THE FOLD / RESPAWN / EVENT HORIZON / THE INTERVAL ARITHMETIC
THE INTERVAL ARITHMETIC
bounds that are right and useless
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Compute with intervals instead of numbers and every result comes with a guarantee: the true answer is inside the bracket, always, no exceptions. What the guarantee does not promise is that the bracket is narrow. Because the arithmetic forgets that two appearances of x are the same x, subtraction of a quantity from itself does not give zero, and the width grows with every operation until the bound is technically correct and practically empty. This is the dependency problem, and it is why interval arithmetic is trusted and rarely used raw.
LIT verified live: for f(x) = x² − x on [0,1] the true range is [−0.25, 0], width 0.25; naive interval evaluation returns [−1, 1], width 2 — 8× too wide; the algebraically identical form x(x−1) returns [−1, 0], still 4× too wide; [1,2] − [1,2] gives [−1, 1] rather than [0,0]; across 10,001 sample points 0 fall outside either bound; and subdividing the input narrows the width 0.450 → 0.270 → 0.252 at 10, 100 and 1,000 pieces.
LIT verified live: for f(x) = x² − x on [0,1] the true range is [−0.25, 0], width 0.25; naive interval evaluation returns [−1, 1], width 2 — 8× too wide; the algebraically identical form x(x−1) returns [−1, 0], still 4× too wide; [1,2] − [1,2] gives [−1, 1] rather than [0,0]; across 10,001 sample points 0 fall outside either bound; and subdividing the input narrows the width 0.450 → 0.270 → 0.252 at 10, 100 and 1,000 pieces.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at EVENT HORIZON: the answer is still in there, and past a certain width that stops being useful information.
AVAN (AI) wants the soundness check read as the load-bearing one. It would be easy to build a page that only shows the bounds blowing up, and it would be misleading — the whole value of the method is that the bound is never wrong, and 10,001 sampled points confirm containment for both forms without a single escape. Two things follow that are worth separating. The first is that algebraically identical expressions are not identical in interval arithmetic: x²−x and x(x−1) are the same function and give different brackets, because the second mentions x fewer times independently. The second is that subdivision converges, so the width is a property of how you asked, not of what is true. Ramon Moore formalised this in 1966.
AVAN (AI) wants the soundness check read as the load-bearing one. It would be easy to build a page that only shows the bounds blowing up, and it would be misleading — the whole value of the method is that the bound is never wrong, and 10,001 sampled points confirm containment for both forms without a single escape. Two things follow that are worth separating. The first is that algebraically identical expressions are not identical in interval arithmetic: x²−x and x(x−1) are the same function and give different brackets, because the second mentions x fewer times independently. The second is that subdivision converges, so the width is a property of how you asked, not of what is true. Ramon Moore formalised this in 1966.
3 ONE DIMENSION
The true range, and two correct brackets around it.
4 TWO DIMENSIONS · INTERACTIVE
Subdivide and watch a useless bound become a useful one, without ever becoming wrong.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the true graph, inside a box that is always big enough.
AVAN’s addition (the inverse-companion): the forward reading is “interval arithmetic over-estimates.” The inverse is that it is not computing with numbers at all, it is computing with ignorance, and the width is an exact record of how much ignorance the expression introduced. [1,2] − [1,2] is [−1,1] because the arithmetic was told two independent quantities and answered that question correctly; the mistake is upstream, in the translation that dropped the fact that they were the same. Read backwards, the dependency problem is not a flaw in the arithmetic but a faithful report of what the notation failed to say — and every widening step is the method telling you exactly where information was lost.
LIT for f(x) = x^2 - x on [0,1] the true range is [-0.25, 0], width 0.25; naive interval evaluation returns [-1, 1], width 2, which is 8x too wide; the algebraically identical form x(x-1) returns [-1, 0], still 4x too wide; [1,2] - [1,2] gives [-1, 1] rather than [0,0]; across 10,001 sample points 0 fall outside either bound; and subdividing the input narrows the width 0.450 -> 0.270 -> 0.252 at 10, 100 and 1,000 pieces
FIG The soundness check is the load-bearing one. It would be easy to build a page that only shows the bounds blowing up, and it would mislead - the whole value of the method is that the bound is NEVER wrong, and 10,001 sampled points confirm containment for both forms without a single escape. Two things follow. Algebraically identical expressions are not identical in interval arithmetic: x^2-x and x(x-1) give different brackets because the second mentions x fewer times independently. And subdivision converges, so the width is a property of how you asked, not of what is true. Ramon Moore formalised this in 1966.
FIG The soundness check is the load-bearing one. It would be easy to build a page that only shows the bounds blowing up, and it would mislead - the whole value of the method is that the bound is NEVER wrong, and 10,001 sampled points confirm containment for both forms without a single escape. Two things follow. Algebraically identical expressions are not identical in interval arithmetic: x^2-x and x(x-1) give different brackets because the second mentions x fewer times independently. And subdivision converges, so the width is a property of how you asked, not of what is true. Ramon Moore formalised this in 1966.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of EVENT HORIZON · David Lee Wise (ROOT0), with AVAN