Anomalies explained · 1 of 12
The bands are there while nothing marks which path was taken, and they go the moment something does; choosing later how to read that mark only sorts dots that have already landed.
From a preprint posted 14 November 2026, not peer reviewed. It proves this account from premises it names and from earlier work, except for three of the statements it still owes (assumed, not yet proved; the third of the three lines below) and other statements it does not prove, listed under hard questions.
The number
V² + D² ≤ 1
V, the fringe contrast (the textbook’s visibility): how deep the bands are. D, the path contrast (the textbook’s distinguishability): how well the mark tells the path. The more the mark tells, the less room is left for bands. The relation is the textbook’s; the paper quotes it.
Measured with single photons, where the exit each photon took, rather than a separate mark, tells the path in part: V² + D² = 0.97 ± 0.03 (Jacques et al., Phys. Rev. Lett., 2008). A pure, lossless setup gives exactly 1.
The film · The eraser erases nothing · 3 min 39 s
This page’s experiment, computed: every dot, mark and count in it drawn from the same model as the figures.
The paradox in three lines
The puzzle
One particle at a time, sent at two slits, builds bands on a screen, as if each one took both paths. Mark which slit it took, and the bands vanish. In the eraser the mark rides on a twin particle; “erase” it later, even after the particle itself has landed, and the bands come back in sorted subsets of the dots (Kim and colleagues, Physical Review Letters, 2000).
On the record
Two paths that nothing tells apart are one history with two parts. What each path brings to a place on the screen, its value (the textbook’s amplitude), adds to the other’s, and values can cancel: where the two arrive out of step there are gaps between the bands. A full mark, a value held by something else, tells the two parts apart: they now sit at different positions of the screen and the marker taken together, so the screen counts each on its own. Counts never cancel, so the bands go, and nothing about the mark has to be written down for that. A landed dot is a posting, an entry in the record, and a posting is never erased, so a later choice at the marker cannot touch it; the choice only sorts dots already there. This uses a premise of the paper, that values add while nothing is written down, and its proof that reads of a register’s positions are squares, from five premises, one of them the owed O2; that the counts of dots are such reads is owed under O1 (line 3); section 03 marks each step as proved in the paper, a premise, owed, or from earlier work.
What comes back, and what is owed
Every number is the textbook’s own: with a full mark, no bands among the unsorted dots, however long the delay, and bands and their exact opposite in the two sorted halves, adding back to the hump; for any mark, V² + D² ≤ 1. Still owed: what a particle, a slit, a marker and a click are in the record’s own terms, and that each landing writes one click, and only one (O1); that a read weighs each position by the value held there and by nothing else (O2); and that the photon and its twin are two parts of one system, each detector acting on its own part, their values combined over one complex unit (O3).
01 · The question
Put a screen behind a wall with two narrow slits in it and send light, or electrons, at the wall one particle at a time. Each one that reaches the screen lands as a single dot. After thousands of dots the screen shows bands (physicists call them fringes): places where many land, and places between them where almost none do. Close either slit and the bands are gone: the dots pile into one broad hump.
So each particle seems to need both slits open to know where not to land. Now mark the path: put something at the slits that marks which one each particle used. The bands vanish, though both slits are open. In the delayed-choice eraser the mark is carried off by a second particle and dealt with only after the first has landed. If that later step mixes the two marks so they no longer say which slit, the bands come back, in sorted subsets of the dots.
That last step is why the eraser is famous. It looks as if a choice made later decides what happened earlier: whether the particle went one way or both ways.
02 · The history
Each step after Young’s made the puzzle sharper.
Thomas Young explains bands of light and dark as two waves that add where crests meet and cancel where a crest meets a trough (presented 1801, printed 1802). He shows them with a sunbeam split by a thin card, the bands in its shadow vanishing when either side is blocked (presented 1803, printed 1804), and describes two narrow slits in his Lectures of 1807.
Geoffrey Ingram Taylor photographs the fringes in the shadow of a needle with light so feeble that it reaches the plate as a trickle over very long exposures. The fringes are unchanged.
Clinton Davisson and Lester Germer scatter electrons off a nickel crystal and find the pattern of a wave, as Louis de Broglie had proposed for matter.
Pier Giorgio Merli, Gian Franco Missiroli and Giulio Pozzi watch on a television monitor as electrons arrive one at a time behind an electron biprism, the electron’s version of two slits. Dot by dot, the bands build up (American Journal of Physics, 1976).
John Archibald Wheeler proposes the delayed choice: decide whether to look for the bands or for the path only after the particle has passed the slits.
Marlan Scully and Kai Drühl propose the quantum eraser: mark the path, then read the mark in a way that no longer says which path (their “eraser”), and the bands return in the right subsets of the counts.
Philippe Grangier, Gérard Roger and Alain Aspect show that light from a source of single photons is caught at one output of a beam splitter or the other, far more rarely at both than any classical wave of light allows, and still makes bands when the two outputs are brought back together (Europhysics Letters, 1986).
Akira Tonomura and colleagues send electrons so sparsely that there is almost never more than one in the instrument at a time. Each lands as one dot, and together the dots make the bands (American Journal of Physics, 1989).
Yoon-Ho Kim, Rong Yu, Sergei Kulik, Yanhua Shih and Marlan Scully run the delayed-choice eraser. A laser photon that passes the slits splits, in a crystal just behind them, into two photons: the first goes on, and its twin carries the mark of which slit. The first photon is caught by a detector moved point by point across the focal plane of a lens, which shows what a distant screen would; its twin is sent by beam splitters either to one of two detectors that keep the path or to one of two behind a beam splitter that mixes the two paths so the mark no longer says which slit (the “eraser”), and the twin’s fate is settled about 8 billionths of a second after the first photon has landed (Physical Review Letters, 2000). Sorted by where the twin went, the dots show bands at one eraser detector and, at the other, the same bands shifted so that their bright bands fall in the first set’s gaps; the path-keeping detectors show none.
Vincent Jacques and colleagues run Wheeler’s experiment with single photons in an interferometer 48 metres long, the choice made at random, space-like separated from the photon’s entry: too far away for any signal at light speed to warn it. With the paths mixed they see interference of 94% contrast; with the paths kept apart they read the path with under 1% error (Science, 2007).
The same group makes the beam splitter that mixes the paths a dial, still switched in or out at random, again space-like separated from each photon’s entry, and measures both sides of the trade at each setting: the number at the top of this page.
The choice moves 144 kilometres away, between two of the Canary Islands (Xiao-song Ma and colleagues, performed 2008, Proceedings of the National Academy of Sciences, 2013), and the light’s path is stretched to satellites and back, up to 3,500 kilometres (Francesco Vedovato and colleagues, Science Advances, 2017); the results match the textbook’s numbers each time.
03 · How the record gets there
How each step stands Proved in the paper Premise Owed From earlier work
The obvious picture fails at once. If each particle went through one slit or the other, and landed as if the other slit were shut, the screen would show one slit’s dots plus the other’s: one broad hump, no bands.
The paper starts from the record. Reality’s record is a ledger: every event is a posting, one entry, and a posting is never erased From earlier work. What is sent at the slits is held in a register, what one cell of the record holds: a value at each of its positions. A register is held while nothing is posted about what it holds. That a particle, its two paths and a click on the screen are a register, its positions and a posting is owed Owed · O1.
The paper takes the step of a held register to be linear, so histories that leave no record add: the record cannot tell them apart, so it does not count them one by one Premise. Four more statements about the step Premise then fix its form, for laws with bounded rates on a finite register, up to one free unit Proved in the paper: nothing is lost, and a state that the law between records would simply shrink is turned instead, at a steady rate in proportion to how fast the law would shrink it. That the unit is not zero, so that anything turns at all, is not proved in the paper; its value is an adopted unit and an identification, not a measured value. So, to section 01’s first question: a particle whose two paths nothing tells apart is one held history with a part through each slit, and at each point of the screen the two parts’ values add.
A read is the weight a posting about some positions carries (the textbook’s probability, once the total is set to one); on the screen, how often a dot lands in each place. The paper proves that the read of a register’s positions is in proportion to the sum of the squares of the values held there Proved in the paper, given that a read weighs each position by the value held there and by nothing else Owed · O2 and that turning every value of the register together changes no total Premise, with three mild premises besides: five in all.
The adding makes the bands, and the square gives their shape. For two values a and b that carry no turn, (a + b)² = a² + b² + 2ab: each path’s own count, plus a cross term. In the double slit each value carries a turn set by the length of its path (through the wavelength; that values are complex at all rests on a premise and a working statement: hard questions), and the read squares sizes: |a + b|² = |a|² + |b|² + 2|a||b| cos φ, with φ the turn between them. The cross term is positive where the two arrive in step and negative where they arrive out of step: bands.
A mark is a value held in a second register that tells the two parts apart, with nothing posted about it. A full mark puts the parts at different positions of the two registers taken together; read at the screen alone, they are then counted separately, |a|² + |b|², with no cross term. The bands go, and the screen shows the hump.
For a mark that only partly tells the paths apart, the trade is the number at the top of the page, V² + D² ≤ 1: the textbook’s relation (Berthold-Georg Englert, Physical Review Letters, 1996), computed with squared weights; the paper quotes it, and no theorem of the paper yields the relation.
The register. A cell of the record has three on/off axes, because space has three dimensions (Pardo-Guerra and Washburn, Three dimensions from a recognition requirement, preprint, 2026), so 2 × 2 × 2 = 8 positions; a clock of eight ticks moves the values from one position to the next (Thapa and Washburn, Reciprocal cost and an eight-tick register: a conditional recognition-operator construction, Mathematics, 2026). Both come from earlier work.
The step. The four statements: the step uses only the law that carries the record from one posting to the next; it loses nothing and gains nothing; it regroups, so three steps taken as one stretch or as three come out the same; and it changes continuously with the law’s factor. Said of the record (no record for a loss to go to or a gain to come from, nothing written in between), the four are a restatement of equal strength, each carried by an identification, not a derivation. With the step linear, and for laws on a finite register whose factors are above zero and at most one (the law never grows a state), they leave exactly this step R(T) under the law T:
R(T) = exp(−icH), H = −log TT is the law between two records; c is one real number that the four statements leave free, the unit of the tick: it converts a rate of decay per recognition into a turn per recognition. Through one identification, light as the long wave of the law, c is the speed of light in cells per recognition, and earlier work adopts c = 1 as its unit (Washburn, The law of motion on the recognition ledger, preprint, 2026). The energy operator is minus the logarithm of the law, not a second choice.
The square. The three mild premises: an empty position weighs nothing, weights add over positions, and a value larger in size never weighs less. The common turn of the premise above is not a tick of the clock, which only moves values from one position to the next. The proof starts from one case. Turning every value together by an eighth of a full turn takes the two halves a and b of one complex value to (a + b)/√2 and (b − a)/√2. A balanced beam splitter applies the same map to the values in its two arms; the likeness is of form only, since the two halves are not two arms. With b = a one half is empty after the turn, and since the turn changes no total, the other half, whose value is √2 times larger, must weigh as much as both did: the read doubles when the value grows by √2. Among powers only the square does that; the value itself grows by √2 and its fourth power by 4. Among all reads, the same balance for unequal values, with an empty position weighing nothing and a value larger in size never weighing less, forces the square.
In the eraser the mark is the twin’s path. When a dot lands, its click is a posting, while the twin still holds the mark, one place for each path, and nothing is yet posted about the mark. That the first photon and its twin are two parts of one system, each detector acting on its own part, is owed Owed · O3. Later the twin is detected. That detection is a new posting, made after the dot’s, and it leaves the dot’s as it was.
Why does a read made later line up with bands? Take a full mark, and a dot at a place where the two paths arrived with values a and b. Send the twin through a beam splitter that mixes its two paths, with its two arms set in step, and read which way it leaves, a read blind to its path (the textbook calls this erasing the mark, though nothing is erased): it comes out + with weight ½|a + b|² and − with weight ½|a − b|², and the two add to |a|² + |b|², the hump at that place. So the + dots are most common where the unmarked screen has its bands, and the − dots fill the gaps; change the setting of the twin’s arms and the + bands shift, but the two piles always sit in each other’s gaps. Read the twin first instead and nothing changes, by the square and O3 (section 04): the screen alone is still the hump, and the read still only sorts.
The same square settles a smaller question, about three slits. With three paths, |a + b + c|² is |a|² + |b|² + |c|² plus one cross term for each pair, so three paths interfere only in pairs, with no term that needs all three at once (Rafael Sorkin, Modern Physics Letters A, 1994), as long as each slit sends the same value whatever the others do. Urbasi Sinha and colleagues bounded that three-path term below one hundredth of the pair terms, chiefly at the brightest place of the three-slit pattern (Science, 2010).
04 · Break it
With the read a square, and given O3 (owed), a split made at the marker leaves the weights at the screen as they were, however far away and however late it is made, so the unsorted screen never shows it. Under any other power of the value, scaled so the weights add to one, a split at the marker, however far away, can change where on the screen the dots land, with no posting passing between marker and screen.
For values whose squares add to one, the paper proves that the square is the only read taken position by position whose weights are never negative, give an empty position nothing and add to one (with three or more positions). So every other power here breaks the position read, O2: dividing by the total makes the weight of one position depend on the others, which is what lets a split at the marker, however far away, move the counts here. That is the paper’s reason for O2, a reason about outcomes, such as where a dot lands, and so about whole complex values; inside one complex value, where O2 asks that the read be a sum over its two halves (the numbers held at two positions of the register, two ticks apart, that together make it), the paper gives no reason beyond the form, and O2 stays owed.
05 · What it means
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). Some things can be told apart, comparing two readings needs both, and reading an old record is a new event (the three facts). The last of them is the eraser’s lesson: sorting the landed dots by a later read of the twin is a new event. Were the eraser the future reaching into the past, as it is often told (section 01), an entry would depend on one written after it. So Recognition Science needed the eraser answered on the record’s own terms.
Reality keeps a permanent record: nothing that happens is ever erased, and the past still exists (why). Each dot on the screen is one of its entries, so nothing done afterwards at the marker, a nanosecond later or a year later, can change it.
Rests on The record’s starting statement, that a posting is never erased, which the paper takes from earlier work; that a dot is a posting is O1, owed.
What the textbook calls a wave is a history nothing has yet told apart: its parts add. What it calls a particle, one dot in one place, is a posting: a count.
Rests on The adding, a premise; the step’s form and the square, proved in the paper from named premises; O1 and O2, owed.
Molecules of two thousand atoms still make bands (Fein and colleagues, Nature Physics, 2019); dust grains have never shown them. A large warm body is told apart all the time: each molecule that bounces off it and each photon it gives off carries some mark away, and enough of them tell its parts apart (measured loss of bands follows the count of collisions: Klaus Hornberger and colleagues, Physical Review Letters, 2003). A full mark held elsewhere takes the bands out of every read of the body alone; once the mark is posted, the bands are gone for good.
Rests on That a posting is never erased, from earlier work; the adding, a premise; the square, proved in the paper for reads of a register’s positions from named premises; and the paper’s working statement that only a contact resolves a history, with no rate of its own. For massive bodies the paper adopts a second working statement, a candidate rule with its unit open; this page does not use it. What a laboratory contact is (O1) and the position read (O2) are owed.
It is the textbook’s probability as a squared length; the arithmetic is in section 04.
Rests on The square, proved in the paper for reads of a register’s positions from five premises, O2 (owed) among them; that it is the only read of the kind set out in section 04, proved; the step’s form, proved from four premises; the fourth-power example, proved; the adding, a premise; O1 and O3, owed.
06 · Hard questions
07 · Scope
08 · Read it
The Five Postulates of Quantum Mechanics on the Record: Theorems, Premises, and What Is Owed
Version of record: Read the preprint (PDF), the institute’s library
The experiments and proposals, as published:
The theory results cited, as published: