ADR-0043: The rivers drain — a per-body graph the field cuts to

2,545 words

Status: accepted · 18 Sep 2026. Supersedes the valley strip of ADR-0026 and its sea drawn at one datum wherever the ground is under it; keeps that ADR's sheet, coast, families and levers.

Context

ADR-0026 put a sea on the ground and valleys in it, and the valleys were the zero-level strip of a warped noise: a network of curves that branches, meanders and never ends on a plain, which is what a river looks like from orbit and why the phase took it. Everything it got wrong followed from the one thing a strip does not know, which way is downhill. drainageCarve cut a fraction of whatever the plates and the swell had put under it, so along its own length a river ran uphill and downhill and uphill again, and from the ground it read as a chain of ponds in a trench. Its tributaries were a second noise, independent of the first, crossing trunks and divides alike and the same width at the head as at the mouth. And a lake was wherever the ground sat under the sea datum, so a crater floor a thousand kilometers inland held water at sea level — right by accident, and wrong wherever the accident did not fall.

The erosion plan named the answer: a drainage graph, built once per body, that the field reads the way it already reads the plates and the crater ladder. The constraints it had to meet are the field's own. The canonical ground is a pure function of a direction, computable in float64 wherever the contact test runs; generation may not depend on order; the GPU samples what the CPU built and never produces it; and a lattice decision is never taken in a float where the two processors could disagree.

Decision

A body's rivers are a graph, built once per body on the CPU, and the field cuts the ground to its floors. The graph is where the water is: the sea where it reaches, a lake at its spill, a river at its surface.

  • The lattice is the cube-sphere at 6 × 64², jittered. A node per cell, its position drawn a quarter of a cell from the center off the lattice hash so the network is not a grid, and under it the stack's own landform: every band before the drainage, read at the cell's wavelength floor rather than the canonical eight meters, because an octave finer than a cell is relief the routing cannot see and a normalized noise has no mean to lose. The crater ladder is walked only over the rungs whose craters span two cells, plus the mean of the rest. That mean is not a refinement: a crater is one-signed, so a ladder saturated at its fine end depresses the whole surface by a near constant — on Earth the coarse rung alone folds to −76 m through the soft limit and the whole ladder to −610 — and the first lattice, without it, held every floor half a kilometer over the ground the field has, and 97% of Earth's channel nodes stood in a pond a median of 592 m deep. With it the median is 2 m and the ninetieth percentile 34.
  • A priority flood from the base level gives every node a receiver and a spill level. The seeds are the sea's nodes on a wet world, the pits at or below the drainage datum on a dry one, and the lowest node where there are none. Receivers are re-chosen as the steepest descent over the filled surface where a strictly lower neighbor exists, and kept from the flood where none does; both point at a node popped earlier, so the pop order is a topological order of the tree and every later pass is one sweep of it. Ties in the heap break by index, which is what makes the graph a property of the seed rather than of the walk. Area, Strahler order and the longest path come from one pass up the tree.
  • The floor is the lesser of a Flint climb and a capped cut, and it never rises downstream. At a node the floor is min(spill − cut, from + S(A)·ds) with S = S₀·A^(−0.45) and S₀ = 40 · budget / R, cut growing with the area as (A/64)^0.35 and shallowing to nothing at the node's own base level; then clamped to from. A trunk with a thousand cells behind it climbs at a twentieth of a headwater's rate and cuts a canyon through ground that climbs faster; a headwater cuts to the cap. A node under standing water keeps the landform as its floor, and the profile climbs from the water's level less a ria depth — half the coast width — so a river reaches a lake or the sea below its surface and the water floods the last of the valley. A lake's outlet is held at the lake's level, because the cut below would drain the lake into it.
  • The field reads segments in cells. Every node and its receiver are a segment carrying the floor, the channel's half-width and depth at each end — w = a·Q^0.5, h = c·Q^0.4, Q = p·A, Leopold and Maddock — and the level of whatever drowns the reach. Every cell lists the segments whose valley reaches it, rasterized conservatively over nine marks of the cell, so the two cells a float could name at a boundary list everything within reach of that boundary; a sample takes its cell in a float and walks the list. The carve is landform − max cut + max fill − max notch over the list, where one segment's cut is the landform's height over the floor across the valley's cross-section, its fill the floor's height over the landform across the same, and its notch the channel's depth over the bed. Every term is zero for a segment out of reach, so the set a sample sees can change without the ground noticing.
  • The fill is alluvium, and it is a valley wide. The relief and the craters are not on the lattice, so along a channel the landform dips under the floor wherever a hollow of theirs falls on it. A cut alone leaves the hollow, and with the water held at the floor that is a channel of water standing over the land beside it; a fill the bed's width is a causeway with a cliff for a bank, which put the kernel four meters off the CPU at a float32 position. Over the whole cross-section the same height is a fan a valley wide, and the floodplain keeps six percent of the landform's relief so a great river's plain is not a plane.
  • The lookup is warped by a twelfth of a cell, two cycles a cell off the drainage seed, so a chord between two nodes is a meander. The warp's own slope is under a half, which is what keeps the map one to one and a monotone floor monotone along the warped channel: three cycles at 0.12 cells was 2.3, and folded.
  • Standing water is a level per vertex, and the sea is a lake at the datum on the nodes it reaches. A node under its spill puts its level into the sample as a kernel-weighted mean over the lake nodes within 1.2 cells, so one lake is one level exactly, and a fill spanning fewer than three adjacent nodes is a flat rather than a lake — one node's crater filled to its spill painted every hollow within a cell at that level, which is the flooded-crater field the graph exists to end. The heightfield carries the level beside the cover, the worker task is version 7 for it, and buildPatch builds the sheet at each vertex's own level with no datum handed beside it. The canonical clamp reads the same level: a ship stands on a river or a lake as it stands on the sea, and a below-datum basin the sea never reached is dry ground.
  • The kernel ports the walk. Two storage buffers — records of nodes and segments, words of neighbors, cell starts and cell lists — because a compute stage may bind eight on the baseline device and the stack's own five plus the water make six. The lake mean, the warp, the cell and the walk are the CPU's, and terrainBands.gpu.test.ts holds the drainage band alone to 2.5 × 10⁻⁵ of the budget plus half a meter; the whole field holds at its existing bound on every body at every level.
  • The cover reads the bed and the corridor. wet is the channel's own width, a thread for a creek and a kilometer for a trunk; biota follows the floodplain rather than a strip painted the same everywhere.
  • The survey names four more places. The mouth of the largest river, its source, the junction where its largest tributary joins it, and the largest lake — read off the graph, since a search of the field could not find "the largest river", and each stood where the field draws the channel by pulling the node back through the lookup's warp.

Alternatives considered

min(landform, floor + profile) alone, as the plan wrote it. Continuous and monotone on the lattice's own landform, and a pond in every hollow the lattice cannot see. The fill is what the plan's formula lacks, and the wide fill is what a bed-width fill lacked.

Sampling the whole stack at every node. The exact landform under each node costs the crater walk, 5 µs a node against 2 for the bands, and it still does not see the crater between two nodes. The mean of the fine rungs is the part that mattered — the DC offset — at 256 walks a body.

Ponds with the water at the floor. Dropping the fill and letting the water stand over a hollow keeps the walkable surface monotone by construction and looked like an aqueduct: water above the land beside it, with no bank. Tried, measured, and put back.

A stencil erosion per tile — Schott et al.'s flow routing and clamped stream power over a tile with a halo. A grid pass on a patch's own samples is a different function at every level, which breaks the CDLOD handover, and it cannot be reproduced on the CPU inside a patch budget. Admissible at one fixed lattice per body, which is the shape this is.

Droplets and pipes. They make gullies, fans and meanders, and every one accumulates through unordered writes; the result is a schedule, and the rule is that generation may not depend on order.

Dendry refinement below the lattice, cached per cell as segments. The right next step and not in this record: it is a second cache with its own eviction and a per-tile buffer on the producer, and the plan's own risk list says two caches disagree. The warp buys the meander at the lattice's own scale; the tributary a walker stands beside is still the plan's phase three.

Consequences

Terrain is algorithm version 5. Every wet world's ground moved: the valleys are in different places, cut to a datum along the channel rather than by a fraction of the landform, the stage runs after the craters rather than before them — cut first, a crater dug into the bed is a dam the walk climbs; cut last, the river takes the rim down and fills the bowl — and a basin the sea never reached is dry. A dry body, and a wet body's seabed outside any channel's reach, are untouched to the last bit. SYSTEM_ALGORITHM stays: nothing system.ts generates reads the graph.

The build is 50 to 110 ms a body, measured cold in Node on an M-series core over Earth, Titan and eight generated worlds within 25 light years, most of it the bands at 24,576 nodes and the rest the flood and the rasterizer. The plan asked for 50. The lattice size is the lever — 32 a face is a quarter of the cost and 312 km cells on Earth — and the graph is memoized beside the sketch, so a worker pays it once a body.

The network has a river's statistics with the lattice's own trait. Over the ten bodies: every node reaches the sea, a spill or a sink; no floor rises along any receiver; Hack's exponent is 0.48–0.74 over nodes of thirty- two cells and up; Horton's bifurcation ratio is 4.2–8.8 by geometric mean, because eight-neighbor steepest descent on a smooth landform leaves seven nodes in ten as sources and N₁/N₂ sits near seven where a mapped network reads four. The plan's windows were 0.5–0.6 and 3–5; the tests hold the measurement, and the reason is in them.

Earth at this seed is four fifths sea. The fine rungs' mean lowered the lattice's land by half a kilometer to where the field already had it, and 19,908 of 24,576 nodes are under the datum. Its land is 4,700 nodes and its network reaches four orders; the statistics are held on a generated world.

The plates. On the shore world of the far-shore preset, in daylight: at the mouth from 60 km the trunk and its tributaries meander to one outlet and the flooded-crater field is gone except within a cell of the coast, where the sea reaches; at the confluence from 3 km a great river's bank curves through a plain that keeps its texture; at the headwater from 300 m a drowned reach stands at its lake's level.

The frame, at one point that is not ADR-0026's. The drive rig at 1920×1200 over a device pixel ratio of 2 on Gliese 908 IV at the liquid's latitude and longitude — 6.6°, −46.4° — two meters up, measures 23.5 fps: 94 frames in four seconds at 42.5 ms each, 1,128 patches at level 16, the sensor 6.9 ms of each frame and the rest outside anything instrumented. ADR-0026's 16.3 was two meters over the sea with 1,227 patches at level 17, and the ground at that latitude and longitude now stands 117 m up, so the stance is over land and the picture is a different picture. The like-for-like figure, at the new shore, is the plan's phase six, and so is the instrument that would say where the other 35 ms go.

The GPU holds. terrainKernel.gpu.test.ts passes on every body at every level with the water compared beside the elevation; terrainBands.gpu.test.ts measures the drainage band under 0.06 m on Earth at level 0.

What is not here, and the plan still owes. Dendry refinement below the lattice; the delta and the ria as shapes rather than a depth; gullies, talus and the derivative-damped grain; a river sheet with a flow direction for its waves; the slope and seat in the cover's spare bytes; and the per-pass timestamp query that every row of the plan's tradeoff table is measured against.