Spin and statistics: why no two electrons share a state
Swap two electrons and their joint state flips sign, so a shared state equals its own negative: zero. Four ticks of reality’s clock flip the sign of every core state, as one full turn flips an electron’s; whether the clock is why a swap flips is not yet derived.
Not resolved. Proved in the paper: that the clock’s core (by definition, the states four ticks turn into their negatives) has four free values and is kept by each tick, and that a joint state which flips sign under a swap is zero wherever its two positions are equal. Not proved: that the clock’s sign, which falls alike on every register the record holds untouched (a working statement), is the one a full turn gives an electron. Owed: that a swap of two identical records (the textbook’s identical particles) is one full turn of one about the other, and with it what turn of a body four ticks are. Preprint, 14 November 2026, not peer reviewed (the three lines).
above zero, outwardbelow zero, inwardwhere it started
The number, on the clock’s core
S4 = −1
Tick 0 of 8: the state as it started
S is one tick of the clock. The clock’s core is, by definition, the states in which each value is minus the value opposite it: there four ticks, half the cycle, turn every value into its negative, and eight bring every value back.
Owed · not yet derived
Computed live in your browser from the clock’s rule: at each tick the value at every position moves to the position before it. The dashed marks are where the values started. At four ticks every bar points the opposite way to its mark; at eight it is back. Tick steps by hand; Pause stops the clock. In the second drawing two identical records change places; the question mark is the statement the paper owes (section 03).
The paradox in three lines
The puzzle
Particles of half-integer spin, such as electrons, protons and neutrons, exclude one another: no two of the same kind share a state. Particles of whole-number spin, such as photons, can crowd together into one (Wolfgang Pauli, Physical Review, 1940). Spin is about how a particle behaves when it is turned. Why should it decide how two of them behave together?
On the record
The record, Recognition Science’s ledger of events, has a clock of eight ticks (from earlier work): each tick moves the eight values of a register one place round. The clock’s core is the set of states whose sign four ticks, half the cycle, reverse, and eight restore. One full turn reverses the sign of a half-integer spin’s state, and two restore it; that the two reversals are one is not proved. Take the textbook’s joint state for two identical records (its identical particles). If a swap reverses its sign, exclusion follows in one line: where the two records’ positions are equal, the joint state equals minus itself, so it is zero. The joint state, the rule that it is even or odd, and the sign are the textbook’s, taken as given.
What comes back, and what is owed
Every number this paradox returns is the textbook’s own, and they are only signs: −1 after half the clock’s cycle and +1 after the whole, the textbook’s signs for a half-integer spin after one full turn and after two. The signs under a swap are the textbook’s, taken as given; given the odd sign, exclusion is exact, as in the textbook, with no violation, and none has been found. The puzzle is not resolved. Still owed, that is, not yet proved: that swapping two identical records is one full turn of one about the other (O4, the fourth of the four statements the paper owes), and with it what turn of a body in space four ticks correspond to.
01 · The question
Why should the way a particle turns decide whether two can share a state?
Every particle has a spin, a built-in amount of angular momentum (the quantity that measures turning) that does not change as it moves; for some particles it is zero. In the natural unit, spin is a whole number or a whole number and a half. Electrons, protons and neutrons have spin ½; photons have spin 1; an atom of helium-4 in its lowest state has spin 0.
The halves and the wholes behave in opposite ways in company. No two electrons share a state: in an atom they stack into shells, two to each orbital, with their spins opposite, instead of all dropping into the lowest one. That stacking gives atoms their size and the periodic table its rows. Photons do the opposite: they pile into one state, which is what the light of a laser is, and in helium-4 cooled far enough, many of the atoms gather in one state and the liquid flows without friction. Physicists call the first kind fermions and the second bosons.
In quantum mechanics the whole difference is one sign. A particle’s state is a list of numbers, its amplitudes, one for each place it could be found and each way its spin could point. Reversing every sign gives the state’s negative. On its own, the negative gives the same chances for every measurement; a sign shows only between parts of one state, or between two states added together.
Two identical particles have one joint state, with a number for each way the pair could be found. In three dimensions, swapping them either leaves it as it was (bosons) or turns it into its negative (fermions). In a plane, a swap can also multiply it by another phase, a number of size one such as i, a number whose square is −1 (Leinaas and Myrheim, 1977); such particles are called anyons. The fermions’ minus sign is what keeps two of them out of one state; the one line of algebra is in section 03. The sign explains the exclusion. What explains the sign?
Spin decides it: every half-integer spin takes the minus sign, every whole-number spin the plus. Markus Fierz in 1939 and Pauli in 1940 proved, for free particles, that relativity leaves no other choice, given two demands set out in section 02, and in 1958 the proof was carried to particles that interact; but no proof is accepted as a simple picture of why. Richard Feynman wrote in his lectures:
“It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation.”
02 · The history
Fifteen years from the rule to the theorem, and a turn of 720°.
The rule came first, as a pattern in atomic spectra; its tie to spin was proved from relativity fifteen years later, and whether a simpler reason exists is still argued over.
1924–1925
Satyendra Nath Bose counts photons as particles that can share a state, and Albert Einstein extends the count to atoms: the counting now called Bose–Einstein statistics (Zeitschrift für Physik, 1924; Sitzungsberichte der Preussischen Akademie der Wissenschaften, 1924–1925).
1925
Wolfgang Pauli reads the shells of the periodic table out of atomic spectra and states the rule: no two electrons in an atom have all their quantum numbers alike (Zeitschrift für Physik, 1925). The same year George Uhlenbeck and Samuel Goudsmit propose that the electron spins, which gives the rule’s fourth number its meaning (Naturwissenschaften, 1925).
1926
Enrico Fermi and Paul Dirac count particles that cannot share a state (Fermi–Dirac statistics), and Werner Heisenberg and Dirac, each on his own, put the difference into the sign of the joint state under a swap (Zeitschrift für Physik, 1926; Proceedings of the Royal Society A, 1926).
1928
Pascual Jordan and Eugene Wigner write the rule with operators that anticommute, so that putting a second electron into a filled state gives zero (Zeitschrift für Physik, 1928).
1939–1940
Markus Fierz (Helvetica Physica Acta, 1939) and then Pauli prove the connection for free particles from relativistic quantum field theory: half-integer spin with the minus sign, from the demand that energy be positive; whole spin with the plus sign, from the demand that measurements at places too far apart for light to connect them do not disturb one another (Physical Review, 1940).
“In conclusion we wish to state, that according to our opinion the connection between spin and statistics is one of the most important applications of the special relativity theory.”
1958
Gerhart Lüders and Bruno Zumino (Physical Review, 1958) and, separately, N. Burgoyne (Il Nuovo Cimento, 1958) prove the connection for particles that interact, from relativity, positive energy and locality; it is now a theorem of quantum field theory.
1968
David Finkelstein and Julio Rubinstein show, for particles made as twists in a field, that swapping two of them can be smoothly reshaped into turning one of them a full turn, so the swap carries the sign of a full turn (Journal of Mathematical Physics, 1968).
1975
Two groups split a beam of neutrons in a crystal interferometer, turn the spins on one path in a magnetic field, and recombine the paths: Helmut Rauch and colleagues (Physics Letters A, 1975) and Samuel Werner and colleagues (Physical Review Letters, 1975). The turned neutrons come back with the opposite sign after one full turn, and with their own sign only after two.
1978
Rauch’s group measures the turn that brings the sign back, within a few degrees of the 720° of two full turns: the measurement drawn in section 03 (Zeitschrift für Physik B, 1978).
1987
Richard Feynman, in remarks added to the printed version of his 1986 Dirac memorial lecture, draws the exchange with a belt, after an idea of Finkelstein’s: “The effect on the wave function of the exchange of two particles is the same as the effect of rotating the frame of one of them by 360° relative to the other’s frame” (Elementary Particles and the Laws of Physics, 1987).
1990–2022
One line of searches for electrons that break the rule looks for the X-rays an electron would give off on dropping into a filled shell of copper. Erik Ramberg and Gregory Snow find the chance that a new electron entering copper ends in a state the rule forbids to be below 1.7 × 10⁻²⁶ (Physics Letters B, 1990). None has been seen since. The VIP-2 experiment puts the same chance below 8.6 × 10⁻³¹ counted the 1990 way, and below 6.8 × 10⁻⁴² (a 90% bound) with a newer model of how the electrons travel through the copper (Symmetry, 2022).
03 · How the record gets there
Two minus signs: one in the clock, one under a swap.
How each step standsProved in the paperWorking statementOwedFrom earlier workA working statement is one the paper uses and does not prove; owed is what the paper names as still to be proved; earlier work is the institute’s earlier papers, cited where used.
Reality’s record is a ledger: every event is a posting, one entry, and a posting is never erased, which is what makes the ledger a record From earlier work. One cell of the record holds a register: a value at each of eight positions (the textbook’s amplitude), eight because space has three dimensions and a cell has one on/off axis for each, 2 × 2 × 2 = 8 (Pardo-Guerra and Washburn, Three dimensions from a recognition requirement, 2026; Thapa and Washburn, Reciprocal cost and an eight-tick register, Mathematics, 2026). The record’s clock moves the values: at each tick the value at every position moves to the position before it, and eight ticks bring the register back as it was From earlier work.
Four ticks turn the core into its negative
Take a register in which each value is minus the value four positions on: a, b, c, d on one half of the ring, −a, −b, −c, −d on the other. These registers are the core. Four ticks carry every value halfway round, onto the position that held its opposite, so each position now holds minus what it held: the whole register has turned into its negative. Write S for one tick; on the core, S4 = −1, and eight ticks give S8 = +1.
That four ticks reverse the core is its definition, not a theorem. What the paper proves Proved in the paper: the core has four free values, a to d, and a tick keeps a core register in the core. So two ticks, a quarter of the cycle, act there as a unit that commutes with the tick and, by the definition, squares to minus one; with it the four values are two complex values. The record identifies that unit with the number i of quantum mechanics; the identification is a working statement (the subject of Why complex numbers). That a register the record holds untouched (a held register) lies in the core, each value answered by its opposite half a cycle later, is a working statement of the record, not proved in the paper Working statement.
Four ticks, on your register. Drag any value, or focus it and use the arrow keys; the opposite position always holds minus it. Each press of Tick applies the clock’s rule to every position, and the readout compares each position with its start. Another register deals a new one; Start over returns to tick 0.
The sign of a full turn
Turn an electron, a proton or a neutron once all the way round, and its state comes back as its negative; turn it twice, and it is back. The neutron interferometers of 1975 saw this by letting a turned path meet an unturned one, and the figure below sets the 1978 measurement of where the sign returns, 716.8 ± 3.8°, beside the textbook’s 720°. On the core, four ticks give −1 and eight give +1; one full turn of a half-integer spin gives −1 and two give +1. The paper sets the two reversals side by side; that they are one is not proved. Nor does it say that four ticks are a full turn of a body in space: what turn of a body in space four ticks correspond to is not settled on the record, and the paper names it beside the owed exchange statement (O4, below) as what this paradox still waits on.
Two turns to come back. The textbook’s share of neutrons at one output of a two-path interferometer whose two paths are equal in strength and add fully at that output when nothing is turned: (1 + cos(a/2))/2, with the spin on one path turned through the angle a along the bottom, computed in your browser from zero to four full turns. After one full turn the turned path carries the sign −1 and the two paths cancel; after two they add again. The band is the 1978 measurement of the turn that brings the sign back (what the 1978 paper calls the periodicity of a spinor; the state of a spin-½ particle, such as the neutron, is what the textbook calls a spinor); the close-up scale sets it beside the textbook’s 720°.
A minus sign under a swap keeps two apart
Now take two identical records (the paper’s phrase, not yet defined on the ledger), each holding a register, and their pair state Ψ(s, t): one value for each pair of positions, s in the first record’s register and t in the second’s (the textbook’s joint state of two identical particles). Suppose a swap reverses its sign: Ψ(t, s) = −Ψ(s, t). Put t = s, and Ψ(s, s) = −Ψ(s, s), so Ψ(s, s) = 0Proved in the paper (the line of Heisenberg and Dirac, 1926): every entry with s = t is zero. Built from two registers f and g in the textbook’s way, the pair state is f(s)g(t) − g(s)f(t), and when g is f every entry is zero. There is nothing left to read: with the odd sign the two cannot hold one register. With the plus sign the two terms add, and when g is f every entry is twice f(s)f(t) instead. The table in section 04 computes all 64 entries. The textbook’s rule is wider: two electrons cannot share a state even when one is the other multiplied by a phase. On the record two ticks are the unit i, so a register and the same register turned by two ticks differ only by that unit, as the textbook’s ψ and iψ do, which the textbook counts as one state; yet on this table, built with real values, their odd pair state is not zero. To exclude that case too, the pair must be built with one unit i for both records. The owed composition statement (O3) asks this of two registers held by two recognizers (the paper’s word): their values composed over one complex unit. The paper owes it for two laboratories. The paper does not yet say which product two identical records take.
What is owed: that a swap is a full turn
So two links are firm. On the core, half the clock’s cycle gives −1, because that is what the core is; and a −1 under a swap gives exclusion, in one line of algebra. What joins them is not proved. That the clock’s sign is a half-integer spin’s is not proved, and it is still owed that swapping two identical records is one full turn of one about the other Owed · O4. If it is, and if that turn also turns the moving record’s own frame once round, a swap carries the textbook’s sign of a full turn: −1 for a half-integer spin, so two cannot share a state; +1 for a whole one, so they can. The record has yet to define its own phrase, the turn of one record about another, and to read the full turn’s sign off the clock itself it must also settle what turn of a body in space four ticks correspond to.
The owed statement, drawn. Left, an exchange: two identical records, drawn as plain discs, each go half a circle round the point between them. Right, the statement’s words taken as a path: one record stays and the other goes once round it. Run both, or step both together with the slider. In the result Finkelstein and Rubinstein proved for their twisted particles, and in Feynman’s belt (section 02), the full turn an exchange matches is one particle’s own frame turned through 360° relative to the other’s. As paths the two differ: in three dimensions, with the two never meeting, an exchange of two identical particles cannot be shrunk to no motion, while a trip once round, with the moving one’s own frame kept still, can be reshaped into two exchanges in a row and shrunk to no motion at all (Jon Magne Leinaas and Jan Myrheim, Il Nuovo Cimento B, 1977). Taken as a path alone, then, the owed statement would give every swap the plus sign, and nothing would be excluded. So if the owed phrase means a motion in space, it must also say how the moving record’s own frame turns; the paper does not yet say.
04 · Break it
Take the minus sign away, and two can share a state.
Exclusion needs three things: a pair state built from the two records’ registers; the rule that, because the records are identical, it is either even or odd under a swap (the textbook’s rule for identical particles); and the odd sign. The algebra does the rest. Below, the pair state is computed entry by entry, all 64 of them, with real values, as in section 03, so it excludes two registers only when one is the other times a plain number. The registers are drawn from the core because held registers lie there (a working statement); the core plays no part in the cancellation. It opens with both records holding the same register under the minus sign: every entry is 0. Switch the sign, or make the registers different.
Break it. Drag a bar to set a record’s register at positions 0 to 3 (positions 4 to 7 follow as their negatives), or press Make them equal. The switch sets the sign under a swap. Each place in the table is one entry, Ψ(s, t) = f(s)g(t) ± g(s)f(t); pick one to see its arithmetic. The large number is the squared length, the sum of the squares of the 64 entries.
So, for two identical records holding the same register, the minus sign is all the algebra needs. The algebra cannot supply the sign: it takes the sign as given, and so does the record today. The sign is where spin has to come in.
05 · What it means
Reality’s clock has a minus sign. Whether a swap carries it is not yet proved.
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). Ordinary matter is built from particles of half-integer spin: electrons, and the protons and neutrons of every nucleus. The textbook takes the minus sign such a particle shows after one full turn from the way turns combine in space. On the record, states are registers moved by a clock. So Recognition Science had to ask whether the clock itself has a sign of that kind.
The mathematics is standard (hard questions). What the paper adds is a reading: on the clock’s core, half the cycle gives −1 and the whole +1, by the core’s definition, and a held register lies in the core (a working statement). That this sign is the spin’s is not proved. Any turn in a plane gives −1 halfway round and +1 at the end, so the pattern alone carries no weight; what would carry it is an account of what turn of a body half the clock’s cycle is, one that also gives a whole-number spin its +1. Three things can be said. The first is the record’s own and rests on a working statement; the second and third take two things from outside the record, the textbook’s minus sign under a swap and Dyson and Lenard’s theorem.
On its core, half a cycle of reality’s clock is minus one, by definition.
The record has a minus sign of its own, by definition: on its core four ticks give exactly −1 and eight exactly +1, the signs a half-integer spin shows after one full turn and after two. That is also why two ticks, done twice, give −1 there and can serve as the unit i. Because a held register lies in the core (a working statement), four ticks reverse every held register alike, so on one held register this sign alone cannot tell a half-integer spin from a whole one; and four ticks only multiply every value of one register by −1, which the paper’s reads cannot see. Like the neutron’s sign, the clock’s could show only against a register that has not turned with it. Whether it is a spin’s sign depends on what turn of a body four ticks are, which is open.
Rests on The core’s definition, with the paper’s proof that the core has four free values and that a tick keeps it; the eight ticks, from earlier work; that a held register lies in the core, a working statement. The open steps are in the three lines.
Exclusion is a cancellation, not a force.
No force is added: nothing pushes two electrons out of one state. Their joint state, minus itself under a swap, sums to zero there, and a state of zero is no state. What keeps an atom’s electrons from all crowding into the lowest shell, and the pressure that holds a white dwarf star up against its own weight, both come from that zero: with the lowest states taken, each further electron must take a higher one.
Rests on The one line of algebra, proved. The minus sign under a swap is the textbook’s rule, proved there in relativistic quantum field theory; where it comes from on the record is still open (section 03).
Without a minus sign, bulk matter would not be stable.
The shells of section 01 are one face of it. Freeman Dyson and Andrew Lenard proved in 1967 and 1968 that matter made of electrons and nuclei is stable, its energy bounded below in proportion to the number of particles, as long as its electrons exclude one another. Dyson showed in 1967 that if every charged particle crowded instead, the energy would fall at least as fast as the number of particles to the power 7/5, and no such bound could hold (Journal of Mathematical Physics, 1967–1968).
Rests on Textbook physics and Dyson and Lenard’s theorem; on the record, the cancellation, proved, with the minus sign under a swap given.
06 · Hard questions
What an expert asks first.
Is any of this different from ordinary quantum mechanics?
No number is: the signs are −1 and +1, and given the odd sign, exclusion is exact, as in the textbook, so no experiment on this page can tell the record from the textbook. What differs is what is taken as given. The textbook takes the full-turn sign from how turns combine in space, and the swap’s sign from the spin–statistics theorem of relativistic quantum field theory (Pauli proved it in 1940 for free particles; Lüders and Zumino, and separately Burgoyne, in 1958 for particles that interact). On its clock’s core the record has, by definition, the pattern of a half-integer spin: −1 after half the cycle, +1 after the whole. It has it for every held register alike, because a held register lies in the core (a working statement). It has not proved that this pattern is the spin’s, and it still owes the swap.
Isn’t this known?
Yes, the firm steps are standard: that half the cycle of an eight-step shift reverses its core is the core’s definition; that the core has four free values and is kept by the tick, with two ticks squaring to minus one on it, is the algebra of a cyclic shift; and the cancellation is the line of 1926. That a swap of two particles carries the sign of a 360° turn of one particle’s own frame relative to the other’s has its own history of geometric accounts, from Finkelstein and Rubinstein (1968) to Feynman’s belt (1987) and Michael Berry and Jonathan Robbins (Proceedings of the Royal Society A, 1997), each tying exchange to a turn by adding something: twisted fields, a belt, a chosen way of carrying the spin along. For point particles in quantum mechanics without relativity, either sign is consistent with either spin, which is why the general proof uses relativity; what the record adds is only the reading set out in section 05.
What is not proved?
That the clock’s sign is a half-integer spin’s. What, on the record, gives a whole-number spin its +1, when four ticks give −1 to every held register alike (a held register lies in the core: a working statement). What makes two records identical: the paper does not define the phrase on the ledger. Which product two identical records take: to exclude two records whose registers differ by the unit i, it must be built over one unit i, as O3 asks of two registers held by two recognizers. Owed: that swapping two identical records is one full turn of one about the other (O4), and with it what turn of a body four ticks are; the ledger has yet to define a turn of one record about another. The eight ticks come from earlier work. So the connection between spin and statistics is not derived here, and the paper marks this paradox not resolved.
07 · Scope
Not claimed by the paper.
No derivation of the connection between spin and statistics (hard questions).
No statement that four ticks are a full turn of a body in space, and no proof that the clock’s sign is a half-integer spin’s.
No violation of exclusion and no number different from the textbook’s (the third line).
08 · Read it
The paper.
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. Spin and statistics is the last of its twelve paradoxes; the clock’s core is in its section on the first postulate.
W. Pauli, Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren, Zeitschrift für Physik 31, 765–783 (1925).
G. E. Uhlenbeck and S. Goudsmit, Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung bezüglich des inneren Verhaltens jedes einzelnen Elektrons, Die Naturwissenschaften 13, 953–954 (1925).
S. N. Bose, Plancks Gesetz und Lichtquantenhypothese, Zeitschrift für Physik 26, 178–181 (1924).
A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzungsberichte der Preussischen Akademie der Wissenschaften (1924), 261–267; Zweite Abhandlung (1925), 3–14.
E. Fermi, Zur Quantelung des idealen einatomigen Gases, Zeitschrift für Physik 36, 902–912 (1926).
P. A. M. Dirac, On the theory of quantum mechanics, Proceedings of the Royal Society A 112, 661–677 (1926).
W. Heisenberg, Mehrkörperproblem und Resonanz in der Quantenmechanik, Zeitschrift für Physik 38, 411–426 (1926).
P. Jordan and E. Wigner, Über das Paulische Äquivalenzverbot, Zeitschrift für Physik 47, 631–651 (1928).
M. Fierz, Über die relativistische Theorie kräftefreier Teilchen mit beliebigem Spin, Helvetica Physica Acta 12, 3–37 (1939).
W. Pauli, The connection between spin and statistics, Physical Review 58, 716–722 (1940).
G. Lüders and B. Zumino, Connection between spin and statistics, Physical Review 110, 1450–1453 (1958); N. Burgoyne, On the connection of spin with statistics, Il Nuovo Cimento 8, 607–609 (1958).
F. J. Dyson and A. Lenard, Stability of matter. I, Journal of Mathematical Physics 8, 423–434 (1967); Stability of matter. II, Journal of Mathematical Physics 9, 698–711 (1968); F. J. Dyson, Ground-state energy of a finite system of charged particles, Journal of Mathematical Physics 8, 1538–1545 (1967).
D. Finkelstein and J. Rubinstein, Connection between spin, statistics, and kinks, Journal of Mathematical Physics 9, 1762–1779 (1968).
H. Rauch, A. Zeilinger, G. Badurek, A. Wilfing, W. Bauspiess and U. Bonse, Verification of coherent spinor rotation of fermions, Physics Letters A 54, 425–427 (1975).
S. A. Werner, R. Colella, A. W. Overhauser and C. F. Eagen, Observation of the phase shift of a neutron due to precession in a magnetic field, Physical Review Letters 35, 1053–1055 (1975).
J. M. Leinaas and J. Myrheim, On the theory of identical particles, Il Nuovo Cimento B 37, 1–23 (1977).
H. Rauch, A. Wilfing, W. Bauspiess and U. Bonse, Precise determination of the 4π-periodicity factor of a spinor wave function, Zeitschrift für Physik B 29, 281–284 (1978).
R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, vol. III, chapter 4, Identical particles (Addison-Wesley, 1965).
R. P. Feynman, The reason for antiparticles, in R. P. Feynman and S. Weinberg, Elementary Particles and the Laws of Physics: The 1986 Dirac Memorial Lectures (Cambridge University Press, 1987).
E. Ramberg and G. A. Snow, Experimental limit on a small violation of the Pauli principle, Physics Letters B 238, 438–441 (1990).
M. V. Berry and J. M. Robbins, Indistinguishability for quantum particles: spin, statistics and the geometric phase, Proceedings of the Royal Society A 453, 1771–1790 (1997).
F. Napolitano et al. (VIP-2 collaboration), Testing the Pauli exclusion principle with the VIP-2 experiment, Symmetry 14, 893 (2022).
S. Pardo-Guerra and J. Washburn, Three dimensions from a recognition requirement, preprint (2026), accepted for publication in Entropy.
A. Thapa and J. Washburn, Reciprocal cost and an eight-tick register: a conditional recognition-operator construction, Mathematics 14(17), 3134 (2026).