Anomalies explained · 11 of 12
There is no one time a particle spends inside the barrier. Time is a count of recorded events; none marks the held particle going in or out, so the record holds no time inside, not even zero, and each measured “tunnelling time” is what some clock’s touch reads.
From a preprint in Recognition Science, posted 14 November 2026, not peer reviewed. That time is the count of recorded events comes from earlier work; the paper takes a form for a contact and proves what a contact of that form does to a held state in two cases; what a laboratory particle and clock are in the record is owed, not yet proved (the three lines below).
The numbers: two different experiments
Electrons leaving hydrogen: any delay at most 1.8 attoseconds
Attoclock (angular streaking) Sainadh et al., Nature, 2019.
Cold rubidium-87 atoms: the spin clock measured 0.61 ± 0.07 milliseconds
Spin clock At the lowest energy at which tunnelling was seen; a barrier of light 1.3 micrometres thick. Ramos et al., Nature, 2020.
An attosecond is a billionth of a billionth of a second. Different particles, barriers and clocks: one number is an upper limit, the other a measured time, and they are not rival answers to one question. Neither is a reading of the crossing computed beside them, which is a picture of neither experiment.
The paradox in three lines
The puzzle
A particle meets a barrier it has too little energy to get over, and sometimes it turns up on the far side anyway: it tunnels. How long was it inside? The ways of defining that time give different answers, and some of them level off as the barrier gets wider, as if a thicker wall took no longer to cross (Thomas Hartman, Journal of Applied Physics, 1962).
On the record
Time is a count of postings, the entries written in reality’s record. While nothing is posted about a particle it is held, and a value held inside a barrier posts nothing: no entry marks it going in or coming out, so the crossing has no count of its own. That is not a time of zero: the record’s count runs on while the particle is held, but no entry marks where the crossing begins or ends within that count, so no stretch of it is a time inside. Every measured “tunnelling time” is the reading of some apparatus making its own contact with the particle, a posting that crosses into it from outside (the textbook’s measurement); what a contact of the paper’s form does to a held state is its contact theorem (section 03 below).
What comes back, and what is owed
The record gives no number of its own here: every number this paradox returns is the textbook’s own, and neither experiment above can tell the record from the textbook. Owed, that is, not yet proved, as the paper’s statement O1: what a laboratory particle, its apparatus and its outcome are in the record, which posting a click is, and that one is posted.
01 · The question
Roll a ball at a hill too high for it and it rolls back, every time. Send an electron at a barrier of energy too high for it and, now and then, it is found on the far side. Quantum mechanics describes the electron by values spread over space (the textbook’s wavefunction). Inside the barrier those values do not stop dead: they fall away steadily, and some value is left at the far side, more the thinner the barrier, and there the particle is sometimes found. That is tunnelling. It is how an alpha particle leaves a nucleus, and it carries the current in a scanning tunnelling microscope.
So how long was the particle inside? Start a clock when it reaches the barrier and stop it when it leaves. The trouble is that nothing reads it inside without touching it: a particle found in the barrier has been touched there, and one found on the far side was not found on the way. In a laboratory, every way of timing the crossing touches the particle somehow.
Two recent clocks, run on real particles, gave the numbers at the top of this page.
02 · The history
George Gamow, and independently Ronald Gurney and Edward Condon, explain alpha decay: the alpha particle tunnels out through the barrier that holds it in the nucleus, which is why some nuclei last less than a microsecond and others billions of years.
Leroy MacColl works out how a wave packet crosses a barrier and finds that the part that gets through shows no appreciable delay (Physical Review, 1932): the first calculation of how long tunnelling takes.
Eugene Wigner, after Leonard Eisenbud, times a scattered wave by how fast its phase changes with energy. Applied to a barrier this is the phase time.
Thomas Hartman computes how long a packet takes to tunnel through a thin film and finds that, for barriers neither very thin nor very thick, the time levels off, hardly growing with the width (Journal of Applied Physics, 1962). This is the Hartman effect.
Alfred Baz’ proposes a clock that rides on the particle: give it a spin, put a weak magnetic field only where you want to time it, and read how far the spin has turned. V. F. Rybachenko applies this spin clock to tunnelling.
Markus Büttiker and Rolf Landauer time the crossing with a barrier whose height oscillates: slow oscillations the particle meets as a still barrier, fast ones as a blur, and the frequency where one turns into the other gives a traversal time (Physical Review Letters, 1982).
Markus Büttiker shows that in a barrier the spin does two things: it turns about the field, and it also tips toward it. One crossing, two spin times (Physical Review B, 1983).
Aephraim Steinberg, Paul Kwiat and Raymond Chiao time single photons through a mirror 1.1 micrometres thick: the peak of the tunnelling pulse arrives 1.47 ± 0.21 femtoseconds earlier than light that crosses the same distance without the mirror, an apparent tunnelling speed of 1.7 ± 0.2 times that of light, which the authors say is not a signal velocity (Physical Review Letters, 1993). Christian Spielmann and colleagues find the Hartman effect with light: the time through thick enough mirrors of this kind does not depend on their thickness (Physical Review Letters, 1994).
Aephraim Steinberg shows that a gentle spin clock reads a weak value of the time spent in the barrier, counted only for particles later found on the far side: a complex number whose two parts are the two spin times (Physical Review Letters, 1995).
Petrissa Eckle and colleagues build the attoclock. A laser field whose direction turns once around every cycle of the light pulls electrons out of helium atoms. An electron freed just as the field peaks would fly off at right angles to the field; the attoclock measures how far the electrons’ most likely direction is turned from that, takes off the turn that the ion’s pull is calculated to add, and reads what is left as a delay. Any delay is at most 34 attoseconds, and at most 12 when averaged over the laser’s intensity (Science, 2008).
U. Satya Sainadh and colleagues run the attoclock on atomic hydrogen, whose single electron can be simulated accurately. They put an upper limit on any tunnelling delay: the first number at the top of this page (Nature, 2019).
Ramón Ramos, David Spierings, Isabelle Racicot and Aephraim Steinberg run Baz’s spin clock for real. Rubidium-87 atoms, cooled into a Bose–Einstein condensate, meet a barrier made of laser light, and a field made by the barrier’s own light turns their spin inside the barrier only: the second number, at the lowest energy at which they saw atoms tunnel (Nature, 2020). The next year Spierings and Steinberg find that at their lowest energy, where at least nine in ten of the atoms that get through have tunnelled, the spin clock reads 0.11 ± 0.03 milliseconds less than for atoms whose energy is near the barrier’s top (Physical Review Letters, 2021).
03 · How the record gets there
How each step stands Proved in the paper Working statement Owed From earlier work
The obvious picture is a small ball that goes in, spends some time inside and comes out. Then a good clock would read that time, and every good clock would agree. They do not, and the textbook’s own definitions do not either: each is drawn from one and the same wave by a different rule, and they do not all agree.
Solve the textbook’s equation for a particle meeting a barrier. Before the barrier, the wave arriving and the wave turned back overlap. Inside, the value falls away steadily with depth and has no wavelength; a steady flow still runs through it, equal to what leaves at the far side. The squared size of the value held inside, summed across the barrier and divided by the flow coming in, is one textbook time: the dwell time. It counts every particle sent in, those turned back as well.
Another is the phase time: how the outgoing wave’s phase shifts with energy, Wigner’s delay. At a single energy it levels off once the barrier is thick, Hartman’s effect, while the share that gets through keeps falling. A packet of many energies keeps that plateau only over a range of widths. The thicker the barrier, the more of what gets through is the packet’s faster parts; any share with more energy than the barrier is high goes over the top, and once the barrier is thick enough for that share to make up most of what gets through, the time grows again.
A third is the spin clock’s. A weak field confined to the barrier turns a spin, and the turn of the spin that comes through, divided by the field’s rate of turning, reads a time. The spin also tips toward the field, and the tilt gives a second spin time. On a flat barrier like these, which is the same seen from either side, the textbook’s spin time, the turn in a vanishingly weak field divided by its rate, equals the dwell time at every energy; at the figures’ energy, half the barrier’s height, both are the hyperbolic tangent of the width (in the figures’ units), 0.96 at width 2. The figure’s field is not vanishingly weak, so its reading differs a little from that, and the spin clock at the top, which averages over its packet’s energies, reads more than both, because the packet’s faster parts get through more easily and, at this width, read longer. The phase time and the tilt give other times.
Reality’s record is a ledger. Every event in it is a posting, one entry, and time is the count of postings From earlier work. The record holds its values in registers: a register is a value at each of its positions (the textbook’s amplitude). A register is held while nothing is posted about what it holds, and it posts only at a contact, a posting that crosses into it from outside (the textbook’s measurement, or an interaction with its surroundings). What a laboratory particle and clock are in the record, a register and a contact, is owed Owed · O1.
In one case, a stronger contact leaves less of the held state exactly as it was, until at one strength it leaves none; in the other, the contact moves nothing out of where it stood: it turns the state’s two sign parts in opposite senses and reads one sign. Write X for what one posting across the register’s boundary applies, an operation that keeps lengths and undoes itself when done twice, and θ for the contact’s strength. The paper calls X the crossing; on this page X, “the crossing X” or “a crossing applied” is that operation, and “the crossing” alone is the particle’s passage through the barrier. The paper takes a contact of strength θ to act as C(θ), the form its theorem assumes, and proves what it does to a held state in two cases Proved in the paper:
C(θ) = cos θ · 1 + i sin θ · X1 leaves the state alone; X crosses it once. Two contacts in a row make one whose strength is the sum: C(θ) C(θ′) = C(θ + θ′).
Where the crossing X moves the state off where it stood, the contact divides it in two parts. In one, nothing was posted, and that part is exactly the state as it was. In the other, X was applied once. Their squared lengths are cos²θ and sin²θ. Where X keeps the state where it stood, nothing leaves where it stood: the contact turns the two parts on which X has a definite sign in opposite senses and changes the size of neither, and a state whose sign is already definite is left alone by a contact of any strength. What the theorem does not give, in either case, is the event: that one of the two parts is posted and the other is not. That is owed Owed · O1.
The attoclock and the spin clock touch the particle in different ways. The attoclock reads only the direction in which the electron leaves, long after and far away, and what that angle measures is still argued over: some attoclock experiments report delays that are not zero (Alexandra Landsman and colleagues, Optica, 2014; Nicolas Camus and colleagues, Physical Review Letters, 2017), others find none within their bounds, and every reading rests on a calculated correction for the pull of the ion. The spin clock’s weak field turns the spin inside the barrier, and the clock is read from the spin of the atoms that come through; which of these touches is the posting is part of O1, owed. In a laboratory, different contacts read different things.
The two measured numbers come from different particles and barriers (see the note under them). The clocks at the top read one computed crossing and still differ, though the stopwatch is the textbook’s calculation, not a laboratory clock, and touches nothing, and the spin clock’s field is weak: they ask the one wave different questions, and the textbook answers each. Section 04 shows what the paper’s theorem says a contact does as its strength grows.
The hypotheses. X is self-adjoint with X² = 1 on a complex inner-product space, and C(θ) = cos θ · 1 + i sin θ · X for real θ. Then C(θ) keeps lengths and C(θ)C(θ′) = C(θ + θ′); and X applied an even number of times is 1, so a state crossed twice has not been crossed.
When X moves the state. Let P be an orthogonal projector with PXP = 0 and ψ = Pψ. Then C(θ)ψ = cos θ ψ + i sin θ Xψ: the first term is cos θ times ψ itself, unchanged, with squared length cos²θ ‖ψ‖²; the second, orthogonal to it, has squared length sin²θ ‖ψ‖².
When X keeps it. Let P commute with X and ψ = Pψ. Then C(θ)ψ = eiθ Π+ψ + e−iθ Π−ψ, with Π± = ½(1 ± X): both parts stay where they were, and their lengths do not depend on θ. If Xψ = sψ with s = ±1, then C(θ)ψ = eisθψ for every θ.
Which case a register is in. On the record’s working statement the crossing X acts on a register by the sign of the register’s own moves times a flip on the channel the posting came from Working statement. Which case holds for the register itself turns on a count the paper leaves open, the number of the register’s own moves in a window of the clock; the record’s working reading puts the register in the second case: a held register stays among its held states, the set the paper calls its code, and a contact reads the sign of its moves and nothing else. On the channel that touched it, when the channel starts in a state that its flip takes to one with no overlap, the first case holds under either reading: with squared length cos²θ the channel stands where it stood and the register is exactly as it was, and with squared length sin²θ the channel is flipped and the register carries the sign. That the case turns on this count, and which case each count gives, is proved in work not yet published; the count itself is open.
The weights. Throughout, the weight of a part is its squared length. For sets of a register’s positions the paper proves it, the square read (the Born rule), from five premises, one of them owed: the position read, that a read weighs each position by the value held there and by nothing else; for parts like these, which are not sets of a register’s positions, it is a hypothesis of the paper’s section.
04 · Break it
Turn up the strength of the clock’s contact and watch how much of the state, here the clock’s own spin, is left exactly as it was, where the contact’s crossing X moves it off where it stood.
The dial shows the case where X moves the state. Where X leaves the state where it stood, the contact removes nothing: it only turns the two parts on which X has a definite sign in opposite senses. For a held register itself, the record’s working reading is this second case: the register stays among its held states, and the contact reads one sign. The weights are squared lengths, a hypothesis of the paper’s section for parts like these, and which part is posted, the event itself, is owed (O1).
05 · What it means
Recognition Science starts from a fact every question already uses, that some things can be told apart, and proves that, under each result’s stated conditions, only one answer is possible (the Overview). Its clock is the record itself: time is the count of postings. Tunnelling asks how long a crossing lasts while the particle is held and posts nothing, so Recognition Science has to answer it on the record’s own terms.
The mathematics is standard: the contact theorem is a short identity about an operation that keeps lengths and undoes itself. The textbook already finds no single true time. What the record adds is a different footing for it: time is a count of postings, and a held crossing has no count of its own. Read on the record, tunnelling says three things about reality.
No clock keeps time apart from the postings. Time runs one way because the record only grows (why). Ask how long a held crossing took and you are asking for a count it does not have of its own; every measured answer is a count kept by some clock that touched it.
Rests on The record’s reading of a clock, that time is the count of postings, from earlier work, used and not proved in this paper; that time runs one way, from earlier work (linked).
Between contacts a held register steps under the law between records: its steady parts turn, each at its own rate (a part at rest does not turn), and its histories add until a posting tells them apart; there is no rate at which it resolves on its own. What a particle held inside the barrier lacks is a posting.
Rests on The step of a held history, proved in the paper for laws on finite-dimensional spaces with every factor above zero (bounded rates; the textbook’s particle meeting a barrier is outside that scope), from a linear step and four statements about it, all premises, up to one free unit; that the unit is not zero, so that states turn at all, is not proved in the paper: the unit is adopted, by an identification, not measured. The paper’s working statement that only a contact resolves, with no rate of resolution that belongs to the body alone. That a laboratory particle is a register is O1, owed.
The contact theorem says what a touch of the paper’s form does to a held state, in the two cases the paper treats (sections 03 and 04), and a touch of strength zero leaves the state exactly as it was; the reading a clock gives is the textbook’s number.
Rests on The contact theorem, proved in the paper for a contact of the form C(θ) that the theorem assumes, in the two cases it treats, with the weights as squared lengths (a hypothesis of that section for parts that are not sets of positions); which case holds for a register is the record’s working reading; that a laboratory clock is such a contact is O1, owed.
06 · Hard questions
07 · Scope
08 · Read it
The Five Postulates of Quantum Mechanics on the Record: Theorems, Premises, and What Is Owed
The experiments and calculations, as published: