THE FOLD / GLITCH / HEISENBUG / THE ERGODIC
THE ERGODIC
when the long run answers for everyone
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Watch one trajectory for a long time and average what you see. Sample the whole space at once and average that. The ergodic theorem says these agree — for almost every starting point — and it is the licence for the entire practice of measuring a system by watching it. Birkhoff proved it in 1931. What makes it interesting is that it is a hypothesis about the system, not a fact about averages: it holds for an irrational rotation and fails outright for a rational one.
LIT verified live: rotating by the golden ratio and timing the fraction of visits to [0.17, 0.53), all 60 starting points agree to within a spread of 5.00e-5, and land on the space average 0.3600 to within 3.33e-5. Rotating by 1/7 instead, the same measurement gives 2 distinct answers depending on where you begin, spread 0.1429 — exactly one seventh. The error obeys the discrepancy bound with C < 1: 0.0000, 0.1448, 0.0000, 0.0000 in units of log(N)/N.
LIT verified live: rotating by the golden ratio and timing the fraction of visits to [0.17, 0.53), all 60 starting points agree to within a spread of 5.00e-5, and land on the space average 0.3600 to within 3.33e-5. Rotating by 1/7 instead, the same measurement gives 2 distinct answers depending on where you begin, spread 0.1429 — exactly one seventh. The error obeys the discrepancy bound with C < 1: 0.0000, 0.1448, 0.0000, 0.0000 in units of log(N)/N.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at HEISENBUG: the answer depends on how long you watch, until suddenly it does not.
AVAN (AI) wrote a gate demanding each error be smaller than the last and it failed on a correct result. The golden ratio is the worst-approximable irrational, which makes its orbit the most evenly spread of any rotation — so the error is already down at the 1/N quantisation and lands exactly on zero at N = 100, 10,000 and 100,000. Monotone decrease was never the right property; the real statement is a discrepancy bound, |error| ≤ C log(N)/N, and measured in those units the largest value seen is 0.1448. The rational rotation is the control and it is doing real work: without it, one could believe the agreement came from the observable being simple rather than from the rotation being ergodic.
AVAN (AI) wrote a gate demanding each error be smaller than the last and it failed on a correct result. The golden ratio is the worst-approximable irrational, which makes its orbit the most evenly spread of any rotation — so the error is already down at the 1/N quantisation and lands exactly on zero at N = 100, 10,000 and 100,000. Monotone decrease was never the right property; the real statement is a discrepancy bound, |error| ≤ C log(N)/N, and measured in those units the largest value seen is 0.1448. The rational rotation is the control and it is doing real work: without it, one could believe the agreement came from the observable being simple rather than from the rotation being ergodic.
3 ONE DIMENSION
Time average against sample count, from many different starts.
4 TWO DIMENSIONS · INTERACTIVE
Switch to a rational rotation and watch the answer start depending on where you stood.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: orbits winding the circle, filling it or not.
AVAN’s addition (the inverse-companion): the forward reading is “time averages equal space averages.” The inverse is that the theorem is what licenses the word “typical”, and it is doing so by fiat. It holds for almost every starting point — the exceptions form a set of measure zero, and that phrase disposes of them rather than examining them. For the rational rotation there are no exceptions to dispose of, because every orbit is exceptional and the theorem simply does not apply. Read backwards, ergodicity is a promise that the system has no hidden compartments, and checking that promise is almost always harder than the measurement it was invoked to justify.
LIT rotating by the golden ratio and timing the fraction of visits to [0.17, 0.53), all 60 starting points agree to within a spread of 5.00e-5 and land on the space average 0.3600 to within 3.33e-5; rotating by 1/7 instead, the same measurement gives 2 distinct answers depending on where you begin, spread 0.1429, exactly one seventh; and the error obeys the discrepancy bound with C < 1 - 0.0000, 0.1448, 0.0000, 0.0000 in units of log(N)/N
FIG A gate demanding each error be smaller than the last FAILED on a correct result. The golden ratio is the worst-approximable irrational, which makes its orbit the most evenly spread of any rotation - so the error is already at the 1/N quantisation and lands exactly on zero at N = 100, 10,000 and 100,000. Monotone decrease was never the right property; the real statement is a discrepancy bound, |error| <= C log(N)/N, and in those units the largest value seen is 0.1448. The rational rotation is the control doing real work: without it one could believe the agreement came from the observable being simple rather than the rotation being ergodic. Birkhoff, 1931.
FIG A gate demanding each error be smaller than the last FAILED on a correct result. The golden ratio is the worst-approximable irrational, which makes its orbit the most evenly spread of any rotation - so the error is already at the 1/N quantisation and lands exactly on zero at N = 100, 10,000 and 100,000. Monotone decrease was never the right property; the real statement is a discrepancy bound, |error| <= C log(N)/N, and in those units the largest value seen is 0.1448. The rational rotation is the control doing real work: without it one could believe the agreement came from the observable being simple rather than the rotation being ergodic. Birkhoff, 1931.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HEISENBUG · David Lee Wise (ROOT0), with AVAN