36. Quantity algebra, coordinate frames, clocks, uncertainty, and equivalence

Exact quantity and uncertainty object
QuantityValue {
  quantityKind: TermRef,
  source: {value: DecimalString, unit: TermRef, resolution?: DecimalString},
  canonical?: {value: DecimalString, unit: TermRef,
               conversionRuleRef: TermRef, conversionReceiptRef: RecordRef},
  dimension: {L, M, T, I, Theta, N, J, angle, information},  # signed small integers
  frameRef?: EntityRef,
  positionRef?: GeometryRef,
  validTime?: HalfOpenInterval,
  uncertainty?: Uncertainty,
  qualityFlags: TermRef[],
  methodRef?: RecordRef,
  calibrationRef?: RecordRef
}

Uncertainty {
  standardUncertainty: DecimalString,
  distribution: normal | uniform | triangular | empirical | unknown,
  coverageFactor?: DecimalString,
  coverageProbability?: DecimalString,
  degreesOfFreedom?: DecimalString,
  covarianceGroupRef?: RecordRef,
  contributors?: [{sourceRef, sensitivityCoefficient, standardUncertainty}],
  evaluationMethod: typeA | typeB | mixed | unspecified
}

Uncertainty propagation profile
For z = f(x1...xn) with covariance matrix Sigma and Jacobian J:
  u(z)^2 = J * Sigma * transpose(J)

For independent multiplication z = x*y, first-order relative form:
  (u(z)/z)^2 = (u(x)/x)^2 + (u(y)/y)^2

Independence may be assumed only when the method declares it. Shared calibration,
clock, environment or source equipment creates covariance and must not be counted twice.
Monte Carlo propagation records sampler, seed, distribution versions, sample count,
convergence checks and output quantiles; a seed is reproducibility metadata, not evidence.

SE(3) frame and transform profile
FrameDefinition {
  frameId, parentFrameId?, handedness: right,
  axisDefinitions[], originDefinition, datumRef?, epoch?,
  bodyFixed: boolean, configurationId?, validInterval
}

RigidTransform {
  parentFrameId, childFrameId, validTime,
  translationM: [DecimalString x3],
  rotationQuaternionWxyz: [DecimalString x4],
  covariance6x6?: [DecimalString x36],
  sourceRefs[], interpolationPolicy, maxExtrapolationNs
}

Rules:
- quaternion norm error <= 1e-12 after decimal-to-working-precision conversion;
- canonical sign chooses the first non-zero component positive;
- composition order is parent_T_child; never inferred from field names;
- a transform cycle at one valid instant is rejected unless it closes within a
  declared tolerance and is stored as a calibration residual, not canonical topology;
- pose is inseparable from frame, configuration and valid time.

Clock mapping and uncertainty
ClockDomain {
  clockId, producerDeviceId, tickUnit, counterBits?, nominalRate,
  monotonicWithinBoot, bootId, oscillatorClass, leapBehavior
}

ClockMappingSegment {
  clockId, referenceScale, deviceTickStart, deviceTickEnd,
  t0Device, t0ReferenceSeconds, t0ReferenceNanos,
  rateRatioA, offsetSecondsB,
  covariance2x2, maxResidualNs, fitMethod, syncObservationRefs[]
}

t_reference = A * (t_device - t0Device) + B
u_time(t)^2 = [dt, 1] * Cov(A,B) * transpose([dt, 1]) + u_sync^2

Counter unwrap is valid only inside one bootId and wrap model. A reset, rate step,
GNSS reacquisition, PTP grandmaster change or residual breach closes the segment.

UTC labels do not create simultaneity
Two channels may both print UTC while carrying different latency, synchronization, buffering and clock uncertainty. Cross-channel fusion must use ClockMappingSegment and the observation/capture time semantics of each channel.