An event can be missed because it occurs between exposures, diluted because its signal is averaged over a long exposure, or unresolved because several events produce the same record. These are properties of an observing procedure, not direct evidence about whether time itself is discrete.
Sampling is more than a frame rate
A camera reporting one frame each second need not collect light throughout that second. Its exposure may last only a fraction of the frame interval. The rest can be a blind gap for the measurement being considered.
Even continuous exposure does not preserve the full time history. A frame records an integrated or averaged response over an interval. Different signals inside that interval can produce the same stored value. Frame rate, exposure duration and response time describe different limitations.
The following calculation isolates the first two effects. It assumes an ideal linear detector with a perfectly rectangular exposure window, no noise, no saturation and no additional response delay. Real instruments may require all of those features to be added.
A pulse and four exposures
Four frames begin at times zero, one, two and three seconds. Each exposure lasts T seconds, with 0 < T ≤ 1. A unit-height rectangular pulse begins at s and lasts w seconds. The whole pulse lies within the displayed four-second interval.
Let Lₖ be the duration of overlap between the pulse and frame k's exposure. The recorded frame average is Lₖ/T. The captured fraction of the pulse's total area is the sum of the overlap durations divided by w:
Lₖ = max[0, min(s + w, k + T) − max(s,k)]; captured fraction = ΣLₖ/w.
Intervals that merely touch have zero overlap duration. Adjacent exposures at T = 1 do not double-count a finite piece of the signal. The model treats time as continuous while calculating exactly how much signal enters each record.
Can the exposures capture the pulse?
Four frames begin one second apart. Each records the average signal during its exposure. A unit-height pulse can fall inside an exposure, across an edge or entirely in a gap.
Purple: unit-height pulseGreen: active exposure windows
Missed, diluted and partly recorded
With T = 0.25 s, s = 0.40 s and w = 0.10 s, the pulse falls entirely between the first two exposures. Every frame average is zero and the captured fraction is zero. A zero record is compatible with this nonzero event under the stated schedule.
Keep the same pulse and increase T to one second. The first frame now records an average of 0.10, and the captured fraction becomes one. The full pulse area is recorded even though the reported average is only one tenth of the pulse's unit height.
For T = 0.25 s, s = 0.20 s and w = 0.10 s, only 0.05 s of the pulse overlaps the first exposure. That frame reports 0.20. Half the pulse area is captured and half is missed. The output average and the captured fraction are different quantities with different denominators.
What can the record reconstruct?
The recorded area is T times the sum of the four frame averages. This equals the total overlap duration for the unit-height pulse. It supplies a useful accounting check, but does not uniquely recover a general signal's peak height, shape or onset.
A short high pulse and a longer low pulse inside the same exposure can have equal areas. Without an assumed pulse family or an additional observation, the average cannot distinguish them. Knowing that the demonstration uses a rectangular unit-height pulse is extra model information supplied to the reader.
Sampling theorems answer related questions under specified bandwidth and reconstruction assumptions. An ideal rectangular pulse is not band-limited, and a gapped exposure is not an instantaneous sample. A frame-rate slogan cannot replace the actual response model.
A detector limit is not a time ontology
Increasing exposure closes gaps but can blur timing information within each frame. Shortening exposure can localize a captured event more narrowly while increasing the chance that an event falls outside the active windows. An observing strategy must choose according to the question and available signal.
The book's account of future-test equivalence helps frame the inference: equal records under one schedule can become distinguishable under another. The lost distinction belongs to the measurement map.
For SMM/DTT, a proposed discrete generative order must be distinguished from an instrument's exposure schedule. A physical claim about fundamental temporal structure needs a prediction that survives an explicit detector model and comparison with conventional explanations. The pulse calculation here assumes continuous time and already produces apparently absent events.
Sources and further reading
- The exposure-overlap calculation is an exactly specified teaching model of rectangular integration; it is not experimental data or a model of a particular commercial camera.
- Mohamed Haj Yousef, Instrument and world, From signal to knowledge and When are two histories the same?.