContinuousMeasurementNoiseIntensity}{\mathbf{R}_{\text{cont}}}
\newcommand{\ContinuousMotionModel}{\mathbf{f}}
\newcommand{\ContinuousOutcome}{\mathbf{x}}
\newcommand{\ContinuousRandomVariable}{\mathbf{X}}
\newcommand{\ContinuousStateTransitionMatrix}{\boldsymbol{\Phi}}
\newcommand{\ContinuousTorque}{\tau_{\text{cont}}}
\newcommand{\ContractedBarrierFunction}{h_{\text{contracted}}}
\newcommand{\ControlAffineDriftField}{\mathbf{f}}
\newcommand{\ControlAffineInputField}{\mathbf{g}}
\newcommand{\ControlCommand}{\mathbf{u}}
\newcommand{\ControlCommandDerivative}{\dot{\mathbf{u}}}
\newcommand{\ControlCommandSpace}{\mathcal{U}}
\newcommand{\ControlConversionRate}{\hat{p}_A}
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\newcommand{\ControlInput}{\mathbf{u}}
\newcommand{\ControlInputDimension}{m}
\newcommand{\ControlInputLowerLimit}{\mathbf{u}_{\min}}
\newcommand{\ControlInputSet}{\mathcal{U}}
\newcommand{\ControlInputUpperLimit}{\mathbf{u}_{\max}}
\newcommand{\ControlInputWindow}{\mathbf{u}_{[t, t+\Delta t]}}
\newcommand{\ControlSampleIndex}{k}
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\newcommand{\ControlSetpointJerk}{\mathbf{j}}
\newcommand{\ControlSetpointPeriod}{\Delta t_{\text{ctrl}}}
\newcommand{\ControlStepIndex}{t}
\newcommand{\ControlTaskIndex}{i}
\newcommand{\ControllabilityMatrix}{\mathbf{C}_{\text{ctrb}}}
\newcommand{\ControlledStopCommand}{\mathbf{u}_{\text{stop}}}
\newcommand{\ControllerReactionDelay}{T_{\text{react}}}
\newcommand{\ConversationTurnIndex}{i}
\newcommand{\ConverterLockoutVoltage}{V_{\text{uvlo}}}
\newcommand{\ConveyorSpeed}{v_{\text{conv}}}
\newcommand{\ConveyorSpeedBound}{v_{\text{conv}}}
\newcommand{\ConvolutionActivation}{H}
\newcommand{\ConvolutionTimeInstant}{t'}
\newcommand{\CookedTokenKeyVector}{\mathbf{k}_{\text{cooked}}}
\newcommand{\CookedTokenQueryVector}{\mathbf{q}_{\text{cooked}}}
\newcommand{\CoolantHeatLoad}{P_{\text{thermal}}}
\newcommand{\CoolantMassFlowRate}{\dot{m}}
\newcommand{\CoolantTemperature}{T}
\newcommand{\CoolingEnergy}{E_{\text{cooling}}}
\newcommand{\CoordinateFrameEmbedding}{\mathbf{e}_{\text{frame}}}
\newcommand{\CoordinationLatency}{L_{\text{lat,coordination}}}
\newcommand{\CoordinationOverhead}{h}
\newcommand{\CopperIronThermalResistance}{R_{\text{th,cu-iron}}}
\newcommand{\CopperReferenceTemperature}{T_0}
\newcommand{\CopperTemperatureCoefficient}{\alpha_{\text{Cu}}}
\newcommand{\CopperThermalCapacitance}{C_{\text{cu}}}
\newcommand{\CopperThermalTimeConstant}{\tau_{\text{th,cu}}}
\newcommand{\CoreAmbientThermalResistance}{R_{\text{th,ca}}}
\newcommand{\CoreRailVoltageDroop}{\Delta V_{\text{droop}}}
\newcommand{\CoreThermalCapacitance}{C_{\text{core}}}
\newcommand{\CoresetBudget}{K}
\newcommand{\CoriolisCentrifugalMatrix}{\mathbf{C}}
\newcommand{\CorrectCandidateTrialPassProbability}{p_{g}}
\newcommand{\CorrectClassIndex}{c}
\newcommand{\CorrectClassLogit}{z_{c}}
\newcommand{\CorrectClassProbability}{\hat{y}_{c}}
\newcommand{\CorrectnessReward}{R_{\text{corr}}}
\newcommand{\CorrelationCoefficient}{\rho}
\newcommand{\CorruptedAttentionScoreMatrix}{\mathbf{S}_{\text{corrupt}}}
\newcommand{\CorruptedModelWeight}{w'}
\newcommand{\CostAnomalyScore}{Z}
\newcommand{\CostBudgetUsedFraction}{\rho_C}
\newcommand{\CostCeiling}{C_{\max}}
\newcommand{\CostPerAcceptedTask}{C_{\text{eff}}}
\newcommand{\CostPerEmittedArtifact}{C_{\text{emit}}}
\newcommand{\CostToGo}{C}
\newcommand{\CoulombFrictionCoefficient}{\mu}
\newcommand{\CoulombFrictionForce}{F_{\text{friction}}}
\newcommand{\CountedFailureIndicator}{e}
\newcommand{\CovariateDriftBound}{\mathcal{O}(S^2 \epsilon)}
\newcommand{\CoverageFactor}{k}
\newcommand{\CpuEnergy}{E_{\text{CPU}}}
\newcommand{\CredibleBrakingDeceleration}{a_{\text{brake}}}
\newcommand{\CreditWindowCapacity}{N_{\text{win}}}
\newcommand{\CriticActionReturnEstimate}{Q_{\phi}}
\newcommand{\CriticApproximationError}{\epsilon}
\newcommand{\CriticModel}{M_{\text{value}}}
\newcommand{\CriticParameters}{\phi}
\newcommand{\CriticReturnEstimate}{V_{\phi}}
\newcommand{\CriticalBatchSize}{B^{*}}
\newcommand{\CriticalConstraint}{c}
\newcommand{\CriticalPathDuration}{T_{\text{crit}}}
\newcommand{\CriticalSectionIndex}{s}
\newcommand{\CriticalSectionWcet}{C^{\text{cs}}}
\newcommand{\CriticalTrackingErrorBound}{\epsilon_{\text{crit}}}
\newcommand{\CriticalTransitDistance}{d_{\text{crit}}}
\newcommand{\CrossEntropyLevelThreshold}{\gamma}
\newcommand{\CrossEntropyLoss}{\mathcal{L}_{\text{CE}}}
\newcommand{\CrossoverFrequency}{\omega_c}
\newcommand{\CruiseSpeed}{v_{c}}
\newcommand{\CumulativeArrivals}{A}
\newcommand{\CumulativeCompletions}{C}
\newcommand{\CumulativeIncidentCount}{N}
\newcommand{\CumulativeLogLikelihoodRatio}{\Lambda}
\newcommand{\CumulativeScheduleStepIndex}{s}
\newcommand{\CumulativeTransmittance}{T}
\newcommand{\CumulativeWorkCost}{C}
\newcommand{\CurrentBinProportion}{A}
\newcommand{\CurrentDistribution}{P_t}
\newcommand{\CurrentFeatureMean}{\mu_{\text{curr}}}
\newcommand{\CurrentFeatureStandardDeviation}{\sigma_{\text{curr}}}
\newcommand{\CurrentLayerWidth}{n_{\ell}}
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\newcommand{\CurrentMonotonicTime}{t_{\text{now,mono}}}
\newcommand{\CurrentNodeIndex}{j}
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\newcommand{\CurrentStateDigest}{h_{\text{state},t}}
\newcommand{\CurrentStateEstimate}{\hat{\mathbf{x}}_{t_{\text{current}}}}
\newcommand{\CurrentTime}{t_{\text{now}}}
\newcommand{\CurrentWallClockTime}{t_{\text{current}}}
\newcommand{\CutCrossingDataVolume}{V_{\text{cut}}}
\newcommand{\CvaeEncoderParameters}{\phi}
\newcommand{\CvaeRecognitionEncoder}{q_\phi}
\newcommand{\CycleAveragePowerCeiling}{\bar{P}_{\max}}
\newcommand{\CycleGuardTime}{t_{\text{guard}}}
\newcommand{\CylinderForce}{F_{\text{cyl}}}
\newcommand{\DailyAccountQueryCount}{q_{\text{daily}}}
\newcommand{\DailyQueryVolume}{Q_{\text{day}}}
\newcommand{\DamageEventCount}{N_{\text{damage}}}
\newcommand{\DamageProbabilityTolerance}{\epsilon}
\newcommand{\DamageProbabilityUpperBound}{p_U}
\newcommand{\DamagingOutcomeIndicator}{Y}
\newcommand{\DampedLeastSquaresPseudoinverse}{\mathbf{J}_{\lambda}^\dagger}
\newcommand{\DampedStepConfiguration}{\mathbf{q}_\lambda}
\newcommand{\DampingRatio}{\zeta}
\newcommand{\DataAcquisitionCost}{C_{\text{acquire}}}
\newcommand{\DataDebtGrowthRate}{r}
\newcommand{\DataEgressCost}{C_{\text{egress}}}
\newcommand{\DataFieldBitCount}{B_{\text{data},i}}
\newcommand{\DataInformationContent}{\mathcal{I}}
\newcommand{\DataLabelingCost}{C_{\text{label}}}
\newcommand{\DataLoaderWorkerCount}{W}
\newcommand{\DataMovementEnergyFraction}{q}
\newcommand{\DataParallelDegree}{N_{\text{DP}}}
\newcommand{\DataPhaseBitRate}{R_{\text{data}}}
\newcommand{\DataPipelineProcessingCost}{C_{\text{process}}}
\newcommand{\DataProcessingCost}{C_{\text{process}}}
\newcommand{\DataScaleCoefficient}{B}
\newcommand{\DataScalingExponent}{\beta}
\newcommand{\DataScalingLossFactor}{D^{-\beta}}
\newcommand{\DataStallFraction}{f_{\text{stall}}}
\newcommand{\DataStorageCost}{C_{\text{store}}}
\newcommand{\DataSupplyRate}{R_{\text{data}}}
\newcommand{\DataTaxonomyAxis}{\mathsf{Data}}
\newcommand{\DataVolume}{D_{\text{vol}}}
\newcommand{\DatasetQuery}{f}
\newcommand{\DatasetQueryResult}{v}
\newcommand{\DatasetShardCount}{K_{\text{shards}}}
\newcommand{\DatasetSize}{D}
\newcommand{\DatasetTransferTime}{T}
\newcommand{\DcBusBulkCapacitance}{C_{\text{bus}}}
\newcommand{\DcBusOvervoltageThreshold}{V_{\text{max\_safe}}}
\newcommand{\DcBusVoltage}{V_{\text{bus}}}
\newcommand{\DcRailVoltage}{V_{\text{rail}}}
\newcommand{\DeadTimeCycleCount}{N_{\text{dtg}}}
\newcommand{\DeadTimeDqVoltageError}{\Delta V_{d,q}}
\newcommand{\DeadTimeSafetyMargin}{t_{\text{margin}}}
\newcommand{\DeadTimeTickPeriod}{T_{\text{dtg}}}
\newcommand{\DeadTimeVoltageError}{\Delta V_{\text{phase}}}
\newcommand{\DeadlineInstant}{t_{\text{deadline}}}
\newcommand{\DebiasingTotalLoss}{\mathcal{L}_{\text{total}}}
\newcommand{\DebounceFrameCount}{N}
\newcommand{\DecisionProcessState}{s}
\newcommand{\DecisionScore}{s_{\text{dec}}}
\newcommand{\DecisionStepIndex}{k}
\newcommand{\DecodeArithmeticIntensity}{I_{\text{decode}}}
\newcommand{\DecodeBatchSize}{B_{\text{dec}}}
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\newcommand{\DecodeRequestSet}{\mathcal{D}}
\newcommand{\DecodeTokenTimeFloor}{T_{\text{token}}}
\newcommand{\DecoderEvaluationCount}{N_{\text{eval}}}
\newcommand{\DecoderPassTime}{t_{\text{pass}}}
\newcommand{\DecodingConstraintState}{q}
\newcommand{\DecomposedTensor}{\mathcal{Y}}
\newcommand{\DecomposedTensorEntry}{y_{ijt}}
\newcommand{\DecouplingCapacitance}{C}
\newcommand{\DecouplingDriftVector}{\mathbf{b}}
\newcommand{\DecouplingMatrix}{\mathbf{E}}
\newcommand{\DecouplingMatrixElement}{E}
\newcommand{\DedicatedHourlyFixedCost}{C_{\text{fixed}}}
\newcommand{\DedicatedVariableTokenCost}{c_{\text{ded}}}
\newcommand{\DefectiveActionProbability}{p_{\text{bad}}}
\newcommand{\DefinitionSerializedSize}{S}
\newcommand{\DeflectionComplianceAllowance}{\delta_{\text{deflect}}}
\newcommand{\DegradedFrictionCoefficient}{\mu_{\text{fault}}}
\newcommand{\DegradedStepTime}{T_{\text{slow}}}
\newcommand{\DelassusOperator}{\mathbf{A}_{\text{Delassus}}}
\newcommand{\DelayPhaseLag}{\Delta \phi}
\newcommand{\DelayTravelDistance}{d_{\text{delay}}}
\newcommand{\DelayWindowIntegrationTime}{\sigma}
\newcommand{\DelayedObservation}{\mathbf{y}_{t_0}}
\newcommand{\DelegatedAllocationAmount}{b}
\newcommand{\DelegationTokenMultiple}{r}
\newcommand{\DeliveredActuatorTorque}{\tau_{\text{act}}}
\newcommand{\DeliveredCapacityFraction}{u}
\newcommand{\DemandCap}{x_{\max}}
\newcommand{\DemandPriceElasticity}{\epsilon_{\text{elast}}}
\newcommand{\DemandThreshold}{x}
\newcommand{\DemonstratedTargetA}{\mathbf{y}_A}
\newcommand{\DemonstratedTargetB}{\mathbf{y}_B}
\newcommand{\DemonstrationActionDistribution}{p}
\newcommand{\DemonstrationActionLabel}{\mathbf{a}}
\newcommand{\DemonstrationArchiveActionDistribution}{p_{\text{demo}}}
\newcommand{\DemonstrationChunkDistribution}{p_{\text{data}}}
\newcommand{\DemonstrationDataset}{\mathcal{D}_{\text{demo}}}
\newcommand{\DemonstrationRetentionRate}{R}
\newcommand{\DemonstratorAction}{\mathbf{a}^{*}}
\newcommand{\DemonstratorActionComponent}{a}
\newcommand{\DenseEvaluationEnergy}{E_{\text{eval}}}
\newcommand{\DenseEvaluationRate}{f_{\text{eval}}}
\newcommand{\DenseVoxelGridMemory}{M_{\text{dense}}}
\newcommand{\DepartmentQuotaLimit}{Q_{\text{department}}}
\newcommand{\DependencyGraphEdgeSet}{E}
\newcommand{\DependencyGraphVertex}{v}
\newcommand{\DependencyGraphVertexSet}{V}
\newcommand{\DepletedBusVoltage}{V_{\text{low}}}
\newcommand{\DeploymentAge}{t}
\newcommand{\DeploymentDistribution}{p_{\text{deploy}}}
\newcommand{\DeploymentDuration}{T_{\text{deployment}}}
\newcommand{\DepthBinCount}{D}
\newcommand{\DepthBinDistribution}{\boldsymbol{\alpha}_{u, v}}
\newcommand{\DepthBinIndex}{k}
\newcommand{\DepthBinLogits}{\mathbf{d}_{u, v}}
\newcommand{\DepthBinProbability}{\alpha_{u, v}}
\newcommand{\DepthCalibrationScale}{s}
\newcommand{\DepthCalibrationShift}{t}
\newcommand{\DepthCriticalPathLatency}{T^{\text{depth}}}
\newcommand{\DepthFrameRate}{f_{\text{cam}}}
\newcommand{\DepthHypothesis}{d}
\newcommand{\DepthModeComponentVector}{\mathbf{c}_s}
\newcommand{\DepthModeFactorMatrix}{\mathbf{C}}
\newcommand{\DepthModeFactorSlice}{\mathbf{c}}
\newcommand{\DepthRayCount}{N_{\text{rays}}}
\newcommand{\DepthScalingRatio}{\alpha}
\newcommand{\DepthStandardDeviation}{\sigma_z}
\newcommand{\DerivativeControlEffort}{\mathbf{u}_{d}}
\newcommand{\DerivativeFilterTimeConstant}{T_{f}}
\newcommand{\DerivativeGainMatrix}{\mathbf{K}_{d}}
\newcommand{\DerivativeOperator}{d}
\newcommand{\DerivativeSplineControlPoint}{\mathbf{v}}
\newcommand{\DerivedMemoryEntry}{c}
\newcommand{\DerivedRecord}{y}
\newcommand{\DesignLatencyQuantile}{Q_{1-\epsilon}}
\newcommand{\DesiredContactPosition}{x_{\text{des}}}
\newcommand{\DesiredContactVelocity}{\dot{x}_{\text{des}}}
\newcommand{\DesiredTaskAcceleration}{\ddot{\mathbf{x}}_d}
\newcommand{\DesiredTaskPosition}{\mathbf{x}_d}
\newcommand{\DesiredTaskSpaceAcceleration}{\ddot{\mathbf{x}}^{\text{des}}}
\newcommand{\DesiredTaskVelocity}{\dot{\mathbf{x}}_d}
\newcommand{\DestinationRankIndex}{j}
\newcommand{\DetectionCount}{n_{\text{det}}}
\newcommand{\DetectionInterval}{t_{\text{detect}}}
\newcommand{\DetectionMeasurementNoiseCovariance}{\boldsymbol{\Sigma}_{\text{meas},j}}
\newcommand{\DetectionRangeShortfall}{\Delta d_{\text{detect}}}
\newcommand{\DetectionTimestamp}{t_{\text{det}}}
\newcommand{\DetectorMissProbability}{p_{\text{miss}}}
\newcommand{\DeterministicPolicy}{\mu_{\theta}}
\newcommand{\DevelopmentIterationCount}{N_{\text{dev}}}
\newcommand{\DeviationDisadvantage}{u}
\newcommand{\DeviceCount}{N}
\newcommand{\DeviceEnergy}{E_{\text{device}}}
\newcommand{\DeviceHardwareTime}{T_{\text{hw}}}
\newcommand{\DeviceMemoryCapacity}{M_{\text{device}}}
\newcommand{\DevicePower}{P_{\text{device}}}
\newcommand{\DeviceRequestCapacity}{Q_{\text{req,max}}}
\newcommand{\DeviceSilentCorruptionProbability}{p}
\newcommand{\DiagnosticCoverage}{C_{\text{diag}}}
\newcommand{\DictionaryCoefficientMatrix}{\mathbf{C}}
\newcommand{\DifferentiabilityClass}{C}
\newcommand{\DifferentialEntropy}{h}
\newcommand{\DiffusionCumulativeSchedule}{\bar{\alpha}}
\newcommand{\DiffusionGaussianNoise}{\boldsymbol{\epsilon}}
\newcommand{\DiffusionMatchingLoss}{\mathcal{L}_{\text{diff}}}
\newcommand{\DiffusionNoiseVariance}{\beta}
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\newcommand{\DiffusionStepIndex}{k}
\newcommand{\DiffusionStepRetention}{\alpha}
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\newcommand{\DirectAxisCurrent}{i_d}
\newcommand{\DirectAxisInductance}{L_d}
\newcommand{\DirectAxisVoltage}{v_d}
\newcommand{\DirectionalCosineMatrix}{\mathbf{R}}
\newcommand{\DirectionalDerivativeStep}{\epsilon}
\newcommand{\DisadvantagedGroupClassificationThreshold}{\gamma_{\text{cls},g=\text{B}}}
\newcommand{\DiscordantPairCount}{n_{\text{disc}}}
\newcommand{\DiscordantPairFraction}{\psi}
\newcommand{\DiscountFactor}{\gamma}
\newcommand{\DiscountedReturn}{G}
\newcommand{\DiscountedReturnStepOffset}{j}
\newcommand{\DiscountedStateVisitationDistribution}{d}
\newcommand{\DiscreteLqrFeedbackGain}{\mathbf{K}_{\text{lqr},d}}
\newcommand{\DiscreteLqrQuadraticObjective}{J_d}
\newcommand{\DiscreteOutcome}{x}
\newcommand{\DiscretePlantDynamics}{f}
\newcommand{\DiscreteRandomVariable}{X}
\newcommand{\DiscreteSampleIndex}{k}
\newcommand{\DiscreteStateSpaceInputMatrix}{\mathbf{B}_{d}}
\newcommand{\DiscreteStateSpaceSystemMatrix}{\mathbf{A}_{d}}
\newcommand{\DiscretizationBinCount}{n_{\text{bin}}}
\newcommand{\DisengagedTakeoverResponseTime}{T_{\Sigma,2}}
\newcommand{\DiskBandwidth}{\text{BW}_{\text{disk}}}
\newcommand{\DisparityStandardDeviation}{\sigma_d}
\newcommand{\DispatchAdmissionCeiling}{\Delta_{\text{da,max}}}
\newcommand{\DispatchAdmissionInterval}{\Delta_{\text{da}}}
\newcommand{\DispatchObservationAge}{t_{\text{age}}}
\newcommand{\DispatchedOperationCount}{N_{\text{ops}}}
\newcommand{\DisplayPipelineLatency}{t_{\text{disp}}}
\newcommand{\DispreferredReferenceRelativeScore}{\Delta_l}
\newcommand{\DispreferredTrajectory}{\tau_l}
\newcommand{\DistanceEstimateStandardDeviation}{\sigma}
\newcommand{\DistanceMeasurement}{z}
\newcommand{\DistanceNormOrder}{p}
\newcommand{\DistanceQueryPoint}{\mathbf{y}}
\newcommand{\DistillationClassProbability}{p^{(T_{\text{distill}})}}
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\newcommand{\DistillationLossWeight}{\gamma_{\text{KD}}}
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\newcommand{\DistortedImageAbscissa}{x_d}
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\newcommand{\DisturbanceBasisFunction}{B}
\newcommand{\DisturbanceBudget}{\Delta_{w}}
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\newcommand{\DisturbanceParameterVector}{\boldsymbol{\psi}}
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\newcommand{\DivergenceMeasure}{\mathcal{D}}
\newcommand{\DivergenceTolerance}{\delta_{\text{tol}}}
\newcommand{\DmaBurstPeriod}{T_{\text{dma}}}
\newcommand{\DmaBurstStallTime}{C_{\text{dma}}}
\newcommand{\DmaQueueTime}{t_{\text{DMA}}}
\newcommand{\DocumentPayloadFraction}{q}
\newcommand{\DomainRandomizationDistribution}{\mathcal{D}_{\text{DR}}}
\newcommand{\DomainRandomizationObjective}{J_{\text{DR}}}
\newcommand{\DominantComponentEnergy}{E_{\text{dominant}}}
\newcommand{\DominantShare}{d}
\newcommand{\DoorOpenReward}{r_{\text{open}}}
\newcommand{\DoubleIntegratorPosition}{p}
\newcommand{\DoubleIntegratorPositionDerivative}{\dot{p}}
\newcommand{\DoubleIntegratorVelocity}{v}
\newcommand{\DoubleIntegratorVelocityDerivative}{\dot{v}}
\newcommand{\DoubleTreeAllReduceTime}{T_{\text{double-tree}}}
\newcommand{\DownstreamStageLatency}{Y}
\newcommand{\DpoLoss}{\mathcal{L}_{\text{DPO}}}
\newcommand{\DraftAcceptanceRate}{\alpha}
\newcommand{\DraftCandidateToken}{\tilde{x}}
\newcommand{\DraftModelDistribution}{q_{\phi}}
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\newcommand{\DraftPositionIndex}{i}
\newcommand{\DraftSamplingDistribution}{q}
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\newcommand{\DraftTokenCount}{\gamma}
\newcommand{\DraftedToken}{T}
\newcommand{\DragonflyGlobalChannelCount}{h}
\newcommand{\DragonflyGroupRouterCount}{a}
\newcommand{\DragonflyTerminalPortCount}{p}
\newcommand{\DrainSourceVoltage}{V_{\text{ds}}}
\newcommand{\DriftElapsedTime}{t_{\text{drift}}}
\newcommand{\DriftImpactTime}{t_{\text{impact}}}
\newcommand{\DriftLieDerivativeOperator}{L_{\mathbf{f}}}
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\newcommand{\DriftTestSampleSize}{n}
\newcommand{\DriftVelocityBound}{v_{\text{drift}}}
\newcommand{\DriveBrakingForceCeiling}{F}
\newcommand{\DriveBrakingTorqueCeiling}{\tau}
\newcommand{\DrivenWheelLoadFraction}{\phi_{\text{drive}}}
\newcommand{\DriverInterruptLatency}{t_{\text{driver\_irq}}}
\newcommand{\DrivetrainAntiresonanceFrequency}{f_{\text{anti}}}
\newcommand{\DrivetrainResonanceFrequency}{f_{\text{res}}}
\newcommand{\DroopedSupplyVoltage}{V_{\text{droop}}}
\newcommand{\DropletDiameter}{d_{\text{droplet}}}
\newcommand{\DropoutProbability}{p_{\text{drop}}}
\newcommand{\DualNumberInfinitesimal}{\epsilon}
\newcommand{\DuplicateEffectProbability}{p_{\text{dup}}}
\newcommand{\DurableAcknowledgmentTime}{t_{\text{ack}}}
\newcommand{\DurableStoreWriteBandwidth}{\text{BW}_{\text{store}}}
\newcommand{\DutyCyclePeriod}{T_{\text{period}}}
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\newcommand{\DynamicReachableSet}{\mathcal{R}}
\newcommand{\DynamicsRealityGap}{\Delta_{\text{dyn}}}
\newcommand{\EagerExecutionTime}{T_{\text{eager}}}
\newcommand{\EarlierBaseFlangeTransform}{\mathbf{T}_{\text{base}, k}^{\text{flange}}}
\newcommand{\EarlierCameraTargetTransform}{\mathbf{T}_{\text{cam}, k}^{\text{target}}}
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\newcommand{\EarliestConflictTime}{t_{\text{conflict}}}
\newcommand{\EarthReferenceFrame}{\mathcal{F}_{\text{earth}}}
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\newcommand{\EchoRoundTripTime}{t_{\text{rt}}}
\newcommand{\EditGrantRights}{\mathcal{R}_{\Gamma_{\text{edit}}}}
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\newcommand{\EffectiveApiOutputTokenPrice}{c_{\text{API}}}
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\newcommand{\EffectiveConfidentialityLabel}{\kappa_{\text{eff}}}
\newcommand{\EffectiveFleetThroughput}{R_{\text{eff}}}
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\newcommand{\EffectiveIntegrityProvenance}{\iota_{\text{eff}}}
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\newcommand{\EffectiveSeriesStiffness}{k_{\text{eff}}}
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\newcommand{\EffectiveWasteFraction}{W_{\text{effective}}}
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\newcommand{\EgressArrivalRate}{\lambda_{\text{arr}}}
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\newcommand{\ElasticMinimumWorkerCount}{N_{\text{min}}}
\newcommand{\ElasticModalFrequencyMatrix}{\boldsymbol{\Omega}}
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\newcommand{\ElasticTorqueSecondDerivative}{\ddot{\mathbf{z}}}
\newcommand{\ElbowSwivelAngle}{\psi}
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\newcommand{\ElectricityPrice}{R_{\$/\text{kWh}}}
\newcommand{\ElementStorageSize}{s_{\text{elem}}}
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\newcommand{\ElementaryOperationCount}{W}
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\newcommand{\EligibleVectorCount}{N}
\newcommand{\EliteSampleFraction}{\beta}
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\newcommand{\EmittedArtifactCount}{N_{\text{emit}}}
\newcommand{\EmittedTokenCount}{S_{\text{out}}}
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\newcommand{\EmpiricalExceedanceFraction}{\hat{\zeta}_{u}}
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\newcommand{\EmpiricalTransferDiscrepancy}{\hat{\epsilon}_{\text{gap}}}
\newcommand{\EmptyByteString}{\epsilon}
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\newcommand{\EncoderLatentRepresentation}{\mathbf{z}}
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\newcommand{\EndEffectorCameraTransform}{\mathbf{T}_{EC}}
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\newcommand{\EndEffectorRollAngle}{\phi_{\text{roll}}}
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\newcommand{\EndEffectorThirdAxisPosition}{z}
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\newcommand{\EndEffectorVelocity}{\mathbf{v}_e}
\newcommand{\EndEffectorYawAngle}{\phi_{\text{yaw}}}
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\newcommand{\EnergyLimitedContactForce}{F_E}
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\newcommand{\EnergyTankBarrierFunction}{h_{\text{energy}}}
\newcommand{\EnergyTankLevel}{E_{\text{tank}}}
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\newcommand{\EnforcedCommandTap}{a_{\text{enf}}}
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\newcommand{\EnforcerDeadline}{T_{\text{deadline,enf}}}
\newcommand{\EngineerHourlyCost}{C_{\text{engineer}}}
\newcommand{\EngineeringTeamSize}{n}
\newcommand{\EngineeringTemperatureCeiling}{T_{\text{ceil}}}
\newcommand{\EnsembleSpreadThreshold}{\sigma_{\text{thresh}}}
\newcommand{\EnsemblingDecayRate}{m}
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\newcommand{\EntityEmbedding}{\mathbf{e}}
\newcommand{\EntityIndex}{j}
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\newcommand{\EnvironmentRolloutStepCount}{h}
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\newcommand{\EpipolarLine}{\mathbf{l}}
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\newcommand{\EpochIndex}{n_{\text{epoch}}}
\newcommand{\EquilibriumWindingTemperature}{T_\infty}
\newcommand{\EquivalentSeriesStiffness}{k_{\text{eq}}}
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\newcommand{\ErasureParityFragmentCount}{m}
\newcommand{\ErroneousPredictionCost}{C_{\text{error}}}
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\newcommand{\ErrorFeedbackResidual}{\mathbf{e}}
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\newcommand{\EscalationLossMatrix}{\mathbf{L}}
\newcommand{\EscalationThresholdProbability}{p_{\text{threshold}}}
\newcommand{\EssentialMatrix}{\mathbf{E}}
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\newcommand{\EstimatedAcceptanceProbability}{\hat{p}}
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\newcommand{\EstimatedCleanActionChunk}{\hat{\mathbf{A}}}
\newcommand{\EstimatedCoriolisMatrix}{\hat{\mathbf{C}}}
\newcommand{\EstimatedDamageProbability}{\hat{p}}
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\newcommand{\EstimatedGravitationalTorque}{\hat{\mathbf{g}}}
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\newcommand{\EstimatedHistoryAdvantage}{\hat{A}}
\newcommand{\EstimatedInertiaMatrix}{\hat{\mathbf{M}}}
\newcommand{\EstimatedPairedAcceptanceDifference}{\hat{\Delta}}
\newcommand{\EstimatedPlantState}{\hat{\mathbf{x}}}
\newcommand{\EstimatedPosition}{\hat{\mathbf{p}}}
\newcommand{\EstimatedRemainingReturn}{\hat{V}}
\newcommand{\EstimatedUtilityStandardDeviation}{s}
\newcommand{\EstimatedVelocity}{\hat{\mathbf{v}}}
\newcommand{\EstimatedWindingTemperature}{\hat{T}_{\text{wind}}}
\newcommand{\EstimationErrorCovariance}{\mathbf{P}}
\newcommand{\EstimatorDrawCount}{N}
\newcommand{\EstimatorDrawStandardDeviation}{\sigma}
\newcommand{\EstimatorState}{\mathbf{x}}
\newcommand{\EstimatorStateVector}{\mathbf{x}}
\newcommand{\EstimatorStepIndex}{k}
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\newcommand{\EulerIntegralSecondShape}{b}
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\newcommand{\EvaluatedExceedance}{y}
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\newcommand{\EvaluationSampleCount}{N}
\newcommand{\EvaluationTrialCount}{N}
\newcommand{\EvaluationTrialIndex}{i}
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\newcommand{\EventIndicator}{\mathbf{1}}
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\newcommand{\EventPolarity}{p}
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\newcommand{\EvidenceLowerBound}{\mathcal{L}_{\text{ELBO}}}
\newcommand{\EvidenceRecordEvent}{e_{\text{obs}}}
\newcommand{\ExactFailureLowerLimit}{p_{L}}
\newcommand{\ExactFailureUpperLimit}{p_{U}}
\newcommand{\ExampleAveragedBatchLoss}{\mathcal{L}_{\text{example}}}
\newcommand{\ExampleIndex}{i}
\newcommand{\ExampleLabel}{y}
\newcommand{\ExceedanceReturnLevel}{z_\epsilon}
\newcommand{\ExceedanceSampleSet}{\mathcal{Y}_u}
\newcommand{\ExceedanceThreshold}{u}
\newcommand{\ExcessDistributionFunction}{F_{u}}
\newcommand{\ExcessForcePenaltyWeight}{\lambda_f}
\newcommand{\ExcessSupportUpperEndpoint}{y_{\max}}
\newcommand{\ExecutedActionStepCount}{T_a}
\newcommand{\ExecutedToolCall}{a}
\newcommand{\ExecutedTrialCount}{N_{\text{trial}}}
\newcommand{\ExecutionHorizon}{H_{\text{exec}}}
\newcommand{\ExecutionNondeterminism}{\xi}
\newcommand{\ExecutionSwitchTime}{T_{\text{switch}}}
\newcommand{\ExogenousDataDistribution}{\mathcal{D}}
\newcommand{\ExpectedCategoryFrequency}{n_i^{\text{exp}}}
\newcommand{\ExpectedCheckingTime}{T_{\text{check}}}
\newcommand{\ExpectedCostLimit}{\delta}
\newcommand{\ExpectedDecisionRisk}{\mathcal{R}}
\newcommand{\ExpectedHistoryActionReturn}{Q}
\newcommand{\ExpectedIrreversibleLoss}{L_A}
\newcommand{\ExpectedNormalMaximum}{m_{N}}
\newcommand{\ExpectedQueryDistribution}{p_{\text{expected}}}
\newcommand{\ExpectedRemainingReturn}{V}
\newcommand{\ExpectedReturnObjective}{J}
\newcommand{\ExpectedRunCompletionTime}{T}
\newcommand{\ExpectedTaskCostSum}{J}
\newcommand{\ExperienceReplayBuffer}{\mathcal{D}}
\newcommand{\ExperimentCount}{N_{\text{exp}}}
\newcommand{\ExperimentUserIndex}{i}
\newcommand{\ExpertCapacityFactor}{C_{\text{cap}}}
\newcommand{\ExpertCount}{N_{\text{experts}}}
\newcommand{\ExpertDiscountedStateVisitationDistribution}{d^{\pi^*}}
\newcommand{\ExpertIndex}{e}
\newcommand{\ExpertSelectionProbability}{p_e}
\newcommand{\ExpertStateVisitationDistribution}{p^{\pi^*}}
\newcommand{\ExpertTokenCount}{B_e}
\newcommand{\ExploitCandidateFraction}{p_{\text{exp}}}
\newcommand{\ExploitTrajectory}{\tau_{\text{hack}}}
\newcommand{\ExploratoryStateVisitationDistribution}{p^{\pi_{\text{expl}}}}
\newcommand{\ExponentBitIndex}{k}
\newcommand{\ExponentialSeriesArgument}{u}
\newcommand{\ExponentialStabilityConstant}{k}
\newcommand{\ExposedCommunicationTime}{T_{\text{comm,exposed}}}
\newcommand{\ExposedUnitCount}{E_{\text{exposure}}}
\newcommand{\ExposureDuration}{t_{\text{exp}}}
\newcommand{\ExposureMidpointTime}{t_{\text{mid}}}
\newcommand{\ExposureMotionDisplacement}{\Delta x_{\text{motion}}}
\newcommand{\ExposureStartTime}{t_{\text{start}}}
\newcommand{\ExposureTravelDistance}{\Delta x}
\newcommand{\ExtendedClassKInfinityFamily}{\mathcal{K}_\infty}
\newcommand{\ExtendedDepthLength}{K_{\text{ext}}}
\newcommand{\ExternalContactTorque}{\boldsymbol{\tau}_{\text{ext}}}
\newcommand{\ExternalDispatchAction}{a_{\text{ext}}}
\newcommand{\ExternalDisturbanceForce}{d}
\newcommand{\ExternalDisturbanceLoad}{\mathbf{d}_{\text{ext}}}
\newcommand{\ExternalJointTorque}{\tau_{\text{ext}}}
\newcommand{\ExternalObjectWrench}{\mathbf{w}_{\text{ext}}}
\newcommand{\ExternalWait}{T_{\text{wait}}}
\newcommand{\ExternalWrench}{\mathbf{F}_{\text{ext}}}
\newcommand{\ExtractionAnomalyClassifier}{f_{\theta}}
\newcommand{\ExtractionQueryCount}{n_{\text{queries}}}
\newcommand{\ExtrinsicCalibrationCovariance}{\boldsymbol{\Sigma}_{\text{ext}}}
\newcommand{\ExtrinsicCovarianceJacobian}{\mathbf{J}_{\text{ext}}}
\newcommand{\ExtrinsicLateralError}{\Delta y}
\newcommand{\ExtrinsicMisalignmentAngle}{\theta_{\text{ext}}}
\newcommand{\EyeInHandCameraMotion}{\mathbf{B}}
\newcommand{\EyeInHandFlangeMotion}{\mathbf{A}}
\newcommand{\EyeInHandUnknownTransform}{\mathbf{X}}
\newcommand{\EyeToHandCameraPose}{\mathbf{B}'}
\newcommand{\EyeToHandFlangePose}{\mathbf{A}'}
\newcommand{\EyeToHandTargetTransform}{\mathbf{Z}}
\newcommand{\EyeToHandUnknownTransform}{\mathbf{X}'}
\newcommand{\FacilityEnergy}{E_{\text{facility}}}
\newcommand{\FacilityHeatRemovalCapacity}{\dot{Q}_{\text{remove,max}}}
\newcommand{\FacilityPower}{P_{\text{facility}}}
\newcommand{\FacilityWaterConsumption}{W_{\text{water}}}
\newcommand{\FactoredIntermediateActivation}{\mathbf{z}}
\newcommand{\FactorizationRank}{k}
\newcommand{\FailureDetectionTime}{T_{\text{detect}}}
\newcommand{\FailureEventCost}{C_{\text{per-failure}}}
\newcommand{\FailureFlagLatency}{\delta_{\text{flag}}}
\newcommand{\FailureMarginFunction}{l}
\newcommand{\FailureModeOccurrenceCount}{n_{f}}
\newcommand{\FailureObservationInterval}{T}
\newcommand{\FailureOverheadFraction}{F_{\text{fail}}}
\newcommand{\FailureRecoveryTime}{T_{\text{recovery}}}
\newcommand{\FailureRegion}{F}
\newcommand{\FailureSet}{\mathcal{F}_{\text{fail}}}
\newcommand{\FailureUpperConfidenceLimit}{p_{\text{fail, upper}}}
\newcommand{\FairShareStabilizingConstant}{\epsilon_{\text{stab}}}
\newcommand{\FairnessGapTolerance}{\epsilon_{\text{fair}}}
\newcommand{\FakeQuantizedOutput}{x_{\text{fake}}}
\newcommand{\FallbackCost}{C_{\text{fallback}}}
\newcommand{\FallbackDeadline}{T_{\text{deadline,fallback}}}
\newcommand{\FallbackHold}{T_{\text{hold}}}
\newcommand{\FalseAcceptHarmCost}{C_{\text{harm}}}
\newcommand{\FalseApprovalCount}{N_{\text{fa}}}
\newcommand{\FalseApprovalOutcomeCount}{k}
\newcommand{\FalseApprovalPropensity}{Q}
\newcommand{\FalseApprovalPropensityLevel}{q}
\newcommand{\FalseDiscoveryRateTarget}{q}
\newcommand{\FalsificationInitialStateSet}{\mathcal{X}_0}
\newcommand{\FanoutBackendCount}{m}
\newcommand{\FarthestCandidateDistance}{D_{\max}}
\newcommand{\FatTreeHostCount}{N_{\text{hosts}}}
\newcommand{\FatigueLifeCycleCount}{N}
\newcommand{\FaultContainmentLatency}{\Delta t_{\text{bounded}}}
\newcommand{\FaultDetectionLatency}{\Delta t_{\text{det}}}
\newcommand{\FaultDomainIsolation}{\mathcal{F}}
\newcommand{\FaultInjectionDuration}{\Delta t_{\text{duration}}}
\newcommand{\FaultStartTimestamp}{t_{\text{fault\_start}}}
\newcommand{\FaultTakeoverLatency}{\Delta t_{\text{takeover}}}
\newcommand{\FaultTakeoverTimestamp}{t_{\text{takeover}}}
\newcommand{\FaultToleranceCost}{C_{\text{ft}}}
\newcommand{\FaultToleranceInvestment}{x}
\newcommand{\FeasibleOutputTokenRate}{\mu_{\text{out}}}
\newcommand{\FeasibleTrajectoryProjection}{\Pi_{\mathcal{K}}}
\newcommand{\FeasibleTrajectorySet}{\mathcal{K}}
\newcommand{\FeatureCategoryCount}{k}
\newcommand{\FeatureCategoryIndex}{i}
\newcommand{\FeatureCellVector}{\mathbf{f}_{u, v}}
\newcommand{\FeatureChannelCount}{C}
\newcommand{\FeatureCount}{N_{\text{features}}}
\newcommand{\FeatureFreshnessLatency}{T_{\text{freshness}}}
\newcommand{\FeatureMapColumnCoordinate}{x}
\newcommand{\FeatureMapGrowthRate}{g}
\newcommand{\FeatureMapHeight}{H_{\text{img}}}
\newcommand{\FeatureMapRowCoordinate}{y}
\newcommand{\FeatureMapWidth}{W_{\text{img}}}
\newcommand{\FeatureRecordTime}{t}
\newcommand{\FeatureRetentionWindow}{T_{\text{retention}}}
\newcommand{\FeatureUpdateInterval}{T_{\text{update}}}
\newcommand{\FeatureUpdateLag}{T_{\text{staleness}}}
\newcommand{\FeatureUpdateTimestamp}{t_{\text{feat}}}
\newcommand{\FeatureViewingDistance}{r}
\newcommand{\FederatedRoundCount}{R}
\newcommand{\FederatedRoundIndex}{t}
\newcommand{\FederatedTrainingEnergy}{E_{\text{federated}}}
\newcommand{\FeedForwardOperations}{O_{\text{FFN}}}
\newcommand{\FeedForwardWidth}{d_{\text{ff}}}
\newcommand{\FeedbackCheckDuration}{t_{\text{check}}}
\newcommand{\FeedbackCriticalPathLatency}{T^{\text{feedback}}}
\newcommand{\FeedbackIntervalCount}{k}
\newcommand{\FeedbackStepIndex}{t}
\newcommand{\FeedforwardJointTorque}{\boldsymbol{\tau}_{\text{ff}}}
\newcommand{\FeedthroughMatrix}{\mathbf{D}}
\newcommand{\FewShotStepCount}{n_{\text{steps}}}
\newcommand{\FewShotSupportSet}{\mathcal{D}_{\text{shot}}}
\newcommand{\FiberPropagationSpeed}{c_{\text{fiber}}}
\newcommand{\FieldbusCycleTime}{T_{\text{cycle}}}
\newcommand{\FilmEncounterRate}{\lambda_{\text{film}}}
\newcommand{\FilterSolveBudget}{T_{\text{budget}}}
\newcommand{\FilterSolveTime}{T_{\text{solve}}}
\newcommand{\FilterTapCount}{N}
\newcommand{\FilterWeight}{w}
\newcommand{\FilteredScreenSpaceCovariance}{\boldsymbol{\Sigma}_{2D}'}
\newcommand{\FinalBusVoltage}{V_{\text{final}}}
\newcommand{\FinalGeneratorLatentVector}{\mathbf{z}_{m}}
\newcommand{\FinalLayerFunction}{f_{N_L}}
\newcommand{\FinalLayerWidth}{n_{N_L}}
\newcommand{\FinalMiniBatchActivation}{a_{B}}
\newcommand{\FinalRecurrentHiddenState}{\mathbf{h}_{S}}
\newcommand{\FinalWatchdogToken}{T_{\text{final}}}
\newcommand{\FingerNormalForce}{f_n}
\newcommand{\FingerTangentialForce}{\mathbf{f}_t}
\newcommand{\FinishedWorkerCount}{j}
\newcommand{\FiniteChange}{\Delta}
\newcommand{\FiniteSampleGeneralizationBound}{\delta_N}
\newcommand{\FiniteTimeHorizon}{T}
\newcommand{\FirstAdjacentCheckIndex}{a}
\newcommand{\FirstAdjacentCheckPassProbability}{p_{a}}
\newcommand{\FirstAdjacentCheckSlotTime}{T_{a}}
\newcommand{\FirstAtomicGradientUpdate}{A}
\newcommand{\FirstCatalogDefinitionCount}{n_{\text{def},A}}
\newcommand{\FirstCatalogDefinitionSize}{S_{A}}
\newcommand{\FirstClassCoverageFloor}{a}
\newcommand{\FirstClientMaskedUpdate}{\mathbf{y}_A}
\newcommand{\FirstClientModelParameters}{\theta_A}
\newcommand{\FirstComparedGroupLabel}{a}
\newcommand{\FirstComparedItem}{i}
\newcommand{\FirstDiscardedSingularNumber}{\sigma_{r+1}}
\newcommand{\FirstErrorStepIndex}{k}
\newcommand{\FirstFloatingPointAddend}{a}
\newcommand{\FirstGroupClassificationThreshold}{\tau_{\text{thr},a}}
\newcommand{\FirstGroupQualifiedFraction}{q_a}
\newcommand{\FirstGroupSampleApprovalRate}{\hat{p}_a}
\newcommand{\FirstGroupSampleSize}{n_a}
\newcommand{\FirstGroupThresholdClassifier}{h_a}
\newcommand{\FirstIncidentTime}{t_1}
\newcommand{\FirstInputNodePrimal}{v_{1-N}}
\newcommand{\FirstJaccardSet}{\mathcal{A}}
\newcommand{\FirstMistakeEvent}{\mathcal{E}}
\newcommand{\FirstRankComputeTime}{T_{\text{rank}_0}}
\newcommand{\FirstRaterCategoryFraction}{p_{y,1}}
\newcommand{\FirstRingTraceElement}{a}
\newcommand{\FirstSummandStandardDeviation}{\sigma_a}
\newcommand{\FirstTwinCriticReturnEstimate}{Q_{\phi_1}}
\newcommand{\FittedSensitivity}{\lambda}
\newcommand{\FixedClearanceAllowance}{\delta}
\newcommand{\FlangeCameraTransform}{\mathbf{T}_{\text{flange}}^{\text{cam}}}
\newcommand{\FlangePose}{\mathbf{T}_{\text{flange}}}
\newcommand{\FlangeReferenceFrame}{\mathcal{F}_{\text{flange}}}
\newcommand{\FlangeTargetTransform}{\mathbf{T}_{\text{flange}}^{\text{target}}}
\newcommand{\FlattenedFilterWeights}{\mathbf{w}_{\text{col}}}
\newcommand{\FleetAcceleratorCount}{N_{\text{fleet}}}
\newcommand{\FleetCommunicationTaxonomyAxis}{C_2}
\newcommand{\FleetComputeTaxonomyAxis}{C_1}
\newcommand{\FleetCoordinationTaxonomyAxis}{C_3}
\newcommand{\FleetEnergyProductivity}{\rho_{\text{energy}}}
\newcommand{\FleetManufacturingDistribution}{\mathcal{M}}
\newcommand{\FleetResourceCapacity}{C_{r}}
\newcommand{\FleetRobotCount}{N_{\text{fleet}}}
\newcommand{\FlippedFloatExponentField}{e'}
\newcommand{\FloatEncodedNumber}{x}
\newcommand{\FloatExponentField}{e}
\newcommand{\FloatMantissaField}{m}
\newcommand{\FloatSignBit}{s}
\newcommand{\FloatingBaseAccelerations}{\ddot{\mathbf{q}}_b}
\newcommand{\FloatingBaseBiasForces}{\mathbf{h}_b}
\newcommand{\FloatingBaseInertiaBlock}{\mathbf{M}_{bb}}
\newcommand{\FloatingBasePose}{\mathbf{q}_b}
\newcommand{\FlopPrice}{c_{\text{FLOP}}}
\newcommand{\FlowMatchingLoss}{\mathcal{L}_{\text{FM}}}
\newcommand{\FlowProbabilityPath}{\rho}
\newcommand{\FlowSampleVector}{\mathbf{y}}
\newcommand{\FlowTimeCoordinate}{t}
\newcommand{\FocalLength}{f}
\newcommand{\FollowerForwardWave}{\mathbf{u}_s}
\newcommand{\FollowerJointPositions}{\mathbf{q}_{\text{follower}}}
\newcommand{\FollowerMechanicalLatency}{t_{\text{mech}}}
\newcommand{\FollowerReturnWave}{\mathbf{w}_s}
\newcommand{\ForceGuidedRelay}{K}
\newcommand{\ForceReadingInnovation}{\nu}
\newcommand{\ForceSensorBias}{b}
\newcommand{\ForceSensorReading}{y}
\newcommand{\ForegroundLatency}{L_{\text{lat,fg}}}
\newcommand{\ForgetDataset}{D_{\text{forget}}}
\newcommand{\ForkBoundarySet}{\mathcal{B}_{\text{fork}}}
\newcommand{\ForkedTrajectoryTokenCount}{N_{\text{total}}}
\newcommand{\FormatComplianceScore}{\Phi}
\newcommand{\FormatEfficiency}{\eta_{\text{format}}}
\newcommand{\FormatRewardWeight}{\lambda_{\text{fmt}}}
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\newcommand{\ForwardEulerTransitionMatrix}{\mathbf{A}_{\text{FE}}}
\newcommand{\ForwardInvariantSafeSet}{\mathcal{C}}
\newcommand{\ForwardKinematicsMap}{\mathbf{FK}}
\newcommand{\ForwardOccupancySet}{\mathcal{R}}
\newcommand{\ForwardPassTokenCount}{n}
\newcommand{\ForwardPosteriorMean}{\tilde{\boldsymbol{\mu}}}
\newcommand{\ForwardPosteriorVariance}{\tilde{\beta}}
\newcommand{\FourthRingTraceElement}{d}
\newcommand{\FrameBitCount}{B_{\text{frame}}}
\newcommand{\FrameBlockingTime}{B}
\newcommand{\FrameExposureStartTime}{t_{\text{start}}}
\newcommand{\FrameIndex}{i}
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\newcommand{\FramePeriod}{T}
\newcommand{\FrameReadoutTime}{t_{\text{readout}}}
\newcommand{\FrameReleaseJitter}{J}
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\newcommand{\FrameSerializationTime}{t_{\text{frame},i}}
\newcommand{\FrameTransmissionTime}{C}
\newcommand{\FrameWireBitCount}{B_{\text{wire},i}}
\newcommand{\FreeConfigurationSpace}{\mathcal{C}_{\text{free}}}
\newcommand{\FreeCreditCount}{N_{\text{free}}}
\newcommand{\FreeSpaceDepth}{d_{\text{free}}}
\newcommand{\FreeSpaceLogOddsIncrement}{L_{\text{free}}}
\newcommand{\FreeSurfaceTiltAngle}{\theta_{\text{tilt}}}
\newcommand{\FreeboardHeight}{h_{f}}
\newcommand{\FreshEvidenceTime}{t_{\text{fresh}}}
\newcommand{\FreshInputTokenCount}{S_{\text{fresh}}}
\newcommand{\FreshnessObjective}{S_{\text{freshness}}}
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\newcommand{\FrictionPyramidMatrix}{\mathbf{F}}
\newcommand{\FrobeniusNorm}{F}
\newcommand{\FrozenBackboneFunction}{f}
\newcommand{\FrozenFeatureEncoder}{f}
\newcommand{\FrozenWeightMatrix}{\mathbf{W}_{\text{frozen}}}
\newcommand{\FrustumFeature}{\mathbf{c}_{u, v, k}}
\newcommand{\FullBatchLoss}{\mathcal{L}_B}
\newcommand{\FullDatasetSize}{D_{\text{total}}}
\newcommand{\FullPrecisionWeight}{w}
\newcommand{\FullSnapshotWriteVolume}{D_{\text{vol,full}}}
\newcommand{\FullSpeedTravelTime}{b}
\newcommand{\FunctionInputAdjoint}{\bar{x}}
\newcommand{\FunctionInputDimension}{N}
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\newcommand{\FunctionInputTangent}{\dot{x}}
\newcommand{\FunctionInputVariable}{x}
\newcommand{\FunctionInputVector}{\mathbf{x}}
\newcommand{\FunctionJacobian}{\mathbf{J}}
\newcommand{\FunctionOutputDimension}{M}
\newcommand{\FunctionOutputElement}{y}
\newcommand{\FunctionOutputVariable}{z}
\newcommand{\FunctionOutputVector}{\mathbf{y}}
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\newcommand{\FusedConvolutionWeight}{\mathbf{W}_{\text{fused}}}
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\newcommand{\FusionInputList}{\mathcal{L}}
\newcommand{\FusionInputRank}{r}
\newcommand{\FusionInputTensor}{\mathbf{X}}
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\newcommand{\FutureTimeInstant}{t'}
\newcommand{\GaeAdvantageEstimate}{\widehat{A}^{\text{GAE}}}
\newcommand{\GateCurrent}{I_g}
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\newcommand{\GateSourceCapacitance}{C_{\text{gs}}}
\newcommand{\GateSourceCharge}{Q_{\text{gs}}}
\newcommand{\GateSourceVoltage}{V_{\text{gs}}}
\newcommand{\GaussianCovarianceMatrix}{\boldsymbol{\Sigma}}
\newcommand{\GaussianDistribution}{\mathcal{N}}
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\newcommand{\GaussianOpacityLogit}{o}
\newcommand{\GaussianPrimitiveCovariance}{\boldsymbol{\Sigma}}
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\newcommand{\GaussianPrimitiveOpacity}{\alpha}
\newcommand{\GaussianPrimitiveScaleMatrix}{\mathbf{S}}
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\newcommand{\GaussianStandardDeviation}{\sigma}
\newcommand{\GaussianThirdAxisScale}{s_{z}}
\newcommand{\GaussianViewDependentColor}{c}
\newcommand{\GearFaceWidth}{b}
\newcommand{\GearModule}{m}
\newcommand{\GearRatedFatigueLife}{L_{\text{gear}}}
\newcommand{\GearReductionRatio}{N_{\text{gear}}}
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\newcommand{\GemmAccumulatorScale}{\beta}
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\newcommand{\GemmLeftOperandEntry}{A_{ik}}
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\newcommand{\GemmOutputMatrix}{\mathbf{C}}
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\newcommand{\GemmTileInnerDimension}{K_{\text{tile}}}
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\newcommand{\GeneralizedAccelerationLowerBound}{\ddot{\mathbf{q}}_{\min}}
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\newcommand{\GevScaleParameter}{\sigma}
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\newcommand{\GlobalDeviationPenalty}{\mathcal{R}}
\newcommand{\GlobalModelParameters}{\theta_{\text{global}}}
\newcommand{\GoalConfiguration}{\mathbf{q}_{\text{goal}}}
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\newcommand{\GoodputRelativeTolerance}{\epsilon_{\mathcal{G}}}
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\newcommand{\GpdScaleParameter}{\tilde{\sigma}}
\newcommand{\GpuAvailability}{A_{\text{GPU}}}
\newcommand{\GpuEnergy}{E_{\text{GPU}}}
\newcommand{\GpuFailureRate}{\lambda_{\text{GPU}}}
\newcommand{\GpuMttf}{\text{MTTF}_{\text{GPU}}}
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\newcommand{\GradientElementCount}{d_{\text{grad}}}
\newcommand{\GradientMemory}{M_{\text{gradients}}}
\newcommand{\GradientPayload}{M_{\text{grad}}}
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\newcommand{\GrantAllowedAccesses}{\mathcal{A}}
\newcommand{\GrantCondition}{\phi_{\Gamma}}
\newcommand{\GrantExpiryTime}{t_{\text{exp}}}
\newcommand{\GrantIssueTime}{T}
\newcommand{\GrantRights}{\mathcal{R}_{\Gamma}}
\newcommand{\GraphEdgeSet}{E}
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\newcommand{\GraphNodeSet}{V}
\newcommand{\GraspContactDisplacement}{\mathbf{d}}
\newcommand{\GraspForceCeiling}{f_{\max}}
\newcommand{\GraspForceFloor}{f_{\min}}
\newcommand{\GraspMap}{\mathbf{G}}
\newcommand{\GraspOrientation}{\mathbf{R}_{\text{grasp}}}
\newcommand{\GraspPose}{\mathbf{T}_{\text{grasp}}}
\newcommand{\GraspPosition}{\mathbf{p}_{\text{grasp}}}
\newcommand{\GraspVerticalAcceleration}{a_z}
\newcommand{\GravitationalTorqueVector}{\mathbf{g}}
\newcommand{\Gravity}{g}
\newcommand{\GravityCenterHeight}{h_{\text{cg}}}
\newcommand{\GravityVector}{\mathbf{g}_0}
\newcommand{\GridFeedCapacity}{P_{\text{grid}}}
\newcommand{\GridSideCellCount}{N}
\newcommand{\GripContactForce}{x}
\newcommand{\GripperContactAngle}{\theta_{\text{contact}}}
\newcommand{\GripperOpeningCommand}{a_{\text{gripper}}}
\newcommand{\GroundPlaneAbscissa}{x}
\newcommand{\GroundPlaneOrdinate}{y}
\newcommand{\GroupCommitWaitTime}{L_{\text{lat}, i}}
\newcommand{\GroupContrastProbability}{p_{\text{contrast}}}
\newcommand{\GroupDisparity}{\Delta_g}
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\newcommand{\GroupRelativeAdvantage}{\hat{A}}
\newcommand{\GroupRewardStandardDeviation}{\sigma_{R}}
\newcommand{\GroupRolloutIndex}{i}
\newcommand{\GroupSize}{G}
\newcommand{\GrpoSampledObjective}{\widehat{\mathcal{J}}_{\text{GRPO}}}
\newcommand{\GuardBandDuration}{t_{\text{guard}}}
\newcommand{\GuardedContactPredicate}{P_{\text{contact}}}
\newcommand{\GuestImageMemory}{M_{\text{guest}}}
\newcommand{\GuestKernelBootTime}{T_{\text{kernel}}}
\newcommand{\HadamardRotationMatrix}{\mathbf{H}}
\newcommand{\HalfPrecisionGemvIntensity}{I_{\text{GEMV, 16-bit}}}
\newcommand{\HammingDataBitCount}{n}
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\newcommand{\HandoffExceedanceProbability}{\epsilon}
\newcommand{\HandoverTimeout}{T_{\text{timeout}}}
\newcommand{\HandoverTimeoutCeiling}{T_{\text{config}}}
\newcommand{\HandshakeProtocolLatency}{T_{\text{handshake}}}
\newcommand{\HardwareCommonCauseRate}{\lambda_{\text{other}}}
\newcommand{\HardwareParameterVector}{\boldsymbol{\xi}_{\text{real}}}
\newcommand{\HardwareServiceLife}{T_{\text{lifetime}}}
\newcommand{\HarmInclusiveTaskCost}{C_{\text{with\ harm}}}
\newcommand{\HarmInstant}{t_{\text{harm}}}
\newcommand{\HarmLeadTime}{t_{\text{harm}}}
\newcommand{\HarmonicDriveTorsionalStiffness}{k_c}
\newcommand{\HarmonicPeriodMultiple}{m}
\newcommand{\HarnessConnectorResistance}{R_{\text{harness}}}
\newcommand{\HashBucketCount}{N_{\text{buckets}}}
\newcommand{\HashEntrySize}{s_{\text{entry}}}
\newcommand{\HashRingServer}{s}
\newcommand{\HashRingServerCount}{N}
\newcommand{\HashRingServerSet}{\mathcal{S}_{\text{srv}}}
\newcommand{\HashedGridMemory}{M_{\text{hashed}}}
\newcommand{\HazardClassIndex}{h}
\newcommand{\HazardOnset}{t_{\text{onset}}}
\newcommand{\HazardOnsetBound}{t_{\text{bound}}}
\newcommand{\HazardousProposalRate}{\lambda_{\text{proposal}, h}}
\newcommand{\HazardousWorldState}{s_B}
\newcommand{\HeadAxisLabel}{h}
\newcommand{\HeadByteAllocation}{M_{\text{head}}}
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\newcommand{\HeadQueryProjectionMatrix}{\mathbf{W}_{Q,i}}
\newcommand{\HeadSamplingProbability}{\theta}
\newcommand{\HeadValuesProjectionMatrix}{\mathbf{W}_{V,i}}
\newcommand{\HeadingAngle}{\theta_{\text{head}}}
\newcommand{\HeadingCurvature}{\kappa}
\newcommand{\HealthyStepTime}{T_{\text{healthy}}}
\newcommand{\HeartbeatAge}{\Delta t_{\text{heartbeat}}}
\newcommand{\HeartbeatInterval}{H}
\newcommand{\HeartbeatReceiptTime}{t_{\text{local}}}
\newcommand{\HeartbeatSequenceCounter}{k}
\newcommand{\HeartbeatTimestamp}{t^{\text{hb}}}
\newcommand{\HeatDissipationPower}{P_{\text{heat}}}
\newcommand{\HermiteBasisFunction}{h}
\newcommand{\HermiteInterpolationFraction}{\lambda}
\newcommand{\HermiteSegmentProgress}{s}
\newcommand{\HertzianContactCoefficient}{k_{c}}
\newcommand{\HertzianContactStiffness}{k_H}
\newcommand{\HiddenActivation}{\mathbf{h}}
\newcommand{\HiddenActivationMatrix}{\mathbf{H}}
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\newcommand{\HighOccupancyBinCount}{m}
\newcommand{\HigherPriorityFrames}{hp}
\newcommand{\HigherPriorityTasks}{hp}
\newcommand{\HighestSignalFrequency}{f_{\max}}
\newcommand{\HnswLayerIndex}{L}
\newcommand{\HnswLevelDraw}{U}
\newcommand{\HnswLevelNormalization}{m_\ell}
\newcommand{\HnswMaximumConnectionCount}{M}
\newcommand{\HnswSearchBeamWidth}{ef_{\text{search}}}
\newcommand{\HoldingTorque}{\tau_{\text{hold}}}
\newcommand{\HomogeneousCameraFramePoint}{\tilde{\mathbf{p}}_C}
\newcommand{\HomogeneousPixelCoordinates}{\tilde{\mathbf{m}}}
\newcommand{\HomogeneousPointCoordinates}{\tilde{\mathbf{p}}}
\newcommand{\HomogeneousTransform}{\mathbf{T}_{AB}}
\newcommand{\HomogeneousWorldFramePoint}{\tilde{\mathbf{p}}_W}
\newcommand{\HomographyColumn}{\mathbf{h}}
\newcommand{\HomographyColumnEntry}{h}
\newcommand{\HomographyConstraintVector}{\mathbf{v}}
\newcommand{\HomographyScaleFactor}{\lambda}
\newcommand{\HomomorphicCiphertext}{c}
\newcommand{\HomomorphicFunction}{f}
\newcommand{\HomomorphicPublicKey}{pk}
\newcommand{\HomomorphicSecretKey}{sk}
\newcommand{\HorizonStepIndex}{k}
\newcommand{\HorizontalFocalLength}{f_x}
\newcommand{\HostArrivalTimestamp}{t_{\text{host}}}
\newcommand{\HostHeartbeatRate}{f_{\text{heartbeat}}}
\newcommand{\HostLinkBandwidth}{\text{BW}_{\text{link}}}
\newcommand{\HostPower}{P_{\text{host}}}
\newcommand{\HostPreparationTime}{T_{\text{prep}}}
\newcommand{\HostRuntimeCount}{N_{\text{host}}}
\newcommand{\HostSoftwareTime}{T_{\text{sw}}}
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\newcommand{\ParentCapabilityGrant}{\Gamma_{\text{parent}}}
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\newcommand{\ParetoTaskDemand}{X}
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\newcommand{\PdnTargetImpedance}{Z_{\text{target}}}
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\newcommand{\PearsonFitStatistic}{\chi^2}
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\newcommand{\PermissionPathParameters}{\kappa}
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\newcommand{\PermittedUsePassProbability}{p_{\text{use}}}
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\newcommand{\PersonalizationVector}{\mathbf{p}}
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\newcommand{\PerturbationNormOrder}{p}
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\newcommand{\PerturbedSignalDimension}{D}
\newcommand{\PerturbedSignalVector}{\mathbf{x}}
\newcommand{\PfcWatchdogTimeout}{T_{\text{watchdog}}}
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\newcommand{\PhysicalStlRobustness}{\rho_{\text{real}}}
\newcommand{\PhysicalTrajectorySet}{\mathcal{S}_{\text{real}}}
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\newcommand{\PixelColumnCoordinate}{u}
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\newcommand{\PlanarSpecialEuclideanGroup}{SE(2)}
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\newcommand{\PlantDynamicsFunction}{\mathbf{f}}
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\newcommand{\PointToPointMessageSize}{n}
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\newcommand{\PoissonSubintervalCount}{n}
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\newcommand{\ReadEndSequenceCount}{c_{\text{end}}}
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\newcommand{\SignificanceLevel}{\alpha}
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\newcommand{\SimplexProjection}{\Pi_{\Delta^{K-1}}}
\newcommand{\SimulatedPlantDynamics}{f_{\text{sim}}}
\newcommand{\SimulatedReturnLevel}{z_{T, \text{sim}}}
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\newcommand{\SimulatedTrajectorySet}{\mathcal{S}_{\text{sim}}}
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\newcommand{\SimulatedUnbrakedImpactForce}{F_{\text{unbraked,sim}}}
\newcommand{\SimulationParameterSet}{\Xi}
\newcommand{\SimulationParameterVector}{\boldsymbol{\xi}}
\newcommand{\SimulatorStepFunction}{\mathbf{F}}
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\newcommand{\SingleLinkBandwidth}{\text{BW}_{\text{link}}}
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\newcommand{\SingleStepImitationError}{\epsilon}
\newcommand{\SingularSpectrumMatrix}{\mathbf{\Sigma}}
\newcommand{\SisaShardCount}{K}
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\newcommand{\SkewDisplacementEnvelope}{d_{\text{skew}}}
\newcommand{\SkillSelectionDistribution}{p}
\newcommand{\SlaveDriveCount}{S}
\newcommand{\SlaveDriveIndex}{s}
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\newcommand{\SlipRegularizationSpeed}{\epsilon_{\text{reg}}}
\newcommand{\SloVector}{\mathbf{S}}
\newcommand{\SlowdownPersistenceUpdateCount}{K}
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\newcommand{\SmallPerturbation}{\delta}
\newcommand{\SmallestJacobianSingularNumber}{\sigma_{\min}}
\newcommand{\SmallestKeySingularNumber}{\sigma_{\min(S, d_{\text{head}})}}
\newcommand{\SmoothedClassifier}{g}
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\newcommand{\SmoothedTargetAction}{\tilde{\mathbf{a}}'}
\newcommand{\SmoothedTopClassProbability}{p_{A}}
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\newcommand{\SmoothingNoiseStandardDeviation}{\sigma}
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\newcommand{\SnapshotStepIndex}{s}
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\newcommand{\SoftExpectedRemainingReturn}{V}
\newcommand{\SoftenedStudentProbabilityVector}{\mathbf{p}_{\text{student}}^{(T_{\text{distill}})}}
\newcommand{\SoftenedTeacherProbabilityVector}{\mathbf{p}_{\text{teacher}}^{(T_{\text{distill}})}}
\newcommand{\SoftmaxFunction}{\sigma}
\newcommand{\SoftmaxInputScore}{x}
\newcommand{\SoftmaxInputVector}{\mathbf{x}}
\newcommand{\SoftmaxNormalizer}{z}
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\newcommand{\SontagStabilizingControl}{\mathbf{u}_{\text{Sontag}}}
\newcommand{\SourceCurrent}{I_{\text{source}}}
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\newcommand{\SourceFramePoint}{\mathbf{p}_B}
\newcommand{\SourceSinkPathSet}{\Pi}
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\newcommand{\SparseMatrixRowCount}{n_{\text{rows}}}
\newcommand{\SparseProductInputComponent}{v}
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\newcommand{\SpatialErrorVector}{\boldsymbol{\epsilon}_{\text{spatial}}}
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\newcommand{\SpatialRowIndex}{i}
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\newcommand{\SpecialEuclideanGroup}{SE(3)}
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\newcommand{\SpecializedEnergyProductivity}{\rho_{\text{energy,specific}}}
\newcommand{\SpecificHeatCapacity}{c_p}
\newcommand{\SpectrallyNormalizedWeightMatrix}{\overline{\mathbf{W}}}
\newcommand{\SpeculativeDecodingSpeedup}{S_{\text{spec}}}
\newcommand{\SpeculativeDraftLength}{K}
\newcommand{\SpeculativeDraftTokenCount}{k}
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\newcommand{\SplineDegree}{p}
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\newcommand{\SpringBrakeEngagementTime}{t_{\text{engage}}}
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\newcommand{\SprtExpectedLogIncrement}{m}
\newcommand{\SprtLowerBoundary}{\Lambda_{\text{lo}}}
\newcommand{\SprtLowerLogBoundary}{b_{\text{low}}}
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\newcommand{\SprtUpperLogBoundary}{u}
\newcommand{\SquareGemmDimension}{N}
\newcommand{\SquaredGaussianSumQuantile}{\chi}
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\newcommand{\StabilizationErrorThreshold}{\epsilon_{\text{stable}}}
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\newcommand{\StackedContactForces}{\mathbf{f}_{\text{c}}}
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\newcommand{\StaleSupportRate}{\epsilon_{\text{stale}}}
\newcommand{\StalenessAdjustedLearningRate}{\eta'}
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\newcommand{\StallCounterTorque}{\tau_{\text{stall}}}
\newcommand{\StallTravelBound}{d_{\text{F1}}}
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\newcommand{\StandardNormalCumulativeDistribution}{\Phi}
\newcommand{\StandardNormalQuantile}{z_q}
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\newcommand{\StandbyReserveCount}{N_{\text{reserve}}}
\newcommand{\StandstillSpeedThreshold}{\epsilon_{\text{vel}}}
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\newcommand{\StateCotangentSpace}{T_{\mathbf{x}}^*}
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\newcommand{\StateEstimationError}{\mathbf{e}}
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\newcommand{\StateTrackingWeightMatrix}{\mathbf{W}_q}
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\newcommand{\StaticBatchingOperations}{O_{\text{traditional}}}
\newcommand{\StaticChassisPower}{P_{\text{chassis\_static}}}
\newcommand{\StaticDataDistribution}{p_{\text{data}}}
\newcommand{\StationaryConditionPolynomial}{P}
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\newcommand{\StatisticalSignificanceLevel}{\alpha_{\text{sig}}}
\newcommand{\StatorCoreTemperature}{T_{\text{core}}}
\newcommand{\StatorIronTemperature}{T_{\text{iron}}}
\newcommand{\StatorThermalCapacitance}{C_{\text{th}}}
\newcommand{\StatusPollCount}{N_{\text{poll}}}
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\newcommand{\SteadyStateAvailability}{A}
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\newcommand{\SteadyStateErrorCovariance}{\mathbf{P}_{\text{est}}}
\newcommand{\SteadyStateIntegratorState}{\mathbf{z}_{\text{ss}}}
\newcommand{\SteadyStateTemperatureRise}{\Delta T_{\max}}
\newcommand{\SteadyStateTrackingError}{\mathbf{e}_{\text{ss}}}
\newcommand{\SteeringTransferTime}{T}
\newcommand{\StepFailureHazard}{a}
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\newcommand{\StereoBaseline}{b}
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\newcommand{\StlIntervalEnd}{b}
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\newcommand{\StockoutNormalQuantile}{z_{1-\epsilon}}
\newcommand{\StockoutProbabilityTarget}{\epsilon}
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\newcommand{\StoredEmbedding}{\mathbf{z}}
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\newcommand{\ToleranceEllipsoidMatrix}{\mathbf{E}}
\newcommand{\ToleratedFailureProbability}{\epsilon}
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\newcommand{\ToolCenterPointFrame}{\mathcal{F}_{\text{TCP}}}
\newcommand{\ToolCenterPointPose}{\mathbf{T}_{\text{TCP}}}
\newcommand{\ToolCenterPointPosition}{\mathbf{p}_{\text{TCP}}}
\newcommand{\ToolClearanceDistance}{d_{\text{tool}}}
\newcommand{\ToolDeclarationTokenCount}{S_{\text{tools}}}
\newcommand{\ToolDescriptorSize}{S_{\text{desc}}}
\newcommand{\ToolPlanarAbscissa}{x}
\newcommand{\ToolPlanarOrdinate}{y}
\newcommand{\ToolPositionError}{e_{\text{tool}}}
\newcommand{\ToolServiceCount}{N_{\text{tool}}}
\newcommand{\ToolSpeedLimit}{v_{\text{TCP}}}
\newcommand{\ToolTargetState}{s}
\newcommand{\ToolTime}{T_{\text{tool}}}
\newcommand{\ToolTipPose}{\mathbf{T}_{\text{tool}}}
\newcommand{\TopClassCount}{k}
\newcommand{\TopMagnitudeIndicator}{\mathbf{1}_{|\mathbf{g}| \geq |\mathbf{g}|_{(K_{\text{top}})}}}
\newcommand{\TopTokenSampleCount}{k}
\newcommand{\TopologicalPlaceNode}{p^{\text{place}}}
\newcommand{\TopologyEndpointCount}{N}
\newcommand{\TorqueCutoffCommand}{\mathbf{u}_{\text{cutoff}}}
\newcommand{\TorqueEffortPenaltyWeight}{\lambda_{\tau}}
\newcommand{\TorqueProducingCurrent}{i_q}
\newcommand{\TorqueRateLimit}{\dot{\tau}_{\max}}
\newcommand{\TorqueRegularizationWeight}{w_\tau}
\newcommand{\TorqueSlewPenaltyWeight}{\lambda_{\dot{\tau}}}
\newcommand{\TorusFirstAxisChipCount}{X}
\newcommand{\TorusSecondAxisChipCount}{Y}
\newcommand{\TorusThirdAxisChipCount}{Z}
\newcommand{\TotalChipPower}{P_{\text{total}}}
\newcommand{\TotalCollisionQueryTime}{T_{\text{query\_total}}}
\newcommand{\TotalDataCost}{C_{\text{total}}}
\newcommand{\TotalDelayPhaseShift}{\phi_{\text{total}}}
\newcommand{\TotalEnergy}{E_{\text{total}}}
\newcommand{\TotalExecutedTrialCount}{N_{\text{total}}}
\newcommand{\TotalFailureRecoveryTime}{T_{\text{recovery}}}
\newcommand{\TotalGateCharge}{Q_g}
\newcommand{\TotalInterventionCount}{N_{\text{int}}}
\newcommand{\TotalKineticEnergy}{E_{k,\text{total}}}
\newcommand{\TotalMemoryFootprint}{M_{\text{total}}}
\newcommand{\TotalPipelineLatency}{L_{\text{lat,total}}}
\newcommand{\TotalTrainingRunCost}{C_{\text{total}}}
\newcommand{\TraceCrashCount}{N_{\text{crash}}}
\newcommand{\TraceCrashIndex}{i}
\newcommand{\TraceDecay}{\lambda}
\newcommand{\TraceIntermediateAdjoint}{\bar{v}}
\newcommand{\TraceIntermediateTangent}{\dot{v}}
\newcommand{\TraceIntermediateVariable}{v}
\newcommand{\TrackIndex}{i}
\newcommand{\TrackLastUpdateTime}{t_{\text{last}}}
\newcommand{\TrackLossTimeout}{T_{\text{lost}}}
\newcommand{\TrackPositionDeviation}{\sigma_{\text{track}}}
\newcommand{\TrackPredictedErrorCovariance}{\mathbf{P}_{t|t-1}}
\newcommand{\TrackPredictedStateEstimate}{\hat{\mathbf{x}}_{t|t-1}}
\newcommand{\TrackedEvidenceHorizon}{\tau_{\text{ev,tracked}}}
\newcommand{\TrackerFrameIndex}{t}
\newcommand{\TrackingErrorAcceleration}{\ddot{\mathbf{e}}}
\newcommand{\TrackingErrorAllowance}{\epsilon_{\text{pos}}}
\newcommand{\TrackingErrorBound}{\epsilon_{\text{track}}}
\newcommand{\TrackingErrorDistance}{e}
\newcommand{\TrackingErrorRate}{\dot{\mathbf{e}}}
\newcommand{\TrackingErrorVector}{\mathbf{e}}
\newcommand{\TrackingRelaxationSlack}{\delta}
\newcommand{\TrackingSlackPenaltyWeight}{p}
\newcommand{\TrackingStartInstant}{t_{\text{start}}}
\newcommand{\TractionLimitedDeceleration}{a}
\newcommand{\TractiveForce}{F_x}
\newcommand{\TrainableStateMemory}{M_{\text{train}}}
\newcommand{\TrainerUpdateTime}{T_{\text{train}}}
\newcommand{\TrainingCompute}{O_{\text{train}}}
\newcommand{\TrainingCost}{C_{\text{train}}}
\newcommand{\TrainingDataset}{D}
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\newcommand{\TrainingEmpiricalCdf}{F_{P_0}}
\newcommand{\TrainingExample}{z}
\newcommand{\TrainingFailureCount}{N_{\text{failures}}}
\newcommand{\TrainingFeatureMean}{\mu}
\newcommand{\TrainingFeatureStandardDeviation}{\sigma}
\newcommand{\TrainingLabelMask}{\mathcal{M}_{\text{train}}}
\newcommand{\TrainingLoss}{\mathcal{L}_{\text{train}}}
\newcommand{\TrainingPathPredictor}{f_{\text{train}}}
\newcommand{\TrainingPipelineFunction}{f}
\newcommand{\TrainingRunCount}{N_{\text{runs}}}
\newcommand{\TrainingSampleThroughput}{R_{\text{train}}}
\newcommand{\TrainingSchemaKeySet}{\mathcal{K}_{\text{train}}}
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\newcommand{\TrainingSplitFraction}{f_{\text{train}}}
\newcommand{\TrainingStepCount}{K}
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\newcommand{\TrainingStepOperations}{O_{\text{step}}}
\newcommand{\TrainingStepTime}{T_{\text{step}}}
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\newcommand{\TrainingTime}{T_{\text{train}}}
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\newcommand{\Trajectory}{\tau}
\newcommand{\TrajectoryAction}{a}
\newcommand{\TrajectoryAdmissionRate}{\lambda_{\text{traj}}}
\newcommand{\TrajectoryBudget}{\mathbf{B}}
\newcommand{\TrajectoryCallCount}{N_{\text{call}}}
\newcommand{\TrajectoryCategoryIndex}{j}
\newcommand{\TrajectoryCheckpointCount}{N_{\text{ckpt}}}
\newcommand{\TrajectoryDivergenceResidual}{e_{\text{div}}}
\newcommand{\TrajectoryDuration}{T_{\text{traj}}}
\newcommand{\TrajectoryGoodput}{\mathcal{G}}
\newcommand{\TrajectoryObservation}{o}
\newcommand{\TrajectoryOptimizationObjective}{\mathcal{U}}
\newcommand{\TrajectoryRecordDuration}{T_{\text{record}}}
\newcommand{\TrajectoryRecordFrame}{\mathcal{F}}
\newcommand{\TrajectoryReturn}{R}
\newcommand{\TrajectoryReward}{R}
\newcommand{\TrajectorySampleCount}{N}
\newcommand{\TrajectoryState}{s}
\newcommand{\TrajectoryStateSpace}{\mathcal{S}}
\newcommand{\TrajectoryTimeLimit}{T_{\max}}
\newcommand{\TrajectoryTokenCost}{C_{\text{session}}}
\newcommand{\TrajectoryTokenLimit}{N_{\max}}
\newcommand{\TrajectoryTurnLimit}{H_{\max}}
\newcommand{\TransceiverDelay}{t_{\text{transceiver}}}
\newcommand{\TransductionInstant}{t_{\text{transduction}}}
\newcommand{\TransferBufferCount}{N_{\text{transfer}}}
\newcommand{\TransferDiscrepancyBound}{\epsilon_{\text{gap}}}
\newcommand{\TransferTimeForPayload}{T_{\text{transfer}}}
\newcommand{\TransferredDataVolume}{D_{\text{vol,transfer}}}
\newcommand{\TransformGraph}{\mathcal{G}}
\newcommand{\TransformGraphEdgeSet}{\mathcal{E}}
\newcommand{\TransformGraphFrame}{\mathcal{F}}
\newcommand{\TransformGraphVertexSet}{\mathcal{V}}
\newcommand{\TransformRotationBlock}{\mathbf{R}_{AB}}
\newcommand{\TransformTranslationBlock}{\mathbf{t}_{AB}}
\newcommand{\TransformationInputCount}{k}
\newcommand{\TransformedCartesianPosition}{\mathbf{p}'}
\newcommand{\TransformedHomogeneousPointCoordinates}{\tilde{\mathbf{p}}'}
\newcommand{\TransistorThresholdVoltage}{V_{\text{th}}}
\newcommand{\TransitionStorageSize}{b_{\text{tr}}}
\newcommand{\TranslationComponent}{v}
\newcommand{\TranslationVector}{\mathbf{v}}
\newcommand{\TranslationalDriftError}{e_{\text{trans}}}
\newcommand{\TranslationalJerk}{j}
\newcommand{\TranslationalKineticEnergy}{E_{k,\text{trans}}}
\newcommand{\TransmissionEnergyPerByte}{E_{\text{tx}}}
\newcommand{\TransmissionInputInertia}{J_{\text{in}}}
\newcommand{\TransmissionTorsionalStiffness}{K_{\text{joint}}}
\newcommand{\TransverseStandardDeviation}{\sigma_\perp}
\newcommand{\TravelPosition}{x}
\newcommand{\TraverseDistance}{d}
\newcommand{\TreatedNeighborFraction}{e}
\newcommand{\TreatmentAssignment}{w}
\newcommand{\TreatmentConversionRate}{\hat{p}_B}
\newcommand{\TreatmentSampleSize}{m_B}
\newcommand{\TreeAllReduceCompletionTime}{T_{\text{tree}}}
\newcommand{\TreeLikeAllReduceTime}{T_{\text{tree-like}}}
\newcommand{\TreeSearchExplorationWeight}{c_{\text{explore}}}
\newcommand{\TreeSearchLeafScore}{V}
\newcommand{\TreeSearchMeanReturn}{Q}
\newcommand{\TreeSearchPriorDistribution}{p_{\text{prior}}}
\newcommand{\TreeSearchVisitCount}{N_{\text{visit}}}
\newcommand{\TrialCampaignWallTime}{T_{\text{wall}}}
\newcommand{\TrialCompletionIndicator}{Y}
\newcommand{\TrialCondition}{c}
\newcommand{\TrialControlStepCount}{N_{\text{steps}, i}}
\newcommand{\TrialCount}{N}
\newcommand{\TrialFailureIndicator}{x}
\newcommand{\TrialFailureProbability}{p_{\text{fail}}}
\newcommand{\TrialIndex}{n}
\newcommand{\TrialInterventionCount}{n_{\text{int}}}
\newcommand{\TrialLoadReversalCount}{n_{\text{rev}}}
\newcommand{\TrialPassProbability}{p_{\text{trial}}}
\newcommand{\TrialPathDistance}{d}
\newcommand{\TrialResetLatency}{T_{\text{reset}}}
\newcommand{\TrialResetTime}{t_{\text{reset}}}
\newcommand{\TrialRolloutTime}{t_{\text{rollout}}}
\newcommand{\TrialRunTime}{t_{\text{run}}}
\newcommand{\TrialSuccessProbability}{p}
\newcommand{\TriggeredWindowArrivalCount}{Q_{\text{arr}}}
\newcommand{\TrueClearance}{C_{\text{true}}}
\newcommand{\TrueLabelComponent}{y}
\newcommand{\TrueOutcome}{Y}
\newcommand{\TrueOutcomeLevel}{v}
\newcommand{\TrueReferentScore}{c_{y_j}}
\newcommand{\TruncatedDistanceSample}{d_{\mu}}
\newcommand{\TruncatedKeyApproximation}{\hat{\mathbf{K}}_r}
\newcommand{\TruncatedKeyLeftSingularMatrix}{\mathbf{U}_r}
\newcommand{\TruncatedKeyRightSingularMatrix}{\mathbf{V}_r}
\newcommand{\TruncatedKeySpectrumMatrix}{\mathbf{\Sigma}_r}
\newcommand{\TruncationBandVolume}{V_{\text{band}}}
\newcommand{\TruncationRank}{r}
\newcommand{\TruthInstrumentSet}{\mathcal{I}_{\text{truth}}}
\newcommand{\TsdfAccumulatedDistance}{D}
\newcommand{\TsdfAccumulatedWeight}{W}
\newcommand{\TsdfObservationWeight}{w}
\newcommand{\TsdfTruncationBand}{\mu_{\text{tsdf}}}
\newcommand{\TsdfWeightCap}{W_{\max}}
\newcommand{\TuckerCoreTensor}{\mathcal{G}}
\newcommand{\TurnAppendLength}{\Delta S_{\text{turn}}}
\newcommand{\TurnBoundaryDelay}{t_{\text{boundary}}}
\newcommand{\TurnBudgetUsedFraction}{\rho_H}
\newcommand{\TurnContext}{c}
\newcommand{\TurnContributionDeviationBound}{\sigma}
\newcommand{\TurnCovarianceBoundScale}{C}
\newcommand{\TurnCovarianceDecayRatio}{\alpha_{\text{corr}}}
\newcommand{\TurnFailureRate}{\lambda_{\text{fail}}}
\newcommand{\TurnGeneratedTokenCount}{n_{\text{gen}}}
\newcommand{\TurnHorizon}{H}
\newcommand{\TurnIndex}{t}
\newcommand{\TurnInputTokenCost}{C_{\text{in}}}
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\newcommand{\TurnOffGateResistance}{R_{g,\text{off}}}
\newcommand{\TurnOnGateResistance}{R_{g,\text{on}}}
\newcommand{\TurnOutputTokenCost}{C_{\text{out}}}
\newcommand{\TurnPayloadProbability}{q_{\text{turn}}}
\newcommand{\TurnReadCount}{r}
\newcommand{\TurnReconstructionTime}{T_{\text{recon}}}
\newcommand{\TurnReexecutionTime}{T_{\text{exec}}}
\newcommand{\TurnReward}{r}
\newcommand{\TurnSandboxCost}{C_{\text{sandbox}}}
\newcommand{\TurnStateGrowth}{s_{\text{turn}}}
\newcommand{\TurnToolFeeCost}{C_{\text{tools}}}
\newcommand{\TustinEigenvalue}{\lambda_{\text{Tustin}}}
\newcommand{\TustinTransitionMatrix}{\mathbf{A}_{\text{Tustin}}}
\newcommand{\TwinCriticReturnEstimate}{Q_{\phi_i}}
\newcommand{\TwistCoordinates}{\boldsymbol{\xi}}
\newcommand{\TwistCovarianceMatrix}{\boldsymbol{\Sigma}_{\boldsymbol{\xi}}}
\newcommand{\TwistMatrix}{\hat{\boldsymbol{\xi}}}
\newcommand{\TwistRotationAngle}{\theta_{\text{rot}}}
\newcommand{\TwistRotationCoordinates}{\boldsymbol{\omega}}
\newcommand{\TwistTranslationMatrix}{\mathbf{V}}
\newcommand{\TwistWedgeOperator}{\wedge}
\newcommand{\UnacceptedTaskMeanCost}{C_{\text{fail}}}
\newcommand{\UnassistedCompletionCount}{N_{\text{unassisted}}}
\newcommand{\UnassistedSuccessIndicator}{A}
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\newcommand{\UncachedPromptLength}{S_{\text{new}}}
\newcommand{\UncertainScenarioParameters}{x}
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\newcommand{\UnconstrainedParseFailureRate}{p_{\text{err}}}
\newcommand{\UndampedNaturalFrequency}{\omega_n}
\newcommand{\UnfoldedConvolutionOutput}{\mathbf{Y}_{\text{col}}}
\newcommand{\UnfoldedPatchMatrix}{\mathbf{X}_{\text{col}}}
\newcommand{\UniformDistribution}{\mathcal{U}}
\newcommand{\UniformLoopTurnCost}{C_{\text{turn}}}
\newcommand{\UnitAliasMultiplier}{k}
\newcommand{\UnitImpulseFunction}{\delta}
\newcommand{\UnitQuaternion}{\mathbf{q}_{\text{quat}}}
\newcommand{\UnitQuaternionScalarComponent}{q_w}
\newcommand{\UnitQuaternionSphere}{S^3}
\newcommand{\UnitQuaternionXComponent}{q_x}
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\newcommand{\UnitQuaternionZComponent}{q_z}
\newcommand{\UnitTemperatureProbabilityVector}{\mathbf{p}_{\tau_{\text{temp}}=1}}
\newcommand{\UnlabeledPool}{U}
\newcommand{\UnloadedTickExecutionTime}{t_{\text{unloaded}}}
\newcommand{\UnmitigatedHardwareFaultRate}{\lambda_{\text{unmitigated}}}
\newcommand{\UnmodeledCommandPathDelay}{\Delta t_{\text{bus}}}
\newcommand{\UnobservableForceOffset}{\Delta}
\newcommand{\UnobservedUnknownSpace}{\mathcal{M}_{\text{unknown}}}
\newcommand{\UnprojectionJacobian}{\mathbf{J}_{\text{unproj}}}
\newcommand{\UnrecoverableDamageCost}{C_{\text{blast}}}
\newcommand{\UnreservedStopDelay}{\delta}
\newcommand{\UnscaledAttentionScoreMatrix}{\mathbf{A}}
\newcommand{\UnshardedLocalExecutionTime}{T_{\text{compute}}}
\newcommand{\UnsquashedLatentAction}{\mathbf{u}}
\newcommand{\UntilPrefixTime}{t''}
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\newcommand{\UpdatedErrorCovariance}{\mathbf{P}_{k|k}}
\newcommand{\UpdatedModelParameters}{\theta_{\text{new}}}
\newcommand{\UpdatedParameterSubset}{\theta_{\text{updated}}}
\newcommand{\UpdatedSoftmaxRunningMaximum}{m'}
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\newcommand{\UpperRatioClippingWidth}{\epsilon_{\text{high}}}
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\newcommand{\UprightRange}{r}
\newcommand{\UpstreamGradientElement}{\bar{y}}
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\newcommand{\UsefulModelFlops}{O_{\text{useful}}}
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\newcommand{\UvloThresholdVoltage}{V_{\text{UVLO}}}
\newcommand{\VacuumLightSpeed}{c}
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\newcommand{\ValidatedMotionDuration}{T_{\text{yield}}}
\newcommand{\ValidatedProposalEvent}{e_{\text{prop}}}
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\newcommand{\VanishingTimeOffset}{\epsilon}
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\newcommand{\VariationalParameters}{\phi}
\newcommand{\VectorAngle}{\phi}
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\newcommand{\VerificationCheckCount}{K}
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\newcommand{\VerificationTrialIndex}{j}
\newcommand{\VerificationTrialTime}{T_{\text{trial}}}
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\newcommand{\VerifierFalseAcceptanceRate}{\epsilon_{\text{FA}}}
\newcommand{\VerifierFalseRejectionRate}{\epsilon_{\text{FR}}}
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\newcommand{\VerifierVerdict}{V}
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\newcommand{\VertexSlotTime}{d}
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\newcommand{\VisualConditioningEmbedding}{\mathbf{E}}
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\newcommand{\WeightMatrix}{\mathbf{W}}
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\newcommand{\WeightVector}{\mathbf{w}}
\newcommand{\WeightedNormArgument}{\mathbf{z}}
\newcommand{\WheelAngularSpeed}{\omega}
\newcommand{\WheelDiameter}{D_{\text{wheel}}}
\newcommand{\WheelInertia}{J_{\text{wheel}}}
\newcommand{\WheelNormalLoad}{F_z}
\newcommand{\WheelRadius}{r}
\newcommand{\WheelRevolutionCount}{N_{\text{rev}}}
\newcommand{\WheelTrackWidth}{b}
\newcommand{\WhiteNoiseProcess}{\mathcal{WN}}
\newcommand{\WholeBodyDecisionVariables}{\mathbf{z}}
\newcommand{\WholeRepositoryShardGrant}{\Gamma_{b}}
\newcommand{\WidthMultiplier}{w_{\text{mult}}}
\newcommand{\WidthScalingRatio}{\beta}
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\newcommand{\WindingCurrent}{I}
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\newcommand{\WithinTaskRunStandardDeviation}{\sigma_{\text{run}}}
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\newcommand{\WorkedTraceProductAdjoint}{\bar{c}}
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\newcommand{\WorkedTraceSineAdjoint}{\bar{b}}
\newcommand{\WorkedTraceSquare}{a}
\newcommand{\WorkedTraceSquareAdjoint}{\bar{a}}
\newcommand{\WorkerBaseCommitHash}{h_{\text{base}}}
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\newcommand{\WorkspaceExtentX}{L_x}
\newcommand{\WorkspaceExtentY}{L_y}
\newcommand{\WorkspaceExtentZ}{L_z}
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\newcommand{\WorldBaseTransform}{\mathbf{T}_{WB}}
\newcommand{\WorldBodyTransform}{\mathbf{T}_{wb}}
\newcommand{\WorldFramePoint}{\mathbf{p}_{\text{world}}}
\newcommand{\WorldLidarTransform}{\mathbf{T}_{wl}}
\newcommand{\WorldMapTransform}{\mathbf{T}_{\text{world}}^{\text{map}}}
\newcommand{\WorldPointCovariance}{\boldsymbol{\Sigma}_{\text{world}}}
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\newcommand{\WorldToCameraRotation}{\mathbf{R}_{CW}}
\newcommand{\WorldToCameraTransform}{\mathbf{T}_{CW}}
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\newcommand{\WornPlantTransitionDistribution}{p_{n \gg 1}}
\newcommand{\WorseArmSuccessProbability}{p_{\text{low}}}
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\newcommand{\WorstCaseInterruptLatency}{t_{\text{irq,lat}}}
\newcommand{\WorstCasePermissionCycleTime}{T_{\text{exec}}}
\newcommand{\WorstCaseResponseTime}{R}
\newcommand{\WristAngularVelocity}{\boldsymbol{\omega}_{\text{wrist}}}
\newcommand{\WristCameraEmbedding}{\mathbf{e}_{\text{wrist}}}
\newcommand{\WristLinearVelocity}{\mathbf{v}_{\text{wrist}}}
\newcommand{\WristObjectTransform}{\mathbf{T}_{\text{wrist} \leftarrow \text{object}}}
\newcommand{\WristSkewDisplacement}{\Delta \mathbf{p}_{\text{skew}}}
\newcommand{\XAxisAngularRate}{\omega_x}
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\newcommand{\XTorqueCeiling}{\tau_x}
\newcommand{\YAxisAngularRate}{\omega_y}
\newcommand{\YAxisRotationCoordinate}{\omega_y}
\newcommand{\YForceCeiling}{F_y}
\newcommand{\YPositionErrorEnvelope}{e_y}
\newcommand{\YTaskToleranceSemiAxis}{r_y}
\newcommand{\YTorqueCeiling}{\tau_y}
\newcommand{\YawRate}{\omega_z}
\newcommand{\YoshikawaManipulability}{w}
\newcommand{\ZAxisRotationCoordinate}{\omega_z}
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\newcommand{\ZPositionErrorEnvelope}{e_z}
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\newcommand{\ZTorqueCeiling}{\tau_z}
\newcommand{\ZTransformVariable}{z}
\newcommand{\ZeroEntryIndicator}{\mathbf{1}_{\{(\mathbf{W}_{\text{sparse}})_{ij} = 0\}}}
\newcommand{\ZeroFailureTrialCount}{n}
\newcommand{\ZeroFailureUpperConfidenceLimit}{p_{\text{fail, upper}}^{95\%}}
\newcommand{\ZeroMatrix}{\mathbf{0}}
\newcommand{\ZeroMomentPoint}{\mathbf{p}_{\text{ZMP}}}
\newcommand{\ZeroVector}{\mathbf{0}}
\newcommand{\ZohHoldTravel}{d_{\text{ZOH}}}
\newcommand{\ZohPhaseShift}{\phi_{\text{ZOH}}}
\newcommand{\ZohTransportDelay}{\tau_{\text{ZOH}}}
\newcommand{\AuthorityLevels}{\AuthorityExposureLevel}
\newcommand{\BreakEvenTaskVolume}{\BreakEvenTaskCount}
\newcommand{\PertokenPricesCache}{\CachedInputTokenPrice}
\newcommand{\ContentionAndCoherencyCoefficientsTwo}{\CoherencyCoefficient}
\newcommand{\CoherencyContentionCoefficient}{\CoherencyCoefficient}
\newcommand{\TaskDeadline}{\CompletionDeadline}
\newcommand{\Deadline}{\CompletionDeadline}
\newcommand{\Confidentiality}{\ConfidentialityLabel}
\newcommand{\ContentionAndCoherencyCoefficients}{\ContentionCoefficient}
\newcommand{\CostPerAcceptedTaskEff}{\CostPerAcceptedTask}
\newcommand{\DuplicateeffectProbability}{\DuplicateEffectProbability}
\newcommand{\EstimatedAdvantageK}{\EstimatedHistoryAdvantage}
\newcommand{\EstimatedValueK}{\EstimatedRemainingReturn}
\newcommand{\ExpectedIrreversibleLossA}{\ExpectedIrreversibleLoss}
\newcommand{\PertokenPricesIn}{\InputTokenPrice}
\newcommand{\CohensKappa}{\InterAnnotatorAgreement}
\newcommand{\KvBlockSizeBlk}{\KvBlockSize}
\newcommand{\MaskedFinetuningLossSft}{\MaskedFinetuningLoss}
\newcommand{\ValidityIntervalOfAMemoryEntryBirth}{\MemoryEntryBirthGeneration}
\newcommand{\ValidityIntervalOfAMemoryEntryDeath}{\MemoryEntryDeathGeneration}
\newcommand{\MemoryHeldDuringAWait}{\MemoryHoldDuringWait}
\newcommand{\OptimalCheckpointIntervalCkpt}{\OptimalCheckpointInterval}
\newcommand{\PertokenPricesOut}{\OutputTokenPrice}
\newcommand{\PertokenDecodeTime}{\PerTokenDecodeTime}
\newcommand{\TokenLossMask}{\PositionLossMask}
\newcommand{\LossMaskT}{\PositionLossMask}
\newcommand{\ValidityIntervalOfAMemoryEntryQ}{\QueryWorkspaceGeneration}
\newcommand{\RequestAndReplylossProbabilitiesTwo}{\ReplyLossProbability}
\newcommand{\RequestAndReplylossProbabilities}{\RequestLossProbability}
\newcommand{\EnvironmentStateK}{\RolloutEnvironmentState}
\newcommand{\StalesupportRate}{\StaleSupportRate}
\newcommand{\StateLevels}{\StateExposureLevel}
\newcommand{\GoalContract}{\TaskContractGoal}
\newcommand{\Environment}{\TaskEnvironment}
\newcommand{\Vocabulary}{\TokenVocabulary}
\newcommand{\ActionProposal}{\TrajectoryAction}
\newcommand{\ConcurrentTrajectoriesTwo}{\TrajectoryAdmissionRate}
\newcommand{\ActionObservation}{\TrajectoryObservation}
\newcommand{\Observation}{\TrajectoryObservation}
\newcommand{\StateAtTurn}{\TrajectoryState}
\newcommand{\PerturnAppend}{\TurnAppendLength}
\newcommand{\Context}{\TurnContext}
\newcommand{\Horizon}{\TurnHorizon}
\newcommand{\VerifierFalseacceptance}{\VerifierFalseAcceptanceRate}
\newcommand{\VerifierFalseacceptanceFr}{\VerifierFalseRejectionRate}
\newcommand{\WorkerCountWorkers}{\WorkerCount}
\newcommand{\AllocationHoldTime}{\AllocatedWallTime}
\newcommand{\CheckpointIntervalCkpt}{\CheckpointInterval}
\newcommand{\AcceleratorsPerNodeNodes}{\ClusterNodeCount}
\newcommand{\CommunicationcomputationRatio}{\CommunicationComputeRatio}
\newcommand{\PercomponentFailureRate}{\ComponentFailureRate}
\newcommand{\Correlation}{\CorrelationCoefficient}
\newcommand{\DataparallelDegreeDp}{\DataParallelDegree}
\newcommand{\NumberOfDevices}{\DeviceCount}
\newcommand{\TypeIAndTypeIiErrorRatesStat}{\HypothesisTestFalseNegativeRate}
\newcommand{\TypeIAndTypeIiErrorRatesSig}{\HypothesisTestSignificanceLevel}
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Fleet Foundations
Single-node performance engineering centers on memory hierarchy latencies and roofline ridge points. The moment a job spans multiple machines, network latency replaces cache latency, component failure becomes a statistical certainty, and interconnect bandwidth dictates scaling efficiency. This appendix gathers the fleet-scale constants, scaling laws, and thermal limits that underpin the book’s multi-node chapters, giving readers the physical baselines needed to reason about cluster-scale infrastructure. In C³ terms, these numbers define the physical scale on which compute capacity, communication bandwidth, and coordination overhead become measurable.
How to Use This Appendix
This appendix is designed as a reference when translating fleet-scale design questions into specific physical constraints and identifying available optimization levers. The five routing bullets below map common engineering questions to the sections that follow.
When evaluating inter-node communication : Consult section 1.2 for the bandwidth and latency hierarchy.When sizing GPU allocations : Consult section 1.3 to analyze why doubling accelerator counts does not halve training time.When calculating cluster reliability : Check section 1.2 for mean time between failures (MTBF) tables and failure probability models.When evaluating power and cooling limits : Consult section 1.4 for power density ceilings and cooling constraints.When budgeting runtime overhead : Use section 1.2 for the four overhead categories that erode useful wall time.
The Three System Paradigms
Machine learning infrastructure at scale inherits design DNA from two distinct computing lineages and then breaks the assumptions of both. Understanding where ML systems borrow from high-performance computing (HPC) and where they borrow from warehouse-scale computing (WSC) is essential for choosing the right design trade-offs. A system architect who treats ML training as “just HPC” will build infrastructure that cannot tolerate failures. One who treats it as “just web services” will build infrastructure that cannot sustain the tight coupling that synchronous training demands.
Warehouse-scale computing
WSC systems descend from the web services tradition, aiming instead to maximize queries per second across loosely coupled services such as search, email, and social media. Because requests are independent, these fleets mix hardware generations over commodity Ethernet and rely on redundancy for fault tolerance: when a node fails, a load balancer reroutes traffic to another replica without the user noticing. Dynamic schedulers such as Kubernetes and Borg bin-pack workloads elastically across interchangeable, expendable “cattle” nodes.
The ML fleet as hybrid architecture
ML systems require the computational throughput of HPC to train massive models with synchronous gradient updates, yet they must operate at the scale and failure rates of WSC, where thousands of accelerators run for weeks. This tension produces a hybrid architecture that borrows selectively from both lineages.
Training workloads are synchronous and bandwidth-hungry like HPC jobs, whereas long-running fleet operations demand WSC-style fault tolerance, and inference workloads reverse the blend by pairing WSC latency sensitivity with HPC compute density. To serve both regimes, the network fuses TCP/IP control planes with InfiniBand or NVLink data planes, while fault tolerance combines checkpoint/restart with elastic training strategies where supported, allowing jobs to adjust resource allocations even when resharding, state restoration, or worker restarts remain necessary, and scheduling pairs all-or-nothing gang allocation for training with dynamic preemption and replacement. ML fleets therefore cannot adopt either HPC or WSC patterns wholesale; they combine elements from each according to the workload phase, as the design trade-offs across the three paradigms in table 1 show.
Five single-machine models reappear throughout the distributed chapters, each extended by a fleet-scale term. The D·A·M Taxonomy
Iron law (\(\Time\approx \DataVolume/\Bandwidth + \Operations/(\PeakRate \cdot \EfficiencyHw) + \Latency\) , Iron Law Mapping Roofline Model (The roofline bound Network-bound workloads are limited by inter-node bandwidth, which the communication-computation ratio in section 1.3.3 measures.Amdahl’s Law : Caps strong-scaling speedup at \(1/\SerialFraction\) for serial fraction \(\SerialFraction\) (Amdahl 1967 ) . At fleet scale, the serial fraction includes synchronization barriers, exposed collective communication, and pipeline bubbles (section 1.3.2 ).Training state rule (Numeric Precision and Training Memory Little’s Law (\(\RequestConcurrency = \ArrivalRate \RequestTimeInSystem\) ) (Little 1961 ) : Relates in-flight requests to arrival rate and time in system. At fleet scale, it sizes the number of serving replicas behind a load balancer (Queuing theory for batched inference
Amdahl, Gene M. 1967. “Validity of the Single Processor Approach to Achieving Large Scale Computing Capabilities.” Proceedings of the April 18-20, 1967, Spring Joint Computer Conference on - AFIPS ’67 (Spring) , AFIPS ’67 (spring), 483–85. https://doi.org/10.1145/1465482.1465560.
Little, John D. C. 1961. “A Proof for the Queuing Formula: \(L = \lambda W\) Operations Research 9 (3): 383–87. https://doi.org/10.1287/opre.9.3.383.
The transition from single-machine to fleet-scale reasoning requires extending these models with new dimensions: network topology, failure probability, and coordination overhead. The numbers in the next section provide the quantitative foundation for that extension.
Numbers Every Fleet Engineer Should Know
Just as single-machine analysis depends on a set of core numbers, fleet-scale engineering is governed by a set of predictable ratios and scaling behaviors. The single-machine numbers still apply within each node, but a new set of numbers governs the spaces between nodes. Hardware ratios vary with generation and provisioning; failure-rate scaling also depends on the assumed failure model. These reference values provide sanity checks while the stated models explain how to substitute different assumptions.
Systems Perspective 1.1: Node-level numbers for fleet reasoning Fleet reasoning depends on a few node-level numbers that directly affect fleet design: the training state rule of about 14 bytes per parameter for common Adam checkpoint state, with larger footprints when gradients, activations, metadata, or extra optimizer buffers are included; NVLink vs. high-bandwidth memory (HBM) bandwidth for intra-node parallelism placement; and peak FLOP/s and HBM capacity for model FLOPs utilization (MFU), effective FLOP/s, batch sizing, and model sharding.
Table 2 and the communication numbers below give inter-node and current-generation values; the memory hierarchy and roofline ridge points within a node (
Accelerator Memory Hierarchy ) provide the necessary baseline. Those node-level baselines make the fleet-level ratios easier to interpret.
Systems Perspective 1.2: Three fleet numbers that matter most
Bandwidth gap : NVLink per-direction bandwidth within a node is ~9× faster than an InfiniBand NDR port between nodes. This ratio determines where parallelism boundaries belong, with model parallelism within a node and data parallelism across nodes.MTBF scaling : A cluster’s mean time between failures is the single-component mean time to failure (MTTF) divided by the number of components (\(1/\RequiredComponentCount\) scaling). At 100,000 GPUs, expect a failure every 30 minutes.Effective utilization : After MFU, scaling efficiency, and overhead losses compound, a cluster with 1,024 GPUs delivers roughly 34.0 percent of its peak FLOP/s as useful training work.
Back-of-envelope checks need only the condensed values in table 2 . Design and debugging work should use the detailed tables in each subsection.
Invariant scaling ratios
These relationships expose architectural trade-offs and model assumptions across the fleet. Because hardware ratios shift across generations, their numerical values must be re-evaluated for the target system.
Network hierarchy ratio
The bandwidth gap between intra-node and inter-node communication is an important architectural comparison. Chip-to-chip links (NVLink, ICI) connect through short, wide, dedicated paths on a shared substrate. Inter-node links (InfiniBand, Ethernet) must traverse cables, switches, and protocol stacks. These design differences often favor intra-node bandwidth, but lane counts, multiple network ports, and topology determine the actual ratio.
Currently, NVLink 4.0 provides 9× more per-direction bandwidth than an InfiniBand NDR port. The ratio compares one accelerator’s aggregate intra-node connectivity with one network port; it is not a universal constant. Parallelism placement must compare the workload’s traffic with the sustained bandwidth actually provisioned at each boundary.
System MTBF scaling
For \(\RequiredComponentCount\) independent components under a constant-failure-rate assumption, system MTBF (equation 1 ) scales inversely with component count:
\[ \SystemMtbf = \frac{\ComponentMttf}{\RequiredComponentCount} \tag{1}\]
Under this model the relation is exact rather than approximate. Doubling the cluster size halves the time between failures. At 100,000 GPUs with a per-GPU MTTF of 50,000 hours, the cluster experiences a GPU failure every 30 minutes. No fault tolerance strategy can avoid this; the question is how quickly the system recovers.
AllReduce overhead scaling
The bandwidth-optimal Ring AllReduce algorithm transfers \(2(\AcceleratorCount-1)\CollectivePayloadSize/\AcceleratorCount\) bytes per participant, where \(\CollectivePayloadSize\) is the collective payload size and \(\AcceleratorCount\) is the number of participating GPUs (Patarasuk and Yuan 2009 ) . As \(\AcceleratorCount\) grows large, this approaches \(2\CollectivePayloadSize\) per GPU, so the per-GPU bandwidth term is nearly independent of the number of GPUs; aggregate cluster traffic still grows with the number of participants. This is why Ring AllReduce scales well in the bandwidth term. The latency term, however, grows as \(2(\AcceleratorCount-1) \times \MessageStartupLatency\) , making latency the bottleneck for small messages on large rings. This trade-off motivates hierarchical AllReduce strategies that use Ring AllReduce within nodes and Tree AllReduce across nodes.
Patarasuk, Pitch, and Xin Yuan. 2009. “Bandwidth Optimal All-Reduce Algorithms for Clusters of Workstations.” Journal of Parallel and Distributed Computing 69 (2): 117–24. https://doi.org/10.1016/j.jpdc.2008.09.002.
Communication numbers
Communication defines the boundaries of parallelism. These tables quantify the bandwidth and latency at each tier of the network hierarchy, from the fastest intra-node links to the slowest wide-area connections. For any distributed ML operation, the relevant question is which network tier its communication crosses.
Table 3 lists the bandwidth tiers that determine where tensor, pipeline, and data-parallel traffic should stay local. Each row pairs a representative interconnect with its typical role in the cluster hierarchy.
One-way latency across interconnect tiers (table 4 ) determines whether synchronous collective operations are viable. Whereas microsecond-scale fabrics support per-step gradient synchronization, wide-area links require asynchronous or amortized updates.
These latency differences favor low-latency fabrics for frequent synchronization. Cross-data-center training must amortize transfers, place pipeline boundaries carefully, or relax synchronization when its learning algorithm permits.
Compute numbers
Raw peak FLOP/s is a necessary but misleading metric for fleet capacity planning. Single-device execution losses and additional scaling losses reduce effective throughput. Multiplying the two is valid only when their measurement windows exclude overlapping costs. Understanding these losses transforms fleet sizing from guesswork into engineering.
MFU measures what fraction of peak FLOP/s a training workload actually achieves. Well-optimized large-model training on current hardware achieves 30–50 percent MFU. The gap comes from memory stalls, kernel launch overhead, pipeline bubbles, and suboptimal operator fusion. These are workload-dependent reference values. Fleet-step MFU already includes in-step communication and pipeline losses; a separate scaling factor applies only to a single-device baseline that excludes those losses.
Scaling efficiency (\(\ScalingEfficiency\) ) measures how much useful computation survives as accelerators are added. Modeled scaling efficiency for the 175B reference job (table 5 ) shows how per-GPU throughput erodes as cluster size grows at a fixed global batch.
Past a few thousand GPUs the fixed per-step costs of pipeline fill and drain dominate, which is why production systems grow the global batch, interleave pipeline stages, and overlap collectives with compute. Systems built this way, such as MegaScale, report near-linear scaling on thousands of GPUs (Jiang et al. 2024 ) .
Jiang, Ziheng, Haibin Lin, Yinmin Zhong, Qi Huang, Yangrui Chen, Zhi Zhang, Yanghua Peng, et al. 2024. “MegaScale : Scaling Large Language Model Training to More Than 10,000 GPUs .” 21st USENIX Symposium on Networked Systems Design and Implementation (NSDI 24) , 745–60.
Coordination numbers
At fleet scale, coordination (failure recovery, checkpointing, and maintenance) consumes a measurable fraction of wall-clock time. These numbers quantify the costs of keeping a large cluster running.
Failure rates by cluster size
Cluster MTBF scaling (table 6 , calculated for a per-GPU MTTF of 50,000 hours) dictates the required checkpoint frequency. As accelerator counts grow from hundreds to tens of thousands, the interval between expected hardware interruptions shrinks from days to minutes.
Failure is no longer an edge case at 8,192 GPUs and above. GPU-only MTBF is about 366.2 min at 8,192 GPUs, and the probability of at least one GPU failure within 24 hours is 98 percent. Fault tolerance is not optional at fleet scale; it is a prerequisite for completing any long-running training job. Fault Tolerance
Checkpoint sizes
Checkpointing is the primary recovery mechanism, and its cost depends on the model size. Table 7 shows checkpoint sizes for common mixed-precision Adam training checkpoints (14 bytes per parameter: 2 bytes for BF16 weights and 12 bytes for FP32 master weights, momentum, and variance). Gradients are normally transient and recomputed after restore rather than serialized as durable checkpoint state.
Overhead budgets
Failure recovery and checkpointing are only part of the wall-clock budget. At fleet scale, table 8 separates four recurring categories of overhead that consume time not spent on useful training.
Power and sustainability numbers
Fleet-scale capacity planning and sustainability reporting require a few power numbers that every fleet engineer should know. At fleet scale, the critical numbers are rack power density , power usage effectiveness (PUE) , and the air-cooling limit , which determine where construction is feasible and what the facility load will be (table 9 ).
A GPU-only power proxy (MW) = (number of GPUs \(\times\) thermal design power (TDP) per GPU)/\(10^6\) excludes host CPUs, memory, storage, and networking. Facility load = total IT load \(\times\) PUE; applying PUE to the GPU-only proxy omits those other IT loads and is not a complete facility budget. A 10,000-GPU H100 cluster at 700 W each gives a 7 MW GPU-only proxy; multiplying by PUE 1.55 gives 10.8 MW before accounting for the other IT loads. Section 1.4 carries this power arithmetic into facility overhead, and Sustainable AI
Current hardware reference (c. 2025–2026)
These numbers (table 10 ) are point-in-time hardware references for back-of-envelope calculations. Compute, memory capacity, and interconnect performance evolve at different rates.
Table 10: Fleet-Scale Hardware Reference (c. 2025–2026) : Per-accelerator specifications for the 2025–2026 generations. The B200, MI300X, and TPU v6e (Trillium) represent the frontier of fleet-scale compute density and memory bandwidth. Intra-node link rates are bidirectional aggregates per accelerator. Google does not publish a per-chip TDP for the TPU v6e.
BF16/FP16 Peak 2,250 TFLOP/s
1,307 TFLOP/s
918 TFLOP/s
Memory Bandwidth 8 TB/s
5.30 TB/s
1.64 TB/s
HBM Capacity 180 GB
192 GB
32 GB
Intra-Node Link 1,800 GB/s (NVLink 5.0)
896 GB/s (Infinity Fabric) (AMD 2023 )
800 GB/s (ICI) (Google Cloud 2026 )
TDP 1,000 W
750 W
Not published
AMD. 2023. AMD Instinct MI300X Accelerators
Google Cloud. 2026. TPU v6e
At the cluster level, a single DGX H100 SuperPOD scalable unit packages 32 DGX H100 nodes (256 H100 GPUs) (NVIDIA 2023 ) , whereas full data-center deployments link dozens of such pods together. Meta’s Grand Teton clusters reach 24,576 H100 GPUs (Meta Engineering 2024 ) , Google’s Tensor Processing Unit (TPU) v5p pods scale to 8,960 chips, and the largest announced clusters as of 2025 exceed 100,000 accelerators.
NVIDIA. 2023. NVIDIA DGX SuperPOD : Next Generation Scalable Infrastructure for AI Leadership
Meta Engineering. 2024. Building Meta ’s GenAI Infrastructure . Engineering at Meta Blog.
Scaling Physics
Quadrupling a training allocation from 1,024 GPUs to 4,096 GPUs often delivers only 2.5\(\times\) the step throughput because network transfers, synchronization barriers, and pipeline bubbles consume a growing share of each step. The numbers in the previous section describe what the hardware can do; scaling physics describes what happens when more of it is brought to bear. Reasoning starts from the ceiling on a single accelerator (the roofline) and then asks what survives as devices are added, governed by three models: Amdahl’s Law extended to fleet overhead, the communication-computation ratio, and weak scaling behavior.
From the accelerator roofline to the fleet
Fleet performance rests on per-accelerator performance. The roofline bound \(\AttainableComputeRate = \min(\PeakRate, \ArithmeticIntensity\times \Bandwidth)\) , and its ridge point \(\RooflineRidgePoint = \PeakRate/\Bandwidth\) , which is about 295.2 FLOP/byte for an H100. A kernel below the ridge is memory bound, and a kernel above it is compute bound.
The roofline is a single-accelerator bound, but its logic is what scales. At fleet level the binding bandwidth is no longer HBM but the interconnect, and the bytes moved are the gradients and activations crossing the network. The ratio of network time to local arithmetic time is the communication-computation ratio (section 1.3.3 ), the roofline of the fleet and the lens the rest of this section applies.
A training job provisioned with 4,096 GPUs achieves only 2.5× the throughput of a 1,024-GPU run. That observation implies about 51.9 percent scaling efficiency at the larger size, so scaling physics provides the diagnostic tools to determine whether the system is performing as physics allows or whether there is an engineering problem to fix.
Amdahl’s Law at fleet scale
When a distributed training loop spends 10 percent of every step waiting on exposed AllReduce collectives, pipeline bubbles, or checkpoint flushes, adding thousands of extra GPUs cannot accelerate the fixed job by more than 10\(\times\) . Amdahl’s Law establishes that the maximum speedup of a parallel system is limited by its serial fraction \(\SerialFraction\) . At fleet scale, the “serial fraction” includes more than sequential code; it covers every operation that forces all \(\DeviceCount\) GPUs to wait:
AllReduce synchronization : All GPUs must complete their gradient computation before any can proceed.Pipeline bubbles : The warmup and cooldown phases of pipeline parallelism leave stages idle.Checkpoint writes : Even asynchronous checkpoints contend for storage bandwidth.Python-level overhead : Single-threaded operations in the training loop (data loading, metric logging).
To see the fleet-scale implications, consider a training workload where 10 percent of wall-clock time is spent in synchronization, communication, and other serial overhead. Amdahl’s Law gives the following speedups:
With 32 GPUs: 7.8× speedup (~24 percent scaling efficiency)
With 256 GPUs: 9.7× speedup (diminishing returns begin)
With 1,024 GPUs: 9.9× speedup (approaching the ceiling)
With 8,192 GPUs: 10× speedup (nearly at the Amdahl limit)
With \(\DeviceCount\to \infty\) : capped at 10×
With just 10 percent serial overhead, no amount of hardware can deliver more than 10× speedup on this fixed workload. This is why fleet-scale training does not simply add more GPUs to the same problem; it scales the problem (weak scaling) to keep the serial fraction small relative to the total work.
Systems Perspective 1.3: The compound overhead trap The 10 percent serial fraction above is optimistic. In practice, fleet-scale serial overhead accumulates from multiple sources: 5 percent pipeline bubbles + 3 percent checkpointing + 10 percent failure recovery + 5 percent maintenance. These are not all strictly serial in the Amdahl sense (some overlap with computation), but they illustrate how quickly small per-category overheads compound into significant throughput loss.
The communication-computation ratio
When 256 GPUs train a 350M-parameter model over InfiniBand, gradient AllReduce takes three times longer than the forward and backward passes combined, leaving the accelerators idle for most of the step. Any distributed training strategy must determine whether computation between synchronization points takes long enough to hide the communication. Equation 2 defines the communication-computation ratio (\(\CommunicationComputeRatio\) ) that answers this directly:
\[ \CommunicationComputeRatio= \frac{\CommunicationTime(\DeviceCount)}{\IronLawComputeTime/\DeviceCount} \tag{2}\]
When \(\CommunicationComputeRatio< 1\) , computation takes longer than communication; hiding that communication still requires independent work and sufficient resources. When \(\CommunicationComputeRatio> 1\) , the system is communication bound, and GPUs spend more time waiting for data than computing on it.
The ratio is the time-domain form of the communication intensity (CI) defined in Communication intensity (the CI ratio) \(\NetworkTrafficVolume\) bytes across the network per step at effective network bandwidth \(\NetworkBandwidth\) and executes its \(\Operations\) local FLOPs at realized rate \(\RealizedDeviceComputeRate\) , then in the bandwidth-dominated regime
\[ \CommunicationComputeRatio\approx \frac{\NetworkTrafficVolume/\NetworkBandwidth}{\Operations/\RealizedDeviceComputeRate} = \CommunicationIntensity \times \frac{\RealizedDeviceComputeRate}{\NetworkBandwidth}, \qquad \CommunicationIntensity = \frac{\NetworkTrafficVolume}{\Operations} \tag{3}\]
Equation 3 shows that exposed communication begins when \(\CommunicationIntensity\) exceeds \(\NetworkBandwidth/\RealizedDeviceComputeRate\) , the network counterpart of the roofline ridge point. Because a device executes far more FLOP/s than its network link moves bytes per second, that threshold is a small fraction of a byte per FLOP, and a faster accelerator lowers it further. An accelerator upgrade alone can therefore turn a compute-bound job into a communication-bound one.
Communication-to-computation ratios across representative scenarios (table 11 ) reveal why the parallelism strategy must match the workload’s arithmetic footprint. Comparing a multi-billion-parameter model against a smaller architecture on the same interconnect shows how quickly collective synchronization can overtake local compute.
The 7B model on 256 GPUs achieves \(\CommunicationComputeRatio=\) 0.11, so communication takes about 11.2 percent as long as computation, which can be partially overlapped. The 340M model on the same cluster has \(\CommunicationComputeRatio=\) 3. Communication dominates, which makes this configuration communication-bound. The solution is either to use fewer GPUs (reduce \(\DeviceCount\) in the AllReduce) or to increase the computation per step (larger batch size, gradient accumulation). For the single 16 MB point-to-point payload, the ideal bandwidth-only lower bound is \(\CommunicationComputeRatio=\) 0.036, which is not a tensor-parallel collective model and cannot by itself establish whether an implementation is compute-bound.
Weak scaling at fleet scale
Provisioning 8,192 GPUs to finish a fixed 7B-parameter training run faster quickly hits Amdahl’s synchronization wall, whereas using those same 8,192 GPUs to train a 70B- or 700B-parameter model keeps per-device arithmetic work high enough to amortize communication. Amdahl’s Law paints a pessimistic picture because it assumes a fixed problem size. In practice, fleet-scale training often follows weak scaling : the problem size (tokens, data, model parameters) grows proportionally with the number of GPUs. Gustafson’s Law captures this more optimistic view (Gustafson 1988 ) .
Gustafson, John L. 1988. “Reevaluating Amdahl’s Law.” Communications of the ACM 31 (5): 532–33. https://doi.org/10.1145/42411.42415.
Weak scaling reflects how fleets are actually used, not merely a mathematical convenience. Engineers do not use 8,192 GPUs to train a 7B model faster; they use them to train a 70B or 700B model in reasonable time. As models and training batches grow, engineers often increase the compute performed between synchronization points. For dense Transformers, compute is commonly estimated as about 6\(\times\) parameters \(\times\) tokens, while gradient communication scales with parameter bytes, so the communication-computation ratio \(\CommunicationComputeRatio\) depends on batch size, sequence length, accumulation, and parallelism layout rather than on parameter count alone.
Systems Perspective 1.4: The compound loss of fleet utilization A 1,024-GPU H100 cluster has a peak aggregate throughput of 1,012,736 TFLOP/s. This illustrative decomposition keeps the factors disjoint: MFU represents per-device kernel and memory efficiency, scaling efficiency represents distributed communication and pipeline losses, and runtime goodput represents checkpointing, failure recovery, and maintenance. The effective throughput is:
\[\EffectiveFleetThroughput = \DeviceCount\PeakRate \times \ModelFlopsUtilization \times \ScalingEfficiency \times \RuntimeGoodput\]
\[= 1,012,736 TFLOP/s \times 0.50 \times 0.83 \times 0.82 \approx 344,634.1 \text{ TFLOP/s}\]
The cluster delivers 34.0 percent of its peak FLOP/s as useful training work. The remaining 66.0 percent is explained by retained factors of 50 percent MFU, 83 percent scaling efficiency, and 82 percent runtime goodput.
These losses reflect the physics of fleet-scale computation rather than a failure of engineering. Additional GPUs can increase useful throughput while reducing efficiency, but the gain must be measured against smaller configurations. The goal is not to reach 100 percent utilization; the goal is to deliver trained models faster than any smaller configuration could.
Thermal and Power Physics
Compute performance ultimately converts to heat, and heat must be removed. At fleet scale, thermal and power constraints are primary concerns. They determine where construction is feasible, how densely accelerators can be packed, and what the operating costs will be. A cluster that is architecturally sound but thermally infeasible cannot be built.
A cluster design calling for 10,000 H100 GPUs at 700 W each (a 7 MW GPU-only power proxy) gives 10.8 MW after applying PUE 1.55. The complete facility budget must also include host, memory, storage, and network power inside the IT boundary before applying PUE. This is the electrical load of a small town. Before optimizing software, the physics of power delivery and heat removal must be feasible.
Power density wall
The shift from traditional data center workloads to AI training has multiplied the power each rack draws, and table 12 quantifies the gap. As accelerator TDPs climb toward a kilowatt per chip, standard air-cooled server rooms quickly run out of thermal headroom.
The 5.8× increase in rack power density between traditional and AI workloads is a fundamental constraint on facility design, not merely an engineering challenge. Existing data centers built for 12 kW racks cannot simply be “retrofitted” with AI hardware. The electrical distribution, cooling infrastructure, and floor loading must all be redesigned. This is why new AI-focused data centers are being built from the ground up with liquid cooling as the baseline assumption.
The energy hierarchy at scale
In a legacy enterprise data center, delivering 1 MW of electrical power to GPU racks requires another 580 kW just to run the chillers, fans, and power conversion equipment that keep the room operating. PUE measures facility energy overhead: total facility energy divided by IT equipment energy (The Green Grid 2007 ; Uptime Institute 2022 ) . A PUE of 1.0 would mean all facility energy is delivered to IT equipment; values above 1.0 represent overhead for cooling, power conversion, lighting, and other facility systems. Cooling technology drives much of that overhead, so PUE falls from legacy enterprise facilities to liquid-cooled AI data centers (table 13 ).
The Green Grid. 2007. Green Grid Data Center Power Efficiency Metrics: PUE and DCIE . The Green Grid.
Uptime Institute. 2022. Uptime Institute Global Data Center Survey 2022 . Uptime Institute.
For fleet-scale cost calculations, PUE directly multiplies the electricity bill. A 10 MW IT load in a legacy data center (PUE 1.58) requires 15.8 MW total, while the same load in a liquid-cooled facility (PUE 1.06) requires only 10.6 MW, a savings of 5.2 MW. For a facility that runs continuously at the reference on-premises electricity price of $0.07/kWh, that difference is worth about $3.2M per year. Sustainable AI
Summary
Fleet scale changes every constraint a single machine imposes. ML fleets combine the tight coupling of high-performance computing for training with the elastic fault tolerance of warehouse-scale computing for serving, and they cannot adopt either tradition wholesale. NVLink provides about 9× more per-direction bandwidth than an InfiniBand NDR port, which is why bandwidth-hungry parallelism stays inside the node. Cluster MTBF scales as \(1/\RequiredComponentCount\) , so at 8,192 GPUs a GPU fails about every 366.2 minutes, with 98 percent probability of at least one failure in 24 hours. After disjoint factors for MFU (about 50 percent), scaling efficiency (about 83 percent), and runtime goodput (about 82 percent), a 1,024-GPU cluster delivers about 34.0 percent of peak FLOP/s as useful work.
Three considerations govern the design response. The communication-computation ratio \(\CommunicationComputeRatio\) indicates the potential for overlap and the pressure from communication; dependencies and available resources determine how much communication can be hidden behind computation. Weak scaling, which grows the problem with the fleet, preserves useful work per device, while strong scaling remains the right choice when the goal is to finish fixed work sooner. Rack power density, about 5.8× the traditional reference for current AI racks, determines the cooling design, subject to the installed heat-removal capacity rather than a universal threshold.