Anomalies Explained · 7 of 12

Bell’s inequality

Measured far apart, with settings chosen at random, two particles give results correlated more strongly than answers they carried could give, unless something passes between them. Nothing does: some answers are made when measured, from one entangled state the two share.

The second sentence is Recognition Science’s account. Its paper, a preprint not yet peer reviewed, proves for one combined system that nothing needs to pass, given the textbook’s rule for probabilities; owed are that two separate laboratories form such a system, and two more statements named below. Every number the account returns, the bounds 2 (Bell) and 2√2 (Tsirelson) among them, is the textbook’s own.

Figure 1, computed live from the cube’s 16 × 16 matrices (four on/off switches, two for each station: section 03). The big number is the sum for the acts set on the dials, not a measurement; the measured values sit under the line, each with one standard error, and the closest approach to 2√2 (Poh et al. 2015, not loophole-free) is 1.6 standard errors below it. Its rounds draw both stations’ answers together, as Figure 2 explains.

Film · Written, not read · 3:56

This page’s experiment, computed on the same cube and 16 × 16 matrices as the figures; the measured values it shows come from the real experiments.

The paradox in three lines

  1. The puzzle

    Two stations, far apart, each choose one of two settings at random and record a result, +1 or −1. If every result were written in advance, carried by the particles like answers on a card, one sum of the four correlations, S, could not pass 2 on average (a correlation is how often the two results agree minus how often they differ). Experiments pass 2.

  2. On the record

    An act is what a station does to get its +1 or −1. Acts that only read values already written cannot pass 2, so passing 2 rules out values held in advance. The record’s account of what is left: some act writes its result. There is one record, and in its account the two stations are two blocks of one cube, sharing one state. Nothing travels: each station’s acts touch only its own block, and acts whose order makes no difference do not move each other’s weights, so no signal passes.

  3. What comes back, and what is owed

    Every number the record returns is the one textbook quantum mechanics gives: at most 2√2, with no signal from either station to the other. Measured in loophole-free tests (each setting chosen at random, too late for a signal at light speed to carry it to the other station, and every pair counted): 2.42 ± 0.20 (Hensen et al. 2015; values written in advance would do at least this well with a chance of 0.039) and 2.0747 ± 0.0033 (Storz et al. 2023, more than twenty error bars above 2).

    Owed, and not yet proved: that two separate laboratories are two blocks of one cube, combined over one complex unit (composition); that a read applies one and the same rule to each value held and adds the results, using nothing else (the position read); and what a station’s apparatus and its click are on the record, and that exactly one outcome is posted (attachment).

Figure 2 Play the game: your cards against the cube’s acts.

Set the two cards with the four buttons, then play; switch the cube’s acts to “Only read” and watch their tally fall back to 3 in 4. Both pairs play the same settings, round for round. The cube’s pair plays the odds quantum mechanics gives at its ceiling, which the experiments approach; it is not a model of how two separated stations do it. In a short run a tally can wander past its mark; the diamond on the scale is each pair’s exact rate.

01 · The question

Were the answers there all along?

In 1935 Albert Einstein, Boris Podolsky and Nathan Rosen described two particles that interact and fly apart. Measure one and you can predict, with certainty, what the same measurement on the other will give, without touching it. If a value can be predicted with certainty without disturbing a thing, they argued, it is real before anyone measures it. Either the position or the momentum of the far particle could be predicted this way, so both would be real at once, and quantum mechanics cannot give both values at once; so, they concluded, it must be incomplete.

Niels Bohr replied within months, under the same title. The arrangement chosen for a measurement, he answered, is part of what defines the quantity being predicted, so their test of reality did not apply, and the description was complete. For almost thirty years the disagreement looked like philosophy: both sides agreed on every number an experiment could return.

David Bohm made the case concrete in his 1951 textbook: two particles of spin one half, sent apart with total spin zero, so that along any one direction their spins come out opposite, and a spin measured at each end along any direction one likes. In 1964 John Bell found the bound that turns it into a question for experiment: if each particle carries its answers with it, one for every direction, and nothing passes between the two ends, the correlations obey that bound, and quantum mechanics breaks it. In 1969 the bound was put in a form real apparatus can test (section 02).

  1. Fifty years of tests

    Real apparatus is imperfect, and every imperfection (a shared state partly mixed with noise, settings slightly off, results misread, particles lost where every pair is counted) pulls S down. The loophole-free tests pay for distance, speed and complete counting with such noise, so they land well below 2√2; a test that gives up those demands can come within a thousandth of it (Poh et al. 2015, in Figure 1). For a test that reports S, what matters is how far above 2 it lands, counted in its error bars: Hensen et al. about two, Storz et al. more than twenty.

  2. 1972

    Freedman, Clauser

    The first test, with pairs of photons from a cascade in calcium atoms, finds the correlations quantum mechanics gives and passes a form of the bound. It detects only a small share of the photons and assumes the detected pairs stand for all, and its polarizers stay fixed through each run, so a signal could have carried one station’s setting to the other.

    Phys. Rev. Lett. 28, 938

  3. 1982

    Aspect, Dalibard, Roger

    The setting at each station switches between two directions while the photons are in flight, too fast for a signal at light speed to carry it across; the switching is regular, not random, and, as in 1972, only a small share of the photons is detected.

    Phys. Rev. Lett. 49, 1804

  4. 1998

    Weihs et al.

    Settings chosen by fast random generators while the photons fly, too late for a signal at light speed to carry one station’s choice to the other.

    Phys. Rev. Lett. 81, 5039

  5. 2015

    Hensen; Giustina; Shalm

    Three groups close the locality and detection loopholes at once: each setting is drawn at random too late for a signal at light speed to carry it to the other station before that station’s result is recorded, and every round counts, so lost particles cannot fake the result. In Delft, electron spins 1.3 km apart are linked through photons sent to a station between them, where a joint click announces each ready pair: S = 2.42 ± 0.20 from 245 pairs. Vienna and Boulder detect photons often enough that lost ones cannot fake the result; they count a missing click as a result and report, instead of S, the chance that values written in advance would do as well: at most 3.74 × 10−31, and 2.3 × 10−7 after allowing for how predictable Boulder’s random settings might have been.

    Nature 526, 682; Phys. Rev. Lett. 115, 250401 and 250402

  6. 2018

    BIG Bell Test; Rauch et al.

    No test can prove that the settings are independent of the cards; it can only push the source of the choices further back: to people’s choices, and to light that left distant quasars billions of years ago, so that any plan fixing the settings would have had to start at least 7.8 billion years ago. The results still break the bound.

    Nature 557, 212; Phys. Rev. Lett. 121, 080403

  7. 2023

    Storz et al.

    Superconducting circuits 30 m apart close both loopholes too; if the results had been written in advance, the chance of a run this strong would be below 1 in 10108.

    Nature 617, 265

02 · The test

Score four pairings. Answers written in advance stop at 2.

A source sends pairs to two stations (in Delft and Munich, two particles held far apart are linked through photons they send to one place where the two photons meet). At each, a setting is chosen at random, 0 or 1, and a result is recorded, +1 or −1. Over many pairs, the correlation E(x, y) of settings x and y is the average of the product of the two results: +1 if they always agree, −1 if they always differ, 0 if they agree half the time. The test that John Clauser, Michael Horne, Abner Shimony and Richard Holt proposed in 1969 adds three correlations and subtracts the fourth:

S = E(0,0) + E(0,1) + E(1,0) − E(1,1)

Now suppose each pair carries a card written in advance: the first station’s answers a0, a1 for its two settings, the second station’s b0, b1, each +1 or −1. For one card, three products added and the fourth subtracted give

a0(b0 + b1) + a1(b0 − b1) = ±2,because one of b0 + b1 and b0 − b1 is 0 and the other is ±2.

Every card gives ±2, so any mixture of cards, in any proportions, gives an average S between −2 and 2; a finite run scatters around its average, which is why every measured S carries an error bar. That is Bell’s inequality in the form Clauser, Horne, Shimony and Holt gave it, the form the tests in Delft, Munich and Zurich used (Hensen et al. 2015, Rosenfeld et al. 2017, Storz et al. 2023). It needs nothing about particles or waves: only that each result depends on what the pair carried and on its own station’s setting, and on nothing at the other station; that the settings are chosen independently of what the pair carried; and that every pair is counted. Answers each station draws by chance on its own when measured change nothing: each station’s draws could have been made in advance and written on the card, so any such account is a mixture of cards and stays between −2 and 2.

The same test as a game

Figure 2 plays the test as a game: a round is won when the two results agree, except when both settings are 1, where they must differ. With the four pairings of settings equally often, the share of rounds won is exactly ½ + S/8. Cards written in advance win at most 3 rounds in 4 on average. Acts that write, at the ceiling 2√2, win ½ + √2/4 ≈ 85.36% of them on average.

03 · Written, not read

On the record, a result is written by the act that makes it.

The record is everything that has happened, written down event by event. Each entry is a posting, and a posting is never erased. To follow one pair, picture four on/off switches, two for each station; one setting of all four is a configuration, and there are 2 × 2 × 2 × 2 = 16 of them, written 00|00 to 11|11. The paper calls such a set of switches a cube (three switches give the eight corners of an ordinary cube), and each station’s two switches a block. In the record’s account the two stations are the two blocks of one cube; for two separate laboratories that is composition, which is owed (section 06).

An act is what a station does to get a +1 or a −1 (in the textbook, a measurement with two outcomes), and each outcome has a weight, the textbook’s probability. Some acts only read: their result was already sitting in the configuration. Others write: for at least some configurations, their result is not there until the act is done. As a drawing of that real difference: a reading act looks at a box already ticked on the card; a writing act asks a question whose answer is not on every card.

If every act only reads, the configuration itself is the card of section 02, and the sum cannot pass 2, whatever the state. That is Bell’s inequality; the paper behind this page, a preprint posted on 14 November 2026 and not yet peer reviewed (section 09), proves it again in the record’s terms (Theorem 27a). The loophole-free tests pass 2, with each setting drawn at random and the stations too far apart for either choice to reach the other in time. So not every result is read off an entry already there. The record’s account of what is left: some act writes its result.

Acts that write can reach 2√2 exactly, on one cube: three of the four correlations come out exactly 1/√2 ≈ 0.707 and the fourth, the one subtracted, exactly −1/√2, so S = 2√2 = 2.828427… (Theorem 28). Figure 1 runs these acts; the construction follows for readers who want it.

The construction, step by step

The bound of 2 on values written in advance holds for any two stations that meet the conditions of section 02; what follows about acts that write is proved on one cube, or on one state, with the textbook’s squared weights (section 06), and not for two separate laboratories.

A state gives each of the 16 configurations a number. Taken together, the 16 numbers form an arrow in sixteen dimensions, of length one. The weight of an outcome is the squared length of the part of that arrow the outcome picks out: the textbook’s probability (section 06 says what the square rests on).

An act’s two results pick out two parts of the state, and an act read twice gives the same result both times. An act that only reads splits the state along whole configurations, so its result was already sitting in the configuration. An act that writes splits it, at least in part, into parts spread over several configurations, for example the sum and the difference of the two settings of one switch, so some configurations do not hold its result. Written as a 16 × 16 table, a reading act multiplies each configuration by +1 or −1, and a writing act moves configurations, for example by flipping a switch: that is how the figures draw them.

Let the state weigh equally the four configurations in which the two stations’ switches match: 00|00, 01|01, 10|10 and 11|11 (in the textbook’s words, a maximally entangled state of the two blocks). Give the first station two acts: flip its first switch; or flip its second switch while carrying the first switch’s sign, a factor −1 whenever the first switch is on. Give the second station the same two acts on its own switches, and let it mix them in equal measure: one setting is their sum divided by √2, the other their difference divided by √2. Three of the four correlations come out exactly 1/√2 ≈ 0.707 and the fourth, E(1,1), exactly −1/√2, so

S = 3/√2 + 1/√2 = 4/√2 = 2√2 = 2.828427…exactly, on this cube of 2 + 2 switches; the paper proves it for every cube of n + n switches with n ≥ 2 (Theorem 28). Figure 1’s “Open the matrices” shows the four acts as 16 × 16 tables.

04 · Why 2√2

The ceiling is a square’s two diagonals.

Boris Tsirelson found in 1980 how far any acts that commute across the stations can go. Let every act be any act at all, reading or writing, with one condition: each of the first station’s acts can be done before or after each of the second station’s with no difference on any state (in the textbook’s words, they commute). Then each correlation is the overlap of two arrows of length one (a state is such an arrow: section 03’s construction): the arrow you get by applying the first station’s act, as a table, to the state, w0 or w1, and the one you get from the second station’s act, v0 or v1. The overlap of two such arrows is the cosine of the angle between them: 1 when they point the same way, 0 at right angles. On one state of length one, with squared weights, the sum is at most 2√2; the paper proves it in the record’s terms (Theorem 27b).

The proof is a picture. S is the overlap of w0 with v0 + v1 plus the overlap of w1 with v0 − v1, and an arrow of unit length overlaps another by at most that other’s length. The sum and the difference of v0 and v1 are the two diagonals of a rhombus with unit sides. The squares of the two diagonals always add to 4 (the parallelogram law), and their lengths add to the most when the rhombus is a square, where each diagonal is √2:

S ≤ |v0 + v1| + |v0 − v1| ≤ √2 + √2 = 2√2The second step is the most the two diagonals can add to, reached when the rhombus is a square.

Drawn to scale. Solid magenta: the sum v0 + v1; dashed: the difference v0 − v1.

Forbidding signals alone does not give it. Sandu Popescu and Daniel Rohrlich wrote down in 1994 a table of probabilities in which neither station’s choice shows in the other’s results alone and S = 4 (Theorem 27c). The three ceilings are strictly apart, 2 < 2√2 < 4, and each is reached: by a card, by the cube, and by Popescu and Rohrlich’s table. Another page in this set, Tsirelson’s bound, asks why the correlation stops at 2√2 when forbidding signals alone would allow 4.

05 · Nothing travels

The agreement is in one state, not in a signal.

The textbook says that two systems together live in “the product of their spaces”. On one cube that rule is simply what sorting the switches into two blocks looks like, and anything that touches only one block can be done before or after anything that touches only the other with no difference, whatever each one is (Theorem 26a); that two separate laboratories divide this way is composition, owed (section 06).

Two acts can be read together exactly when they commute, and then reading one first leaves the other’s weights unchanged (Lüders’ theorem of 1951, proved again in the paper’s terms as Theorem 24). So the first station’s own weights are the same whichever setting the second station picks: no act that touches only one block shows in the other block’s results alone. The agreement is already in the one shared state, and it appears only when the two lists are set side by side.

Figure 3 Turn the second station’s setting. The first station’s own weights stay put.

Drag the needle or use the arrow keys. The column “own weight, second station read first” adds up the joint weights; the column “own weight, nothing read at the second station” is the first station’s weight on its own; the two always agree. “A random state” draws new random values on all sixteen configurations: the first station’s own weights change with the state, never with the dial.

06 · Break it

Break it two ways.

Take writing away. Make every act read only, a sign for each configuration, and the sum falls back to 2 at most, whatever the state: the configuration is the card of section 02. Figure 5 tries it on a thousand random cases at a time.

Leave the first station’s signs on. The cube’s own acts each carry the signs of the switches before them (a factor −1 for each of those switches that is on), so the second station’s acts, used as they come, carry the first station’s signs. Then the two stations’ acts no longer commute (their order makes a difference), and they cannot be read together: on some states, whether the second station reads first shows at the first, which would be a signal (Theorem 26b; Figure 4 shows one). With those signs taken off, each station’s acts touch only its own switches (Theorem 26c), and the two can be read together on every state.

Both breaks are proved. What is owed here is composition: that two separate laboratories are two blocks of one cube, each station acting only on its own block with the other’s signs taken off, read as one cube and combined over one complex unit. The square in each weight rests on the textbook’s rule, taken as given, and on a second owed statement, the position read, one of the five premises the paper names for it; the complex unit for two laboratories is part of composition. The square and the complex unit each have their own page in this set: The Born rule and Why complex numbers. The details are below for readers who want them.

The signs, the square and the complex unit, in detail

The cube comes with one set of its own acts, its generators: flip one switch while carrying the sign of every switch before it (the construction of Jordan and Wigner). They are not sorted into stations: any two of them anticommute (done in the other order, they give minus the result), and by Theorem 24 acts that do not commute cannot be read together. Among the cube’s own generators, the two stations can be read together, on every state, exactly when the second station’s acts have the first station’s signs taken off (Theorem 26b, c).

The square: for reads of the record’s register (what one cell of the record holds: one real value at each of eight positions), it is proved from five premises (Theorem 16). Two are principal: the position read (owed), and that the total weight does not change when all the values are turned together, as one. Three are mild: reads add over positions, an empty position reads nothing, and a value larger in size never reads less. Without any one of the five, the square does not follow. The idea at the heart of the proof, that a weight taken value by value and unchanged when all the values are turned together goes as the square of the value, is in additive form the argument by which Maxwell found the law of velocities in a gas, in 1860. In the paper’s proof the key step turns all the register’s values together by an eighth of a full turn (not a tick of the record’s eight-tick clock, which moves every value to the next position): a map of the same form as a balanced beam splitter, where two equal values go in and one output stays empty. On the record it is a turn of the register, not a device. A single position’s read is only a term of the total weight, not an outcome, since turning all the values together moves it; the smallest outcomes the register offers are its complex values, each held by two positions. For the outcomes of acts that write, the squared weight is the textbook’s rule, taken as given; so, for every act, is the rule that the state after a result is the part that result picks out. For the cube’s sixteen configurations, the paper takes the weights from the position read, applied to the whole cube (owed).

Every statement about the correlation sum on this page holds for real numbers and for complex ones alike, and this test cannot tell them apart. A test with three stations and two independent sources, proposed by Renou et al. (2021), separates real blocks whose two sources share nothing: such blocks reach at most 7.66, a bound found numerically and not known to be tight, while complex ones reach 6√2 ≈ 8.49. Chen et al. (2022), with everything on one superconducting chip, measured 8.09 ± 0.01; Wu et al. (2022), keeping the sources and the measurements apart, measured 7.83 ± 0.03, which they report as 5.3 standard errors above 7.66. So real blocks whose sources share nothing are excluded. Real blocks whose sources may share a suitable state give the complex values: in this network, sharing one entangled bit is enough, and less is not (Weilenmann, Gisin and Sekatski 2025), and when the independence of the sources is defined by what can be observed, real blocks reach every correlation of the complex theory in any finite network (Hoffreumon and Woods, two preprints, 2026). On the record, that shared state is the part of the two registers’ product that two ticks of the record’s eight-tick clock reverse (proved, Proposition 29). On the record’s picture of two registers, the measurements are evidence that two registers in two laboratories agree on that unit; they do not measure the owed statement itself. Owed for two laboratories, in the paper’s words: two registers held by two recognizers are two blocks of one cube, and their joint states lie in the part of the product that two ticks of the clock reverse, which is their product over one complex unit.

Figure 4 Leave the first station’s signs on, and the two stations cannot be read together.

Two switches: the first station’s signs on the second station’s acts, taken off or left on, and the size of the cube. The grid shows which pairs of acts commute. Below it, on a state where one of the first station’s results is certain, reading the second station first leaves that weight alone when the signs are off and moves it when they are on.

Figure 5 One switch: reading or writing.

Each draw computes 1,000 sums from the cube’s matrices with random acts, half on Figure 1’s state and half on random states. With “Only read” no dot passes 2; with “Write” the dots spread past 2 but never past 2√2. The second line gives the largest sum over every state for Figure 1’s dials.

07 · What it means

One record, written when things happen.

Recognition Science starts from three facts every question already uses and proves that, under each result’s stated conditions, only one answer is possible (the Overview). The mathematics of the three ceilings is standard (the first hard question says whose). Recognition Science needed it on its own record, because the record keeps every event as a written entry, and Bell’s question asks of any such record whether results are stored in it and read, or written when they happen.

Each square below says what a statement rests on: green, proved (standard results included, proved again in the paper’s terms), with the measured violation where it says so; an open red square, owed.

01 Proved in the paper, and measured

No card of results is waiting to be read.

If an entry made before the act held the results, in the particles, the source or a station, and each station’s result depended only on that entry and its own setting, that entry would be a card of section 02. In the loophole-free tests each setting is drawn at random during the run, with the stations too far apart for either choice to reach the other in time; and the tests pass the card’s limit. So, if the settings are independent of any such entry, as section 02 requires, no such entry holds the card of results.

02 Owed for two laboratories

The agreement lives in one record, not in two places.

A long-standing unease about quantum mechanics: how the agreement between results here and results there can depend on the settings chosen at both ends, when neither end’s setting shows in the other’s results. Sort one cube’s switches into two blocks: the blocks compose by the product, and acts that touch only one block are blind to acts that touch only the other (proved, section 05). That two laboratories are two blocks of one cube is composition, which is owed (section 06). Given composition, the agreement between distant stations needs no messenger: it was never in two places.

03 Proved in the paper

On one cube, the limit is a square’s two diagonals.

The √2 in the ceiling is the diagonal of the unit square, from the parallelogram law of squared lengths (section 04). The same law makes a read of the record’s register the sum of the squares of the values it reads (Theorem 16); in additive form the argument is Maxwell’s, of 1860. It rests on five premises, listed in section 06; one of them, the position read, is owed. For the parts that acts which write pick out, the squared weight is the textbook’s rule, taken as given.

08 · Hard questions

What an expert asks first.

Isn’t this just Bell’s theorem?
Yes: all three ceilings are standard mathematics, Bell’s bound of 1964 in the form Clauser, Horne, Shimony and Holt gave it in 1969, Tsirelson’s of 1980 and Popescu and Rohrlich’s of 1994. The paper adds no new bound and no new number: it places them on one record, reading the textbook’s product of two spaces as one cube sorted into two blocks (section 05), and it uses a standard fact about the cube’s own generators, those of Jordan and Wigner: generators of different blocks anticommute, and with the first block’s sign taken off they commute (section 06).
Is any of this different from ordinary quantum mechanics?
No number is, and no Bell test can tell the record from the textbook. What differs is what is taken as given: the textbook postulates the product of two spaces, while on the record it holds by construction for two blocks of one cube and is owed, by name, for two laboratories.
Is the record’s account local?
Not in Bell’s sense: Bell showed that no account in which each result is settled at its own station, from what is there and what the pair shared beforehand, passes 2 when the settings are chosen independently of what the pair shared. The experiments pass 2, and physicists call such correlations nonlocal. What is proved on one cube is that no act at one station touches the other’s block and no choice at one station shows in the other’s results alone; the agreement is in the one state of both blocks, and in neither block alone.
What is not proved?
The three owed statements in “The paradox in three lines”: composition, the position read and attachment; besides them, the premises and the textbook’s rules named in section 06. Without composition, nothing the record proves about two stations goes beyond “probabilities, and no signals”, whose ceiling is 4.

What this page does not claim

  • That anything travels faster than light, or that a signal can be sent this way.
  • That hidden values, set in advance, decide the results.
  • An account of why two real laboratories stop at 2√2: that waits on composition.

09 · Read it

The paper and the sources.

The Five Postulates of Quantum Mechanics on the Record: Theorems, Premises, and What Is Owed

Jonathan Washburn · Recognition Physics Research Institute

Preprint, 14 November 2026, not peer reviewed. Bell’s inequality is paradox 7 of the twelve in its Section 8; the theorems used here are in Sections 5 to 7. The results this page cites by number carry these names in the paper: Theorem 16, “The read is the square”; Theorem 24, “Acts can be read together exactly when they commute”; Theorem 26, “Blocks of one cube”; Theorem 27, “Three ceilings”; Theorem 28, “One cube reaches the middle ceiling”; Proposition 29, “The complex product inside the real one”.

Read the preprint (PDF), the institute’s library

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