$$
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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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$$
The C\(^3\) Taxonomy
In a distributed training cluster, “it is slow” is even less informative than on a single machine. A job can stall because accelerators are underutilized (Compute), gradient synchronization saturates interconnects (Communication), or checkpointing and recovery delay progress (Coordination). This appendix extends single-node diagnostics to fleet-scale infrastructure, providing the C\(^3\) scorecard used throughout the book to isolate the binding bottleneck across distributed systems.
How to Use This Appendix
This appendix is designed as a reference. Start with the diagnostic summary table, form a hypothesis about which C\(^3\) axis dominates, and then pick the tool that can confirm (or falsify) that hypothesis.
When training throughput is low, check model FLOPs utilization (MFU), communication fraction, and runtime goodput, then map each to its compute, communication, or coordination axis. When scaling efficiency drops below expectations, use the Fleet Law decomposition to identify which term grew. When cost is exploding, use the C\(^3\) scorecard to ensure that effort targets the dominant term, not a nonbottleneck.
The C\(^3\) taxonomy is the diagnostic framework for fleet-scale ML systems engineering. Where the single-machine D·A·M framework (The D·A·M Taxonomy \(^3\) taxonomy diagnoses bottlenecks across the distributed fleet. Most fleet-scale performance problems can be diagnosed by identifying the dominant axis: Compute (are the accelerators doing useful math?), Communication (is the network moving data fast enough?), or Coordination (is the system spending too much time on synchronization, failure recovery, and scheduling?). Many production bottlenecks sit at intersections between these axes.
From D·A·M to C\(^3\)
The C\(^3\) taxonomy does not replace D·A·M; it extends it. When a workload moves from one machine to a fleet, each D·A·M axis acquires new failure modes that the single-machine framework cannot capture, and table 1 traces each axis to the C\(^3\) dimension that inherits it.
The most important row in table 1 is the last one. On a single machine, the overhead term (\(\Latency\) ) in the iron law is typically small, consisting of kernel launch latency, Python dispatch, and synchronization barriers. At fleet scale, coordination becomes an axis in its own right: checkpoint writes, failure detection and recovery, pipeline bubble overhead, scheduler preemptions, and maintenance windows collectively consume a significant fraction of wall time. Coordination is the axis that single-machine diagnosis lacks, and much of this book is devoted to it.
Diagnostic Summary
With the mapping in place, diagnosis becomes a matter of matching symptoms to the axis that constrains the fleet. Each C\(^3\) axis maps to a physical constraint, observable symptoms, measurable metrics, and engineering levers, and table 2 collects these mappings as the main reference for fleet-scale diagnosis.
The Fleet Law
Under ideal partitioning of fixed work, the same classification can be written as a time budget for each distributed step. The Fleet Law, introduced in The Fleet Law
\[ \DistributedStepTime(\DeviceCount) = \frac{\IronLawComputeTime}{\DeviceCount} + \CommunicationTime(\DeviceCount) + \SynchronizationTime(\DeviceCount) - \OverlappedTime \]
This equation is the fleet-scale counterpart of the iron law. Where the iron law decomposes single-machine execution into data movement, compute, and overhead, the Fleet Law decomposes distributed execution into local arithmetic, network data transfer, and synchronization logic. The diagnostic strategy is identical: measure each term, identify which dominates, and direct engineering effort at the dominant term.
Component decomposition
Each Fleet Law term maps to specific measurable activities that can be isolated in a timeline trace or cluster telemetry log. By attributing every millisecond of step duration to one of four terms, engineers can tell whether hardware is doing useful math, moving bytes across wires, or waiting on barriers:
\(\IronLawComputeTime/\DeviceCount\) \(\DeviceCount\) devices. Governed by MFU and per-GPU kernel efficiency. Improvements come from better kernels, mixed precision, and operator fusion.\(\CommunicationTime(\DeviceCount)\) \(\SynchronizationTime(\DeviceCount)\) \(\OverlappedTime\)
Operational events may be amortized into the step budget or accounted for separately in runtime goodput. The same checkpoint or recovery delay must not appear in both. The compute-time fraction follows:
\[ \UsefulComputeFraction = \frac{\IronLawComputeTime/\DeviceCount}{\DistributedStepTime(\DeviceCount)} \]
When \(\UsefulComputeFraction\) drops below 0.5, the fleet spends more time on communication and coordination than on useful arithmetic. This compute-time fraction is not total fleet efficiency; useful fleet efficiency also depends on MFU, scaling efficiency, and goodput. The C\(^3\) taxonomy identifies which noncompute term is responsible.
C\(^3\) Intersections
In a production cluster, a stalled training step rarely comes from a single isolated resource; a slow collective can idle tensor cores while a synchronous checkpoint write blocks both the GPUs and the network fabric. The Venn diagram in The C^3 Intersection Landscape \(\FleetComputeTaxonomyAxis\) ), the networking backbone (\(\FleetCommunicationTaxonomyAxis\) ), and the orchestration layer (\(\FleetCoordinationTaxonomyAxis\) ). Because real bottlenecks arise at the pairwise and central boundaries where two or more of these dimensions compound, diagnosing them requires mapping observed symptoms back to the governing intersection before selecting an engineering fix. Table 3 expands the three pure dimensions, the three pairwise boundaries, and the central convergence into concrete engineering techniques and their primary book chapters.
Compute \(\cap\) communication
When a GPU finishes its backward pass in 40 ms but waits 60 ms for an AllReduce collective to return gradients over the network, its tensor cores sit idle even though both the compute kernels and the links are working as specified. This intersection governs whether the system can hide communication behind computation. The communication-to-compute ratio (\(\CommunicationComputeRatio= \CommunicationTime(\DeviceCount)/(\IronLawComputeTime/\DeviceCount)\) ) is the key diagnostic metric (The communication-computation ratio \(\CommunicationComputeRatio< 1\) , computation takes longer than communication and there is enough arithmetic duration to hide the transfer in principle; dependencies and resource contention determine whether that overlap is achievable. When \(\CommunicationComputeRatio> 1\) , GPUs complete their local work before the network delivers the next round of operands, and the system becomes communication-bound.
In a GPU trace, this bottleneck appears as exposed communication intervals where tensor cores sit idle awaiting collective completion (such as ncclKernel_AllReduce or ncclKernel_AllGather). Engineering at this intersection focuses on overlap and compression: launching AllReduce during the backward pass before all gradients are finalized, pipelining activation transfers in 1F1B schedules, increasing microbatch sizes or gradient accumulation steps to dilute communication frequency, and applying wire quantization to reduce payload bytes, while accounting for metadata, codec time, and numerical error (Collective Communication Distributed Training
Compute \(\cap\) coordination
A 1,024-GPU cluster that pauses every thirty minutes to flush a multi-terabyte checkpoint to shared storage turns expensive arithmetic capacity into idle wait time. This intersection captures the idle compute caused by orchestration and resilience mechanisms. While compute engines should remain saturated with arithmetic, coordination overhead repeatedly forces hardware into idle states. Three distinct mechanisms govern this intersection:
Pipeline bubbles : In pipeline parallelism (PP), stages idle during pipeline fill (warmup) and drain (cooldown). For \(\PipelineParallelDegree\) pipeline stages and \(\MicrobatchCount\) microbatches, the bubble fraction \(\PipelineBubbleFraction = (\PipelineParallelDegree - 1)/(\PipelineParallelDegree - 1 + \MicrobatchCount)\) (Bubble fraction Checkpoint stalls : Synchronous checkpoint writes that block the training loop to serialize terabytes of weights and optimizer states convert expensive compute time into storage wait time. Asynchronous checkpointing mitigates this by snapshotting host-pinned or local Non-Volatile Memory Express (NVMe) memory in milliseconds, delegating remote persistence to background I/O threads.Recovery rollback : When a node fails in an \(\DeviceCount\) -device cluster, training halts and rolls back to the last valid checkpoint, squandering the arithmetic executed since that snapshot. Minimizing the failure blast radius through fast health heartbeats and dynamic node replacement bounds this compute loss (Fault Tolerance Fleet Orchestration
Communication \(\cap\) coordination
When a single GPU or network link drops packets or throttles its clock during an AllReduce collective, every other rank in the group must wait at the dependency boundary before the next step can proceed. This intersection captures the synchronization cost embedded in network communication. A collective operation (AllReduce, AllGather, ReduceScatter) moves data over physical wires (\(\FleetCommunicationTaxonomyAxis\) ) while creating dependencies among its participating ranks (\(\FleetCoordinationTaxonomyAxis\) ). It need not impose a separate cluster-wide barrier, but a delayed contribution can prevent dependent outputs from becoming ready.
The governing failure mode is that one slow participant paces the whole collective. If a single GPU or link experiences packet drops, thermal throttling, or ECC retry delays, every other GPU in the collective waits, transforming a localized degradation into a cluster-wide halt. At fleet scale (\(\DeviceCount\ge 10{,}000\) ), tail latency dominates the barrier time, because the probability that at least one node straggles approaches unity (\(1 - (1 - \NodeStragglerProbability)^\DeviceCount\to 1\) ). Engineering at this intersection focuses on decoupling synchronization from raw communication: using hierarchical collectives that isolate high-bandwidth intra-node NVLink domains from inter-node InfiniBand fabrics, adopting bounded-staleness asynchronous training to absorb transient tail latency, and enforcing topology-aware gang scheduling to align collective communication rings with physical spine-leaf switch rails (Collective Communication Fault Tolerance
Fleet-scale ML systems engineering
Partitioning a 405-billion-parameter model across 16,384 GPUs requires fitting shard states into high-bandwidth memory (HBM) (\(\FleetComputeTaxonomyAxis\) ), matching tensor and pipeline traffic to intra-node NVLink and inter-node InfiniBand bandwidths (\(\FleetCommunicationTaxonomyAxis\) ), and keeping pipeline bubbles and checkpoint stalls from consuming wall-clock time (\(\FleetCoordinationTaxonomyAxis\) ). At the center (\(\FleetComputeTaxonomyAxis\cap \FleetCommunicationTaxonomyAxis\cap \FleetCoordinationTaxonomyAxis\) ), all three constraints converge simultaneously. This is the discipline of fleet-scale ML systems engineering . Systems engineers at this center do not optimize a single accelerator or a single network switch; they manage the trade-offs of the distributed machine as an integrated whole. The central engineering challenges encompass three coupled design problems:
4D parallelism co-design : Balancing data parallelism (DP), tensor parallelism (TP), pipeline parallelism (PP), and expert parallelism (EP) so that memory consumption fits within accelerator HBM (\(\FleetComputeTaxonomyAxis\) ), high-frequency communications map to intra-node NVLink and low-frequency communications map to inter-node fabrics (\(\FleetCommunicationTaxonomyAxis\) ), and pipeline bubbles and barrier dependencies are minimized (\(\FleetCoordinationTaxonomyAxis\) ).Memory sharding vs. communication volume : Frameworks like PyTorch FSDP2 and DeepSpeed ZeRO-3 eliminate memory redundancy (\(\FleetComputeTaxonomyAxis\) ) by partitioning model states across workers, but pay for it with repeated AllGather communication (\(\FleetCommunicationTaxonomyAxis\) ) and state coordination metadata (\(\FleetCoordinationTaxonomyAxis\) ).ML productivity goodput : ML productivity goodput (MPG), the fraction of a fleet’s capacity-time that becomes useful work, is the product of scheduling goodput (\(\SchedulingGoodput\) ), runtime goodput (\(\RuntimeGoodput\) ), and program goodput (\(\ProgramGoodput\) ) (One word, several measures \(^3\) axes rather than mapping one to one onto them. Communication overlap and shorter pipeline bubbles improve program goodput, while checkpointing and recovery, which are Coordination costs, determine runtime goodput.
C\(^3\) coordination and overlap
Just as single-node execution transforms the additive iron law into a maximum bounded by dependencies (D·A·M coordination from sum to max equation 1.3 \(\OverlappedTime\) . Pipeline schedules, separate communication streams, asynchronous checkpoints, hierarchical collectives, and double-buffered data streaming each hide one C\(^3\) axis behind another (table 4 ).
Storage as the persistent substrate
A common diagnostic question is why Storage does not appear as a fourth explicit coordinate in the C\(^3\) taxonomy or as an independent \(\Glyph{T_{\text{storage}}}\) term in the Fleet Law. The physical data center is procured around Compute, Network, and Storage (Data Storage \(^3\) dimensions:
In Compute (\(\FleetComputeTaxonomyAxis\) ) : High-throughput local NVMe drives function as the tier-3 extension of the accelerator memory hierarchy (below SRAM and HBM), serving parameter offloading (ZeRO-Offload) and activation spilling.In Communication (\(\FleetCommunicationTaxonomyAxis\) ) : Remote dataset streaming from parallel file systems (Lustre, GPFS) or object stores (S3, Ceph) shares cluster network bandwidth and network interface card (NIC) capacity with collective communication traffic.In Coordination (\(\FleetCoordinationTaxonomyAxis\) ) : Durable distributed storage provides the persistent state anchor required for checkpoint serialization, node failover, and job rescheduling.
When storage latency stalls a training run, profilers identify the symptom not as raw disk latency, but either as a network communication bottleneck (\(\FleetCommunicationTaxonomyAxis\) , dataset starvation) or as a blocking coordination barrier (\(\FleetCoordinationTaxonomyAxis\) , synchronous checkpoint flush). Storage is the persistent substrate supporting all three dimensions, rather than a separate term in the step execution budget.
In the middle of a production incident, fast heuristics narrow the search space before a profiler is needed. These thresholds provide that first line of defense.
The C\(^3\) traffic light
Each C\(^3\) axis has one headline metric, and its value against two thresholds sorts the axis into green, yellow, or red (table 5 ). Comparing observed MFU, communication fraction, and runtime goodput against these bands immediately isolates which subsystem warrants a deeper trace.
The bottleneck diagnostic table
An optimization pays only when it shortens the term that binds, so the diagnosed axis decides which levers are worth pulling. Table 6 pairs each axis with the optimizations that help and those that are wasted.
C\(^3\) Case Studies
Theoretical constraints manifest as confusing symptoms in production. These scenarios illustrate how to apply the C\(^3\) taxonomy to fleet-scale performance problems. Each case isolates one dominant axis before showing which optimization levers follow from that diagnosis.
Case 1 on the underutilized compute fleet
Symptom
An engineering team provisions 4,096 H100 GPUs for a large language model training run. The training loop runs without errors, but the PyTorch Profiler shows MFU of only 15 percent. The network profiler shows communication accounts for less than 10 percent of step time. The cluster is running but barely working.
Diagnosis
The Compute axis is the leading hypothesis to investigate. With 15 percent MFU, 85 percent of the fleet’s peak capacity is not credited to useful model FLOPs. Low exposed communication makes local execution a useful starting point; kernel traces and host-side waits must distinguish inefficient arithmetic from data starvation or unobserved coordination gaps.
The fix
If traces confirm inefficient local execution, target the compute axis across 4,096 accelerators to raise per-device kernel utilization. Three adjustments recover most of the missing arithmetic throughput:
Mixed precision : Ensure BF16/FP8 Tensor Cores are engaged. A common culprit is FP32 fallback in normalization layers or loss computation.Operator fusion : Use torch.compile or similar just-in-time (JIT) compilation to fuse element-wise operations and reduce kernel launch overhead.Batch size tuning : If per-GPU batch size is too small, the matrix multiplications have insufficient arithmetic intensity to saturate the Tensor Cores.
Raising MFU from 15 percent to 50 percent on the same hardware delivers 50 percent/15 percent = 3.3× more useful work, the equivalent of tripling the fleet without buying a single GPU. Because communication is already below 10 percent of step time, all of that per-device gain translates directly into faster step completion.
Case 2 on the communication wall
Symptom
A 512-GPU data-parallel training run achieves 45 percent illustrative compute-phase efficiency during compute kernels on each GPU, which is good per-device kernel efficiency. Wall-clock MFU over the full step (including AllReduce) is only 20.2 percent because AllReduce consumes 55 percent of every training step. Scaling from 64 GPUs to 512 GPUs yields only 4× speedup instead of the expected 8×, or 50 percent scaling efficiency.
Diagnosis
The Communication axis dominates. Each GPU has healthy compute-phase efficiency, but more than half the step time is spent synchronizing gradients across the network. The system is Communication-Bound : adding more GPUs can worsen the exposed communication fraction. Under fixed-work scaling, local computation shrinks while the collective retains startup and transfer costs.
The fix
Target the communication axis without touching the per-GPU computation. Three network-level interventions shrink or hide the exposed collective time:
Compute-communication overlap : Launch AllReduce during the backward pass rather than waiting until it completes. Modern frameworks (FSDP, DeepSpeed) support this natively.Gradient compression : Apply Top-k sparsification or quantization to reduce the bytes crossing the network by 10–100\(\times\) .Hierarchical collectives : Use intra-node NVLink for the first reduction stage, then inter-node InfiniBand only for cross-node aggregation, reducing inter-node participants relative to a flat rank-level algorithm. Actual traffic savings depend on the collective layout, NIC allocation, and topology.
If communication were eliminated entirely, throughput would increase by 2.2× (Amdahl’s Law applied to the 45 percent compute fraction). Realistically, reducing communication from 55 percent to 20 percent of step time would recover most of the lost scaling.
Case 3 on coordination overhead
Symptom
A 10,000-GPU training run shows 40 percent compute-phase model FLOPs utilization per device, measured only while kernels execute, and communication accounts for only 15 percent of step time. Both figures look healthy. Yet only 60 percent of wall time is left after the run’s coordination overheads. That share is not a goodput factor, because it mixes losses that belong to different factors. Pipeline bubbles lie inside retained steps and are a program goodput loss, while checkpoints, recovery, maintenance, and preemption fall outside the steps (Accounting boundaries
Diagnosis
The Coordination axis dominates. Although per-GPU computation and inter-node communication are both efficient, nonproductive overhead consumes 40 percent of wall time: scheduler preemptions take 17 percent, failure recovery takes 10 percent (because at 10,000 GPUs, GPU failures occur about every 5 hours), pipeline bubbles take 5 percent, checkpoint writes take 3 percent, and maintenance windows take 5 percent. Because faster arithmetic alone cannot recover this lost time, checkpoint bandwidth, recovery mechanisms, and scheduling must be evaluated against the specific overhead each can reduce.
The fix
Target the coordination axis so that orchestration and fault-recovery events stop idling the healthy accelerators. Four operational changes attack both the in-step bubble losses and the out-of-step downtime:
Asynchronous checkpointing : Overlap checkpoint writes with the next training step, reducing visible checkpoint overhead from 3 percent to near zero.Elastic training : When a node fails, shrink the job and continue rather than halting all 10,000 GPUs for recovery. This converts the 10 percent failure recovery cost into a smaller throughput reduction.Pipeline schedule optimization : Switch from GPipe to an interleaved 1F1B schedule to reduce bubble fraction, or increase microbatch count per pipeline flush.Faster failure detection : Reduce heartbeat timeout from 30 seconds to 5 seconds with hardware-level health monitoring, cutting the idle time between failure occurrence and recovery initiation.
Production Troubleshooting
Each common fleet-scale failure mode first shows up as a user-visible symptom, and that symptom points to the C\(^3\) axis most likely responsible (sometimes two), a measurement that confirms it, and an action (table 7 ). Operators can match an observed cluster symptom to its primary C\(^3\) axis, run the listed diagnostic check, and apply the corresponding engineering fix.
C\(^3\) Scorecard
The C\(^3\) scorecard grades fleet efficiency against known thresholds, extending the single-machine scorecard (D·A·M Scorecard Table 9 defines the three metrics that characterize fleet health.
C\(^3\) Infrastructure Diagnostic Decision Tree and Fleet Diagnostic Scorecard
When step time suddenly doubles in the middle of a training run, on-call operators see raw hardware counters—elevated GPU temperatures, PCIe retry alerts, or InfiniBand pause frames—rather than a neatly labeled C\(^3\) axis. Cluster operations teams need a systematic procedure for mapping those low-level hardware signals to the binding C\(^3\) axis before taking remedial action.
Telemetry signal mapping to C\(^3\) axes
A degraded PCIe riser or a clogged cooling loop first surfaces in hardware counters across silicon, PCIe buses, thermal sensors, and network switches. Monitoring daemons collect four primary classes of low-level telemetry to catch these faults early:
MFU Drops : Sudden or sustained drops in model FLOPs utilization below expected baselines.PCIe AER (Advanced Error Reporting) Retries : Non-fatal bus errors causing packet retries, link downgrades (e.g., PCIe Gen 5 \(\to\) Gen 3), or host-to-device direct memory access (DMA) throttling.Thermal Throttling : GPU SM clock downclocking (e.g., 1.9 GHz \(\to\) 1.1 GHz) triggered by junction thermal limits or cooling fan degradation.InfiniBand Credit Stalls and PFC Pause Frames : Priority flow control (PFC) pause frames, congestion notification packets (CNP), or switch buffer credit starvation.
Diagnostic decision tree
Triage starts from whichever telemetry source shows the anomaly and narrows to the specific signal before naming the binding axis (figure 1 ). A node-side signal points to the compute axis, while a network-side signal separates fabric congestion, a communication cost, from a single straggler that holds every rank at the barrier, a coordination cost.
Structured diagnostic telemetry table
Table 10 maps hardware-level telemetry counters to root causes, C\(^3\) axes, and concrete operational remediations. Pairing each counter with its physical subsystem lets automated health checks route alerts to the right repair workflow without manual guesswork.
Fleet diagnostic scorecard
The fleet diagnostic scorecard in table 11 carries each single-node Data, Algorithm, and Machine (D·A·M) concern to the fleet telemetry that exposes it, the C\(^3\) axes it affects, and a target benchmark. Each row links a single-node resource category to the cluster-level telemetry signal and target benchmark that govern distributed execution.
Scaling Laws Through the C\(^3\) Lens
Scaling laws are usually written in algorithmic FLOPs, but a training fleet delivers only the useful FLOPs that survive utilization, communication, and coordination losses. Translating a scaling-law target into C\(^3\) terms means removing its hidden perfect-systems assumption and replacing peak FLOP/s with effective FLOP/s, the usable throughput that connects model-quality forecasts to real cluster capacity.
The ideal-system assumption in scaling laws
Scaling laws (Kaplan et al. 2020 ; Hoffmann et al. 2022 ) and their successors predict model quality as a function of algorithmic training compute. They usually abstract away systems efficiency: wall-clock time, accelerator peak FLOP/s, compute-phase efficiency, communication overhead, and scheduler losses appear later when teams provision hardware to deliver the target compute budget. In C\(^3\) terms, scaling-law FLOPs must be converted into raw fleet capacity after compute-phase, distributed-step, and runtime-goodput losses.
Kaplan, Jared, Sam McCandlish, Tom Henighan, Tom B. Brown, Benjamin Chess, Rewon Child, Scott Gray, Alec Radford, Jeffrey Wu, and Dario Amodei. 2020. “Scaling Laws for Neural Language Models.” ArXiv Preprint abs/2001.08361.
Hoffmann, Jordan, Sebastian Borgeaud, Arthur Mensch, Elena Buchatskaya, Trevor Cai, Eliza Rutherford, Diego de Las Casas, et al. 2022. “Training Compute-Optimal Large Language Models.” Advances in Neural Information Processing Systems (NeurIPS) 35: 30016–30. https://doi.org/10.52202/068431-2176.
The gap between scaling-law predictions and observed training outcomes is, in large part, a C\(^3\) gap. A team that budgets \(10^{24}\) peak hardware FLOPs of capacity will deliver fewer useful model FLOPs after three multiplicative losses: per-GPU compute-phase efficiency, inter-node scaling efficiency (\(\ScalingEfficiency\) ), and runtime goodput.
The effective FLOP/s concept
A fleet’s effective FLOP/s is the usable throughput after compounding three disjoint C\(^3\) efficiency factors across the cluster:
\[\EffectiveFleetThroughput = (\DeviceCount\PeakRate) \times \underbrace{\ModelFlopsUtilization}_{\text{Compute}} \times \underbrace{\ScalingEfficiency}_{\text{Distributed step}} \times \underbrace{\RuntimeGoodput}_{\text{Runtime goodput}}\]
MFU, the single-device baseline, measures useful model arithmetic against peak throughput only while local computation is active. This compute-phase factor is not the step-averaged fleet MFU of table 9 , whose \(\DistributedStepTime\) denominator already includes the in-step losses that \(\ScalingEfficiency\) counts, so the product does not count them twice. Scaling efficiency captures communication and other distributed-step losses, including pipeline bubbles. Runtime goodput captures time outside those measured steps, including checkpointing, failure recovery, scheduler preemptions, and maintenance. These disjoint measurement windows prevent the same loss from being counted twice, while a concrete estimate makes the multiplicative loss visible at fleet scale.
Systems Perspective 1.1: Compound C³ loss on a 100,000-GPU cluster Consider a 100,000-GPU H100 cluster with 98,900 PFLOP/s of peak aggregate throughput. After the three disjoint C
\(^3\) efficiency factors:
\[\text{Effective} = 98,900 PFLOP/s \times 0.50 \times 0.56 \times 0.65 \approx 17999.8 PFLOP/s\]
The fleet delivers 18.2 percent of its peak capacity as useful training work. The compound C\(^3\) loss , the ratio of peak to effective compute, is 5.5×, so achieving a given effective compute budget requires 5.5× the raw hardware: the compute phase retains a 50 percent efficiency factor, the distributed step retains a 56 percent scaling factor (using the modeled 8,192-GPU tier as an illustrative proxy that includes pipeline losses), and runtime goodput retains a 65 percent factor after checkpoints, failures, scheduler preemptions, and maintenance.
These losses reflect the physics of fleet-scale computation rather than a failure of engineering. The C\(^3\) taxonomy quantifies where the losses occur so that optimization effort targets the dominant term.
Summary
The C\(^3\) taxonomy provides a systematic framework for diagnosing fleet-scale bottlenecks. Each axis maps to a distinct physical constraint: arithmetic throughput and MFU bound compute; network bandwidth and collective algorithm efficiency bound communication; and synchronization overhead, failure recovery, and operational losses bound coordination. The Fleet Law quantifies these constraints, enabling systematic diagnosis. Use the C\(^3\) traffic light for quick triage, the bottleneck diagnostic table to choose the right lever, and the C\(^3\) scorecard to grade fleet health.
Key Takeaways: Where to look first at fleet scale
Every fleet-scale bottleneck has a dominant C\(^3\) axis : Compute, Communication, or Coordination, with many real bottlenecks spanning intersections. Identify the dominant axis before optimizing.Measure the C\(^3\) scorecard : Compare MFU, scaling efficiency, and goodput with workload-specific baselines. The displayed thresholds screen for problems; traces must confirm the cause before selecting an optimization.The compound C\(^3\) loss is multiplicative : Peak FLOP/s \(\times\) compute-phase efficiency \(\times\) scaling efficiency \(\times\) runtime goodput = effective FLOP/s. The illustrative 100,000-GPU case retains ~18.2 percent of peak.Coordination is the new axis : Single-node overhead remains relevant, while distribution adds cross-machine dependencies. The coordination case attributes 40 percent of its modeled budget to checkpoints, failures, pipeline bubbles, scheduling, and maintenance.Optimizing the wrong C\(^3\) axis yields limited improvement : Faster GPUs cannot fix a communication-bound fleet by themselves; faster networks help coordination only where transfer time, such as checkpoint I/O, contributes to that overhead.
