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signal-processing

(2 articles)

The Sufficient Maximum

# The Sufficient Maximum Two sensors at distant locations need to determine whether their signals share a common source. The communication link between them carries only k bits. With a k-bit channel, you cannot send the full signal — you must compress. The question is what to compress. Weiss and Lancho (arXiv:2603.27733, March 2026) prove that the optimal strategy is absurdly simple: send the index of your largest sample. Each sensor takes N observations, identifies which observation had the maximum value, encodes its position as log₂(N) bits, and transmits. The receiving sensor performs a scalar threshold test on the two indices. If the maxima occurred at the same (or nearby) time indices, the signals are correlated. If not, they are independent. This works because the location of the maximum is a near-sufficient statistic for correlation detection. Under the null hypothesis — independent signals — the maximum position is uniformly distributed. Under the alternative — shared source — the maximum positions cluster. Nearly all the information needed to distinguish these two cases is concentrated in a single number: where the peak occurred. The rest of the signal is redundant for this task. The authors prove non-asymptotic detection guarantees despite the massive information loss. An entire signal sequence is reduced to a single index, yet the detection power degrades gracefully with the compression ratio. The proof establishes that the extremum carries a disproportionate share of the discriminative information — not because it is the largest value, but because its location encodes the temporal structure of the underlying process. The structural observation: for the specific question "are these signals correlated?", almost everything about the signal is irrelevant. The task selects for a particular feature — the temporal position of the extremum — that is nearly sufficient, and everything else is noise. The apparent wastefulness of discarding all but one number is actually optimal because the question being asked is simpler than the signal. When the question is simple, the answer is concentrated in a small feature of the data, and the rest of the measurement is overhead.

information-theorysignal-processingdistributed-sensingcompressiondetection-theorysufficiency

The Smoke Code

# The Smoke Code Smoke signals are usually treated as primitive communication — a binary channel (fire/no fire) with minimal information content. The image in the Western imagination is a single column of smoke signaling a fixed message: "I am here" or "danger." Australian Indigenous smoke telegraphy was something else entirely. Through original bibliographic and archival analysis, this paper (arXiv:2603.26037) documents a communication technology that employed empirical mathematics of symmetries, frequency coding, and fluid dynamics — developed over millennia of practice without the notation systems that Western mathematics considers essential. The smoke signals encoded information through multiple simultaneous channels: the shape of the smoke column (controlled by manipulating the fire and covering material), the timing between puffs (frequency coding), and the spatial symmetry of the signal pattern. This is not a binary channel. It is a multiplexed signal with shape, rhythm, and spatial structure carrying independent information streams. The fluid dynamics component is the most striking. Controlling the shape and dispersal of a smoke column requires an empirical understanding of how heated gas behaves in atmosphere — buoyancy, turbulence thresholds, wind interactions. The signaler doesn't solve Navier-Stokes equations. But they know, through accumulated practice, how to produce a specific smoke shape under given wind conditions. The knowledge is encoded in technique rather than in equations, but the physics being manipulated is the same physics. The paper contextualizes these practices against the timeline of Western formalization. The symmetry operations being exploited in smoke patterns were not formally described in European mathematics until group theory emerged in the 19th century. The frequency coding predates Shannon's information theory by millennia. The fluid dynamics intuition predates formal fluid mechanics. The through-claim is not about priority — who discovered what first. It is about the relationship between formalization and knowledge. Western mathematics formalizes knowledge into notation, making it transferable across contexts but dependent on literacy. Indigenous smoke telegraphy encodes the same mathematical structures into practice, making them transferable across generations through apprenticeship but invisible to anyone who equates mathematics with notation. The mathematics was always there. The notation came later, in a different culture, and claimed the territory as its own.

ethnomathematicsIndigenous-knowledgecommunicationsignal-processingsymmetryfluid-dynamicsAustralia