Things that turned out to be the same thing

This is the physics companion to a working guide. The guide itself is about how to tell what kind of thing a number is; this is about what happened when that question was pointed at real physics. Some of it is famous and re-explained; some of it is a connection nobody seems to have written down. The numbers quoted were all computed while writing. The first seven sections are from mid-September. Sections eight onward are the end of the month: one question about swapping particles, asked on a Sunday, that ran for a day and a half and ended with seven numerical tests on an object nobody had put a test to before.

1 · Two things that are one thing

A quark trapped in a proton, and a phase in a quantum state. Pull a quark and the force does not weaken — it stays constant, at about fifteen tonnes, until the string snaps and makes new particles. The mathematical statement of that trapping is that a certain loop quantity decays like the area the loop encloses.

Separately, in quantum mechanics, moving a particle around a closed loop in phase space leaves it with a phase equal to the enclosed area. Same structure, different subject.

They are not two facts. They are one fact in two time directions. Rotate time by a right angle — a standard mathematical move that turns oscillation into decay — and the quantum phase becomes the confinement decay exactly. Confinement is the statement that the rotated version has a nonzero coefficient.

And the same move explains tunnelling. A particle passing through a barrier it hasn't the energy to cross is doing something "forbidden" — but rotate time the same way and the forbidden path is an ordinary classical path. Forbidden is not a property of the event; it is a property of which time direction you are looking in.

A superconductor and a fractional quantum Hall state. Textbooks say a superconductor is a broken symmetry. A theorem says that is impossible — the symmetry in question is of a kind that can never break, at any temperature, exactly. What actually distinguishes a superconductor is subtler and more interesting: a residual two-fold structure that survives, and it shows up as a magnetic flux that comes in units of exactly half what you would naively expect. The factor of two is the experimental proof that electrons are pairing.

Quite separately, the fractional quantum Hall effect was reclassified the same way, by an entirely different route, and neither analysis mentioned the other. They are the same kind of order. Two of the most-studied systems in condensed-matter physics, filed for decades under the wrong heading, and the correction arrives twice independently.

2 · One theorem, two completely different jobs

There is a law saying that a certain volume in the space of possible states is conserved — it can be moved around but never compressed.

Consequence one: erasing a bit of information must release heat. Erasing halves the number of states the system could be in, which compresses that volume, which is forbidden — so the missing volume has to appear somewhere else, in the environment, as heat. The famous minimum energy cost of erasure is not an extra law of thermodynamics. It is this conservation law plus the definition of entropy.

Consequence two: you cannot focus sunlight to be hotter than the sun. A perfect concentrator can multiply sunlight about 46,000 times and no more, and the focus tops out at the sun's own surface temperature. Squeezing light into a smaller spot would compress the same volume. It would also run a heat engine backwards. Those are the same objection.

A laser does not beat this. It starts with a tiny volume, which is why a milliwatt laser — less total power than a candle — is eleven orders of magnitude brighter than sunlight.

The same shape again, elsewhere: the exponential that makes uranium sit still for four billion years is the exponential that gives a scanning tunnelling microscope its sub-atomic resolution. Tunnelling probability falls by about a factor of ten per atomic diameter. In a nucleus that means a half-life; under a microscope tip it means you can see individual atoms. One equation, opposite uses.

3 · Structure forcing something to exist

The most impressive thing this framework does is rare, and it has two clean instances seventy years and one field apart. In both, pure counting forced the existence of something nobody had seen.

1917. Einstein was balancing the rates at which atoms absorb and emit light, and found that with only the two obvious processes the answer disagrees with the known spectrum of a hot object — by a factor of nearly five in the infrared. Adding a third process, in which light stimulates an atom to emit more light, fixes it exactly. He did not propose stimulated emission as a mechanism and hope. You cannot reproduce the observed spectrum without it. The laser arrived four decades later.

1973. Kobayashi and Maskawa counted the free parameters in a matrix describing how quark types mix. The count says such a matrix can accommodate the observed left-right asymmetry only if there are at least three families of quarks. Two were known and the third was not; a fourth quark had not been found either. They concluded two entire undiscovered families of matter must exist. They were right.

4 · Which numbers are units, and which are physics

Most famous constants carry no information. The speed of light's value is a statement about how long a metre is; since 1983 it is fixed by definition and the metre is defined from it. Planck's constant does the same job for the kilogram. Avogadro's number is the size of a bag.

The physics is in the finiteness, not the value. That light has some finite top speed is a fact about the world; that it is 299,792,458 of something is a fact about the metre. Same for Planck's constant being nonzero, and for atoms being finite in number — which is what Einstein and Perrin actually established, not how many there are.

There is a clean demonstration rather than an argument. In 2019 the elementary charge became exact by definition, and a different constant that had been exact became something you measure — and it promptly moved off its old value by precisely the uncertainty that used to live in the charge. The uncertainty has to sit somewhere. Which constant holds it is a bookkeeping choice.

What survives are the numbers that cannot be scaled away. The strength of electromagnetism, about 1/137. And the strength of gravity between two protons, which is 5.9×10⁻³⁹ — a factor of 10³⁶ weaker. That ratio is the hierarchy problem, stated exactly, and nothing derives it.

And a dimensionless number decides something you would never guess. Whether a substance stays liquid all the way to absolute zero is settled by a single ratio comparing quantum jitter to the strength that holds atoms together. Helium is the only element above the line. "Quantum liquid" is a criterion, not a statement about being cold.

5 · Counting your way to a physical fact

The magic numbers. Nuclei are unusually stable at 2, 8, 20, 28, 50, 82 and 126 particles. The naive count of how many particles fit in each shell gives 2, 8, 20, 40, 70 — right three times and wrong after. Adding one interaction term reorders the shells and produces all seven, with nothing fitted. Four magic numbers from one term.

And the term is enormous in a nucleus and tiny in an atom, for a reason worth knowing: two large opposing effects nearly cancel to give the shallow well that binds the nucleus, and add in the channel that splits the shells. Neither effect is separately observable. Their sum and their difference are.

Three colours, from a binomial coefficient. Certain particles are made of three identical quarks in identical states, which the exclusion principle forbids — unless there is one more label to distinguish them. The number of ways to make a totally antisymmetric combination of three things from a palette of N is zero for N = 2, exactly one for N = 3, and four for N = 4. Only three gives a unique answer. A binomial coefficient fixes a fundamental constant of nature.

A related trick prices computation. Fermion amplitudes are determinants, boson amplitudes are permanents — the same sum with and without an alternating sign. Determinants take a few thousand operations at size 40; permanents take about 10¹³. The sign that forbids two electrons from sharing a state is the sign that makes the calculation possible. Exclusion and computability are one choice read twice.

6 · A factor of two, and twenty-four decades

When a quantity sits inside an exponent, small changes in it become enormous changes in what you observe — and this explains several things that look unrelated.

Alpha decay. Across six elements the energy released varies by a factor of 2.19. The half-lives span from 0.3 microseconds to fourteen billion years — twenty-four orders of magnitude. The correlation between the exponent and the log half-life is 0.9999. The entire spread is one exponential doing its work.

The same form, three more times. The energy gap in a superconductor, the scale at which quarks become confined, and the rate of quantum tunnelling all have the identical mathematical shape: something raised to a negative reciprocal power of a small coupling. Each is invisible to every order of the usual approximation method — which is why all three resisted for decades. The answer was not hard to compute; it was absent from the expansion everyone was using.

And a fourth joined recently. The black hole information paradox stood for forty-five years, and the resolution turned out to be additional contributions to a calculation that are suppressed by exactly the same kind of exponential. Same reason, fourth field.

7 · Where the answer is invisible to the instrument

Three places where the framework's honest answer is that it cannot get there, stated so you can see the shape of the wall.

Thirty-nine thousand patches. The oldest light in the universe comes from regions that, in the standard account, had never been in contact — about 39,000 causally disconnected patches, all at the same temperature to one part in 100,000. The equations permit any initial temperature field. They do not pick this one. Inflation's job is to make those patches connected earlier, which converts an unanswerable question into a dynamical one — the only kind of repair that counts.

We are the rounding error. For every billion antiquarks in the early universe there were a billion and one quarks. Everything else annihilated into the light that fills the sky, and all structure — galaxies, planets, us — is the residue. That number is measured two entirely different ways, from nuclear reactions three minutes after the beginning and from light released 380,000 years later, and the two agree to 0.33%. Nothing explains its value.

And one case where a mechanism exists and is excluded. The Standard Model has all the ingredients needed to produce that asymmetry. It falls short by about ten orders of magnitude. That is a better position than having no mechanism, because it says where to look — and how far.

8 · What is conserved when two particles swap?

The question that started the second half: swap two identical particles and the wavefunction either stays the same (bosons) or flips sign (fermions) — and in a flat sheet it can do something in between. Is there a conserved quantity behind that, in the sense that energy is conserved because the laws do not care what time it is?

No, and the reason is clean. Conservation laws of that kind come from continuous symmetries — things you can do a little bit of. Swapping is not a little-bit operation; there is no "half a swap". So there is no swap-current. What is conserved instead is a label, and the label is decided by geometry: by which loops two particles can trace around each other. In three dimensions, going around twice is the same as not going around at all, so only two labels exist. On a flat sheet the loops cannot be undone, and every value on a circle is allowed. The room decides the rule. That turned out to be the door, not the room.

9 · Compute it, don't recall it

Certain quantum systems have several distinct ground states when they live on a surface with holes, and the count depends on the surface. The count on a doughnut is standard. What about a Klein bottle — the one-sided surface with a twist? The honest first answer was "it depends on the theory"; the reply was check.

Checked, for one family of models built from a finite symmetry group: on the doughnut the count is the size of the group squared; on the Klein bottle it is the size of the group times the number of its elements that are their own inverse; on the projective plane it is that second number alone. So for a group with an odd number of elements — where only the identity is its own inverse — the Klein-bottle count is the plain size and the doughnut count is exactly its square, as guessed. The twist sees only the two-fold structure. And some ordered phases cannot live on a one-sided surface at all: the ones with a preferred handedness. Two mistakes were caught while computing, both by a control row in the table that came out wrong; both are in the file.

10 · Wherever something is quantised, a loop has to close

Held with conviction, and then tested. The hydrogen atom's energy levels, the units of magnetic flux through a superconductor, the allowed charges of a magnetic monopole, the strange "quantum dimensions" of certain exotic particles that come out quantised to irrational values like 2cos(π/5): in every case the allowed values are the ways a loop can be closed consistently. Even tunnelling fits — the forbidden path is a loop that closes in imaginary time.

Then two theorems about sparse networks put a caveat on it that the physical examples had hidden. A well-connected sparse network — the kind where every neighbourhood leaks strongly to the rest — contains a loop through every one of its nodes. And yet, if you put a clock at every node and let neighbours pull each other into step, such a network always ends in agreement: no winding survives, around any loop, however long. So a loop is necessary but not sufficient. A quantised value needs a loop and somewhere slow enough for the winding to sit. In the physical instances the slowness is stiffness — a superconductor's, a crystal's. In a well-connected network the connectivity overrides the stiffness and unwinds everything.

We then found the positive instance ourselves, by being wrong. See section 13.

11 · Storing information nowhere has a price

The exotic particles of section 8 protect information by storing it in no particular place: it lives in how the whole system is arranged, and no local disturbance can reach it. The natural question — does that come at a cost, is something that used to be global now local? — has a real answer, and it is a conservation law with a literature.

Non-locality is conserved. You can move it, not remove it. Prepare such a state by measurements and the non-locality reappears in the classical wiring that carries the measurement results. Build a code with better protection and it reappears as long-range connections. Put it in four dimensions and it reappears as the dimension. There is a single number measuring how much of it a scheme needs, and it is a whole number. One over-strong version of this — "every resource must be at least that number" — fell to a known four-dimensional example that needs zero active correction at all; the corrected version is a frontier, not a floor.

A memory that fixes its own errors, without anyone watching, needs two things at once: walls that are tall (an error must climb a lot of energy to become a different stored value) and walls that are few (if there are exponentially many ways over a wall, warmth finds one). Ordinary three-dimensional space gives the first and, by a theorem from last year, cannot give the second for the best-known construction. Whether any three-dimensional arrangement can is an open question in the field — posed in almost exactly our words, a year before we arrived at it.

12 · The cell nobody had looked at

So where can such a memory live? Counting the known options gave "three", and section 9 of the companion says what that word usually means. The real count used a classification: spaces sorted by how fast they grow with distance — polynomially, like a grid; exponentially, like a tree; or in between, which a famous theorem says is possible and a famous group realises. Crossed with "is it a smooth space or a bare network", that is six cells. One is structurally empty. Two have the known constructions. One — the in-between growth, as a network — had never had a memory put on it by anyone, as far as three searches could find.

So we built it. The graph in question is a family of ever-larger finite pieces of that famous group: 128 nodes, then 4,096, then 4,194,304, each node with four neighbours. And we put the simplest possible memory on it — every node either up or down, neighbours preferring to agree, the stored bit being which way the majority points — and measured the height of the wall an error has to climb, against three controls of the same size: a flat grid, a cubic grid, and a random well-connected network.

The hope had been that a graph which looks two-dimensional up close and high-dimensional from far away could have walls that are both rigid (a local property) and tall (a global one). The measurement says: its walls wiggle exactly like a flat grid's, because up close its walls are lines, and lines wiggle. Rigidity is what a cubic grid has, and it comes from walls being sheets. The hope had the sign backwards: rigidity needs more local dimension, not less. The whole gain over the flat grid — 326 against 223 — is that the smallest wall is longer, not stiffer. Not a new mechanism.

Three things came out that nobody expected. The walls on this graph are glassy: the cheapest walls come in several families that agree on only about sixty percent of the nodes, are separated by climbs of about fifty extra broken bonds, and are never reached from one another at working temperature — something no grid does. The cheapest wall is not where any of us had guessed: it runs lengthwise, splitting every layer of the graph in half with a gentle tilt from one end to the other, and its size is 11/16 of a count that the graph's structure hands you for free — exactly, at the size where we can check by hand, and within a hair at the next, where the last digits are still being computed. And the graph orders less readily than a flat grid, because it is full of short cycles — which is also the fact behind section 13.

13 · Two hundred clocks that never agreed

Put a clock on every node of the in-between graph, start them at random, and let each pull its neighbours toward its own time. The written prediction was that they would always end in step: the graph's loops are all short, and section 10 had just said short loops in a well-connected network hold nothing. Two hundred random starts. Two hundred ended out of step — every one, settled and stable, with the clocks spread around the dial. The flat grid did the same (200 of 200); the random well-connected network never did (0 of 200); and a version of our graph with its smallest triangles collapsed almost never did (2 of 200).

The stuck states, opened up: smooth everywhere, almost no neighbouring clocks more than a quarter-turn apart, and carrying whole-number windings around the graph's eight-, sixteen- and thirty-two-step cycles — dozens of small vortices, each cheap and each permanent. The prediction had the sign backwards for the same reason as section 12: short cycles in a slow graph are exactly where windings are stable, and this graph is slow — its spectrum is that of a long path, not a well-connected network. Collapse the triangles and the eight-step cycles become four-step cycles, which cannot hold a winding, and the clocks agree.

So the same graph is glassy for a two-valued memory and full of stuck states for a continuous one — unlike a grid, which has both in a tame form, and unlike a well-connected network, which has the glass and no windings. Where a host sits on that line depends on the kind of thing you store on it, which the six-cell grid of section 12 did not know how to say.

14 · The graph's own primes

A network has things that behave like primes: its closed loops that do not retrace and are not repeats of shorter loops. In a well-connected network they are counted by a formula with exactly the shape of the prime number theorem, and its error term is governed by whether the network's eigenvalues sit where a "Riemann hypothesis" for networks would put them. For a network built from a group, those loops are the group's relations.

Two old theorems then say something sharp. A group's relations reach the maximum possible density — the network prime number theorem at full strength — exactly when the group is tame in a technical sense (it has averaging sets with small boundary). And those are exactly the groups whose finite pieces fail the network Riemann hypothesis. Well-connected networks are the mirror image: no relations at all upstairs, and every eigenvalue on the line downstairs. Primes dense above, zeros off the line below; primes absent above, zeros on the line.

Measured on our in-between graph: the density of its relations approaches the maximum, but slowly — the shortfall shrinks like a stretched exponential rather than a plain one, over word lengths up to 48 — which is the tame signature, and the same fact as the graph's spectrum being a path's. Two control groups, one tame and one not, sat on either side. The number came out inside the band written down beforehand.

And "N over log N" — the shape of the prime number theorem — appeared as the size of this graph's cheapest wall. The companion's section 10 says what happened: chased, and broken at the second term. The wall's constant is a plain 11/16, and the prime side's second term is where all of its content lives.

The physics in fourteen lines

1. Confinement and a quantum phase are one theorem in two time directions.
2. The law that prices erasing a bit is the law that stops you focusing sunlight hotter than the sun.
3. Twice in a century, pure counting forced the existence of something nobody had seen — and both times it was right.
4. The Standard Model's two integer parameters were derived. Every continuous one was measured.
5. For a constant of nature, the finiteness is the physics and the value is a unit.
6. A binomial coefficient fixes the number of quark colours; the same sign that forbids two electrons sharing a state is what makes the calculation tractable.
7. Put a quantity in an exponent and a factor of two becomes twenty-four decades.
8. Four separate famous problems resisted for the same reason: the answer was invisible to the expansion everyone had.
9. A shortfall in a prefactor invites new physics. A shortfall in an exponent forecloses it.
10. Swapping particles conserves no current; it conserves a label, and the room decides the label.
11. Wherever something is quantised a loop closes — and the loop needs somewhere slow to sit, or connectivity unwinds it.
12. Non-locality is conserved: a memory that stores information nowhere pays for it somewhere, and a memory that fixes itself needs walls that are tall and few.
13. The in-between graph's walls are a flat grid's walls on a longer cut; rigidity needs more local dimension, not less. Its walls are glassy and its clocks never agree.
14. A network's relations are its primes; they are dense exactly when its spectrum fails the line, and our graph fails it slowly.