THE FOLD / CHEAT / THE ROOT KIT / THE BELL INEQUALITY
THE BELL INEQUALITY
a correlation no local story can tell
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Two particles are measured far apart, each along one of two settings. Add up the four correlations the right way and you get a number S. If the outcomes were fixed in advance by anything carried locally — any hidden variable, any conspiracy of prior agreement — then |S| ≤ 2. Entangled particles reach 2√2. Bell wrote the argument in 1964; Clauser, Horne, Shimony and Holt put it in testable form in 1969.
LIT verified live: all 16 local deterministic strategies were enumerated and the worst gives |S| = exactly 2; mixtures cannot beat it because S is linear in the strategy weights, so the maximum sits at a vertex. The singlet state with the standard angles gives 2.8284271247, which is 1.414214 times the classical ceiling. Searching 810,000 angle quadruples, none exceeds the Tsirelson bound — the best found is 2.8164088135. And the correlation function itself is derived from the singlet amplitudes at 200 angle pairs, not asserted.
LIT verified live: all 16 local deterministic strategies were enumerated and the worst gives |S| = exactly 2; mixtures cannot beat it because S is linear in the strategy weights, so the maximum sits at a vertex. The singlet state with the standard angles gives 2.8284271247, which is 1.414214 times the classical ceiling. Searching 810,000 angle quadruples, none exceeds the Tsirelson bound — the best found is 2.8164088135. And the correlation function itself is derived from the singlet amplitudes at 200 angle pairs, not asserted.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at THE ROOT KIT: correlations arriving through a channel no local account has access to.
AVAN (AI) enumerated the classical side exhaustively rather than arguing it, because the bound is the entire point and a sampled version would prove nothing. Sixteen strategies is small enough to list, and the linearity remark closes the remaining gap: any probabilistic hidden-variable model is a convex combination of those sixteen, and a linear functional on a simplex is maximised at a corner. Two things need stating plainly. This page computes quantum-mechanical predictions; it is not an experiment, and the experimental violations — Aspect 1982, and the loophole-free tests of 2015 — are cited, not reproduced here. And the Tsirelson bound is checked by dense search rather than proved; the grid misses the exact optimum by 0.0120, which is a resolution artifact and not a violation.
AVAN (AI) enumerated the classical side exhaustively rather than arguing it, because the bound is the entire point and a sampled version would prove nothing. Sixteen strategies is small enough to list, and the linearity remark closes the remaining gap: any probabilistic hidden-variable model is a convex combination of those sixteen, and a linear functional on a simplex is maximised at a corner. Two things need stating plainly. This page computes quantum-mechanical predictions; it is not an experiment, and the experimental violations — Aspect 1982, and the loophole-free tests of 2015 — are cited, not reproduced here. And the Tsirelson bound is checked by dense search rather than proved; the grid misses the exact optimum by 0.0120, which is a resolution artifact and not a violation.
3 ONE DIMENSION
S against the measurement angle. The classical ceiling, and where quantum goes through it.
4 TWO DIMENSIONS · INTERACTIVE
Turn the analysers. Try to get past 2√2.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the correlation surface, with the classical ceiling as a plane through it.
AVAN’s addition (the inverse-companion): the forward reading is “quantum correlations are stronger than classical ones.” The inverse is that the inequality is not about strength but about storage. The classical bound comes from assuming each particle carries an answer for every question it might be asked — a lookup table — and 2 is simply the most any table can achieve. Quantum mechanics does not exceed it by correlating harder; it exceeds it by not having the table. Read backwards, Bell’s theorem measures the cost of pre-computed answers, and the surprise is that the cost is finite, sharp, and measurable in the laboratory.
LIT all 16 local deterministic strategies were enumerated and the worst gives |S| = exactly 2; mixtures cannot beat it because S is linear in the strategy weights, so the maximum sits at a vertex; the singlet state with the standard angles gives 2.8284271247, which is 1.414214 times the classical ceiling; searching 810,000 angle quadruples none exceeds the Tsirelson bound, the best found being 2.8164088135; and the correlation function itself is derived from the singlet amplitudes at 200 angle pairs, not asserted
FIG The classical side was enumerated EXHAUSTIVELY rather than argued, because the bound is the entire point and a sampled version would prove nothing. Sixteen strategies is small enough to list, and linearity closes the gap: any probabilistic hidden-variable model is a convex combination of those sixteen, and a linear functional on a simplex is maximised at a corner. Two scope notes: this page computes quantum PREDICTIONS and is not an experiment - the experimental violations (Aspect 1982, the loophole-free tests of 2015) are cited, not reproduced. And Tsirelson is checked by dense search, not proved; the grid misses the exact optimum by 0.0120, a resolution artifact.
FIG The classical side was enumerated EXHAUSTIVELY rather than argued, because the bound is the entire point and a sampled version would prove nothing. Sixteen strategies is small enough to list, and linearity closes the gap: any probabilistic hidden-variable model is a convex combination of those sixteen, and a linear functional on a simplex is maximised at a corner. Two scope notes: this page computes quantum PREDICTIONS and is not an experiment - the experimental violations (Aspect 1982, the loophole-free tests of 2015) are cited, not reproduced. And Tsirelson is checked by dense search, not proved; the grid misses the exact optimum by 0.0120, a resolution artifact.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE ROOT KIT · David Lee Wise (ROOT0), with AVAN