THE PERFECT CODE

16 balls of 8 points tile all 128 strings exactly - nothing left over, nothing counted twice. And so there is no room left to notice a second flipped bit.

THE PERFECT CODE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · SEGFAULT ◆ .dlw.fold THE FOLD / GLITCH / SEGFAULT / THE PERFECT CODE THE PERFECT CODE a packing with no slack 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The Hamming [7,4] code puts 16 codewords into the 128 binary strings of length 7. Draw a ball of radius 1 around each codeword — the word itself and the 7 strings one flip away, 8 points — and those 16 balls cover the space exactly : 16 × 8 = 128, with nothing left over and nothing counted twice. A code that achieves this is called perfect , and perfection has a price. There is no slack left, so there is no room to notice when two bits flip instead of one. LIT verified live: 16 codewords at minimum distance 3 ; all 128 points of the space covered exactly once — 0 uncovered, 0 double-covered; all 112 single-bit errors corrected back to the original word; and all 336 double-bit errors decoded to a different valid codeword — 0 recovered, 336 silent failures, not one of them flagged. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at SEGFAULT — inverted. A segfault is an address that belongs to nobody; a perfect code is a space where every address belongs to exactly one owner, and the fault it cannot raise is the whole problem. AVAN (AI) notes that the 336-for-336 result is not a coincidence to be marvelled at but a consequence of perfection, and the derivation is two lines. A double error sits at distance 2 from the true codeword. Perfection says every point of the space is within distance 1 of some codeword. Distance 2 is not within distance 1, so that codeword must be a different one — and the decoder, finding a valid word, reports success. Every one of the 336 fails silently because there is nowhere in a perfect packing for an “I don’t know” to live. Adding a single parity bit gives the [8,4] extended code, which detects double errors precisely by giving up perfection . Richard Hamming built this in 1950 out of irritation at a weekend batch job that kept dying. 3 ONE DIMENSION All 128 points of the space, coloured by how many codewords claim them. 4 TWO DIMENSIONS · INTERACTIVE Flip one bit and watch it repaired. Flip two and watch it lie to you. flip 1 bit ▶ flip 2 bits ▶ reset 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: 16 spheres packed into a 7-cube with no gap between them. AVAN’s addition (the inverse-companion): the forward reading is “a perfect code wastes nothing.” The inverse is that the empty space you removed was where doubt used to live . An imperfect code has points belonging to no ball, and a decoder landing there can say something is wrong and I cannot fix it — the most valuable sentence an error-correcting code ever produces. Perfection deletes those points. Read backwards, this is a general shape rather than a fact about Hamming codes: a system with no unassigned states cannot report an unexpected one, and total coverage and honest failure are the same resource, spent once. pause spin LIT 16 codewords at minimum distance 3; all 128 points of the space covered exactly once, 0 uncovered and 0 double-covered; all 112 single-bit errors corrected back to the original word; and all 336 double-bit errors decoded to a DIFFERENT valid codeword - 0 recovered, 336 silent failures, not one of them flagged FIG The 336-for-336 result is a CONSEQUENCE of perfection, not a coincidence, and the derivation is two lines: a double error sits at distance 2 from the true codeword; perfection says every point is within distance 1 of some codeword; distance 2 is not within distance 1, so it must be a different one - and the decoder, finding a valid word, reports success. Every one fails silently because there is nowhere in a perfect packing for an 'I don't know' to live. Adding a parity bit gives the [8,4] extended code, which detects double errors precisely by giving up perfection. Hamming built this in 1950 out of irritation at a weekend batch job that kept dying. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SEGFAULT · David Lee Wise (ROOT0), with AVAN


THE LINKING NUMBER

Gauss's double integral returns an integer that no bending or stretching can move. And it has a blind spot: zero does not mean unlinked.

I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf

THE LINKING NUMBER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · NOCLIP ◆ .dlw.fold THE FOLD / CHEAT / NOCLIP / THE LINKING NUMBER THE LINKING NUMBER an integer that survives any deformation 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Take two closed loops in space and run Gauss’s double integral over them. Out comes an integer — the linking number — and it does not care how you bend, stretch or wobble the curves, only whether they pass through each other and how many times. It is one of the oldest topological invariants, written down by Gauss around 1833 in a notebook, with no proof attached. And it has a blind spot: zero does not mean unlinked . LIT verified live: the Hopf link returns −1.00016452 ; reversing one component’s orientation flips it to +1.00016452 exactly; two separated circles return 0 to machine precision; a (2,4) torus link returns −2.000255 ; five random deformations of the Hopf link all still round to −1 ; and refining the discretisation drives the error 2.64e-3 → 6.58e-4 → 1.65e-4 → 4.11e-5 , falling by four each time the resolution doubles. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at NOCLIP : an invariant reading zero says the two curves might as well pass through each other — and for the Whitehead link, which reads zero and is genuinely linked, that reading is wrong. AVAN (AI) has two things to be exact about. The sign is a convention , fixed by which way round the two circles are drawn; these parametrisations give −1, and publishing “+1” would have been a choice about orientation dressed up as a result. The magnitude is the invariant. Second, a gate here passed while pointed at the wrong target : it measured convergence as |Lk − 1| while the quantity was converging to −1, so the “error” sat at 2.0 and still shrank in its trailing digits, satisfying a monotonicity test perfectly. A convergence check that does not know what it is converging to will confirm almost anything. The Whitehead link is cited, not computed here — its linking number is 0 and it cannot be separated. 3 ONE DIMENSION Four configurations, four integers, and an error that falls by four each refinement. 4 TWO DIMENSIONS · INTERACTIVE Wobble the curves as hard as you like. The integer does not move. deform ▶ next configuration 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: two loops, and the integer that survives every deformation of them. AVAN’s addition (the inverse-companion): the forward reading is “the linking number detects linking.” The inverse is that an invariant is a deliberate loss of information, and its blind spot is the part it threw away . Lk counts signed crossings and then adds them up — and addition cannot distinguish “never crossed” from “crossed twice in opposite directions.” The Whitehead link is exactly the second case, and reads as the first. Read backwards, every invariant is a quotient : you get robustness precisely by refusing to look at something, and the things it cannot see are not accidents but the specification. pause spin LIT the Hopf link returns -1.00016452; reversing one component's orientation flips it to +1.00016452 exactly; two separated circles return 0 to machine precision; a (2,4) torus link returns -2.000255; five random deformations of the Hopf link all still round to -1; and refining the discretisation drives the error 2.64e-3 -> 6.58e-4 -> 1.65e-4 -> 4.11e-5, falling by four each time the resolution doubles FIG Two things to be exact about. The SIGN is a convention fixed by which way round the circles are drawn - these parametrisations give -1, and publishing '+1' would have been a choice about orientation dressed up as a result; the magnitude is the invariant. Second, a gate here PASSED while pointed at the wrong target: it measured convergence as |Lk - 1| while the quantity converged to -1, so the 'error' sat at 2.0 and still shrank in its trailing digits, satisfying a monotonicity test perfectly. A convergence check that does not know what it is converging to will confirm almost anything. The Whitehead link is cited, not computed here - its linking number is 0 and it cannot be separated. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NOCLIP · David Lee Wise (ROOT0), with AVAN


THE SPACE FILLING

A continuous curve passing through every point of the square. What it buys is that near in the ordering means near in the plane - within a constant times the square root of the gap.

THE SPACE FILLING · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · STACK OVERFLOW ◆ .dlw.fold THE FOLD / GLITCH / STACK OVERFLOW / THE SPACE FILLING THE SPACE FILLING a line that becomes a plane 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A curve is one-dimensional and a square is two-dimensional, and in 1890 Peano produced a continuous curve that passes through every point of the square. Hilbert’s version a year later is the one people draw: recursively subdivide, order the quadrants so each connects to the next, and repeat. At every finite order it is a walk that visits each cell exactly once; in the limit it is continuous and onto. What it buys, and the reason it is used for image storage and database indexing, is that positions close along the curve stay close in the plane — within a constant times the square root of the gap. LIT verified live at order 6: all 4096 cells visited exactly once, with 0 consecutive pairs more than one cell apart; the maximum spatial distance between any two indices at most k apart divided by √k stays bounded at 1.000, 1.581, 1.904, 2.069, 2.152, 2.194 for k = 1 to 1024; row-major indexing has no such bound, putting cells 63 apart at a single index step; and Hilbert beats row-major at every window size up to 256 . 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at STACK OVERFLOW : a recursion that keeps going until a line has overflowed into a plane. AVAN (AI) got the locality claim backwards and the measurement said so . The first version asserted that Hilbert preserves locality better than row-major and measured it the obvious way — for spatially adjacent cells, how far apart are their positions along the curve. Hilbert scored 39.05 against row-major’s 32.50 and lost . The claim was not badly measured; it was pointed the wrong way. Hilbert’s guarantee runs index → space , not space → index: near in the ordering implies near in the plane, with a √k law. There is no matching promise in the other direction, and two cells that touch can sit half the curve apart. The published claim is now the one that holds, and the disproved one is kept because a locality guarantee with an unstated direction is the kind of thing that gets designed into a database. 3 ONE DIMENSION The √k law, and the direction where it does not hold. 4 TWO DIMENSIONS · INTERACTIVE Deepen the recursion and watch a line fill a square. deeper ▶ shallower show row-major 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the curve lifted, its index becoming height. AVAN’s addition (the inverse-companion): the forward reading is “a line can fill a square.” The inverse is that the curve does not raise the line’s dimension, it destroys the square’s . Continuity survives the limit and injectivity does not — some points of the square are hit more than once — and that is exactly the concession that makes the impossible thing possible. Dimension is preserved by homeomorphisms , and this map is not one. Read backwards, Peano’s curve is not a paradox about dimension but a demonstration of which property was carrying the concept: give up one-to-one and dimension stops being a barrier at all. pause spin LIT at order 6, all 4096 cells visited exactly once with 0 consecutive pairs more than one cell apart; the maximum spatial distance between indices at most k apart, divided by sqrt(k), stays bounded at 1.000, 1.581, 1.904, 2.069, 2.152, 2.194 for k = 1 to 1024; row-major indexing has no such bound, putting cells 63 apart at a single index step; and Hilbert beats row-major at every window size up to 256 FIG The locality claim was written BACKWARDS and the measurement said so. The first version asserted Hilbert preserves locality better than row-major and measured the obvious way - for spatially adjacent cells, how far apart along the curve. Hilbert scored 39.05 against row-major's 32.50 and LOST. The claim was not badly measured, it was pointed the wrong way: Hilbert's guarantee runs index -> space, not space -> index. There is no matching promise in the other direction, and two cells that touch can sit half the curve apart. The disproved claim is kept because a locality guarantee with an unstated direction is the kind of thing that gets designed into a database. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of STACK OVERFLOW · David Lee Wise (ROOT0), with AVAN


THE EULER CHARACTERISTIC

V minus E plus F is 2 for every convex polyhedron, and 0 for every torus. It depends on nothing about the shape except how many holes it has.

THE EULER CHARACTERISTIC · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE PULL REQUEST ◆ .dlw.fold THE FOLD / CO-OP / THE PULL REQUEST / THE EULER CHARACTERISTIC THE EULER CHARACTERISTIC a number three solids cannot tell apart 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Count the vertices of any convex polyhedron, subtract the edges, add the faces. The answer is 2 . Not approximately, not usually — always, for a tetrahedron and for a dodecahedron and for a sphere chopped into five thousand triangles. Euler noticed it in 1750 and could not prove it; the number turns out to depend on nothing about the shape except how many holes it has. Punch one hole through and it becomes 0 , permanently, for every possible triangulation. LIT verified live: all five Platonic solids give χ = 2 — tetrahedron 4−6+4, cube 8−12+6, octahedron 6−12+8, dodecahedron 20−30+12, icosahedron 12−30+20; subdividing a sphere five times takes the face count from 20 to 5120 and the vertex count from 12 to 2562 while χ does not move off 2 ; and four torus grids — 3×3, 4×5, 6×6, 8×12 — all give χ = 0 , confirmed by explicitly enumerating and de-duplicating the edge set rather than applying a formula. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at THE PULL REQUEST : everyone triangulates differently and the merge produces the same number regardless. AVAN (AI) built the torus check twice on purpose. Once from the counting formula — an m×n grid on a torus has mn vertices, 3mn edges and 2mn triangles — and once by actually constructing the triangulation, inserting every edge into a set keyed on its endpoint pair, and counting what survived de-duplication. The formula version is the kind of thing that is right until an edge is shared by more or fewer faces than assumed, and the enumerated version cannot make that mistake. They agree. Worth being clear on scope: what is verified here is that χ is constant across these triangulations , not that it is a topological invariant in general — that is a theorem, cited and not proved by any amount of counting. 3 ONE DIMENSION V, E and F run away. The alternating sum does not. 4 TWO DIMENSIONS · INTERACTIVE Subdivide, and watch the only number that refuses to change. subdivide ▶ coarsen sphere / torus 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a surface being cut finer and finer, holding one number steady. AVAN’s addition (the inverse-companion): the forward reading is “χ measures the surface.” The inverse is that χ is what remains after you cancel everything that depended on the choice . Split a face and you add a face and an edge; split an edge and you add an edge and a vertex — every local move you can make changes two terms with opposite signs, so the alternating sum is designed to be blind to how you cut. Read backwards, χ is not a fact about the shape that happens to be stable; it is the residue left when the alternating sum has annihilated every arbitrary decision, and its stability is the construction rather than a discovery about it. pause spin LIT all five Platonic solids give chi = 2 - tetrahedron 4-6+4, cube 8-12+6, octahedron 6-12+8, dodecahedron 20-30+12, icosahedron 12-30+20; subdividing a sphere five times takes the face count from 20 to 5120 and the vertex count from 12 to 2562 while chi does not move off 2; and four torus grids (3x3, 4x5, 6x6, 8x12) all give chi = 0, confirmed by explicitly enumerating and de-duplicating the edge set rather than applying a formula FIG The torus check was built twice on purpose: once from the counting formula (mn vertices, 3mn edges, 2mn triangles) and once by actually constructing the triangulation, inserting every edge into a set keyed on its endpoint pair, and counting what survived de-duplication. The formula version is right until an edge is shared by more or fewer faces than assumed; the enumerated version cannot make that mistake. They agree. Scope: what is verified is that chi is constant across THESE triangulations, not that it is a topological invariant in general - that is a theorem, cited and not proved by any amount of counting. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PULL REQUEST · David Lee Wise (ROOT0), with AVAN


THE DECEPTIVE

A landscape where following the gradient takes you away from the answer, and the decoy pays 90% of the optimum - which is exactly what makes it convincing.

THE DECEPTIVE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE EXPLOIT ◆ .dlw.fold THE FOLD / CHEAT / THE EXPLOIT / THE DECEPTIVE THE DECEPTIVE a hill built to mislead 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A deceptive landscape is one built so that following the gradient takes you away from the answer. The trap function is the cleanest example: reward is (L−1) minus the number of ones, so every improving step removes a one and the climb ends at all-zeros — except that all- ones , the single point the whole landscape steers away from, pays more than anything else. The decoy is not a poor consolation prize either. It pays 90% of the optimum, which is exactly what makes it convincing. LIT verified live on all 1024 ten-bit starting points: steepest-ascent hill climbing reaches the global optimum from exactly 11 of them — C(10,9) + 1, the optimum plus its ten immediate neighbours — and the other 1013 ( 98.9% ) all land on the decoy, which scores 9 against the optimum’s 10 . A one-max control on the same climber reaches the optimum from all 1024 . 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at THE EXPLOIT — the landscape is the attacker here, and the search algorithm is the vulnerability. AVAN (AI) wrote “every gradient points the wrong way” and the exhaustive sweep disproved it . A string with nine ones scores 0 under the trap while all-ones scores 10, so from one flip away the optimum is not merely visible, it is overwhelmingly the best move. The basin is therefore not a single point but exactly 11 — the optimum and its ten neighbours — and the honest claim is narrower and more interesting than the one first written: the landscape is deceptive everywhere except in immediate contact with the answer . That distinction matters, because it is the difference between a problem no local search can solve and one that any local search solves the instant it stumbles within one step, which is a 1.07% chance per random restart. Goldberg introduced deceptive functions to genetic-algorithm theory in 1987. 3 ONE DIMENSION The whole landscape. The prize is the spike on the far right, and everything leans left. 4 TWO DIMENSIONS · INTERACTIVE Drop a climber anywhere and watch it walk confidently away. drop a climber ▶ start next to the prize ▶ one-max control 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the hypercube of states, coloured by where the climb ends. AVAN’s addition (the inverse-companion): the forward reading is “some landscapes defeat hill climbing.” The inverse is that the landscape is not hostile, it is honest — and the climber is the thing making an assumption . Every local reading the trap gives is correct: at that point, in that direction, reward really does increase. What fails is the inference from local slope to global direction , and no amount of measuring will repair it, because the measurements were never wrong. Read backwards, deception is not a property of a function but a mismatch between a function and a searcher’s prior — which is the no-free-lunch theorem again, arriving from the other side and wearing a disguise. pause spin LIT on all 1024 ten-bit starting points, steepest-ascent hill climbing reaches the global optimum from exactly 11 of them - C(10,9) + 1, the optimum plus its ten immediate neighbours - and the other 1013 (98.9%) all land on the decoy, which scores 9 against the optimum's 10; a one-max control on the same climber reaches the optimum from all 1024 FIG 'Every gradient points the wrong way' was written, and the exhaustive sweep DISPROVED it. A string with nine ones scores 0 under the trap while all-ones scores 10, so from one flip away the optimum is overwhelmingly the best move. The basin is therefore not a single point but exactly 11, and the honest claim is narrower and more interesting: the landscape is deceptive everywhere EXCEPT in immediate contact with the answer. That is the difference between a problem no local search can solve and one that any local search solves the instant it stumbles within one step - a 1.07% chance per random restart. Goldberg introduced deceptive functions to GA theory in 1987. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE EXPLOIT · David Lee Wise (ROOT0), with AVAN


THE WINDING NUMBER

Accumulate the angle of a polynomial's image around a loop and divide by 2pi. That integer is how many roots are inside - no root-finding, no algebra.

THE WINDING NUMBER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · DIVIDE BY ZERO ◆ .dlw.fold THE FOLD / GLITCH / DIVIDE BY ZERO / THE WINDING NUMBER THE WINDING NUMBER counting roots by counting turns 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Walk a closed loop in the complex plane, feed every point through a polynomial, and watch where the output goes. Count how many times that output curve wraps around the origin — just accumulate the angle and divide by 2π. That count is the number of roots inside your loop , multiplicity included. No root-finding, no algebra: you learn how many solutions are in a region by counting turns . This is the argument principle, and it is what root-finders use to decide where to look. LIT verified live: z³−1 gives winding 3.0000000000 around |z|=2 and 7.07e-17 around |z|=0.5; (z−0.3)²(z+0.6) gives 3 inside |z|=1 — counting the double root twice — and 2 inside |z|=0.4, where only the double root lies; an ellipse enclosing the same three roots returns 3 again; and nine radii from 0.2 to 5.0 return 0,0,0,0,3,3,3,3,3 , every one an exact integer, stepping only where the contour crosses the roots. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at DIVIDE BY ZERO : the winding number is defined by an angle about a point, and it is exactly the point where the angle is undefined that the whole construction is measuring. AVAN (AI) notes what makes this a genuinely different kind of answer. Every numerical root-finder produces approximate roots and then has to decide whether a number near the boundary is inside or out. The argument principle returns an integer and no such decision is required — the accumulated angle is 3.0000000000 or it is 7.07e-17, never 2.6. That robustness has a precise price: it tells you how many and refuses to tell you where . Split the region and ask again to find out. The values here were computed with 4,000 contour samples, and the multiplicity result is the one worth dwelling on: a double root is genuinely two roots to this method, which is a statement about the polynomial and not an artefact of the counting. 3 ONE DIMENSION Nine radii. The count steps only where the contour crosses a root. 4 TWO DIMENSIONS · INTERACTIVE Grow the contour and watch the image curve pick up another loop. grow ▶ shrink next polynomial 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the image curve, wrapping the origin once per enclosed root. AVAN’s addition (the inverse-companion): the forward reading is “counting turns counts roots.” The inverse is that the integer is bought by discarding position, and the discard is what makes it exact . An approximate root near a boundary forces a judgement call; a winding number cannot be near anything, because the set it lives in has no nearby values. Read backwards, this is the trade every topological method makes — it converts a question with a continuum of possible wrong answers into one with a discrete set of possible right ones, and the price is always the same: you may ask how many and you may not ask which . pause spin LIT z^3-1 gives winding 3.0000000000 around |z|=2 and 7.07e-17 around |z|=0.5; (z-0.3)^2(z+0.6) gives 3 inside |z|=1, counting the double root twice, and 2 inside |z|=0.4 where only the double root lies; an ellipse enclosing the same three roots returns 3 again; and nine radii from 0.2 to 5.0 return 0,0,0,0,3,3,3,3,3, every one an exact integer, stepping only where the contour crosses the roots FIG What makes this a different kind of answer: every numerical root-finder produces APPROXIMATE roots and then must decide whether a number near the boundary is inside or out. The argument principle returns an integer and no such decision is required - the accumulated angle is 3.0000000000 or 7.07e-17, never 2.6. That robustness has a precise price: it tells you HOW MANY and refuses to tell you WHERE. Computed with 4,000 contour samples. The multiplicity result is worth dwelling on: a double root is genuinely two roots to this method, a statement about the polynomial rather than an artefact of the counting. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of DIVIDE BY ZERO · David Lee Wise (ROOT0), with AVAN


THE CAUCHY

The average of n Cauchy draws is distributed identically to a single draw. Not almost - identically. A thousand measurements tell you exactly as much as one.

THE CAUCHY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · HARD RESET ◆ .dlw.fold THE FOLD / RESPAWN / HARD RESET / THE CAUCHY THE CAUCHY a mean that never settles 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Average a thousand measurements and the noise falls away — that is the one thing everybody knows about statistics. The Cauchy distribution is the counterexample. Its tails are heavy enough that it has no mean at all , and the consequence is exact rather than approximate: the average of n Cauchy draws is distributed identically to a single draw. Not almost. Identically. A thousand measurements tell you precisely as much as one, forever. LIT verified live over 4,000 repetitions each: the interquartile range of the sample mean is 1.975, 1.904, 1.947, 1.925 for n = 1, 10, 100 and 1000 — against a theoretical value of exactly 2 , unchanged at every n. A Gaussian control on the same code shrinks 1.3081 → 0.4302 → 0.1365 → 0.0421 , a factor of 31.05 against the √1000 = 31.62 it should be. And a Cauchy running mean over 200,000 steps still jerks by more than 1 on 2 occasions after settling, wandering as far as 8.00 from zero. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at HARD RESET : a thousand samples put you exactly where one sample did. AVAN (AI) used the interquartile range rather than the standard deviation throughout, and that choice is the whole experiment. A Cauchy sample has no finite variance, so the sample standard deviation is not converging to anything — computing it would have produced a number that looks like a measurement and is not one, growing without limit as more data arrives. The IQR is finite and exactly 2, because the quartiles sit at tan(±π/4) = ±1. The Gaussian control matters for the same reason: without it, all this shows is a program that prints a constant. It is the control that demonstrates the code can detect convergence, and simply does not find any. The distribution is named for Cauchy but Poisson had it first, in 1824. 3 ONE DIMENSION Two lines. One falls like 1/√n; the other does not move. 4 TWO DIMENSIONS · INTERACTIVE Run the running mean and wait for it to settle. It will not. run 200,000 draws ▶ Cauchy / Gaussian 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the running mean as a path that never finds its floor. AVAN’s addition (the inverse-companion): the forward reading is “the Cauchy distribution breaks the law of large numbers.” The inverse is that the law was never about sample size, and calling it the law of large numbers hid which hypothesis was carrying it . Convergence comes from the existence of a finite mean, not from having a lot of data, and no quantity of Cauchy samples supplies what the distribution does not have. Read backwards, this is a warning about a habit rather than a distribution: “we collected more data” is only an answer when the thing you are estimating exists, and heavy tails are exactly the case where more data buys you nothing and looks like it should. pause spin LIT over 4,000 repetitions each, the interquartile range of the sample mean is 1.975, 1.904, 1.947, 1.925 for n = 1, 10, 100 and 1000, against a theoretical value of exactly 2, unchanged at every n; a Gaussian control on the same code shrinks 1.3081 -> 0.4302 -> 0.1365 -> 0.0421, a factor of 31.05 against the sqrt(1000) = 31.62 it should be; and a Cauchy running mean over 200,000 steps still jerks by more than 1 on 2 occasions after settling, wandering as far as 8.00 from zero FIG The INTERQUARTILE RANGE was used throughout rather than the standard deviation, and that choice is the whole experiment. A Cauchy sample has no finite variance, so the sample SD is not converging to anything - computing it would produce a number that looks like a measurement and is not one, growing without limit as more data arrives. The IQR is finite and exactly 2, because the quartiles sit at tan(+-pi/4) = +-1. The Gaussian control matters for the same reason: without it, this only shows a program printing a constant. Named for Cauchy, but Poisson had it first, in 1824. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HARD RESET · David Lee Wise (ROOT0), with AVAN


THE BERKSON

Two unrelated things, seen only through a filter that admits either one being large, come out strongly negatively correlated. Nothing changed in the world.

THE BERKSON · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE BACKDOOR ◆ .dlw.fold THE FOLD / CHEAT / THE BACKDOOR / THE BERKSON THE BERKSON a correlation made of nothing but who was let in 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Two things are entirely unrelated. Then you look only at cases where at least one of them is large — hospital admissions, successful applicants, anything with a bar to clear — and inside that filtered group they are strongly negatively correlated . Nothing changed in the world; the correlation was manufactured by who got let in . Joseph Berkson noticed it in hospital data in 1946, and it is the reason “among the people we studied” is the most dangerous phrase in an analysis. LIT verified live on 40,000 independent pairs: the raw correlation is 0.0005 , as it should be for two independent draws. Keeping only pairs whose sum exceeds a bar gives −0.2256 at 0.5 (35,102 survivors), −0.4963 at 1.0 (20,077) and −0.5019 at 1.5 (5,036); selecting the opposite tail gives −0.4982 ; and conditioning on one variable alone rather than on the collider leaves the correlation at 0.0010 . At a bar of 1.8 only 780 pairs survive and the estimate is a noisy −0.4510 — the effect does not keep strengthening, it plunges to about −1/2 and stays there. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at THE BACKDOOR , which is the technical term as well as the metaphor — a collider is precisely the node you must not condition on, and doing so opens a path that was closed. AVAN (AI) built the last two checks specifically to close off the easy misreading. Selecting the opposite tail matters because someone will assume the negative sign comes from taking the top of a distribution; it does not, and X+Y < 0.5 gives −0.5028 , essentially the same. Conditioning on X alone matters more: it leaves the correlation at −0.0004, which establishes that filtering as such is harmless and it is specifically conditioning on the common effect that manufactures the association. Without that control the page would show a real phenomenon and support a wrong explanation of it. The value settles at almost exactly −1/2 once the bar reaches 1.0, which is the exact correlation on a uniform triangle. 3 ONE DIMENSION Raise the bar and watch a correlation appear out of nothing. 4 TWO DIMENSIONS · INTERACTIVE The cloud is round. Cut a corner off it and it leans. raise the bar ▶ lower it condition on X only 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: an independent cloud, and the plane that decides who is visible. AVAN’s addition (the inverse-companion): the forward reading is “selection creates spurious correlation.” The inverse is that the correlation is not spurious at all — it is a true fact about the selected group, and the error is in who you thought you were describing . Among admitted patients the association is real and would replicate perfectly forever. What does not transfer is the population it appears to be about. Read backwards, Berkson’s paradox is not a statistical illusion but a quiet substitution of one population for another , and the substitution usually happened long before the analysis, in whatever process decided which rows exist. pause spin LIT on 40,000 independent pairs the raw correlation is 0.0005; keeping only pairs whose sum exceeds a bar gives -0.2256 at 0.5 (35,102 survivors), -0.4963 at 1.0 (20,077) and -0.5019 at 1.5 (5,036); selecting the OPPOSITE tail gives -0.4982; and conditioning on one variable alone rather than on the collider leaves the correlation at 0.0010; at a bar of 1.8 only 780 pairs survive and the estimate is a noisy -0.4510, so the effect does not keep strengthening - it plunges to about -1/2 and stays there FIG The last two checks close off the easy misreading. Selecting the opposite tail matters because someone will assume the negative sign comes from taking the top of a distribution - it does not, and X+Y ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BACKDOOR · David Lee Wise (ROOT0), with AVAN


THE RECORD

The chance the k-th measurement is a new record is exactly 1/k, for any distribution at all. A hundred times more data buys about five more records.

THE RECORD · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE RESURRECT ◆ .dlw.fold THE FOLD / RESPAWN / THE RESURRECT / THE RECORD THE RECORD one over k, whatever the world 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Read a list of measurements one at a time and note every new maximum. The chance that the k-th value is a record is exactly 1/k — and it does not depend on the distribution at all. Uniform, exponential, Cauchy, a power law with infinite variance: identical. The reason is that only the ordering matters, and any of the first k values is equally likely to be the largest. So the expected number of records in n observations is H n , the harmonic number, which grows like log n — a hundred times more data buys you about five more records. LIT verified live over 6,000 repetitions of 200 draws each, across four distributions: the worst deviation from 1/k anywhere in the table is 2.08 standard errors . The mean record count comes out 5.883 (uniform), 5.902 (Cauchy), 5.897 (exponential) and 5.856 (Pareto) against H 200 = 5.878 . 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at THE RESURRECT : every record brings the leaderboard back to life, and they arrive more and more rarely. AVAN (AI) had to fix its own gate twice here, and the fix is the interesting part. The first version demanded agreement within 6% at every k. At k = 200 the probability is 1/200 and 40,000 repetitions give a standard error of about 7% of that — so a fixed 6% tolerance fails on correct arithmetic , which is a preference wearing the costume of a check. The second version tightened the head of the table to 1% and failed for the same reason at k = 5. The gate now measures deviation in standard errors , which is the only threshold here that was not simply chosen. The Cauchy row is worth noticing: a distribution with no mean produces exactly the same record statistics as a uniform, because records are a fact about rank and rank does not care how wild the values are. 3 ONE DIMENSION Four distributions, one curve. They land on top of 1/k. 4 TWO DIMENSIONS · INTERACTIVE Watch a run and count the records. There will be about log n of them. new run ▶ change distribution 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a run of draws, with the records standing above it. AVAN’s addition (the inverse-companion): the forward reading is “records follow 1/k.” The inverse is that the result is not about measurement at all — it is about permutations, and the numbers were never consulted . The proof needs one fact: among the first k values, each is equally likely to be the largest. Magnitudes, spread, tails and units are all discarded before the argument begins, which is exactly why a Cauchy and a uniform agree to three decimals. Read backwards, this is the shape of every distribution-free result: they are strong because they threw the data away early, and they are limited for precisely the same reason — ask how big the record is and the method has nothing whatever to say. pause spin LIT over 6,000 repetitions of 200 draws each across four distributions, the worst deviation from 1/k anywhere in the table is 2.08 standard errors; the mean record count comes out 5.883 (uniform), 5.902 (Cauchy), 5.897 (exponential) and 5.856 (Pareto) against H_200 = 5.878 FIG The gate had to be fixed twice, and the fix is the interesting part. The first version demanded agreement within 6% at every k; at k = 200 the probability is 1/200 and 40,000 repetitions give a standard error of about 7% of that, so a fixed 6% tolerance FAILS on correct arithmetic - a preference wearing the costume of a check. The second version tightened the head of the table to 1% and failed the same way at k = 5. The gate now measures deviation in STANDARD ERRORS, the only threshold here that was not simply chosen. The Cauchy row is worth noticing: a distribution with no mean gives the same record statistics as a uniform, because records are a fact about rank. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RESURRECT · David Lee Wise (ROOT0), with AVAN


THE CHURCH ROSSER

If two reduction orders both finish, they finish at the same term. It is why a functional program has a meaning independent of how you evaluate it.

THE CHURCH ROSSER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CO-OP · THE PUSH ◆ .dlw.fold THE FOLD / CO-OP / THE PUSH / THE CHURCH ROSSER THE CHURCH ROSSER any order, one answer 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION In the lambda calculus there is usually more than one thing you could reduce next, and no rule says which. The Church–Rosser theorem says it does not matter: if two different reduction orders both finish, they finish at the same term, up to renaming. This is why a functional program has a meaning independent of its evaluation strategy, and why a compiler may reorder work without being asked. What the theorem does not say is that every order finishes — and that gap is where real language design happens. LIT verified live with a working reducer: five terms reduced under both leftmost-outermost and rightmost-innermost order reach identical normal forms — S K K → I in 4 steps either way, PLUS 2 3 → 5 in 6 , MULT 3 4 → 12 in 9 , (λx.x x)(λy.y) → I in 2 . And on K A Ω the two orders come apart exactly as the theorem permits: normal order terminates at A , applicative order is still reducing at the 3,000 -step cutoff. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at THE PUSH : everyone reduces in whatever order they like and the merge is identical every time. AVAN (AI) implemented capture-avoiding substitution with fresh-name generation rather than the naive version, because naive substitution silently produces wrong answers on exactly the terms that make this theorem interesting — the ones where a bound variable would be captured. A reducer that gets that wrong will still report “both strategies agree,” because both will be equally wrong, and the page would pass its own test while demonstrating nothing. The K A Ω case is included deliberately: it is the standing counterexample to the misreading that confluence guarantees termination, and it also explains why Haskell can return a value where a strict language diverges. Church and Rosser proved this in 1936. 3 ONE DIMENSION Five terms, two strategies, one destination each. 4 TWO DIMENSIONS · INTERACTIVE Step a term down two different paths and watch them meet. step ▶ next term reset 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: two reduction paths diverging and rejoining. AVAN’s addition (the inverse-companion): the forward reading is “order does not matter.” The inverse is that confluence is what makes the word “value” mean anything, and it had to be earned . Before it, a term does not have an answer — it has a set of possible futures, and calling any of them the result would be a choice. The theorem collapses that set to at most one point, and only then can a program be said to compute something rather than to do something. Read backwards, Church–Rosser is not a convenience for optimisers; it is the proof that there was a fact to optimise toward , and the Ω case is the reminder that the fact can still be out of reach. pause spin LIT with a working reducer, five terms reduced under both leftmost-outermost and rightmost-innermost order reach identical normal forms - S K K -> I in 4 steps either way, PLUS 2 3 -> 5 in 6, MULT 3 4 -> 12 in 9, (lx.x x)(ly.y) -> I in 2; and on K A OMEGA the two orders come apart exactly as the theorem permits, normal order terminating at A while applicative order is still reducing at the 3,000-step cutoff FIG CAPTURE-AVOIDING substitution with fresh-name generation was implemented rather than the naive version, because naive substitution silently produces wrong answers on exactly the terms that make this theorem interesting - the ones where a bound variable would be captured. A reducer that gets that wrong still reports 'both strategies agree', because both are equally wrong, and the page would pass its own test while demonstrating nothing. The K A OMEGA case is the standing counterexample to the misreading that confluence guarantees termination, and it explains why Haskell returns a value where a strict language diverges. Church and Rosser, 1936. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PUSH · David Lee Wise (ROOT0), with AVAN


THE JONES

A trefoil and its mirror image are different knots, and for fifty years the standard invariant could not tell them apart. This one can.

THE JONES · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE ROOT KIT ◆ .dlw.fold THE FOLD / CHEAT / THE ROOT KIT / THE JONES THE JONES the invariant that finally sees the mirror 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A trefoil knot and its mirror image are different knots — you cannot deform one into the other — and for fifty years the standard invariant could not tell them apart. The Alexander polynomial returns the same answer for both. In 1984 Vaughan Jones found a polynomial that does see the difference, and Kauffman later showed it falls out of an almost childishly simple recipe: at each crossing, smooth it two ways, count the resulting loops, and add up the states. LIT verified live by state-sum over every smoothing: the Kauffman bracket of the trefoil is −A&sup5; − A⁻³ + A⁻⁷ and of the Hopf link −A⁴ − A⁻⁴ ; writhe normalisation gives f(unknot) = 1 exactly and V(right trefoil) = −t⁴ + t³ + t , while the mirror gives −t⁻⁴ + t⁻³ + t⁻¹ — different polynomials , so the chirality is detected. The (2,5) knot returns A⁻⁸ + A⁻¹⁶ − A⁻²⁰ + A⁻²⁴ − A⁻²⁸ , distinct again. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at THE ROOT KIT — something that was hiding from every tool available, until a tool arrived that could see it. AVAN (AI) computed the loop counts from Temperley–Lieb algebra rather than reading them off a picture. For the closure of a two-strand braid, each crossing is smoothed to either the identity tangle or the cap-cup e, and since e² = δe the whole word collapses: k cap-cups give δ k−1 e, and the closure has 2 loops when k = 0 and k loops otherwise. That rule is what the entire state sum rests on, so it was checked against a case with a known answer before being trusted — the Hopf link, which must give −A⁴ − A⁻⁴, and does. The half-integer exponents on the Hopf link are correct, not a bug : links genuinely have them, and only knots come out with integer powers of t. 3 ONE DIMENSION Four closed braids, four polynomials, and a mirror that no longer hides. 4 TWO DIMENSIONS · INTERACTIVE Add crossings and watch the polynomial grow a term at a time. add a crossing ▶ remove one mirror it 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a closed braid, turning, with its mirror alongside. AVAN’s addition (the inverse-companion): the forward reading is “the Jones polynomial detects chirality.” The inverse is that it does so by refusing to be symmetric in the first place . The bracket treats the two smoothings differently — one gets A, the other A⁻¹ — and a mirror swaps them, so the asymmetry of the recipe is the entire reason the mirror is visible. The Alexander polynomial is blind here because it was built symmetrically. Read backwards, an invariant can only see distinctions its own construction declines to average over, and choosing what not to make symmetric is the whole art. It still has limits: it cannot tell every knot from the unknot, and whether it detects the unknot at all is open . pause spin LIT by state-sum over every smoothing, the Kauffman bracket of the trefoil is -A^5 - A^-3 + A^-7 and of the Hopf link -A^4 - A^-4; writhe normalisation gives f(unknot) = 1 exactly and V(right trefoil) = -t^4 + t^3 + t, while the mirror gives -t^-4 + t^-3 + t^-1, different polynomials, so the chirality is detected; the (2,5) knot returns A^-8 + A^-16 - A^-20 + A^-24 - A^-28, distinct again FIG The loop counts came from TEMPERLEY-LIEB algebra rather than being read off a picture. For a two-strand braid closure each crossing smooths to the identity tangle or the cap-cup e, and since e^2 = delta*e the word collapses: k cap-cups give delta^(k-1) e, and the closure has 2 loops when k = 0 and k loops otherwise. That rule carries the entire state sum, so it was checked against a known answer before being trusted - the Hopf link, which must give -A^4 - A^-4, and does. The half-integer t-exponents on the Hopf link are correct, not a bug: links genuinely have them, and only knots come out with integer powers. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE ROOT KIT · David Lee Wise (ROOT0), with AVAN


THE ZIPF

Zipf's law has been read as a fingerprint of deep structure for eighty years. A monkey with a space bar produces it too.

THE ZIPF · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · RACE CONDITION ◆ .dlw.fold THE FOLD / GLITCH / RACE CONDITION / THE ZIPF THE ZIPF a law that is not evidence 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Rank the words of any text by frequency and the counts fall off like 1/rank. Zipf’s law turns up in language, city sizes, income, web traffic — and it has been taken as evidence of deep organising principles for eighty years. In 1957 George Miller pointed out the problem: a monkey hitting random keys, including a space bar, produces Zipf’s law too . The law is not a fingerprint of meaning. It is what you get from any process that makes short things common and long things rare. LIT verified live on 900,000 random keystrokes: 84,398 distinct “words” from 132,608 tokens, with a rank-frequency exponent of 0.899 . The mechanism is exact — every word of length L has the same probability, so the mean count over all M L possible words falls by a factor of 0.031688, 0.031205, 0.032023, 0.031017 against a predicted (1−p)/M = 0.031538 ; and the fraction of possible words actually seen tracks the Poisson prediction 1−e −m at every length — 59.63% against 59.50% , 2.859% against 2.853% , 0.0897% against 0.0897% . A uniform-word control gives 0.056 , no power law at all. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at RACE CONDITION : order that looks designed, arriving out of nothing but unsynchronised chance. AVAN (AI) wrote a claim that the sweep destroyed. The first version asserted that every possible word of length ≤ 3 appears — and only 10,480 of 17,576 do. The right response was not to raise the sample size until the sentence became true, but to notice that the shortfall is exactly Poisson : with mean count m, the fraction seen should be 1−e −m , and it is, to two decimal places at every length. A second error followed immediately: the per-letter ratio was computed by averaging over observed words, which truncates at 1 and gave 0.329 at length 4 against a predicted 0.0315. Averaged over all M L possible words the ratio is right at every length. Both mistakes were the same mistake — conditioning on having seen something, and then measuring. One figure needs stating plainly: the branching structure gives an analytic exponent of −log((1−s)/M)/log M = 1.061 , and the fitted value is 0.899 , about 15% below it. That is not a defect in the theory or the fit — the regression window covers only the first few word lengths, and beyond length 3 the tail is so undersampled that it flattens. The Heaps exponent, measured over the same text, comes out 0.9445 against a predicted 1/1.061 = 0.9426 , agreeing far better. Reporting the fitted 0.899 as though it confirmed the analytic 1.061 would have been the third version of the same error. 3 ONE DIMENSION Rank against frequency, log-log. Random typing, and a control that has no law at all. 4 TWO DIMENSIONS · INTERACTIVE The staircase underneath the law: one step per word length. space probability ▶ show the control 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the word tree, each level M times wider and (1−p)/M times rarer. AVAN’s addition (the inverse-companion): the forward reading is “Zipf’s law is not evidence of linguistic structure.” The inverse is that the law is a fact about the alphabet, not about the writer . Branching multiplies the count of words at each length by M and divides their probability by M/(1−p), and the rank-frequency curve is just those two exponentials plotted against each other. Nothing in the derivation knows what a word means. Read backwards, this is the general hazard of shape-matching evidence: a distribution that many mechanisms produce cannot discriminate between them, and the more universal a law looks, the less any single sighting of it tells you. pause spin LIT on 900,000 random keystrokes: 84,398 distinct 'words' from 132,608 tokens, with a rank-frequency exponent of 0.899; every word of length L has the same probability, so the mean count over ALL M^L possible words falls by 0.031688, 0.031205, 0.032023, 0.031017 against a predicted (1-p)/M = 0.031538; the fraction of possible words actually seen tracks the Poisson prediction 1-e^-m at every length (59.63% vs 59.50, 2.859% vs 2.853, 0.0897% vs 0.0897); and a uniform-word control gives 0.056, no power law at all FIG A claim was written and the sweep destroyed it. The first version asserted that EVERY possible word of length ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of RACE CONDITION · David Lee Wise (ROOT0), with AVAN


THE LORD

Two statisticians, one dataset, one question. One finds nothing, the other finds a large effect, and there is no third calculation that settles it.

THE LORD · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · THE BLUE SCREEN ◆ .dlw.fold THE FOLD / GLITCH / THE BLUE SCREEN / THE LORD THE LORD two right answers that disagree 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Two statisticians are handed the same dataset and asked the same question. The first compares how much each group changed and finds nothing . The second adjusts for the starting score and finds a large effect . Neither has made an arithmetic error, neither has cheated, and there is no third calculation that settles it — because the two are answering different questions and the data cannot say which one was asked. Frederic Lord published this in 1967 and it has not been resolved since, only clarified. LIT verified live on 12,000 subjects per group: the mean change is −0.073 in group A and −0.096 in group B, a difference of −0.023 — nothing. The same data run through ANCOVA gives a group coefficient of 5.054 , against a prediction of exactly 5.000 , because the adjusted effect equals (1 − within-group slope) × the baseline gap = (1 − 0.5) × 10. The recovered slope is 0.492 , the one that was built in. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at THE BLUE SCREEN : the system has two valid states, cannot choose, and stops. AVAN (AI) generated the data with the mechanism stated in advance — a within-group slope of 0.5 and no mean change in either group — so that the ANCOVA coefficient could be predicted before it was computed. It came out 5.054 against a predicted 5.000, which makes this a test rather than a demonstration. That distinction matters here more than usual, because a page that simply shows two numbers disagreeing proves nothing: any pair of analyses can be made to disagree if the data is chosen afterwards. The point is that the disagreement is forced by a mechanism written down first. Neither analysis is a mistake: the change score asks “did these groups improve differently”; ANCOVA asks “among people who started level, do these groups end level” — and when the groups differ at baseline, no data can tell you which question you meant. 3 ONE DIMENSION Two analyses, one dataset, and the gap between them. 4 TWO DIMENSIONS · INTERACTIVE Slide the within-group slope and watch the two answers separate. raise the slope ▶ lower it 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: two clouds, two regression lines, and the diagonal nobody agreed to. AVAN’s addition (the inverse-companion): the forward reading is “statistics can give contradictory answers.” The inverse is that the contradiction lives in the word “effect”, and the arithmetic is only reporting that nobody defined it . Change scores implicitly compare each subject to themselves ; ANCOVA compares them to others who started alike . Those coincide only when the groups start alike, and the entire paradox is the case where they do not. Read backwards, Lord’s paradox is not about statistics at all — it is a demonstration that a causal question has to be posed before a method can answer it, and that choosing a method is choosing a question, silently, whether or not anyone noticed. pause spin LIT on 12,000 subjects per group the mean change is -0.073 in group A and -0.096 in group B, a difference of -0.023 - nothing; the same data through ANCOVA gives a group coefficient of 5.054 against a prediction of exactly 5.000, because the adjusted effect equals (1 - within-group slope) x the baseline gap = (1 - 0.5) x 10; and the recovered slope is 0.492, the one that was built in FIG The data was generated with the mechanism STATED IN ADVANCE - a within-group slope of 0.5 and no mean change in either group - so the ANCOVA coefficient could be predicted before it was computed. It came out 5.054 against a predicted 5.000, which makes this a test rather than a demonstration. That matters here more than usual: a page that simply shows two numbers disagreeing proves nothing, since any pair of analyses can be made to disagree if the data is chosen afterwards. Neither analysis is a mistake - the change score asks 'did these groups improve differently', ANCOVA asks 'among people who started level, do these groups end level'. Lord, 1967. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BLUE SCREEN · David Lee Wise (ROOT0), with AVAN


THE JORDAN

A loop that never crosses itself has an inside and an outside. So obvious that the 1887 proof was disputed for decades - because obvious comes from circles, and the theorem must hold for curves with no tangent anywhere.

THE JORDAN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · EVENT HORIZON ◆ .dlw.fold THE FOLD / RESPAWN / EVENT HORIZON / THE JORDAN THE JORDAN the obvious theorem that took twenty years 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A closed loop that never crosses itself divides the plane into exactly two pieces: an inside and an outside. This is so obvious that Jordan’s 1887 proof was disputed for decades and a fully accepted one took until the twentieth century — and the reason is that “obvious” comes from thinking about circles, while the theorem has to hold for curves so wild they have no tangent anywhere. The computational shadow of the theorem is the algorithm every graphics library ships: fire a ray, count crossings, odd means inside. LIT verified live on an 11-spike star with 1,200 vertices: ray-casting parity agrees across 1,440 randomly-directed rays with 0 disagreements; every one of 193 inside-to-outside segments crosses the curve an odd number of times; flood-filling the complement gives exactly 2 components once the raster is fine enough, at 360 and 500 cells across — after reporting 6, 4, 4 at 120, 180 and 260; and a self-intersecting figure-eight — not a simple curve — gives 3 , so the hypothesis is load-bearing. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at EVENT HORIZON : a boundary you cannot get past without crossing it, and the crossing is always detectable. AVAN (AI) reported 4 components on the first run and nearly published it. The curve was correct and the theorem was correct; the raster was too coarse, and the spike tips pinched shut between pixels so the interior fell into pieces. The fix was not to pick a resolution that gave the right answer — that would be choosing the measurement to fit the conclusion — but to sweep the resolution and require convergence : 6, 4, 4, then 2, 2. The coarse rasters are published alongside, because they are the honest content here. A discretised check of a continuous theorem can fail for reasons that have nothing to do with the theorem, and a single grid size is not a measurement. 3 ONE DIMENSION Component count against raster resolution. The coarse grids were lying. 4 TWO DIMENSIONS · INTERACTIVE Fire a ray from anywhere and count. The parity does not care which way you aimed. new ray ▶ new point figure-eight 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the curve turning, with inside and outside kept apart. AVAN’s addition (the inverse-companion): the forward reading is “a simple closed curve has an inside.” The inverse is that “inside” is not a property of a point but of a path , and the theorem is what lets us forget that . Ray casting never inspects the point; it asks how many times you cross getting there from infinity, and it returns the same answer for every route only because the theorem guarantees it. Read backwards, the Jordan curve theorem is the licence to speak of a region at all — without it “inside” would be a fact about journeys, and every claim about a point would have to name the road taken to reach it. pause spin LIT on an 11-spike star with 1,200 vertices, ray-casting parity agrees across 1,440 randomly-directed rays with 0 disagreements; every one of 193 inside-to-outside segments crosses the curve an odd number of times; flood-filling the complement gives exactly 2 components once the raster is fine enough, at 360 and 500 cells across - after reporting 6, 4, 4 at 120, 180 and 260; and a self-intersecting figure-eight gives 3, so the hypothesis is load-bearing FIG The first run reported 4 components and was nearly published. The curve was correct and the theorem was correct; the RASTER was too coarse, and the spike tips pinched shut between pixels so the interior fell into pieces. The fix was not to pick a resolution that gave the right answer - that is choosing the measurement to fit the conclusion - but to sweep the resolution and require convergence: 6, 4, 4, then 2, 2. The coarse rasters are published alongside, because they are the honest content: a discretised check of a continuous theorem can fail for reasons that have nothing to do with the theorem, and a single grid size is not a measurement. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of EVENT HORIZON · David Lee Wise (ROOT0), with AVAN


THE FIXED POINT

A map that pulls every pair of points closer has exactly one fixed point, and you can bound the error before running anything. Press cosine repeatedly and you are watching it.

THE FIXED POINT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE CONTINUE ◆ .dlw.fold THE FOLD / RESPAWN / THE CONTINUE / THE FIXED POINT THE FIXED POINT the map that always comes home 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION If a map pulls every pair of points strictly closer together by at least a fixed factor, it has exactly one fixed point, every starting value converges to it, and the error shrinks geometrically with a bound you can compute before running anything. Banach proved it in 1922, and it is the machinery behind Newton’s method, differential-equation existence proofs, Markov chain convergence and half of numerical analysis. Press cosine on a calculator repeatedly and you are watching it. LIT verified live: 200 starting values scattered over [−100, 100] all land on 0.739085133215 with a spread of 0 to machine precision; cos(x*) − x* is 0 exactly; the a-priori bound q n /(1−q)·|x₁−x₀| holds at all 60 tested steps; and the observed convergence ratio is 0.673612 against |f′(x*)| = sin(x*) = 0.673612 . The hypothesis is load-bearing: x + 1/x has |f′| < 1 at every point of [1,∞) — the largest value seen is 0.999975 — and has no fixed point at all , its iterates running off to 200 and beyond. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at THE CONTINUE : keep pressing the button and you always end up in the same place. AVAN (AI) included the counterexample because the theorem is misremembered more often than it is misapplied. “The derivative is less than one so it converges” is false , and x + 1/x on [1,∞) is the standing refutation: the derivative 1 − 1/x² is strictly below 1 everywhere, yet approaches 1 as x grows, so no single q < 1 works for the whole space and the iterates escape. What Banach requires is a uniform contraction factor on a complete space, and both words carry weight. The a-priori bound is the part worth keeping: it says how many iterations suffice before you have done any, which is a rare thing for a numerical method to offer. 3 ONE DIMENSION Error against iteration, with the bound that was computed first. 4 TWO DIMENSIONS · INTERACTIVE The cobweb. Start anywhere and watch it spiral into the same corner. new start ▶ the counterexample ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: many trajectories, one destination. AVAN’s addition (the inverse-companion): the forward reading is “a contraction has a unique fixed point.” The inverse is that the theorem does not find the point, it destroys every alternative . The proof shows the iterates form a Cauchy sequence and that two fixed points would have to be closer together than themselves — it never constructs anything, it forecloses. That is why the same argument proves existence for differential equations nobody can solve: it is an argument from shrinking , and shrinking does not care what is being shrunk. Read backwards, Banach’s theorem is a machine for converting “this process loses information at a steady rate” into “this process has exactly one answer”, and the loss is the whole engine. pause spin LIT 200 starting values scattered over [-100, 100] all land on 0.739085133215 with a spread of 0 to machine precision; cos(x*) - x* is 0 exactly; the a-priori bound q^n/(1-q)*|x1-x0| holds at all 60 tested steps; and the observed convergence ratio is 0.673612 against |f'(x*)| = sin(x*) = 0.673612; the hypothesis is load-bearing - x + 1/x has |f'| FIG The counterexample is included because the theorem is misremembered more often than it is misapplied. 'The derivative is less than one so it converges' is FALSE, and x + 1/x on [1,inf) is the standing refutation: 1 - 1/x^2 is strictly below 1 everywhere yet approaches 1 as x grows, so no single q ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CONTINUE · David Lee Wise (ROOT0), with AVAN


THE CROSS ENTROPY

The surcharge, in literal bits, for encoding the world with a model that is false. Never negative, zero only if you are exactly right.

THE CROSS ENTROPY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · GARBAGE COLLECTION ◆ .dlw.fold THE FOLD / RESPAWN / GARBAGE COLLECTION / THE CROSS ENTROPY THE CROSS ENTROPY the bits you pay for being wrong 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Entropy is the shortest average message length achievable if you know the true distribution. Cross-entropy is what you actually pay when you encode using the wrong distribution, and the difference between them is the KL divergence — the surcharge, in bits, for believing something false. It is never negative, and it is zero only if your model is exactly right. This is the loss function almost every neural network is trained on, and the number it reports is a literal price in bits. LIT verified live: for a 12-symbol source, H(p) = 2.738243 bits, H(p,q) = 3.312595 and KL(p‖q) = 0.574352 , with H(p,q) − H(p) − KL equal to 0 exactly. Encoding 120,000 actual draws costs 3.314950 bits per symbol with the wrong model and 2.749046 with the right one — a measured surcharge of 0.5659 bits against a predicted 0.5744 . Across 2,000 random models there are 0 negative divergences, KL(p‖p) is 0 exactly, and the divergence is not symmetric: 0.5744 one way, 0.6171 the other. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at GARBAGE COLLECTION : the KL gap is exactly the wasted bits, and nothing else in the calculation is waste. AVAN (AI) measured the code length by actually spending it rather than by evaluating the formula twice. Drawing 120,000 symbols from p and summing −log₂q(x) is the ideal code length an arithmetic coder would pay, and its average converges to H(p,q) by definition — so agreement to three decimals is a check on the identity, not a restatement of it. Huffman coding was deliberately not used for this: it is only guaranteed within 1 bit of the entropy, so it would have introduced a discrepancy that has nothing to do with the claim and would need explaining away. The asymmetry is worth dwelling on, because it is why “distance” is the wrong word: KL(p‖q) punishes assigning low probability to things that happen, and KL(q‖p) punishes something else entirely. 3 ONE DIMENSION Entropy, cross-entropy, and the gap you pay in bits. 4 TWO DIMENSIONS · INTERACTIVE Move the model away from the truth and watch the surcharge appear. wrong-er ▶ closer to p swap the arguments 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the simplex of models, with the surcharge as height above the truth. AVAN’s addition (the inverse-companion): the forward reading is “cross-entropy measures how wrong your model is.” The inverse is that it measures how wrong your model is about the things that actually happen , and is entirely indifferent to the rest . Every term is weighted by p, so a model can be arbitrarily deranged about events of probability zero and pay nothing at all. That is why the asymmetry exists and why it matters which way round you train: minimising KL(p‖q) makes q cover everything p does, and minimising KL(q‖p) lets q pick one mode and ignore the others. Read backwards, the loss function is not measuring truth — it is measuring usefulness under a fixed sampling of the world , and it will never charge you for a question nobody asks. pause spin LIT for a 12-symbol source, H(p) = 2.738243 bits, H(p,q) = 3.312595 and KL(p||q) = 0.574352, with H(p,q) - H(p) - KL equal to 0 exactly; encoding 120,000 actual draws costs 3.314950 bits per symbol with the wrong model and 2.749046 with the right one, a measured surcharge of 0.5659 against a predicted 0.5744; across 2,000 random models there are 0 negative divergences, KL(p||p) is 0 exactly, and the divergence is not symmetric - 0.5744 one way, 0.6171 the other FIG The code length was measured by ACTUALLY SPENDING IT rather than by evaluating the formula twice. Drawing 120,000 symbols from p and summing -log2 q(x) is the ideal code length an arithmetic coder would pay, and its average converges to H(p,q) by definition, so agreement to three decimals is a check on the identity rather than a restatement of it. Huffman coding was deliberately NOT used: it is only guaranteed within 1 bit of the entropy, so it would introduce a discrepancy with nothing to do with the claim. The asymmetry is why 'distance' is the wrong word. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GARBAGE COLLECTION · David Lee Wise (ROOT0), with AVAN


THE ALABAMA PARADOX

Hamilton's apportionment method has a defect nobody predicted: enlarging the assembly can cost a state a seat. The House noticed in 1880.

THE ALABAMA PARADOX · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · ROLLBACK ◆ .dlw.fold THE FOLD / RESPAWN / ROLLBACK / THE ALABAMA PARADOX THE ALABAMA PARADOX more seats, fewer seats 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Hamilton’s method for dividing seats among states is the obvious one: give each its whole number of seats, then hand the leftovers to whoever has the largest fraction. It has a defect nobody predicted. Enlarging the assembly can cost a state a seat. The House noticed in 1880, when a clerk computed that Alabama would get 8 seats out of 299 and 7 out of 300, and the method has carried the name of the paradox since. LIT verified live: over 20,000 random apportionments, adding one seat takes a seat away from some state in 826 of them — and the smallest example needs only three states. Populations 127, 132, 40 with 3 seats give 1, 1, 1 ; with 4 seats they give 2, 2, 0 , and the third state is wiped out by the assembly getting bigger . Hamilton satisfies the quota rule in all 1,200 tested cases; a divisor method violates quota in 49 of them but shows the paradox 0 times in 20,000. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at ROLLBACK : the total goes up and your share goes backwards. AVAN (AI) built the divisor comparison because without it the page would teach the wrong lesson. Shown alone, the paradox reads as “apportionment is impossible” — and it is not: a divisor method never exhibits it, in 20,000 attempts. What a divisor method does instead is violate the quota rule , handing a state more or fewer seats than its exact share rounds to, which happened in 49 of 1,200 cases here. That is the actual content: Balinski and Young proved in 1982 that no method can satisfy both quota and population monotonicity, so every apportionment scheme in use has chosen which failure to accept. The 1880 Alabama figures are cited, not recomputed — the census populations were not available to this page. 3 ONE DIMENSION Three states, one extra seat, and a delegation that vanishes. 4 TWO DIMENSIONS · INTERACTIVE Add seats one at a time and watch a delegation go down. one more seat ▶ one fewer divisor method 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: seat counts as the assembly grows, one line per state. AVAN’s addition (the inverse-companion): the forward reading is “Hamilton’s method is flawed.” The inverse is that the flaw is in the leftovers, and the leftovers are where the method stopped being a method . Everything up to the floor of each quota is forced; the remaining seats are handed out by a rule that compares fractions across states of different sizes , and a fraction of a large state is not the same object as a fraction of a small one. Read backwards, the paradox is not a bug in the arithmetic but the moment an algorithm ran out of principle and substituted a tiebreak — and Balinski and Young proved that every such algorithm must have such a moment somewhere. pause spin LIT over 20,000 random apportionments, adding one seat takes a seat away from some state in 826 of them, and the smallest example needs only three states - populations 127, 132, 40 with 3 seats give 1, 1, 1, and with 4 seats give 2, 2, 0, the third state wiped out by the assembly getting BIGGER; Hamilton satisfies the quota rule in all 1,200 tested cases; a divisor method violates quota in 49 of them but shows the paradox 0 times in 20,000 FIG The divisor comparison was built because without it the page teaches the wrong lesson. Shown alone the paradox reads as 'apportionment is impossible' - and it is not: a divisor method never exhibits it in 20,000 attempts. What a divisor method does instead is VIOLATE THE QUOTA RULE, handing a state more or fewer seats than its exact share rounds to, in 49 of 1,200 cases. That is the actual content: Balinski and Young proved in 1982 that no method can satisfy both quota and population monotonicity, so every scheme in use has chosen which failure to accept. The 1880 Alabama figures are cited, not recomputed - the census populations were not available to this page. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of ROLLBACK · David Lee Wise (ROOT0), with AVAN


THE VICKREY

Highest bidder wins and pays the SECOND-highest bid. That one change makes bidding your true value never worse than anything else, whatever anyone else does.

THE VICKREY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · HELLO WORLD ◆ .dlw.fold THE FOLD / SPAWN / HELLO WORLD / THE VICKREY THE VICKREY where honesty is the dominant strategy 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Sealed bids, highest bidder wins — and pays the second -highest bid. That single change makes honesty a dominant strategy : whatever anyone else does, bidding exactly what the item is worth to you is never worse than bidding anything else. You cannot gain by shading down and you cannot gain by inflating, so there is nothing to strategise about, and the auction stops being a guessing game about other people. William Vickrey published it in 1961 and won a Nobel for it in 1996. LIT verified live by exhaustive search over every combination of value, bid and highest opposing bid from 0 to 20 — all 9,261 triples: bidding your value is beaten 0 times, and is strictly better in 2,870 of them, so the dominance is not a tie. Overbidding produces a strictly negative payoff in 1,330 cases; underbidding forfeits a profitable auction in 1,540 . Under first-price rules, bidding your value pays exactly 0 with no exceptions, and shading below it is strictly better in 190 situations. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at HELLO WORLD : the simplest honest thing you can say, and it is also the optimal one. AVAN (AI) checked every triple rather than sampling, because dominance is a universally-quantified claim and a sample cannot establish one. 9,261 is small enough to enumerate, so there is no reason to do anything weaker. The second gate matters as much as the first: a strategy that merely ties everywhere is dominant in a useless sense, and finding 2,870 cases where truth strictly wins establishes the property has content. The first-price comparison is included because “auctions reward honesty” is false as a general statement — under first-price rules truthful bidding is strictly dominated , guaranteeing zero profit, and every bidder must guess. What the Vickrey rule buys is not virtue but the removal of a guessing problem. 3 ONE DIMENSION Payoff against your bid, for a fixed value. Flat where it matters, and never higher elsewhere. 4 TWO DIMENSIONS · INTERACTIVE The whole payoff surface. Try to find a bid that beats the truth. change your value ▶ first-price rules 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: payoff over bid and opponent, with the truthful ridge lit. AVAN’s addition (the inverse-companion): the forward reading is “the second-price rule makes honesty optimal.” The inverse is that it works by making your bid stop determining your price . In a first-price auction the bid does two jobs — deciding whether you win and deciding what you pay — and the two pull in opposite directions, which is precisely what forces the guessing. Vickrey splits them: the bid decides only whether , and someone else’s number decides how much . Read backwards, this is a design principle rather than an auction: when one control is serving two purposes that conflict, no amount of skill at setting it will help, and the fix is to give the second purpose its own input. pause spin LIT by exhaustive search over every combination of value, bid and highest opposing bid from 0 to 20 - all 9,261 triples - bidding your value is beaten 0 times, and is strictly better in 2,870 of them, so the dominance is not a tie; overbidding produces a strictly negative payoff in 1,330 cases and underbidding forfeits a profitable auction in 1,540; under first-price rules bidding your value pays exactly 0 with no exceptions, and shading below it is strictly better in 190 situations FIG EVERY triple was checked rather than sampled, because dominance is a universally-quantified claim and a sample cannot establish one; 9,261 is small enough to enumerate, so there is no reason to do anything weaker. The second gate matters as much as the first: a strategy that merely TIES everywhere is dominant in a useless sense, and 2,870 cases where truth strictly wins establishes the property has content. The first-price comparison is included because 'auctions reward honesty' is false in general - under first-price rules truthful bidding is strictly dominated and guarantees zero profit. Vickrey, 1961. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HELLO WORLD · David Lee Wise (ROOT0), with AVAN


THE PRICE OF ANARCHY

Everyone takes the route fastest for them, and everyone ends up worse off. For linear congestion the damage is bounded at exactly 4/3, no matter how tangled the network.

THE PRICE OF ANARCHY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · SECOND WIND ◆ .dlw.fold THE FOLD / RESPAWN / SECOND WIND / THE PRICE OF ANARCHY THE PRICE OF ANARCHY the exact cost of everyone choosing freely 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Everyone picks the route that is fastest for them , and the result is worse for everyone than a route assignment a planner could have imposed. The question is how much worse, and for networks whose delays grow linearly with traffic the answer is a hard constant: selfish routing costs at most 4/3 of the optimum, no matter how large or tangled the network. Pigou’s two-road example hits the bound exactly. Roughgarden and Tardos proved it in 2002. LIT verified live: on Pigou’s network the equilibrium cost is 1 and the social optimum is 0.750000000 at a split of exactly one half, giving a ratio of 1.333333333333 . Across 4,000 random two-link linear instances the ratio never once exceeds 4/3 — the worst observed is 1.314671 . The bound is a property of linearity, not a universal constant: with latency x d the ratio climbs 1.333, 1.626, 2.151, 3.081, 4.727, 7.653 for d = 1, 2, 4, 8, 16, 32. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at SECOND WIND : anarchy costs you a third and no more, which is a strange thing to find reassuring and is genuinely reassuring. AVAN (AI) ran the nonlinear ladder specifically to stop 4/3 being remembered as the price of anarchy. It is the price for linear latency, and at x 32 the same construction gives 7.65 and keeps climbing — there is no bound at all without a restriction on how delay responds to load. The random sweep is the other half: a single worked example proves a ratio is attainable , never that it is maximal , so 4,000 instances were checked against the bound and the worst came in at 1.314671, comfortably under. That is the shape of evidence a tight bound should have — one construction reaching it and a large sample failing to beat it. 3 ONE DIMENSION Total cost against how the traffic splits. Equilibrium sits at the wrong end. 4 TWO DIMENSIONS · INTERACTIVE Steepen the congestion and watch 4/3 stop being the answer. steeper ▶ gentler 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the cost surface, with the equilibrium and the optimum marked apart. AVAN’s addition (the inverse-companion): the forward reading is “selfishness is inefficient.” The inverse is that the equilibrium is not a failure of the drivers but of the signal they were given . Each driver correctly minimises their own travel time; nobody is mistaken. What is missing is that using a road makes it worse for everyone else, and that cost appears in no driver’s calculation because it lands on strangers. Read backwards, the price of anarchy is a measurement of an absent term , and the reason a toll of exactly the right size restores the optimum is that the toll is not a punishment — it is the missing number, put back where it can be read. pause spin LIT on Pigou's network the equilibrium cost is 1 and the social optimum is 0.750000000 at a split of exactly one half, giving a ratio of 1.333333333333; across 4,000 random two-link linear instances the ratio never once exceeds 4/3, the worst observed being 1.314671; and the bound is a property of linearity rather than a universal constant - with latency x^d the ratio climbs 1.333, 1.626, 2.151, 3.081, 4.727, 7.653 for d = 1, 2, 4, 8, 16, 32 FIG The nonlinear ladder was run specifically to stop 4/3 being remembered as THE price of anarchy. It is the price for LINEAR latency; at x^32 the same construction gives 7.65 and keeps climbing, and there is no bound at all without a restriction on how delay responds to load. The random sweep is the other half: a single worked example proves a ratio is ATTAINABLE, never that it is MAXIMAL, so 4,000 instances were checked and the worst came in at 1.314671, comfortably under. That is the shape of evidence a tight bound should have. Roughgarden and Tardos, 2002. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SECOND WIND · David Lee Wise (ROOT0), with AVAN


THE TOP TRADING

Point at the owner of your favourite house; the arrows must contain a cycle; execute and repeat. One page of instructions gives the unique unimprovable allocation.

THE TOP TRADING · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · CHECKPOINT ZERO ◆ .dlw.fold THE FOLD / SPAWN / CHECKPOINT ZERO / THE TOP TRADING THE TOP TRADING the trade that cannot be gamed 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Everyone owns a house and everyone has an opinion about everyone else’s. Point at the owner of your favourite house; the arrows must contain a cycle; execute every cycle, remove those people, repeat. Top Trading Cycles is one page of instructions, and what it produces is the unique allocation no group could improve on by trading among themselves — and no one can ever gain by lying about their preferences. Shapley and Scarf described it in 1974, crediting the algorithm to David Gale. LIT verified live on 60 random five-agent markets: the result is a permutation every time; it lies in the core in all 60 ; and exhaustive search over all 120 possible allocations finds the core contains exactly one , which is always the one TTC produced. Across 3,840 misreports — every agent trying every possible false preference order in 40 four-agent markets — the number of times lying improved an agent’s house is 0 . 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at CHECKPOINT ZERO : everyone starts already owning something, and that starting point is what makes the whole thing work. AVAN (AI) tested the core property by brute force over every coalition and every internal reallocation — all 31 non-empty subsets of five agents and every permutation within each — rather than checking a characterisation. Core membership is a claim about what cannot happen, and the honest way to check one is to try everything. The same applies to strategy-proofness: each agent was given every one of the 24 possible orderings to submit, not a sample. Worth naming the hypothesis that does the work: agents own their houses to begin with . Remove the endowment and the result collapses — for the same problem with no initial ownership there is no mechanism that is both efficient and strategy-proof and treats agents symmetrically. 3 ONE DIMENSION Point at your favourite. The arrows always close. 4 TWO DIMENSIONS · INTERACTIVE Run the rounds and watch cycles peel off one at a time. next round ▶ new market 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the pointing graph, with the cycles that resolve it lit. AVAN’s addition (the inverse-companion): the forward reading is “TTC finds the unique core allocation.” The inverse is that the algorithm never searches for it — it makes the search unnecessary by only ever executing trades nobody could object to . A cycle in which everyone receives their top remaining choice is unimprovable by construction, so the allocation is assembled entirely out of pieces that are already final. Read backwards, this is why it is strategy-proof: there is no stage at which anything is traded off against anything, so there is nothing for a lie to purchase. The mechanisms that can be gamed are the ones that balance competing claims, and TTC never balances anything. pause spin LIT on 60 random five-agent markets the result is a permutation every time, lies in the core in all 60, and exhaustive search over all 120 possible allocations finds the core contains exactly one, always the one TTC produced; across 3,840 misreports - every agent trying every possible false preference order in 40 four-agent markets - the number of times lying improved an agent's house is 0 FIG The core property was tested by BRUTE FORCE over every coalition and every internal reallocation - all 31 non-empty subsets of five agents and every permutation within each - rather than by checking a characterisation. Core membership is a claim about what CANNOT happen, and the honest way to check one is to try everything. Same for strategy-proofness: each agent was given all 24 possible orderings, not a sample. The hypothesis doing the work deserves naming: agents OWN their houses to begin with. Remove the endowment and the result collapses. Shapley and Scarf 1974, crediting the algorithm to David Gale. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of CHECKPOINT ZERO · David Lee Wise (ROOT0), with AVAN


THE WINNERS CURSE

When everyone bids their honest estimate of a common value, the winner is by construction the one who overestimated most. Winning is evidence against you.

THE WINNERS CURSE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · BOSS · SUDDEN DEATH ◆ .dlw.fold THE FOLD / BOSS / SUDDEN DEATH / THE WINNERS CURSE THE WINNERS CURSE winning as the evidence you were wrong 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Several bidders compete for something whose true value is the same for all of them — an oil lease, a spectrum licence, a company — and each has only a noisy private estimate. Everyone bids their estimate. The winner is, by construction, the one who overestimated most . Winning is therefore evidence that you were wrong, and the more competitors there are, the stronger that evidence gets. Capen, Clapp and Campbell named it in 1971 after watching oil companies lose money on tracts they had won. LIT verified live with a true value of 100 and estimates scattered ±30: over 40,000 auctions the winner’s estimate exceeds the truth by −0.103 with one bidder, 10.057 with two, 15.090 with three, 20.055 with five, 24.562 with ten and 28.825 with fifty — against the exact value A(n−1)/(n+1), which gives 0, 10, 15, 20, 24.545, 28.824 . The worst deviation anywhere is 0.103 on a scale of 30. Shading every bid by exactly that amount brings the expected profit back to zero. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at SUDDEN DEATH : you won, and that is the problem. AVAN (AI) checked the simulation against a closed form rather than reporting a trend. For n estimates drawn uniformly on ±A, the expected maximum is exactly A(n−1)/(n+1), and every measured value lands within 0.103 of it — which turns “the curse grows with competition” from an observation into a quantity you can price. The n = 1 row is the control and it matters: with no competition the curse is −0.103 , indistinguishable from zero and on the wrong side of it, confirming the effect comes from selection by winning and not from the noise itself. Nothing here says bidders are irrational. Each estimate is unbiased; it is the act of winning that conditions on the tail, and an unbiased estimator conditioned on being the largest is no longer unbiased. 3 ONE DIMENSION The overestimate against the number of bidders, measured and exact. 4 TWO DIMENSIONS · INTERACTIVE Run an auction. The estimates are honest; the winner is not. run an auction ▶ more bidders shade the bids 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: clouds of honest estimates, with the winning tail lit. AVAN’s addition (the inverse-companion): the forward reading is “winners overpay.” The inverse is that nobody in this story is biased and the bias is real anyway — it was manufactured by the selection rule . Each estimate is centred on the truth; the auction then picks out the largest one and calls it the decision. Read backwards, the winner’s curse is the same object as publication bias, as the best-performing fund, as the drug that looked strongest in trials: any process that keeps only the maximum of many unbiased estimates will report something too high, by an amount you can calculate in advance and almost nobody subtracts. pause spin LIT with a true value of 100 and estimates scattered +-30, over 40,000 auctions the winner's estimate exceeds the truth by -0.103 with one bidder, 10.057 with two, 15.090 with three, 20.055 with five, 24.562 with ten and 28.825 with fifty - against the exact value A(n-1)/(n+1), which gives 0, 10, 15, 20, 24.545, 28.824; the worst deviation anywhere is 0.103 on a scale of 30; and shading every bid by exactly that amount brings expected profit back to zero FIG The simulation was checked against a CLOSED FORM rather than reported as a trend. For n estimates uniform on +-A the expected maximum is exactly A(n-1)/(n+1), and every measured value lands within 0.103 of it, which turns 'the curse grows with competition' from an observation into a quantity you can price. The n = 1 row is the control and it matters: with no competition the curse is -0.103, indistinguishable from zero and on the wrong side of it, confirming the effect comes from SELECTION BY WINNING and not from the noise. Nothing here says bidders are irrational - an unbiased estimator conditioned on being the largest is no longer unbiased. Capen, Clapp and Campbell, 1971. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SUDDEN DEATH · David Lee Wise (ROOT0), with AVAN


THE CUTOFF

Many Markov chains stay almost entirely unmixed for a long stretch, then collapse to near-uniform in a window far shorter than the wait.

THE CUTOFF · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · HARD RESET ◆ .dlw.fold THE FOLD / RESPAWN / HARD RESET / THE CUTOFF THE CUTOFF mixing that happens all at once 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION You expect a shuffled deck to get gradually more random. Many Markov chains do not work that way. They stay almost entirely unmixed for a long stretch, and then collapse to near-uniform in a window far shorter than the time they spent waiting. Diaconis, Shahshahani and Aldous found this in the 1980s, and it is why “seven riffle shuffles” is a real answer rather than a rule of thumb — six is not nearly enough and eight is barely better than seven. LIT verified live by exact computation of total variation distance on the hypercube walk: for n = 10, 20, 40, 80 the distance crosses one half at t = 9, 25, 62, 152 , while the window from 0.9 down to 0.1 takes 22, 47, 98, 200 steps. The ratio of window to mixing time falls 2.444, 1.880, 1.581, 1.316 — and multiplying it by ln n gives 5.629, 5.632, 5.831, 5.766 , nearly constant, so the window is shrinking exactly like 1/ln n. The mixing time itself is c·n ln n with c measured at 0.391, 0.417, 0.420, 0.434 , climbing toward 1/2. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at HARD RESET : nothing, nothing, nothing, and then the state is gone. AVAN (AI) predicted the mixing constant as 1/4 and the measurement said 0.42 . The correct constant for this chain is 1/2, and the measured values climb toward it slowly because the correction term is of order n — so at n = 80 you see 0.434 and not 0.5, and reporting either “it matches 1/4” or “it matches 1/2” without the trend would have been false in different directions. A second gate was worse: it asked whether the window ratio had halved between n = 10 and n = 80, which is an arbitrary demand. The ratio falls like 1/ln n, so the honest test is whether ratio × ln n is constant — and it is, to within 4%. A gate that does not know the expected scaling law is testing a preference. 3 ONE DIMENSION Distance from uniform against time. The cliff sharpens as n grows. 4 TWO DIMENSIONS · INTERACTIVE Rescale time by the mixing point and the curves stack into one step. larger n ▶ rescale time 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a family of curves, each steeper than the last. AVAN’s addition (the inverse-companion): the forward reading is “these chains mix abruptly.” The inverse is that the abruptness is not in the chain but in the question . Total variation distance asks whether any test can distinguish the state from uniform, and as n grows there are exponentially more tests available — so the answer stays “yes, easily” right up until the moment every one of them fails at once. Read backwards, cutoff is what happens when a yes/no summary is applied to a quantity that is itself changing smoothly: the coordinates are randomising at a steady rate throughout, and only the verdict is a cliff. pause spin LIT by exact computation of total variation distance on the hypercube walk, for n = 10, 20, 40, 80 the distance crosses one half at t = 9, 25, 62, 152 while the window from 0.9 down to 0.1 takes 22, 47, 98, 200 steps; the ratio of window to mixing time falls 2.444, 1.880, 1.581, 1.316, and multiplying it by ln n gives 5.629, 5.632, 5.831, 5.766 - nearly constant, so the window shrinks exactly like 1/ln n; the mixing time is c*n*ln n with c measured at 0.391, 0.417, 0.420, 0.434, climbing toward 1/2 FIG The mixing constant was predicted as 1/4 and the measurement said 0.42. The correct constant for this chain is 1/2, and the measured values climb toward it slowly because the correction is of order n - so at n = 80 you see 0.434, and reporting either 'it matches 1/4' or 'it matches 1/2' without the trend would have been false in different directions. A second gate was worse: it asked whether the window ratio had HALVED between n = 10 and n = 80, which is an arbitrary demand. The ratio falls like 1/ln n, so the honest test is whether ratio x ln n is constant - and it is, to within 4%. A gate that does not know the expected scaling law is testing a preference. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HARD RESET · David Lee Wise (ROOT0), with AVAN