A property of the drawing, mistaken for one of the world

“Field lines are things that exist in space.”

The same dipole drawn twice: once as a vector at every point of a grid, which is what the field is, and once as lines, which is a choice about which of those vectors to join up. How many lines there are is set by whoever draws them. The essay keeps the convention and states the three facts it gets right and the ones it hides.

Tested in Field lines are a choice, not a discovery · the the field concept reading path

“A particle has a definite position and momentum; measurement is what cannot get both.”

The bound is σx·σk = 0.500, read off two brackets in two panels through two separate axes, and it is a property of the wave rather than of the apparatus: a short packet is *made of* a spread of wavelengths. What the relation forbids is the state, not the measurement of it.

Tested in Sharpness has to be paid for · the uncertainty reading path

“The surface of water is a thin elastic film.”

A film stores more energy the further it is stretched, because stretching it strains bonds that are already there. A liquid surface costs the same energy for every square millimetre created, however much has been created already — the molecules just rearrange. The figure makes the distinction usable: the energy of a drop is proportional to its AREA, with no reference to any unstretched state.

Tested in The skin that is not a skin · the surface tension reading path

“Viscosity is friction between sliding layers of fluid.”

There are no layers — the division is drawn by whoever is describing the flow. What the fluid does is carry momentum across the flow by molecular motion, and the constant that results has units of metres squared per second, which is a DIFFUSIVITY and not a friction coefficient. The figure draws a suddenly started sheet spreading its motion as an error function of y/√(4νt), which is the diffusion solution and not a friction one.

Tested in Momentum going sideways · the viscosity reading path

“A superfluid is a liquid with two components mixed together.”

The two-fluid model is a bookkeeping device that predicts correctly and describes nothing physical. There is no way to take a sample and separate the normal from the super part, and no atom is a member of either — it is one fluid, and the split is a decomposition of its behaviour into two responses, in the same way a wave can be resolved into components that are not separately present.

Tested in The liquid that will not slow down · the superfluidity reading path

“Light bends because gravity pulls on the photon's mass.”

The bending follows from the acceleration alone, with no property of the light entering: the far wall moves while the pulse is in flight, so the pulse lands low. The drop is ½a(L/c)², which contains nothing about the photon. What the photon's own nature does decide is the size of the answer near a real mass, where the equivalence-principle calculation gets exactly half of it.

Tested in The floor that cannot be told from gravity · the equivalence principle reading path

“Something happens to the light on the way up.”

Nothing does. Emitted and received frequency are both measured locally, each by a clock at rest beside the apparatus, and the disagreement is between the two clocks rather than in the beam between them. The figure prints the fractional rate difference as nanoseconds per day, which is a statement about clocks and does not mention light at all.

Tested in The clock that runs slow lower down · the gravitational redshift reading path

“Something goes wrong at the horizon — physics breaks down there.”

The Schwarzschild coordinates break down there; the geometry does not. Every scalar built from the curvature is finite at r = rs, and the tidal stretching an infalling observer feels goes as 1/M², so at a large enough mass it is undetectable. What is infinite at the horizon is a coordinate, and a change of coordinates removes it — which is the definition of an artefact of the drawing.

Tested in The surface that only lets things in · the horizons reading path

“Time stops at the horizon.”

The quantity that diverges is the Schwarzschild time coordinate, which is a distant bookkeeper's label. The faller's own clock records a finite interval — 4.8 milliseconds from ten Schwarzschild radii for a ten-solar-mass hole, computed here two independent ways that agree to a part in a thousand — and nothing marks the crossing on it.

Tested in Two clocks that disagree about the fall · the horizons reading path

“A gravitational wave stretches space, so the ruler measuring it stretches too and nothing can be detected.”

The masses in the figure are free — nothing holds them — and their separation changes. A ruler is not free: it is held together by electromagnetic forces with their own length scale, and it resists. The measurement compares a light travel time against an atomic clock, and neither is carried along by the wave.

Tested in The wave that stretches one way and squeezes the other · the gravitational waves reading path

“Field lines are dragged along by the fluid.”

Field lines are a drawing convention with no identity from one instant to the next — there is no experiment that follows a particular line. What is conserved is the flux through each comoving loop, which is a number, and the figure counts it: seven lines through a loop of radius R and seven through the same loop at R/2. The dragging picture gives the right answers and is not a statement about anything real.

Tested in The field that cannot get out · the flux freezing reading path

“The Planck length is the pixel size of the universe — space is made of little blocks that size.”

Nothing in the argument produces a lattice. What it produces is a length at which the two descriptions being combined stop being separable, since localising a mass to that scale needs an energy whose own Schwarzschild radius exceeds it. That is a statement about where two theories overlap, not about a structure, and no experiment has probed within fifteen orders of magnitude of it.

Tested in Where every model runs out at once · the planck scale reading path

“A shadow's edge is where the light stops.”

There is no edge. The intensity rises through fringes outside the geometrical boundary — the first of them 1.37 times brighter than with no obstacle at all — and fades smoothly to darkness inside it with no fringes whatever. The asymmetry between the two sides is not a property of the obstacle; it is the difference between adding a chord to a spiral that is still turning and one that has already wound up.

Tested in The spiral that says how much light arrives · the huygens reading path

“Circularly polarised light is a third kind of light, distinct from linear.”

It is two linear components of equal amplitude with a quarter cycle between them, and the figures trace the field's path over a cycle to show it. A quarter-wave plate absorbs nothing and rejects nothing — the two components leave with exactly the amplitudes they arrived with — and the only thing that has changed is how far one has been delayed against the other. Which description is 'the' state of the light depends on the basis chosen to describe it, and no measurement settles that.

Tested in The crystal that answers twice · the polarisation reading path

“Field lines are cut by a moving conductor, and cutting them is what generates the voltage.”

Field lines have no identity from one instant to the next — nothing in Maxwell's equations labels a line so that it can be said to be the same line later — so 'how many were cut' is a question about a drawing. The rule of thumb works because it happens to compute the flux change in simple geometries, and it gives the wrong answer for the disc, where the lines and the disc are both axially symmetric and no counting distinguishes turning from standing still.

Tested in The rule that is two laws wearing one coat · the induction reading path

“Because a photograph does not show contraction, length contraction is not real.”

Length contraction is a comparison of two positions at one time in one frame, and a photograph compares two positions at two different times. They are different measurements of different things and both are correct. A ruler laid alongside the cube at β = 0.9 reads 43.6 per cent of the proper length, and the camera records a rotation, and nothing has to be reconciled because the two are not answers to the same question.

Tested in The contraction no photograph shows · the length contraction reading path

“The rotation is an artefact of choosing to write the transformation one way rather than another.”

A decomposition can be written as a boost followed by a rotation or as a rotation followed by a boost, and the two rotations have the same angle. What is not a choice is that a *pure* boost cannot reproduce the product at all — the composed matrix is not symmetric, and every pure boost matrix is. The rotation is present in the transformation before anybody decides how to factor it.

Tested in The turn that two pushes leave behind · the velocity addition reading path

“Light slows down near a massive body.”

Its local speed is c, measured by any local observer with their own clock and their own ruler, everywhere and always. What is delayed is the arrival, computed in the coordinates of a distant observer, and the delay is a statement about how those coordinates relate to local clocks and rulers rather than about the light. The two accounts give the same measured echo time, and only one of them is about a speed.

Tested in The delay that is not a bend · the light deflection reading path

“A stationary path is a single line, so light travels along a line.”

The stationary point is where a *band* of paths agrees in phase, and the width of that band is what actually carries the light — of order the square root of the wavelength times the distance, which is millimetres for visible light over a kilometre. Draw the ray as a line and that width is invisible; block a zone of it a metre wide and the intensity at the detector changes, which is the measurement that says the line was a summary.

Tested in The path that takes the longest time · the fermat reading path

“The bound charges are on the surface, so the interior is unaffected.”

The interior is where the effect is. Surface charge of density P·n̂ makes a field throughout the volume, uniform for an ellipsoid, and that field is what the interior molecules actually sit in. The picture of charge accumulating at the edges invites the conclusion that something happened at the edges; what happened at the edges is a boundary condition, and its consequence is everywhere inside.

Tested in The field the matter takes away · the dielectrics reading path

“Buoyancy is an upward force the fluid exerts.”

There is no upward-pointing agent anywhere. Every pressure force on a submerged body points *inward*, perpendicular to the surface, and the upward resultant appears only because the inward pushes on the bottom are stronger than those on the top. Drawing buoyancy as a single upward arrow at the centre of the displaced volume is a summary of an integral, and the summary fails precisely when the integral does — at a sealed face, at a free surface, and at a body straddling two fluids.

Tested in The block the water does not lift · the buoyancy reading path

“The length of a compound pendulum is the distance from the pivot to the bob.”

There is no bob and no such length. What has a period is the whole body, and the length that reproduces its period as a simple pendulum is (k² + h²)/h — larger than the pivot-to-centre distance, smaller than the bar, and not a distance between any two parts of the object. The figures plot it against pivot position and it has a minimum: no compound pendulum can be equivalent to a simple one shorter than twice its radius of gyration.

Tested in The length nobody has to measure · the pendulum reading path

“The light travels in the core.”

Some of it does. The fraction depends on how far the mode is above cutoff, and near the point where a mode is about to be lost most of its power is in the cladding — which is why a fibre's bend loss, its sensitivity to what is painted on the outside, and every evanescent-field sensor exist at all. The ray drawing shows a line inside the core because a ray has no width, and the width is where the physics is.

Tested in The channel with no walls · the guided waves reading path

“A spin pointing up and a spin pointing down are opposite states.”

They are opposite *directions* and orthogonal *states*, and the two words mean different things. The probability of getting the first answer when a spin prepared along one axis is measured along another is cos²(θ/2), so orthogonality — never giving that answer — arrives at 180 degrees rather than at 90. The half-angle is the whole difference between a spin and an arrow, and the figures show the classical cosine recovered exactly as the average of the two outcomes.

Tested in The angular momentum that is not a rotation · the spin reading path

“The photon must have gone down one arm or the other.”

If it had, there would be no dark port. The two amplitudes for reaching that port cancel exactly — to seven parts in 10¹⁶ across the whole phase sweep the figures compute — and a cancellation between two contributions is only available if both are present. The interesting outcome, the click at the port that should never click, is caused by an object in an arm the photon demonstrably did not take, because taking it means absorption.

Tested in The measurement that never touched it · the measurement reading path

“The stored angular momentum must be largest where the fields are strongest.”

It does not depend on either field at the ring. The figures compute it for ring radii from five centimetres to eighty, over which the electric field at the ring falls by a factor of 256 and the magnetic field at the ring is exactly zero throughout, because the ring is outside the solenoid. The stored quantity is the same at every radius to the last digit: it is qΦ/2π, and Φ is the flux the ring *encloses* rather than anything present where the ring is.

Tested in The angular momentum that is in nothing at all · the field energy reading path

“The working fluid is at the reservoir temperatures.”

It cannot be, or no heat would flow. Every p–V diagram of a Carnot cycle draws isotherms labelled Th and Tc, and in a running engine the fluid is below Th while absorbing and above Tc while rejecting, by whatever margin the heat flow requires. The figures solve for those internal temperatures and the whole difference between the ceiling and the achievable efficiency lives in them.

Tested in The engine that has to finish · the heat engines reading path

“A gravitational lens produces a magnified image of the source.”

It produces two images and, at perfect alignment, a ring — and the word magnification names an increase in total brightness rather than in resolution. The figures compute both image positions against alignment and their product is exactly −1 in units of the Einstein radius, so the two are always on opposite sides and always one inside the ring radius. Neither image resembles the source's shape more closely than the lens's own distortion permits.

Tested in The lens with no focal length · the light deflection reading path

“A material has a refractive index, in the way it has a density.”

It has a complex function of frequency, of which the index is one part evaluated at one place. The figure of n and κ across a resonance shows the index rising, falling, and passing below one within a few per cent of the resonant frequency, while κ rises from nothing to the largest value on the axis. Which of those numbers gets called 'the index' is a decision about which frequency is interesting.

Tested in The constant that depends on how fast it is asked · the dielectrics reading path

“Entanglement is a correlation, like two gloves posted in separate boxes.”

The gloves are exactly what a pre-agreed list of answers is, and they reproduce every one-sided statistic the entangled state gives — each analyser alone sees a fair coin either way. The figures separate the two by what happens when the analysers are turned relative to each other: the shape of the correlation is a cosine where the gloves give a sawtooth, and at three particles the gloves are forced into a prediction the state contradicts with certainty.

Tested in The disagreement that one run settles · the entanglement reading path

“A spectral line is split by the instrument, and a better instrument would show it split further still.”

It would not. The two sodium D lines stop separating: at a resolving power of a few thousand they are two clean peaks, and every increase past that leaves them where they are. The figure resolves them with 1,200 illuminated rulings and shows the dip; the splitting is a property of the atom's levels, and the instrument's job is only to be good enough to see a gap that is already there.

Tested in The line that is really two · the atomic spectra reading path

“A vortex is a whirlpool: fluid going round a hole.”

The hole is a consequence rather than a cause. The circulation is fixed by the winding of the phase, at h/m, and the core exists because a velocity going as 1/r would need infinite energy at the centre — so the fluid density falls to zero over about an angstrom. The figure of speed against radius shows 2πr·v equal to the quantum at every radius drawn, which is a statement about a loop's winding and not about a hole's size.

Tested in The whirlpool that comes in one size · the superfluidity reading path

“The centre-of-mass frame is a mathematical convenience with no physical content.”

It is the frame in which the products are made, and its speed decides how much of the beam's energy they have to keep. The figures compute the same collision in both frames and get the same invariant mass; what changes is how much energy is spent carrying the wreckage forward. In a fixed-target collision at the LHC's beam energy the centre of mass moves at 0.999993 c, and everything made in it has to move at very nearly that.

Tested in The collision that wastes most of the energy · the relativistic dynamics reading path

“A photon has a relativistic mass E/c², so light is heavy after all.”

The quantity E/c² is not a mass in any invariant sense; it is an energy in mass units, and it depends on the observer. Two photons have E/c² each and an invariant mass between them of anything from zero to 2E/c² depending on their angle, which no per-particle attribution can reproduce. What appears on a set of scales is the system's invariant mass, and it is the same number for every observer.

Tested in The box of light that weighs something · the Mass-energy reading path

“Field lines break and reconnect.”

A field line is a curve tangent to a vector field, and which curve is which is a labelling. What is physically defined is the connectivity of the plasma — whether a given parcel of fluid is magnetically joined to another — and that is what freezing preserves and reconnection changes. The figure draws the four quadrants of a neutral point as four flux systems for exactly that reason: the quadrants are the objects, and the lines drawn inside them are a convention.

Tested in The knot the field cannot untie · the flux freezing reading path

“A dark-field micrograph shows the specimen's own structure, more clearly.”

It shows the specimen's spectrum with a term removed. The figure computes both: with every order admitted the image repeats with the grating's period, and with only the first pair admitted it repeats with half of it — the same specimen, twice as many lines, and the period in the photograph is a property of the filter.

Tested in The image that is a diffraction pattern twice · the imaging reading path

“A constraint force does no work, so it carries no energy and can be ignored.”

It does no work on the constrained motion, which is a statement about the direction it points and not about its size. The same reaction that does no work on a bead sliding on a wire is what the wire has to be strong enough to exert, is what determines where the bead leaves, and is what a rotating hoop supplies at mω²R however fast it is spun. A quantity that is invisible in the energy budget is not thereby small.

Tested in The force a coordinate cannot see · the least action reading path

“The safety parabola is a boundary in space, so something changes physically as it is crossed.”

Nothing at the boundary is different from anywhere else. What changes is the number of solutions to an equation about the *launcher*: two inside, one on it, none outside. The same point in space is inside the boundary for a faster gun and outside it for a slower one, which is a statement about guns and not about that place.

Tested in Everywhere a throw can reach · the projectile reading path

“The wavelength is what carries across a boundary unchanged, since the wave is the same wave.”

The wavelength is the quantity in the table that changes most. Frequency crosses unchanged, because the boundary does not move and cannot invent a beat; the tangential wavenumber crosses unchanged, because the boundary is uniform along itself. The wavelength, the phase speed, the normal component and the direction all change by the ratio of the indices — 633 nm becomes 422 nm on entering glass, and it is the colour that is preserved rather than the length.

Tested in The law that only asks about one component · the refraction reading path

“The slab is pulled in by the field between the plates.”

The field between the plates is perpendicular to the direction the slab moves, so its force on the induced charges has no component along that direction at all. Inside the idealised model there is no force anywhere. The pull comes entirely from the fringing field at the plate edges, where the lines bow outward and acquire a component along the plates — the part of the picture textbooks label 'edge effects, neglected'.

Tested in The force that lives where the model is not · the dielectrics reading path

“The daily and annual waves are different phenomena, since one is felt in centimetres and the other in metres.”

They are one solution with one number changed. The penetration depth goes as the square root of the period, so the annual wave reaches √365 = 19.1 times deeper than the daily one — 2.24 metres against 11.7 centimetres in the same soil. A single equation with a single diffusivity produces a spade's depth and a cellar's, and the ratio between them is the square root of the number of days in a year.

Tested in The summer that reaches the cellar in December · the diffusion reading path

“Each observer sees the other's clock running slow, which is a contradiction unless one of them is really moving.”

What each *sees* is a Doppler factor: k on recession and 1/k on approach, and those are not equal and not reciprocal-looking. What each infers after subtracting light travel time is a dilation of γ = (k + 1/k)/2, and that is symmetric. The apparent contradiction lives entirely in conflating a raw observation with a corrected inference, and the k factor makes the difference explicit because it is the raw one.

Tested in Everything from an exchange of pulses · the doppler reading path

“The circle traced by the axis of a released top is the precession circle.”

The traced path is drawn here by integrating the equations from rest, and it is a chain of cusps rather than a circle: the axis falls, turns sideways, rises to the tilt it started at, stops, and falls again. The circle is the average of that path, and the average is the slow root of the quadratic. A top let go with a backward flick traces loops instead, and one launched at exactly the slow rate traces the circle — three different paths with the same top, the same field and the same tilt.

Tested in The top that nods before it settles · the rotation reading path

“Magnetisation rises smoothly with the applied field.”

It rises in jumps. The staircase drawn here has fourteen hundred steps in the length of crystal simulated, and the smooth curve is that staircase drawn small. Barkhausen heard the jumps in 1919 with a coil and an amplifier, which is why the phenomenon is a century older than any theory of it.

Tested in The curve that is really a staircase · the magnetisation reading path

“An asymmetric spectral line indicates an instrumental defect or an unresolved second line.”

It indicates two interfering paths, and it has a definite shape with one parameter. The figures fit that shape and locate the zero at minus the parameter and the peak at its reciprocal, which no instrumental broadening produces and no pair of overlapping peaks reproduces — a sum of two peaks cannot go to zero.

Tested in The resonance with a zero in it · the resonance reading path

“The moving observer's axes are longer, so their units are bigger.”

The unit tick on a boosted axis has an interval of exactly one, checked at forty speeds. Its length on the page is √((1+β²)/(1−β²)), which grows without bound and is a property of the sheet of paper rather than of spacetime. It is not even the Lorentz factor, which is a different curve on the same axes.

Tested in The diagram a ruler cannot read · the spacetime diagram reading path

“A gravitational wave stretches the arm, so the light takes longer to cross it.”

The wavelength of the light is stretched by the same factor at the same instant, so a picture in which one thing changes and the other does not proves nothing. What the calculation gives is a difference in the light's round-trip time, and the response is an average of the wave over that whole trip — which is why it has zeros at frequencies where the average is nothing.

Tested in What the instrument actually hears · the gravitational waves reading path

“The hole radiates like a hot sphere the size of its horizon.”

The peak wavelength emitted is about sixteen times the horizon radius, at every mass, checked across fourteen decades. A body radiating waves longer than itself is not a hot surface, and the black-body formula with the horizon area in it is bookkeeping that happens to give the right answer to within factors the grey-body corrections supply.

Tested in The hole that outlives everything and then does not · the horizons reading path

“A wave sent into a plasma at an angle to the field travels in the direction it was sent.”

Its crests do. Its energy does not. Differentiating the phase speed v_A cos θ with respect to angle gives a group velocity of exactly (v_A, 0) for every θ — checked here at six angles to seven decimal places — so the energy runs along the field whatever direction the wavevector points. A polar diagram of phase speed is a picture of crests, and reading it as a picture of where the wave goes is reading the wrong surface.

Tested in The wave that does not know what the gas is made of · the flux freezing reading path

“An antenna's near field is set by how big the antenna is.”

It is set by the frequency. The three terms cross at kr = 1, which is r = λ/2π, and no dimension of the source appears in that condition. A one-millimetre loop and a one-metre loop driven at 13.56 MHz have the same radian sphere of 3.52 metres. What the antenna's size decides is how much is radiated, through a radiation resistance going as the square of the length — a separate quantity, drawn separately here.

Tested in The distance where a field changes its mind · the retardation reading path

“More elements in an array make a narrower beam.”

The aperture makes the beam. The half-power width is 0.886 λ/L with L the length of the array, measured here off the computed pattern across two and a half decades, and the number of elements does not appear. Adding elements within a fixed length changes nothing about the main beam — it only pushes the grating lobes out of real angles. An eight-element array and an eighty-element array of the same length have the same resolution.

Tested in When the source is not heard all at once · the retardation reading path

“Lee-wave crests sit above and downwind of the ridge that made them.”

They lean upstream as they rise, and the tilt is the signature of energy travelling upward. The figure locates the maximum displacement a quarter of a vertical wavelength above the ground and finds it well upstream of the ridge crest. A drawing with the crests leaning downwind describes energy propagating into the ground, which is the one direction it cannot go.

Tested in The wave that is required to stand still · the stratification reading path

“Inertial circles drift because the Earth's rotation rate varies with latitude, and the drift is the physics.”

The drift is real and it is smaller than what a careless integrator supplies. Explicit Euler on a rotation gains amplitude every step and produced a smooth spiral three quarters of a radius across, pointing north — the wrong way — with nothing in the picture to say so. The figure uses fourth-order integration, checks that the speed is conserved to a part in ten⁸, and runs the identical integrator with β set to zero as a control.

Tested in The deflection that closes on itself · the circular motion reading path

“The endpoint of the spectrum is where the electron carries all the energy, so the spectrum reaches it.”

It approaches it and essentially nothing gets there. The density of decays falls as the square of the energy left over, so the fraction within one electronvolt of tritium's 18.6-keV endpoint is 2.9 × 10⁻¹³ — computed here as an integral rather than quoted. The measurement that matters is made where almost nothing is happening, which is why the apparatus is enormous.

Tested in The energy that did not all arrive · the decay reading path

“Charge conservation means charge cannot be created or destroyed.”

It means something stronger and more local: charge cannot disappear here and appear there, even simultaneously, because what the equation constrains is a flux through a boundary. The spacetime box drawn here makes the difference explicit — a global bookkeeping law would allow a charge to vanish in one place and reappear a metre away, and continuity forbids it while a mere conservation of the total would not.

Tested in Charge and current are one thing · the field transformation reading path

“The electric and magnetic fields are two fields that happen to be related.”

They are six components of one object, and the boost acts on the object. Written as a four-by-four antisymmetric array the sixteen slots hold six independent numbers — exactly the three of E and three of cB — and the transformation mixes the array's top row into its lower block. The array is verified antisymmetric entry by entry here, before and after the boost, which is what forces the count to be six rather than eight.

Tested in Six numbers, one object · the field transformation reading path

“Radiation pressure is an optical effect, and the acoustic version is an analogy.”

It is the same statement with a different speed in the denominator, and the acoustic version is by far the larger. A watt of light gives 3.3 nanonewtons; a watt of sound absorbed in water gives 675 micronewtons, because the speed of sound is 200,000 times smaller. That is why acoustic power is measured by weighing and optical power is not.

Tested in Where the loudness goes · the attenuation reading path

“There must be some force between spins that the magnetic one leaves out.”

There is no force between spins in any of the calculations here. The two-site Hamiltonian diagonalised in the second figure contains a hopping amplitude and a Coulomb repulsion and nothing else, and its singlet lies below its triplet by 4t²/U. What the spins decide is which spatial state two electrons are permitted, and the two permitted states cost different amounts of electrostatic energy — 2.5 per cent apart in the third figure, which integrates the repulsion over both.

Tested in What holds a magnet together is not magnetism · the magnetisation reading path

“A domain wall is the boundary between two domains.”

It is a region, and on the scale of the domains it is not thin. Iron's wall turns over across sixty-six nanometres — about two hundred and fifty atomic planes — and permalloy's across nine hundred and twenty-five, which is wider than many of the particles such material is made into. The profiles are drawn here on one scale of nanometres, and on that scale the wall is most of the picture.

Tested in The first length that belongs to the substance · the magnetisation reading path

“A mode occupies a definite region of an aperture and a definite range of angles.”

Only the area of a phase-space cell is fixed, not its shape. The hero figure tiles one étendue four cells wide and two high; two wide and four high would be the same count of the same modes, and a beam tightly confined in position with a broad angular spread is the same single mode as its Fourier partner. What an optical system does is change a cell's shape at constant area.

Tested in The invariant that is a count · the etendue reading path

“A mirror can raise sunlight's temperature.”

A mirror changes no photon's energy, so the temperature its spectrum belongs to is the same before and after — the flat line in the first figure. What concentration raises is the flux, and therefore the temperature a body placed in the beam settles at. The two coincide only at the geometric limit, where the concentrator has reproduced the source's own radiance and can go no further, which the figure checks by recovering the Sun's temperature.

Tested in The work a diluted beam will not do · the etendue reading path

“Light leaks from one guide into its neighbour, so a coupler transfers whatever fraction the leakage allows.”

Nothing leaks. Two identical guides side by side have two modes, symmetric and antisymmetric, and neither of them lives in either guide — the antisymmetric one has a node exactly between them, checked here to be exactly zero. Light launched into one guide is those two modes in equal parts, they travel at slightly different speeds, and what looks like a transfer is the beat. The two powers sum to one at every distance to twelve figures.

Tested in Two tails that swap everything · the guided waves reading path

“A null result in a clock comparison says nothing, since nothing was seen.”

A null result is a number. A synthetic three-year record of weekly comparisons at 5 × 10⁻¹⁶ each, with drift fitted out, returns Δβ = (−2.6 ± 3.4) × 10⁻⁷; daily comparisons at 10⁻¹⁷ for a year would bound it at 4.5 × 10⁻⁹. Each factor of ten in clock quality tightens the bound tenfold, and each factor of ten in duration only threefold.

Tested in The clocks that must all slow together · the gravitational redshift reading path