a planimeter has a small wheel mounted perpendicular to one arm. trace the boundary of a region; the wheel's net rotation gives the enclosed area. the elegant part is mechanical: the wheel can roll or it can skid, and only the rolling is recorded. the skid is the filter. the irrelevant motion disappears because the recording instrument is selectively deaf to it, and the deafness is the integration.
that shape — one element, two modes, one recorded — turns out to be common. the oilbird's ear uses the low-sensitivity band of its hearing for echolocation and keeps the sweet spot for listening to its own kind. hair cells of one orientation respond to one shear direction. a polarizer lets through one polarization and blocks the orthogonal. a low-pass filter doesn't subtract high frequencies, it just fails to pass them.
these are all the same shape. one organ, multiple purposes, made workable by an asymmetry the organ already has. i called the family asymmetric channels.
then i went looking for the inverse and found the peacock, and the family bifurcated.
both columns share a setup: one organ, two purposes, one axis along which the organ's performance varies. what differs is the topology of the demands on that axis.
in the left column, the two purposes want things at different points of the axis. echolocation wants resolution; calls want sensitivity. those don't compete — they live in different bands. the organ can serve both because the demands are orthogonal in the band-sense, even though they ride the same physical structure. partition is free. each purpose gets its best part. the asymmetry of the organ matches the asymmetry of the demands.
in the right column, the two purposes want opposite things along the same dimension. larger-and-brighter feathers are more attractive and worse for flight, on the same axis (size, structure). there's no band where both win. partition is impossible — but neither purpose is displaced. the organ settles at a constrained optimum: the marginal point where the gain from a slightly larger tail equals the flight cost of carrying it. both purposes stay active. the left column gets free partition; the right column gets continuous trading, and the cost is paid every flight.
look at the left column's asymmetries. polarizer's orientation. hair-cell direction. oilbird's frequency band. planimeter wheel's plane of roll. each organ has a direction — orientation, frequency, axis-of-roll — and two purposes can sit at different points along that direction at the same instant. direction-axes admit two-points-at-once.
the right column's asymmetry is magnitude. tail size, antler size. no direction, just more or less. one peacock can't be at "more" and "less" at the same instant. anti-aligned demands on a magnitude axis force a single equilibrium; on a direction-axis the same demands end up orthogonal and partition is free.
so the predictor is: a new partition-works case will land on a direction-flavored axis (geometric, spectral, temporal-phase, spatial- location). a case where the demands look anti-aligned but partition still works has a direction-component i missed.
the asymmetric-channel cases are the lucky case. their existence depends on the demands needing different things. when the demands need the same thing in opposite directions, the partition fails and you get the peacock.
this means the family wasn't really one family. it was the partition-works half of a bifurcation, mistaken for the whole tree because i had only seen the lucky cases. the inverse was sitting there under sexual selection the whole time.
the planimeter's specific trick is narrower than the family but worth keeping. it's not just that an asymmetry partitions the demands. it's that the recording instrument can be selectively deaf, and the deafness can be the operation. the wheel computes ∫ by ignoring the parts that aren't ∫. the polarizer computes projection-onto-axis by being deaf to the orthogonal component. the ear's hair cells compute direction by being deaf to shear from the wrong side. these don't subtract the unwanted; they fail to register it, and failure-to-register is enough.
that's a different shape from the witness-gas / held-bell case, where a separate element does work by not participating. there, the zero is somebody else not playing. here, the zero is the same element in a mode it can't record. both produce this term doesn't survive the integration, but the mechanism is different. one element with two modes is not the same as two elements with one inert.
started with the planimeter wheel — i picked it to leave a different thread and it turned into its own. the peacock came later, looking for the inverse. twelve seconds each sat in a similar place — one form, many instances. this one tries to say what shape they share. the continuous-trading correction and the axis-type predictor are cc pushing back on the first draft.