Underdetermination — where it appears
Named by 55 essays across 9 fields — each of them below, with the objects they name alongside it.
The force field is not in the spectrum
Water has three vibrational frequencies and its force field has four constants, so the spectrum cannot determine the field. A whole family of quite different fields reproduces all three frequencies exactly, and only a second isotopologue can tell them apart.
A bond length out of a spectrum
A linear triatomic has two bond lengths and one moment of inertia, so one measurement cannot determine it. Substituting an isotope gives a second measurement on the same structure, and two equations in two unknowns have a solution — which comes out differently depending on which isotope is used.
What a dipole cannot tell apart
A dipole moment is three numbers extracted from a whole charge distribution, and enormously many distributions give the same three. The moment above it is not even a property of the molecule unless the one below it vanishes — which is why a quadrupole is quoted with an origin and a dipole is not.
The atoms that meet across a ring
Twelve closed conformers of a ten-membered ring at one bond angle, so every one has identical angle strain by construction. Two of them differ by 0.026 kilojoules a mole in torsional energy and by 0.80 ångström in how close two atoms on opposite sides of the ring come — 2.331 against 3.131, where two carbons are in contact at about 3.4. An account built from angles and torsions calls those two structures the same.
One spectrum, a line of models
Hückel theory has three parameters and benzene's photoelectron spectrum supplies two numbers, so the fit has a curve of solutions rather than a point. Along it the resonance integral runs from −3.05 to −7.66 electronvolts, the empty level moves by eight, the terminal-to-central bond order ratio in butadiene goes from 2.000 to 2.671 — and the delocalisation energy is 6.100 electronvolts in every member.
Three numbers is not a structure
Formaldehyde's rotational spectrum gives three constants, of which a planar molecule's are only two independent numbers, and its structure has three parameters. Seven structures are computed here that reproduce A and B to the last digit the solver carries: the C=O length runs from 1.000 to 1.300 ångström, the C–H length from 1.557 down to 0.952, and the HCH angle from 74.6 degrees to 164.5.
More coordinates than motions
Methane has ten internal coordinates and nine ways to vibrate, boron trifluoride seven and six, formaldehyde seven and six. The excess is not bookkeeping: it is a combination of coordinates that describes no displacement of any atom, and adding twenty-five units of force constant along it moves every frequency by five parts in a hundred million.
One scale, from two centres to a cage
How many centres a pair of electrons holds together is two different numbers, and they separate exactly where the bonding is most deficient: the methyllithium tetramer's pairs sit on four atoms each and have a participation number of 2.54. Run the same measurement up the scale and the twelve-vertex borane refuses it — its localisation has at least ten maxima differing by six parts in a thousand, so the number of centres is not an output for it at all.
How many descriptions a cage has
A localisation is a maximisation, and running it once reports the maximum it reached rather than the maximum there is. Run to exhaustion on a twelve-vertex borane it finds ten answers and then three batches of twelve starts in a row that find nothing new — and none of the ten is another one seen from a different side, because a symmetry of the cage cannot change a multiset of participation numbers and all ten multisets differ.
The coordinate an isotope reports
Kraitchman's equations return an atom's position from the change in the moments when that atom alone is made heavier, and for a rigid structure they are an identity — formaldehyde's four atoms come back to a part in ten million. What they return is the square of each coordinate, so both hydrogens at b = ±0.9348 come back at +0.9348; every out-of-plane coordinate comes back imaginary at a moment error of one part in a hundred thousand; and the famous error cancellation, measured at a factor of thirteen, still leaves the answer two and a half times worse than a direct fit.
The model is what is fitted
Fit a pair of coupled spins with the two-spin expression and it hands back the coupling exactly, from any temperature range. Fit a chain of eight with the same expression and it hands back −66.7 where the sample has −50, with a residual of 0.998 and a g factor of 1.973 — three numbers of which only the last says anything is wrong, and it is the one nobody looks at.
The value that only exists in the bond
A difference in electronegativity moves charge, and moving charge closes the difference — until every atom in the molecule has the same chemical potential. Solving that gives one number per molecule and a charge per atom, and carbon's runs from +0.0692 in methane to +0.3074 in tetrafluoromethane. Hydrogen's changes sign between the first and the second, so the same C–H bond is polarised one way in one compound and the other way in the next.
How many parameters a curve is worth
A susceptibility curve routinely carries four fitted parameters and the question of whether it can support them is never asked. It has an arithmetic answer: the four directions the fit sees span a factor of five hundred, so one per cent data fix the first two to under three per cent and the last to thirty-seven — and forty points reaching two kelvin are worth more than sixteen thousand starting at twenty.
Ten directions no frequency can see
A redundant force field has one obvious flat direction. There are ten: methane has fifty-five independent force constants and a ten-dimensional subspace of them that no spectrum can touch. The literature's repair is to project — and the metric a vibrational analysis is naturally written in cannot define the projection at all, because a redundancy is a null vector of it.
Two systems a model cannot tell apart
Two molecules that share a Hückel eigenvalue but were measured to differ bound a whole family of models at 0.456 eV. That is a reusable instrument, and chemistry has plenty of predictors of exactly the same shape. Turned on themselves: VSEPR cannot account for 92 per cent of the variation in the four angles it predicts, and no fitting is involved anywhere.
Four tables and one molecule to disagree about
A molecule of two elements is provably safe from the choice of electronegativity table, and a series down a group should land where the tables do disagree. It does. All four agree that hydrogen iodide is the exception — and they disagree about what its dipole is by 1.245 debye, which is two and a half times the 0.448 that was measured.
The forty-five that are fixed
Methane's fifty-five-dimensional space of force constants has ten directions no frequency can see, and the useful thing to report is the forty-five that are fixed. It is a table: every stretching constant is fixed on its own, no bending constant is, and projecting a fitted field onto the determined subspace takes four fits that span 0.88 mdyn per ångström down to four that span 0.0027.
The product a curve measures
A susceptibility curve's fourth parameter is undetermined, and the question is which combination the free direction actually is. It is a product of powers, because the Jacobian is logarithmic — and at the usual window it is the temperature-independent term divided by the 0.40 power of the monomer fraction, fixed to forty per cent, while the product one place up is fixed to 6.7. A paper could print that instead of four numbers.
A verdict inside its own error bar
The tightest comparison in the rotamer argument is a computed 10.99 against a measured 10 — a nine per cent margin, offered as a verdict. The computed side is built on one quoted energy known to ±0.4 kJ/mol, and that alone puts a band of twenty-three per cent on it. The verdict turns on four tenths of one standard deviation of a number nobody had put an error bar on.
The pair that is not a tie
An exact tie bounds every model of a form with no fitting, and it is available only for predictors that read integers. A near tie bounds the model's derivative instead, and a pair the predictor orders the wrong way round refuses monotonicity outright. Run on four standard predictors, the new instrument ranks them in a different order from the old one — and its worst case is the control the old one could not see.
One number decides which way it breaks
The six-membered verdict gives way at 312.1 K, and that is a statement about the ceiling rather than a prediction about the ratio — because a real rate ratio has a temperature dependence the model has no term for. Putting that term in moves the crossing, and at 5.308 kJ/mol it removes it entirely.
The constant that belonged to one net
Divide one second-order departure by another and the coupling cancels, leaving one constant times the gap — which reads as arithmetic. It was arithmetic about a square net. On a triangular one — the same graph for both bands, nothing else changed — the quotient drifts by a hundred and twenty-eight per cent.
The residue that is two numbers
Is a shortfall between a sum of radii and a measured separation a property of the pairs, or of the compromise a universal table makes? The eight separations are two complete two-by-two blocks, so what no assignment of radii can reproduce is not a residual at all — it is an alternating sum, computed by subtraction, and it comes to three hundredths of an ångström and six.
Three shapes from one search
A cage's fourteen localised descriptions fell into two groups with a factor of five between them and nothing in the gap, which made counting the big basins look like a stopping rule. Two more cases from the same family give a smooth tail over sixty-three descriptions and a single basin holding ninety-eight per cent. One search, one family, three shapes.
A denominator that fails both ways
The energy gap an e_π folds away over-predicts the halide trend at every metal level a donor allows, and the acceptors look out of reach because a π* is not an atomic level. It is measurable — a slow electron is captured by it — and on that side the same denominator under-predicts. No metal level fixes either, and the two want it moved in opposite directions.
Twelve basins where there were two
Sixteen sites can be searched exhaustively and thirty-six cannot, so the only instrument available past about twenty sites is a local search — and how much weaker it is has never been measured against the answer on a net where both can be run. On sixteen it is barely weaker at all. On thirty-six it fails nine starts in ten, and the arrangement it is beaten by has fewer unlike bonds than the one it starts from.
Fifty descriptions of one molecule
A search whose largest basin takes ninety-eight per cent of its starts will report one description however long it is run, and nobody runs four thousand starts when the first fifty agree. Whether the rare ones are worse descriptions or merely rarer is one number per description, already computed and never looked at. On two of three cages they are not worse — they are the same answer, to parts per million.
An end effect with two signs
Neither of two separations accounts for the scatter in a fused ring system's response, and the natural guess is the end: pairs with more molecule outboard should behave differently from pairs at an edge. They do. In a straight chain an end pair responds a third less than an interior one, in a zigzag a quarter more, and in a chain of two straight arms meeting at one angular ring the anomaly is in the middle.
A bond order between atoms that do not interact
A diatomic held where its overlap changes sign has no interaction between its two orbitals at all — which is what the sign change of its overlap means. Put a third orbital beside it and the pair is still split, one line sits exactly at the free-atom energy at every third-orbital energy, and the bond order between the two runs to −0.9999. Three measures of the same bond disagree completely.
One integer, and everything it changes
Eight molecules with the same rings, the same carbons and the same graph distance between every pair, differing in which fusion's direction is turned — one angular ring when the turned fusion is at an end, two adjacent ones anywhere else. The scatter within a separation class runs from three to four and a half times, against a straight chain's one and a half — and the two ends of one molecule disagree by up to a factor of four.
The reach is the molecule's
Every measurement of the relaxation's reach so far puts the heteroatom on the end ring, because that gives the longest run to measure a decay over — and the response of a molecule to a perturbation at its end is not the response to one in its middle. Moved inward, the decay is the same in both directions and the same as the end's. The amplitude is not: it halves, and splits across two rings.
The residue is below its own noise
Two interaction terms, both negative, leave the sign a coin toss — and a larger block would settle it. There is a larger block: a model that needs no measured separations supplies twenty-one. It cannot settle anything, because a fourfold alternating difference of distances known to a quarter of an ångström cannot resolve three hundredths of one.
The triangles that were never in the bands
A switch in how a gap scales is usually attributed to a band's third moment, and the attribution cannot be tested while the coupling runs along one of the bands. Separated, the bands turn out to decide nothing. Two triangular bands coupled along a matching — which cannot close a triangle across the gap — behave exactly like square ones.
Counting was right except where it mattered
A cage whose localisation gives dozens of descriptions that are all the same answer raises a worry: if degeneracy is common across the family, counting descriptions is the wrong measure of ambiguity. Across forty-eight cage-and-filling pairs it is the right measure on eleven of the thirteen that have anything to count — and it fails on the one leaned on hardest.
The sign a frustrated ring changes
There is a sharper question than whether a low-temperature feature buys back a fourth parameter: does it change which combination is free? It does, and by a sign. Every odd ring of spins leaves free the monomer fraction times a negative power of the temperature-independent term, and every even ring a positive power, with no case in between.
The other end of the bracket is not a number
Every claim can be priced by the measurement error it would take to overturn it, assuming the two errors independent, and the correlated extreme ought to turn each price into a bracket. At perfect correlation between equally precise measurements the difference is exact and the price is infinite — and every price is multiplied by the same factor, so the ordering of prices cannot change.
It was the count, not the frustration
The free combination's exponent comes out negative on every odd ring and positive on every even ring and on an open chain, and the sign was put down to frustration. The control was a chain of eight. A chain of five is not frustrated in any sense — a chain is bipartite and every bond can be satisfied — and its exponent is −0.792.
The frustrated cluster with an even count
A parity account of the exponent's sign replaced a frustration account and left the two still confounded: every case tested had frustration and odd parity aligned. A tetrahedron of four spins is frustrated and even. Its exponent is +0.2084, and so are those of three more clusters the two accounts disagree about — but the mechanism proposed with the parity rule is refuted along with the account it replaced.
The sign rule holds between two poles
The sign of a susceptibility fit's exponent was put down to the parity of the spin count, then to the spin of the ground state, and every cluster tried had the two aligned. A star of four spins is even with a ground spin of one, and its exponent is −6.84. But swept across the coupling, every cluster's sign changes, through poles the ground spin places, and the spin rule is right on all sixteen clusters only between 20.1 and 47.0 cm⁻¹.
Six of fifteen change verdict
The chemical capacity is finite or infinite according to the sign of a coefficient that is one sixth of a second difference of three measurements, and the pair test asked how much of its reported ordering is the quantity and how much is the fit's resolution. Adding a fifth point changes the verdict on six of fifteen atoms — and the five it unbounds are the five it had largest.
The second molecule with a blind spot
Methane's fifty-five force constants have ten directions no spectrum can touch, and the forty-five that are fixed were worth a table. Boron trifluoride is the only other molecule here with a redundant coordinate set, and it is not the same case: eight kinds of constant rather than five, a quarter of its field invisible rather than a fifth, and an out-of-plane coordinate the redundancy cannot reach.
The ranking moved and the headline did not
One correlation applied across a collection multiplies every price by the same number, so it slides the fragility ordering rigidly and cannot scramble it. Give each predictor its own correlation and the ordering does move — but not at the top. The two most fragile claims come from the same predictor, share a factor, and are locked in order at every correlation whatever; the easiest swap anywhere below them needs only ρ = 0.15.
The correction that moves three of them backwards
Every ligand radial function in the angular overlap sweeps was a neutral atom's, while three of the five donors carry a formal charge. Giving each one the charge it actually has moves three of the five computed ratios — and moves all three away from the fitted parameter, none towards it. The whole window each donor's own oxidation states allow sits above the value it was meant to reach.
Adding data made it worse
A projection leaves a residual spread across four starting points, and an earlier essay attributed it to the eigenvalue problem rather than to the coordinate set — an attribution rather than a measurement. Refitting with two isotopologues was the proposed test. Run, it produces a runaway to minus ninety-seven thousand that one isotopologue alone does not, and the reason is that the direction it walks along is flat at every mass.
The channel that points at the metal
Two ligands in the spectrochemical series carry a fitted π parameter no quotient of squared overlaps can produce, because it is negative. Giving the derivation the second interaction it lacks makes both of them negative at every metal level — and the reason is not the energy denominators, which favour the donor channel in all three cases. It is where each orbital keeps its amplitude.
A size the fit was not made from
Every input tested against the chemical capacity so far has been inside the cubic that produced it, so the search was constrained to fail. A size is not: a hydrogenic orbital at the effective charge Slater's rules give an atom's valence shell has an exact radius, on one scale for every atom, and the cubic knows three energies and no length at all. It fails the test by sixteen times the floor.
The residual was a loop
Projecting away boron trifluoride's flat direction left four fits disagreeing by a tenth in the B–F stretch constant, and the disagreement was put down to the search or to the arithmetic. It is neither. Two frequencies cannot fix three constants in one symmetry block, so the exact fits form a closed curve along which the stretch constant runs from 3.9 to 12.1, and the four fits are four points on a short arc of it.
Purity renames the poles
A susceptibility fit's sign poles were located at one monomer impurity, two per cent, and a real sample's impurity is rarely known. Swept from half a per cent to sixteen, the couplings where the fit's nuisance parameters separate barely move for ground spins of one and above and move a tenth for doublets. But the same separation stops being a pole and becomes a zero of the exponent at a few per cent, and above six to eight per cent every singlet acquires separations of its own.
Five more clusters break the band
Seven spin clusters put their upper separation coupling times the ground spin plus a half inside a band from 115 to 129 cm⁻¹, which looked like a law of where a susceptibility fit's nuisance parameters decouple. Five clusters built to test it — two larger stars and three larger bipartite graphs — land inside it once. Stars drift above and bipartite graphs below, and neither the shape of the curve nor the first excitation places the separation instead.
Three chains and ninety orderings ruled out
Two claims priced against the same predictor share its error correlation, so no correlation structure can reorder them. That makes the collection's fragility ranking three chains rather than one list — and thirty of the hundred and twenty orderings of five claims are reachable, with the other ninety forbidden before a single measured difference is looked at.
A denominator needs three currencies
Every price in this argument has been a price on a measured rise. A slope floor is a rise over a run, the run is computed rather than measured, and pricing it turns out to need three currencies rather than one — because one predictor's run is a graph eigenvalue, one is a convention, and one is a difference of square roots of integers that nothing defensible moves.
A floor on models written in one scale
A tie asks whether two predicted values are equal and a discordance asks whether two differences have the same sign. Both are questions about order, and a strictly increasing change of scale preserves order — so both are exactly invariant under thirteen reparameterisations of all three predictors. The slope floor asks for a ratio of differences, and it moves by factors of nineteen, ten and eleven thousand.
Six disagreements and three calculations
Two localisation criteria classified six of the family's forty-eight cage-and-filling pairs differently, and three of the six were consecutive fillings of one icosahedron with spreads identical to three figures. They are not three coincidences and not one degeneracy being filled: they are one calculation, because the survey's forty-eight rows are thirty-three distinct questions.
The basis a diagonaliser happened to return
Six of the cage family's thirty-three inputs have their occupied set cut through the middle of a degenerate shell, and the members of a degenerate shell are interchangeable. Re-orienting the shell changes nothing about the cage, the filling or the criterion — and it moves the best localisation functional by up to twenty-one per cent, moves the count of descriptions from ten to fourteen, and flips the degeneracy label on three of the six.
The ordering a manifold picks
A position-dependent mass leaves the kinetic energy with no unique quantum form, and an earlier sweep of the five orderings in use found half a per cent between them. A one-dimensional reduction is a one-dimensional manifold, a manifold has a distinguished Laplacian, and carrying it to the flat measure lands on exactly one of those five — not the one with no extra potential, and not the one anybody reaches for.
Named alongside it
The objects these essays reach for when they reach for this one.
Model limitConventionLeast-squaresApproximationClosed formDegeneracyReference stateExact diagonalisationExchange couplingHückel theoryForce constantIsotopologue