ANOMALIES EXPLAINED · 10 OF 12

The Leggett–Garg inequality

Is a large object in one definite state between looks? Not until something records it. A look writes an event into the record, and a yes-or-no look made first moves the next one’s chances by an exact amount, at most one half.

The exact move (a standard identity) and its limit of one half are proved in the paper behind this page (a preprint, posted 14 November 2026, not peer reviewed), given two standard rules of quantum mechanics: chances are squared lengths, and a look leaves the part of the state that matches what it found. Owed, that is, stated but not yet proved: attachment (what a laboratory look is in the record) and the position read, both set out under “What comes back, and what is owed” below.

What three yes-or-no looks can score

3/2

the textbook’s value, at a turn of π/3 (60°), for looks that keep the whole matching part; at most 1 if settled between looks and left as it was by each look. Measured on a superconducting loop, as a move, not a score: an uninspected middle look shifted the last look’s average beyond what control runs allow for a settled loop, by about 84 standard deviations (Knee et al. 2016).

Figure 1, running in your browser: three looks at a superconducting loop, each reading its current as clockwise (+1) or anticlockwise (−1), with the same turn between looks. Runs land one look at a time and fill the three tallies; the big number is K from the runs, beside K computed exactly. The K axis marks the bound 1, the ceiling 3/2 and one measured score, from neutrons (Kreuzgruber et al. 2024), taken in its own arrangement and so placed on the axis, not at a turn. The slider sets the turn, the loop’s oscillation over the time between looks. The runs are ideal: every read is sharp and the loop does not decohere (its surroundings pick up a trace of which state it is in); real data are pulled toward 0 and fall short of the peak.

When Nobody Looks · film, 3 min 57 s

The paradox in three lines

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). In it, every event is an entry written into one record. The middle of the three lines below says what that record makes of the puzzle.

  1. The puzzle

    Look at one object at three times, each look giving +1 or −1: a current running one way round a loop, or the other. Give each pair of times its own runs, so that no third look sits in the runs that score a pair, and score the pair: how often its two looks agree, minus how often they differ. If the object is in one definite state between looks, and a careful look leaves it as it was, the scores for times 1 and 2 and for times 2 and 3, minus the score for times 1 and 3, K, cannot pass 1. Is a large object like that?

  2. On the record

    No. What is definite is what has been written. Between two looks, an object that writes nothing holds values that add, as waves add, not one settled state; and a look is itself a writing: a posting, an event written into the record. Reading one yes-or-no look first moves the chances of another by an exact amount: never more than one half, and zero, whatever the state, exactly when the order of the two looks makes no difference. A turn of the object between looks (the steady swing of its state) makes the order matter unless it is none or a half turn, and a turn between 0 and 90° carries the score past 1; an object whose configurations are only shuffled between looks, like cards, with each look reading which configuration it is in, never passes 1. On the ideal turned loop of Figure 1 the score can pass 1 only because the runs that score looks 1 and 3 leave the middle look out: made in every run, it would keep the score at most 1.

  3. What comes back, and what is owed

    Every number the record returns here is the textbook’s own. The 3/2 at a turn of π/3 is the textbook’s computation with squared weights; the paper recomputes it, and its theorems do not produce it. The bound 1 comes from counting over triples of values (section 01), the three-look form of Leggett and Garg’s inequality. No experiment on this page can tell the record from the textbook; what the record adds is a place for the rules to come from, and a short list of what is left.

    Owed, and not yet proved: what a laboratory object and a look at it are in the record, and that exactly one outcome is posted (attachment); and that a read weighs each position of a register (each place where it holds a value) by the value held there and by nothing else (the position read).

01 · The question

Is the flux there when nobody looks?

Quantum mechanics lets a small thing, an electron or an atom, between looks, be in none of the states the next look could find: the textbook calls it a superposition. In 1985 Anthony Leggett and Anuj Garg asked whether a large object can be like that, and turned the question into a test. They called the opposite view macroscopic realism, now macrorealism: a large object is always in one of its distinct states. Their test adds a second assumption, that a careful enough look can find out which state without changing it, and takes for granted a third: no later choice of looks changes an earlier result. Their title asked it of a superconducting loop, where a current can run one way round or the other: is the flux there when nobody looks? (The flux is the amount of magnetic field threading the loop; a look finds it at one of two values, one for each way the current runs.)

The test needs nothing but a score. Look at the object at three times, t₁, t₂ and t₃; each look gives +1 or −1. Each pair of times gets its own runs, with no third look in them. For each pair, the correlation C is the fraction of runs in which its two looks agree minus the fraction in which they differ. Now suppose the object has a value at every time, q₁, q₂ and q₃, each +1 or −1, whether or not anyone looks. Every run then carries a triple, and each triple scores either 1 or −3:

Every triple of values, scored
q₁q₂q₃+++++−+−++−−−++−+−−−+−−−
score11−311−311

The score of a run is q₁q₂ + q₂q₃ − q₁q₃.

No triple scores more than 1, so no average does: K = C₁₂ + C₂₃ − C₁₃ is at most 1, provided no look changed what a later look would find, and no later choice changed an earlier result. Everything turns on that proviso, and section 03 gives the exact amount a look changes.

Quantum mechanics passes the bound. Draw the loop’s state as an arrow on a dial, and a look as a line across the dial, +1 at one end and −1 at the other. The chance that the look finds +1 is the square of the cosine of half the angle between the arrow and the +1 end: an arrow on the line gives a sure answer, an arrow at right angles to it even odds. Left alone, the arrow turns steadily. If it turns by the same angle θ between looks, looks one interval apart act like lines θ apart, and looks two intervals apart like lines 2θ apart. A look leaves the arrow on the end it found, so for two lines whose +1 ends are an angle α apart the second look agrees with chance cos²(α/2): agree minus differ is cos α, whatever the arrow’s start. So the three correlations are cos θ, cos θ and cos 2θ, and

K = 2 cos θ − cos 2θ, above 1 for every turn between 0 and π/2 (90°), and 3/2 at θ = π/3 (60°): the most that three yes-or-no looks can score when each look keeps the whole part of the state that matches its answer, the standard rule (the Lüders bound). Looks that resolve more than two values, on an object with more than two states, can pass 3/2, approaching 3 as the number of states grows (Budroni and Emary 2014).

  1. 02 · The tests

    The inequality and its tests

    Every test has to answer the macrorealist’s reply: perhaps the look was clumsy, and it, not the object, broke the bound. The strict tests are built to narrow that door. Leggett and Garg aimed the test at large objects, but the same score works on anything with two values, down to single particles, where it tests realism about a microscopic quantity instead. Real tests fall short of the ideal: the object decoheres between looks (its surroundings pick up a trace of which state it is in), readouts are imperfect, particles are lost, and runs made at different times can drift (Emary, Lambert and Nori 2014 review the tests); the neutron test quotes its statistical and systematic errors separately.

  2. 1985

    Leggett and Garg

    The inequality: anything in one definite state between looks, unmoved by careful looks and unaffected by later choices, keeps a score built from looks at different times within a fixed bound; for three looks, K at most 1.

    Phys. Rev. Lett. 54, 857

  3. 2010

    Palacios-Laloy et al.

    The first violation in a superconducting circuit, which was watched continuously but only gently (a weak measurement) while its state swung: a Leggett–Garg inequality broken by 5 standard deviations. With weak measurement the readout is a noisy signal rather than ±1, non-invasiveness rests on the weakness of the coupling rather than on a separate check, and the circuit’s two states are not two currents.

    Nature Physics 6, 442

  4. 2016

    Knee et al.

    A strict test on a superconducting loop whose persistent current, about 170 nanoamperes, can run either way round; its look reads two states that are mostly, not wholly, the two directions (how macroscopic they are is discussed in that paper). It measures a two-time move, not the score K. Some runs include a look at the middle time whose result nobody inspects, and some do not; the difference is how much that look moved the last look’s average. Control runs start from the two states an ideal look would read for sure (the real readout tells them apart only partly: its contrast is about 0.28 of an ideal look’s) and measure how much the look moves those, as Leggett, and Wilde and Mizel, proposed. Started in an even superposition of the two, the loop’s average moved 0.063 beyond what the control runs allow: about 84 standard deviations (84 times the uncertainty of that number), from seven million trials per point. The test assumes a look disturbs a loop in a definite state no more than it disturbs the controls.

    Nature Communications 7, 13253

  5. 2024

    Kreuzgruber et al.

    Neutrons in an interferometer, each split between two paths and recombined, looked at by blocking each path in turn with an absorber, so that a neutron that gets through was, on macrorealism’s own picture, never on the blocked path and never touched; the correlations come from neutron counts across those arrangements: K = 1.120 ± 0.007 ± 0.019, past 1.

    Phys. Rev. Lett. 132, 260201; statistical and systematic errors

03 · The price of a look

A look writes, and what it moves has an exact price.

In plain words: without the first look, the second look’s answer can come by two routes, and unless the two looks commute the routes can interfere; looking first tells the routes apart, the interference term is lost, and losing it is the move. Figure 2 shows it with numbers. The rest of this section says it exactly.

In Recognition Science’s terms, an object that writes nothing into the record between two looks is a held register. It carries values that add, and nothing in it is settled. A look is a posting: one unit written into the record. That a laboratory loop and a look at it are such a register and such a posting, and that one outcome is posted, is attachment, which is owed. The paper makes a look at a register exact. An act is a yes-or-no look: two outcomes, +1 and −1. The paper takes the state after it to be the part that matches its answer (the textbook’s rule), so it gives the same answer if repeated at once. Write ψ for the state, drawn as the arrow of section 01; the drawing doubles angles, so an arrow φ from the +1 end is a state whose part along +1 has length cos(φ/2). The part of ψ that matches outcome s is written Psψ. Its weight, by the standard rule taken as given for these parts (section 05), is its squared length: a read (the textbook’s probability, once the total is set to one). On the dial, that weight is the squared cosine of section 01.

Read one act, B, first, then another, A. The weight of outcome s of A changes by exactly

Δs = (s/4) ⟨ψ, B(AB − BA) ψ⟩, The corollary “What reading first costs” (Corollary 23), proved in the paper. Its size is never more than one half of the total weight, and one half is reached, for example when the two acts anticommute (AB = −BA) and the state already gives outcome s for A.

The reason is the theorem “The price of disturbance” (Theorem 22), and it is interference. Inside outcome s of A sit two parts of the state, one for each outcome of B. With B not looked at, they add as values, and the weight of outcome s carries a cross term, twice the real part of their overlap; that real part is exactly −(s/8)⟨ψ, B(AB − BA)ψ⟩. Reading B first tells the two parts apart, and the cross term drops out, as interference does when the path is marked. Its loss is the shift.

So two acts can be read together, with no shift whatever the state, exactly when their order makes no difference: AB = BA, the acts commute (the theorem “Acts can be read together exactly when they commute”, Theorem 24; in standard quantum mechanics, Lüders’ theorem of 1951). When they do not commute, the shift is fixed by the two acts and the state alone, given squared weights and the standard rule for the state after a look; no property of an apparatus enters it, for an ideal look. A real look can also disturb in its own ways; the 2016 test measures that with control runs.

Figure 2

What looking first costs

Drag the two acts and the state around the dial, or use the sliders. The bars give the chance that A reads +1, alone and with B read first; the change computed from the parts sits beside the paper’s closed form.

Figure 3

Read together, or not

The four parts two acts pick out from a state. Turn one act against the other and watch the overlap appear, on most states, the moment they stop commuting.

04 · Why a turn passes 1

On the loop, the middle look, made or not, is the whole difference.

Instead of turning the arrow between looks, turn the looks: look 2’s line sits at θ from look 1’s, and look 3’s at 2θ. Call the three looks Q₁, Q₂ and Q₃: the acts whose answers are the ±1 of looks 1, 2 and 3.

Now shuffle instead of turning: between looks the object’s configurations are only swapped among themselves, like cards, and each look reads which configuration it is in. Then every Qk reads the configuration, and such acts commute. No look moves another, so the three pair scores come from one table of chances for triples of answers, and K stays at most 1.

Any turn other than none or a half turn makes Q₂ and Q₃ fail to commute. Then the look at t₂, had it been made, would move the chances at t₃: the kind of move Knee et al. measured, in their own arrangement (section 02). After a settled result at t₁, the corollary “What reading first costs” gives its size for these acts: (sin²θ)/2, up for one outcome and down for the other, so 3/8 each way at θ = π/3. A look that moves the chances of a later one is what the literature calls a violation of no-signalling in time (Kofler and Brukner 2013); for the turned loop it happens at every turn except none and a half turn, and turns between 90° and 180° break another inequality of the same family instead: the count of section 01 with its signs changed, 1 + C₁₂ + C₂₃ + C₁₃ ≥ 0.

In the runs that score C₁₃ nobody looks at t₂, so nothing is moved, and C₁₃ = cos 2θ. Had all three looks been made in every run, each run would post a full triple; on the loop, C₁₂ and C₂₃ stay cos θ and C₁₃ would be cos²θ, giving

K = 2 cos θ − cos²θ = 1 − (1 − cos θ)², never above 1, at any turn.

The gap between the two scores is sin²θ: C₁₃ is the chance of agreeing minus the chance of differing, so a move of (sin²θ)/2 from one to the other changes it by twice that. At θ = π/3 that is 3/8 + 3/8 = 3/4, and it carries the score from 3/4 to 3/2.

Figure 4

Shuffle and read

Random shuffles of a register, random reads and random starting weights, thousands of them, computed in your browser. How many pass the bound 1, and the largest score found, are shown live.

05 · Break it

Replace the turn by a shuffle, and the score never passes 1.

With sharp yes-or-no looks, the score past 1 needs some look left out of some runs: with all three looks made in every run, each run yields a full triple of answers and K stays at most 1, for any object. On a two-state object turned between looks, like the loop, the look that matters is the middle one, left out of the runs that score C₁₃ (section 04; Figure 5’s second switch puts it back). Beyond that the score past 1 rests on three things. The first, the turn between looks, is a setting of the experiment, and Figure 5’s first switch replaces it: with a turn of π/3 the score climbs past 1 to 3/2; with a shuffle, each look reading the configuration, the looks commute again and the score never passes 1, at any setting. The other two are rules.

The second is the weight: the shift of section 03 is a statement about squared lengths. The paper proves that reads of a register’s positions are squared lengths from five premises about a read: two principal ones, the position read, which is owed, and a total that does not change when every value is turned together; and three mild ones, that reads add over positions, that an empty position reads nothing, and that a value larger in size never reads less (the Born rule page takes this step). For the parts that the looks of a turned loop pick out, squared weights are not proved: they are the standard rule, taken as given.

The third is the state after a look: the part that matches the outcome. That too is the standard rule; the paper takes it as the definition of a part, and deriving it belongs to attachment.

Figure 5

Turn, or shuffle

Three looks, every number computed exactly from the reads. Two switches: what happens between looks (a turn or a shuffle), and what each run looks at (two looks or all three). Only a turn, with two looks per run, carries K past 1.

06 · What it means

Definite is what is written.

The two results this page uses, Lüders’ theorem and the exact shift, are standard mathematics (card 01); the paper proves both from its definition of a yes-or-no look and its parts, given squared weights. Recognition Science needed them because in the record a look is a posting, and they give the exact price one look sets on the next, at a register. In the record, Leggett and Garg’s question becomes: does a look read an entry already there, or write a new one?

Each square says what a statement rests on: green, proved in the paper; gold, Recognition Science’s account of the record; an open red square, owed.

01 Proved in the paper

Lüders’ theorem, with the price of disturbance made exact.

For sharp yes-or-no looks that keep the whole part of the state matching their answer, standard quantum mechanics already ties complementarity, that some quantities cannot be read together, to the disturbance of one look by another: two such looks can be read together, with no move on any state, exactly when they commute (Lüders’ theorem). The paper proves this for its acts and also gives the exact price when they do not: there is then a state on which reading one first moves the other’s chances, by an amount fixed by the two acts and the state, at most one half, given squared weights and the standard rule for the state after a look. There is no third case, for a look that does no more than keep the matching part.

02 The record

A look writes. It does not uncover.

Macrorealism, joined to non-invasive looks as the test assumes, pictures a look as uncovering a value that was already there, so that a careful enough look would change nothing. On the record a look is a posting, an event that writes a new entry, and section 03 prices what the writing moves. The tests of section 02 find moves of this kind and scores past 1, as the textbook says they should. That a laboratory look is such a posting is owed (card 04).

03 The record

Nothing is settled until it is written, and what is written stays.

A held register (section 03) carries no answer until a look writes one, so “is it there when nobody looks?” has a plain answer: what is there is the register’s values, not a result. Once a look has written its result, the result is permanent. Reality keeps a permanent record: nothing that happens is ever erased, and the past still exists (why).

04 Owed

What a look at a loop is, in the record, is owed.

The theorems give the algebra of a look at a register. What a superconducting loop and a look at it are on the record, which posting a detector’s click is, and that one outcome is posted, is attachment: the wider of this paradox’s two owed statements, and the one that eleven of the paper’s twelve paradoxes wait on. It asks one question: what, in a laboratory, is an entry?

07 · Hard questions

What an expert asks first.

Is any of this different from ordinary quantum mechanics?
No number is, as the three lines say. What differs is what is taken as given: the textbook takes its rules for a measurement as given; here a look is a posting; its algebra follows from the definition of a yes-or-no look (as Lüders’ theorem does in the textbook); the square follows, for reads of a register’s positions, from five named premises, one of them owed; the weights of this page’s looks and the state after a look are still the standard rules; and what is not yet proved is named.
Isn’t the violation just a clumsy measurement?
It is disturbance, not clumsiness. On the textbook’s rules the middle look’s disturbance is exactly what separates the score from one that stays at most 1 (section 04), and given squared weights and the standard rule for the state after a look, no better-built look at the same sharp quantity leaves every state unmoved (a real look may disturb in other ways too). The strict tests were built against clumsy looks: in 2016 control runs measure the look’s own disturbance, and in 2024 a neutron that reached the detector while one path was blocked was, on the picture in which its path is all there is, on the other path and never touched. The 2016 test finds what macrorealism forbids if a look disturbs a loop in a definite state no more than the control runs show; the 2024 test finds what it forbids if a look that touches nothing disturbs nothing. A macrorealist who lets a look disturb something hidden is not ruled out.
What is not proved?
The two owed statements named in the three lines, attachment and the position read; and, as section 05 says, the squared weights of a turned loop’s looks and the state after a look, which are the standard rules, taken as given.
Does it say where the quantum world ends?
Not as a size. The paper names no size, mass or time at which a body becomes settled, and no limit on how large an object can be and still break the bound. On the record a body’s histories, the ways it could have gone, add until a posting tells them apart, and only a contact, a posting that crosses into the body from outside, settles which: a working statement, not a theorem. For heavy bodies the paper’s one candidate rule, also a working statement, has the form of a rule Roger Penrose proposed, in which a heavy body’s two histories, once apart, stop being one when the gravitational energy of their difference has built up one unit, but its unit, one entry of the record, is open, so it gives no number. Which laboratory event posts is attachment, which is owed. In standard physics, decoherence and coarse looks are the usual reasons a violation fades (Emary, Lambert and Nori 2014).

08 · The paper

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, posted 14 November 2026, not peer reviewed. Leggett and Garg’s question is paradox 10 of the twelve in the paper’s Section 8; the theorems used here are in Sections 5 and 6.

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

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