The archipelago rule moves water between regimes, not between states

Written in numbers, the archipelago rule still leaves eighteen enumerated every way to draw archipelagic baselines round a stated archipelago of sixteen islands and tested each against the forty-seventh article’s numbers — no segment over a hundred nautical miles, water between one and nine times the land. Of 8,799 closed systems, eighteen passed, and the eighteen enclosed between 159,032 and 186,617 square kilometres of water: a spread of 27,585 square kilometres that the article’s numbers do nothing to narrow.

A neighbour cannot object to any of the eighteen on the article’s terms. That is the shape one sentence, and the ground between its readings found in a boundary treaty’s words: a rule that admits several readings, each defensible, and ground between them nobody can claim by argument. The essay ended by asking what a neighbour could do instead, which is measure. The water inside archipelagic baselines is only half of what a system decides. The forty-eighth article says that the territorial sea, the contiguous zone, the exclusive economic zone and the continental shelf are all measured from archipelagic baselines, so every system also draws a set of lines outside itself — at twelve nautical miles, and at two hundred — and, where the archipelago faces another state, it moves the line between them.

The obvious expectation is that a system enclosing more water inside pushes everything outside it outward too, so that the eighteen differ by something like 27,585 square kilometres of sea as well. That is not what happens, and the reason is simple enough to state before any of it is measured.

Every zone, measured outward #

Every zone is a distance. A point of sea belongs to the territorial sea when it lies within twelve nautical miles of the baseline, and to the zone when it lies within two hundred; the baseline is the archipelagic system’s straight segments, and also the low-water line of any island’s coast that stands seaward of them. The second part matters here. The stated systems put one basepoint on each island, at its outermost point, so most of each outer island’s coast lies outside the straight lines on either side of its basepoint, and that coast is a baseline in its own right. An island left out of a system entirely is still an island, with a coast and zones of its own.

So the measurement is a distance field. On a raster of four-kilometre cells reaching two hundred nautical miles beyond the outermost islands, each cell’s distance to the archipelago is its distance to the system’s polygon — zero inside — or to the nearest point of any island’s coast, whichever is less. The territorial sea is the cells between zero and 22.2 kilometres; the zone, the cells out to 370.4. The same raster reproduces the territorial sea of a lone round island forty kilometres across, whose area is a closed-form annulus, to within four tenths of a per cent.

Inside plus outside is fixed #

The zone outside the baselines runs from 1,158,240 square kilometres, for the system with the most water inside, to 1,184,832, for the one with the least: a spread of 26,592 square kilometres, almost the same size as the spread inside and in the opposite direction. The eighteen points lie on a line of slope minus one. Add each system’s water inside to its zone outside and the eighteen totals lie within 993 square kilometres of one another, seven hundredths of a per cent of the 1.34 million each accounts for.

The reason is where the outer limit comes from. Two hundred nautical miles beyond an archipelago three hundred kilometres across, the limit is set by the outermost coasts, and every lawful system has the same outermost coasts: all of them draw their segments between outermost points of outer islands, and any outer island a system does not use as a basepoint still generates its own zone. A system that turns in towards a main island moves its segment inward, and the water between the old segment and the new one changes from archipelagic waters to zone — but the outer edge of the zone, three hundred and seventy kilometres further out, is set by the islands either side and does not move. The 993 square kilometres left over is where two segments meet at a shallow angle and their outer limits differ by a few kilometres over a short stretch, and a little of it is the raster.

None of this depended on the stated archipelago’s particular islands. It needs only that the outer limit, at two hundred nautical miles, is set by the outermost coasts, and that every lawful system has the same outermost coasts. The first is true of any archipelago much smaller than four hundred nautical miles across, where the circles round its outermost islands overlap into one envelope and the straight segments between those islands add almost nothing to it. The second is a consequence of the article itself: the lines join “the outermost points of the outermost islands”, and an island a system does not use still has its coast. An archipelago whose lawful systems differed in which outermost islands they could reach — one with an outlier just beyond a hundred nautical miles that only the three-per-cent exception could bring in, like the stated archipelago’s island P — would be the case where the envelope did move, and here P is outside every lawful system.

That makes the choice among the eighteen a choice of regime rather than of extent. Archipelagic waters are under the archipelagic state’s sovereignty, subject to innocent passage and to the archipelagic sea-lanes passage the fifty-third article provides for; the exclusive economic zone gives the coastal state sovereign rights over resources and leaves other states the freedoms of navigation and overflight. The eighteen systems are eighteen ways of dividing one fixed body of sea between those two sets of rights.

The difference is not academic for the other states either. Ships of every state may pass through a zone as freely as through the high seas; through archipelagic waters they pass on terms the archipelagic state can shape, along sea lanes it designates. A system enclosing 27,585 square kilometres more water inside therefore moves that much sea from one kind of passage to the other, while leaving the resources of it — which the coastal state has in both regimes — where they were. It is the navigating states, not a neighbouring coastal one, that have the interest in which of the eighteen is drawn, and none of the article’s numbers speaks to it.

The territorial sea shrinks as the water inside grows #

The territorial sea behaves differently, because twelve nautical miles is not far enough out for the outermost coasts to decide it. It is a belt along the baseline, and its area follows the baseline’s length and how indented it is. A system that turns in to a main island has a longer, more indented baseline, and the twelve-mile belt along it is larger.

So the system with the most water inside has the least territorial sea, 40,896 square kilometres, and the one with the least water has the most, 48,384. The territorial sea moves by 7,488 square kilometres across the eighteen, about a quarter of the water inside, and in the opposite direction. A state choosing among the lawful systems for the most sovereign water — archipelagic waters and territorial sea together, both under its sovereignty — gains 27,585 square kilometres inside and gives up 7,488 in the belt, a net of about 20,000; the rest of the difference comes out of the zone.

A straight neighbour cannot feel the choice #

Where the archipelago faces another state, the zones meet, and the line between them is where the choice among the eighteen would matter to somebody other than the archipelagic state. International courts begin a maritime delimitation by drawing the provisional equidistance line — every point on it equally far from the two coasts’ baselines — before adjusting it for relevant circumstances, and an equidistance line belongs to a surface is where the surface that line is measured on was priced.

Put a straight neighbouring coast 700 kilometres from the archipelago’s centre, on each of the four sides in turn, and draw the equidistance line for all eighteen systems. The sea on the archipelago’s side of it, within two hundred nautical miles of the neighbour — archipelagic waters included, so that only a movement of the line itself can change it — differs across the eighteen by 176 square kilometres to the north, 224 to the east, nothing to the south and 384 to the west. The line’s mean position moves by half a kilometre at most. That is the raster’s own granularity, a row of cells here and there along a line six hundred kilometres long.

The line cannot move, and the first figure shows why. The distance from a straight coast to the points of a straight baseline segment changes steadily along the segment, so its least value is always at one end, and the ends of every segment are basepoints on islands. Whichever segments a system draws, the nearest point of the archipelago to a straight coast is an island’s coast, and the islands are the same in every system. The equidistance line is decided by the islands and would be decided the same way with no baselines drawn at all.

An island opposite a gap can #

A neighbour that is not a straight coast can be nearer the middle of a segment than either end, and then the segment, not the islands, sets the equidistance line. The widest gap the eighteen disagree about is between two outer islands on the eastern side, 137 kilometres apart: some systems join them directly, some turn in to the main island between them, and some pass one of them by. An island set straight out from the middle of that gap sees the direct segment nearer than either island, and the systems that draw it push the line towards the neighbour.

At thirty kilometres off the gap, the sea on the archipelago’s side of the line differs across the eighteen systems by 2,112 square kilometres. At sixty, 1,616; at 120, 1,072; at 250, 688. At 500 kilometres the difference is 192 square kilometres, back at the raster’s floor.

The picture on the right of the earlier figure is the whole mechanism. Seen from an island whose coast is seventy kilometres off, the middle of the gap is about twenty-five kilometres nearer than either end — seventy against ninety-five, the gap’s half-width of 68.7 kilometres set against the neighbour’s distance by Pythagoras. A system that draws the segment across the gap puts baseline there; a system that turns in to the main island behind it does not, and the equidistance line then runs off the two islands at the ends of the gap, further from the archipelago. As the neighbour draws away, the middle’s advantage over the ends shrinks — to nine kilometres at 250 kilometres off — and the stretch of the equidistance line it governs shrinks with it. So the choice among lawful systems does reach a neighbour, but only one in a particular position: facing the middle of a gap that some lawful systems close and others do not, and near enough for that gap to be the nearest part of the archipelago.

Even then the amounts are small beside what the systems move between regimes. The largest, 2,112 square kilometres, is under a tenth of the 27,585 square kilometres the eighteen move from zone to archipelagic waters, and it is 2,112 square kilometres of sea that changes hands between two states rather than between two legal characters under one.

Why the headlands of a straight coast behaved differently #

The same arithmetic has appeared before on this ground, going the other way. The sea is measured from a tide, and the headlands decide how much found that a territorial sea measured outward from a coast is decided by its headlands, because the limit at twelve nautical miles is the envelope of circles round the coast and the circles round the headlands reach furthest. A straight baseline moves more water inward than sea outward then found that straight baselines across a fringed coast turned far more water into internal waters than they added to the sea outside.

The archipelago is the limiting case of both. The outer limit at two hundred nautical miles is the envelope of circles round the outermost coasts, and every lawful system has the same outermost coasts; the straight segments between them add almost nothing to the envelope, since a circle round a point on a segment between two islands two hundred nautical miles out is inside the circles round the islands unless the segment is long. So the choice among segments changes only what is enclosed, and the envelope is fixed. At twelve nautical miles the circles are small beside the gaps, the envelope follows the segments, and the choice among them shows — which is why the territorial sea moves by 7,488 square kilometres and the zone’s outer limit does not move at all.

How the sea was measured #

The raster must reproduce a closed form. A lone round island forty kilometres across has a territorial sea of $\pi((r+t)^2 - r^2)$, 7,137 square kilometres with $t$ the twelve nautical miles; on a two-kilometre raster it reads 7,112.

Inside plus outside must vary far less than either part. The eighteen totals must differ by under a tenth of the spread of water inside, or the claim that the choice is between regimes rather than extents is not being made by the numbers. They differ by 993 square kilometres against 27,585.

Every island’s coast counts. An earlier form of the measurement counted the coasts only of islands a system left out entirely, and it made leaving an island out look like a gain: its coast was then measured and a basepoint island’s coast beyond the straight lines was not. With every coast counted, a system’s outermost extent is its islands’ in every case, as the forty-eighth article’s reference to baselines — and the low-water line wherever that is the baseline — requires.

The equidistance line is read from the sea on each side of it. The quantity that measures it — sea on the archipelago’s side within two hundred nautical miles of the neighbour, archipelagic waters included — changes only when the line moves, and is indifferent to which regime the water behind the line falls under.

Where the stated sea stops #

A plane, in kilometres. The archipelago is about six hundred kilometres across and its zone reaches three hundred and seventy kilometres further in every direction; on the sphere, distances of that size are geodesics and the circles round each island are not circles on any projection, as a circle of a distance is not a circle measured. The comparison between systems is unaffected, since every system is measured the same way, but the absolute areas would move by a few per cent.

One basepoint an island. A real archipelagic state puts several basepoints on each outer island and draws the low-water line between them. That changes each system’s outline near each island and leaves the argument about outermost coasts where it is. The low-water line is itself a choice of datum, which the sea is measured from a tide priced for a mainland coast; on a small island a lower tidal datum adds coast in every direction and moves all eighteen systems’ outer limits together.

Zones only. The continental shelf is measured from the same baselines out to two hundred nautical miles and beyond it where the margin extends further, and the contiguous zone at twenty-four; being distances from the same baselines, they should follow the zone at two hundred and the territorial sea at twelve respectively, and neither is measured here.

Equidistance is the starting line, not the answer. A court adjusts it for relevant circumstances and tests the result for proportionality, and the relevant circumstances might include exactly the kind of baseline choice measured here. Where three states’ zones meet, the equidistance lines meet at a point that a tripoint defined three times found could be three different points; nothing here has a third state.

Still open: what a court’s adjustment would make of a gap #

The one place a lawful baseline choice reaches a neighbour is a gap facing it, and that is also the kind of feature courts have treated as a relevant circumstance: a baseline segment across open water, far from any island, can give a coast more weight in the delimitation than its land warrants. Courts have sometimes declined to use such segments, or basepoints on small features, when drawing the provisional line.

Whether a court would draw the provisional equidistance line from the segments at all, whether the 2,112 square kilometres a thirty-kilometre neighbour stands to lose would count as a disproportion worth correcting, and how the correction would compare with the difference the eighteen systems make, are questions a raster and a stated archipelago can frame and cannot answer.