Comparing the same measured time intervals
All 34 interval-association and prediction-bound checks pass. These are mathematical controls, with no original recordings or model fits. They address two ways an acceleration-only comparison could accidentally compare different quantities: uncertain synchronization and incomplete packet coverage.
An IMU (inertial measurement unit) records increments accumulated during intervals. A packet timestamp does not, by itself, identify that interval's duration or its alignment with the receiver clock.
Whole packets, explicit clocks
The support helper takes explicit integration intervals under a named clock hypothesis, receiver timestamps, a comparison window and bounds on one common time offset. It does not infer physical integration periods from packet-header differences. Nominal-period and header-period interpretations remain separate.
A packet is retained only if its entire integration interval lies inside the window for every allowed offset. Boundary packets with uncertain membership are listed separately. Receiver extrapolation and excessive receiver gaps reject the comparison. Reversed, overlapping or zero-duration integration intervals are rejected rather than repaired.
Packet gaps are preserved. For example, two retained intervals lasting 0.5 and 1.0 seconds with a gap between them provide 1.5 seconds of support. Neither the gap nor the discarded boundary packets become measurements. Every candidate must predict the average over that same interval union, using the same timing allowance. No interpolation of scientific measurements is performed.
The current helper covers a common clock offset, not frequency error or time-varying latency. Its integration intervals are exact under the supplied hypothesis; uncertain physical durations need additional envelopes or separately evaluated hypotheses. A support check does not calibrate the instrument clock.
Bounding prediction averaging and synchronization
Let the predicted continuous force magnitude be \(q(t)\). Suppose a declared bound \(L\) limits \(|dq/dt|\) over every packet's full possible time support. For packet durations \(T_i\), midpoint evaluation differs from the continuous, duration-weighted average by at most
Moving all intervals by an unknown common offset within \([a,b]\), relative to a reference offset \(d\), adds at most
The proof uses \(|q(t)-q(m_i)|\leq L|t-m_i|\) in each interval and integrates the distance from its midpoint. No independence or statistical averaging is assumed. Tests include an analytic tent function that reaches the averaging bound, a linear history under several offsets, unequal packet durations and gaps.
These bounds concern the predicted scalar force magnitude. The separate direction-cone bound concerns the loss of magnitude when the measured body-force vector turns inside an IMU packet. Both effects matter.
What remains before recording comparisons
The rate envelope \(L\), body-force direction envelope, integration hypotheses and offset allowance need a declared, defensible common policy. An excluded gyro or fused attitude cannot silently supply them. Receiver-derivative error, trajectory truncation, calibration and nonlinear uncertainty are not included in these time-support bounds. They still require a combined error budget before the recorded-data comparison. No statistical coverage or new scientific decision is established by passing these controls.