True as far as it goes, and silent on what decides — page 2 of 2

“The end effect is a fixed amount of energy at the two ends, so it has no length in it.”

It is a fixed amount of energy, which is the end-energy result and survives. It also has a length: the excess alternation decays over 1.51 bonds at a gap of 1.150 and 5.29 bonds at a gap of 0.143, going as the gap to the power −0.60 over the measured range.

Tested in The chain distorts hardest where it stops · the series on Peierls distortion

“The substitution method works because the zero-point errors of the parent and the substituted species nearly cancel.”

They cancel by less than assumed. The parent's and the deuterated species' fractional corrections differ by 20 to 65 per cent, so the correlation buys a factor of 2.5 rather than the thirteen an invented five per cent mismatch gave.

Tested in The correction that was invented · the series on rotation

“The identity fixing the mean ligand charge holds for any arrangement.”

It holds where every central orbital has a partner. A planar arrangement leaves the central p perpendicular to the plane unmatched and exactly degenerate with the orphan ligand combinations, and the mean charge comes back at 10⁻¹⁷ instead of the identity's −0.2 — the arithmetic saying it has been asked an ill-posed question.

Tested in Two models that disagree about the shape · the series on hypervalency

“Freezing the correction at one geometry costs only the change in its size.”

It costs the change in the division too. The share moves by more than half its range across the separations computed, so a frozen number is wrong about how much correction there is and wrong about which atom it belongs to.

Tested in A correction that is two functions · the series on basis

“A distortion's symmetry species tells you which vibrations soften it.”

Only where the species appears once, which is nowhere in this census. All seventeen molecules have a species appearing more than once, including water at 2A₁ + B₂ and methane at A₁ + E + 2T₂, and 71.4 per cent of their vibrations on average belong to such a species.

Tested in A label that prices nothing · the series on point group

“A highly symmetric molecule is the safe case for a species argument.”

It is the least bad case rather than a safe one. The two octahedral molecules have the smallest ambiguous share at 40 per cent, and the two with no symmetry at all have 100 — but 40 per cent is still two of every five distortions the label cannot price.

Tested in A label that prices nothing · the series on point group

“A correlation energy measures how strongly the electrons avoid each other.”

It measures how strongly they avoid each other where the interaction is. A Hubbard correlation energy is 100 per cent on-site as an identity — the interaction is exactly zero at every larger separation — so the enhancement at one site away, which is nearly as large as the depletion at zero, contributes nothing to it at all.

Tested in Half of it is given back at one bond · the series on correlation

“A finite ring cannot show a sharp transition, so its exponent cannot mean anything.”

What is computed is where a variational minimum leaves its boundary, which is a property of a free energy analytic in one order parameter, and that is obliged to give one half. It does: the closest pair of reduced temperatures gives 0.5017 to 0.5058 on every ring and stiffness tried.

Tested in The exponent was the window's · the series on metal

“A model with a good correlation against measurement has explained the measurement.”

It has explained at most what it can distinguish. VSEPR assigns NH₃ and PH₃ the same arrangement and they were measured 14.5° apart, out of a range of 15.7° across all four molecules — so 92.4 per cent of the variation is outside anything the model can see, whatever correlation it reports.

Tested in Two systems a model cannot tell apart · the series on models

“A model that gets a quantity right to a tenth of a unit is a good model of that quantity.”

The spin-only moment is right to about a tenth for the first five ions and it assigns Cr³⁺ and Co²⁺ the same number, which were measured 0.940 μB apart — 22.5 per cent of the range the nine ions span. The accuracy and the resolution are different quantities and only the first is usually quoted.

Tested in Two systems a model cannot tell apart · the series on models

“A response that falls with distance has a decay length.”

Only if there is a scale to set one. The bare acene's fitted length is 1.397, 1.669, 1.896, 2.104 and 2.302 rings at four to twelve rings, growing monotonically with no sign of a limit, while its own gap falls from 0.590 to 0.110.

Tested in A reach that has no length · the series on aromaticity

“The disagreement is a matter of the one constant, which could be refitted.”

Refitting it is a refit, and the constant carried over from the noble gases gives a mean error of 0.242 ångström where the radii give 0.183. What a sweep of the constant buys is reported rather than argued about, and it is not a prediction whatever it buys.

Tested in A size a confound cannot supply · the series on contour

“Choosing the best of the four tables would settle the matter.”

The best of them misses the halide series by 0.254 debye in the root mean square and the worst by 0.505, on a series whose largest member is 1.826. A factor of two between the tables sits inside a common error of the same size as the disagreement.

Tested in Four tables and one molecule to disagree about · the series on dipole

“Bent bonds are the localised description of a double bond.”

They are when |Δ| < 2|d|. With the π orbital 74.4 per cent on oxygen, the bent description wins for σ populations from 0.380 to 1.000 and loses below that — so a double bond polarised the wrong way round has σ and π as its localised orbitals, not bananas.

Tested in The angle that does not have to be searched for · the series on hybrids

“The undetermined parameter in a four-parameter susceptibility fit is the temperature-independent term.”

Over the usual window, the free direction is the temperature-independent term divided by the 0.40 power of the monomer fraction, with a largest component of 0.92 rather than 1. Over a window starting at 2 K it *is* nearly the term's own axis; over one starting at 80 K the monomer fraction dominates it instead.

Tested in The product a curve measures · the series on magnetism

“The second root of the cubic's fixed-point equation is either spurious or the ionic solution.”

It is neither, and which it looks like depends on the element. Carbon's sits at −6.396 electrons, which is not a state anything has; lithium's at −0.146, which is inside the range molecules use. What both are is the partner of a collision: the two roots meet at −η/3γ and past that there is no root at all.

Tested in Where a closed form stops being one · the series on electronegativity

“The number of ligand pairs with no central-atom partner is n + L − 4.”

It is, in twenty-three of the twenty-six arrangements computed here. It is not for methane in a square, phosphorus pentafluoride in a plane or sulfur hexafluoride in a plane: those give one, two and three orphans against a formula saying nothing, one and two.

Tested in The square that wastes an orbital · the series on hypervalency

“A length measured on a long enough chain is the chain's physics.”

At an elastic constant whose true length exceeds every chain here, the number returned is a fixed fraction of the chain: 0.0872, 0.0848, 0.0833, 0.0824, 0.0812, 0.0799 and 0.0786 of it at 64, 96, 128, 160, 224, 320 and 448 sites.

Tested in A decay that keeps slowing down · the series on Peierls distortion

“The correction is a property of the vibrations.”

It is the product of a vibrational quantity and a structural one, and across these four the structural one has the wider range: the moments span a factor of 55 and the softnesses a factor of 7.2, in opposite directions.

Tested in An expression for what was a warning · the series on rotation

“A single coefficient hides the differences between the molecules.”

The coefficients are 0.518, 0.647, 0.667 and 0.705 — a factor of 1.36, against a factor of 11.4 in the thing being predicted. What it hides is a real fifth of the answer for water, which has the fewest modes for a mean curvature to average over.

Tested in An expression for what was a warning · the series on rotation

“Each half of a counterpoise correction can be interpolated from a few computed points.”

The heavier centre's can — its second differences are 0.27 per cent of its value. The lighter centre's cannot: its second differences are 7.1 per cent at the median and 588 at the worst, because it is a difference of two energies of order a hartree whose difference is a millionth.

Tested in The half that cannot be computed · the series on basis

“A search that stops finding new answers has found them all.”

It has found them all with a probability its own basin sizes give. A search of 200 starts on this cage has a 63 per cent chance of missing at least one of the fourteen, and one of 480 has 12 per cent — both of which found the right number, and neither of which could have known.

Tested in Where the count stops being an effort · the series on multicentre

“A search that finds many maxima is finding structure in the problem.”

Only if a control finds one. Two isolated double bonds have exactly one localised description and 1,000 starts find exactly one, so the cage's fourteen are the cage's rather than the search's.

Tested in Where the count stops being an effort · the series on multicentre

“No real ligand comes near the threshold.”

Iodide is at eπ = 0.20 eσ, which is 80 per cent of the way to 0.25; bromide is at 72 per cent and chloride at 64. None crosses, so the protection holds for every ligand parameterised here — but the margin is a fifth rather than a factor.

Tested in The orbital a ligand cannot reach · the series on electron count

“Every orbit of internal coordinates contributes one totally symmetric combination.”

Only if the coordinate survives its own stabiliser. Boron trifluoride's out-of-plane coordinate is fixed by the horizontal mirror and sent to minus itself by it, so its orbit contributes nothing: two of its three orbits are symmetric, not three.

Tested in A formula that predicts minus eleven vibrations · the series on point group

“The sign of an overlap integral says which combination is bonding.”

It says so only in a model that ties the interaction to the overlap, which every extended-Hückel model does. The sign here is a difference between two regions — the lobes facing across the gap contribute negatively, the outer lobes positively — so it is set by a cancellation and not by anything happening between the nuclei.

Tested in A bond with nothing in the middle · the series on overlap

“Multiplying by the zero-point amplitude makes the soft coordinates dominate.”

It helps and is not enough. Boron trifluoride's ordering reverses — the soft pair departs 0.031 against 0.012 — and ammonia's does not: the stiff pair still departs 0.046 against 0.020.

Tested in One number was one direction · the series on spectrum

“The sign of the π term is all the denominator accounts for.”

The sign is exactly the sign of the denominator at every ligand in the series, and the denominator also over-accounts for the size — so it carries the sign and more than the whole of the trend, with the overlap required to give some back.

Tested in The gap that would have to be smaller · the series on ligand field

“Weighting a correlation hole with a longer-ranged interaction estimates what that interaction would cost.”

It does while the neighbour term is a perturbation: at V = 1 and V = 2 on a ring of six at U = 8 the weighted and solved give-backs are 10.99 against 10.92 per cent and 21.98 against 21.50. At V = 4 they are 43.97 per cent and −45.57 per cent, which is not a correction to an estimate but the opposite sign.

Tested in The give-back that turned into a saving · the series on correlation

“Defect levels become a defect band when there are enough defects.”

Not enough defects — enough chain. The typical spacing between runs of a given length does not depend on the chain length at all; what does is the closest pair, which is about 1/(Nρ²) apart and shrinks as the chain grows. The onset is where that closest pair's coupling overtakes the mean level spacing.

Tested in The length at which levels become a band · the series on defect

“A four-hundred-site chain is long enough to show what a random alloy does in the gap.”

It is long enough to show isolated box states, which is what a four-hundred-site chain shows and measures correctly. It is shorter than the onset length for every run length computed here, so nothing in it could have shown a band.

Tested in The length at which levels become a band · the series on defect

“What the boundary tracks is a property of the one-electron problem.”

It tracks the amount of correlation the filling admits. The satellite share of the removal weight at U = 64 is 0.21 to 0.33 below half filling and 0.55 to 0.68 at half filling or above, and the systems with no boundary are exactly the ones with the small shares.

Tested in A satellite that never loses its place · the series on photoelectron

“The gapless case was a special case of the gap ordering.”

It survives the change of filling and the ordering does not. Both systems with an exactly zero gap — a ring of four half filled and a ring of six two-thirds filled — have a contrast of 1.0 at every repulsion including the smallest, so a degenerate ground state has no distinguishability rather than a large boundary.

Tested in A satellite that never loses its place · the series on photoelectron

“It decays in the number of fused rings between them instead.”

Better, and not enough. The per-step decay length changes by six per cent between the linear and angular families where the per-unit length changes by twenty-two — but four pairs at two steps in one molecule span a factor of 1.74, which is a third of a step's worth of decay.

Tested in Neither of the two separations · the series on aromaticity

“The six-membered verdict gives way at 312.1 K.”

312.1 K is where the ceiling meets a horizontal line at ten. A rate ratio is not a horizontal line. With the substituted transition state lower by 1, 2, 3 and 4 kJ/mol the crossing moves to 315.7, 321.7, 334.3 and 388.6 K; at 5.308 there is no crossing at any temperature between 150 and 900 K.

Tested in One number decides which way it breaks · the series on strain

“Raising the temperature is what would refute the rotamer account for the six-membered closure.”

Only below 5.308 kJ/mol. Above it the measurement falls faster than the ceiling does, the crossing reappears below room temperature — 273.3 K at 8 kJ/mol, 288.0 K at 12 — and it is cooling that refutes the account. The direction of the test reverses at one number.

Tested in One number decides which way it breaks · the series on strain

“The excess gap divided by a band's shape departure is one constant times the separation of the two bands.”

True to 0.00% across a factor of four in the separation when both bands hop on a square net, and false by 128% on the same range when both hop on a triangular one. The quotient is a property of the net, not of the perturbation theory.

Tested in The constant that belonged to one net · the series on Bands in a solid

“A second measurement of the same kind is the one that fails to buy anything.”

The band shape and the excess gap are measurements of different kinds and invert at a condition number of 6.74 on one graph. On two they invert at 160.5, and one per cent measurements then fix the gap to 45% instead of 1.5%.

Tested in The constant that belonged to one net · the series on Bands in a solid

“Two independent localisations agreeing is evidence that the answer is well defined.”

It is evidence about the system. Two runs agree 95.32 per cent of the time on one of these cases and 3.51 per cent of the time on another, and the two are the same molecule shape with a different electron count.

Tested in Three shapes from one search · the series on multicentre

“The finding therefore does not survive.”

The threshold that misclassifies fewest on all 153 pairs is 0.05065 — the same line, to five figures, that the small set drew. It gets six pairs wrong. Over all 2,192 disputed-against-agreed comparisons the disputed pair has the smaller difference 97.45 per cent of the time. A boundary with a gap and a rule with exceptions are different claims, and the second is supported.

Tested in A gap that was a choice of bonds · the series on dipole

“A molecule adopts the arrangement that minimises the repulsion between its ligands.”

Four of the eight do, exactly, and all four have no lone pairs. The other four sit 4.20, 5.15, 10.52 and 26.47 per cent above it, and all four have lone pairs — which this energy does not count.

Tested in Expensive is not the same as unadopted · the series on hypervalency

“The alternation a Peierls chain settles at goes as exp(−π K/2), up to a constant.”

It is that expression's asymptote. Against the exact stationarity condition the expression is 10.46 per cent low at K = 1.0, 6.49 per cent at 1.2 and 0.0034 per cent at 4.0, and the error falls as the alternation to the power 1.72 — so it is an expansion in the distortion, and it is worst wherever the distortion is large enough to see.

Tested in The amplitude the collapse left behind · the series on metal

“A finite-size error in a lattice calculation is controlled by the number of sites.”

Forty-two rings across seven elastic constants and six sizes fall on one curve when the departure is plotted against the number of sites multiplied by the limiting alternation. A ring of 400 at K = 3.0 and a ring of 40 at K = 2.0 are the same distance from the limit, at ten times the site count.

Tested in The amplitude the collapse left behind · the series on metal

“A relation that holds between a molecule's moments need not hold between their corrections.”

For water it does, exactly: ΔI_c − ΔI_a − ΔI_b comes out at −9 × 10⁻¹⁶ amu Ų, because all three of its vibrations keep it planar. For boron trifluoride it fails by 0.046859 amu Ų, and the molecule has exactly one out-of-plane mode.

Tested in The axis that goes the other way · the series on rotation

“A better basis makes the counterpoise correction less important, so its assembly matters less.”

The correction does collapse — by a factor of 2,717 at a separation of two across the same sweep. But the fraction by which a pairwise assembly gets it wrong rises over the same range, so the regime where the correction is small is the regime where assembling it is proportionally worst.

Tested in The correction that gets harder to assemble · the series on basis

“A structural reach and a magnetic reach are the same length seen twice.”

Once the geometry is held rigid they agree to 0.7 per cent at the widest gap tested and to seven per cent everywhere, and their exponents against the gap are −0.6798 and −0.6803. Let the geometry relax and the structural one is 6 to 16 per cent longer, so the extra length belongs to the feedback rather than to the π system.

Tested in The floor was in the bookkeeping · the series on delocalisation

“Two crossovers in one shell differ because the levels they join differ.”

Two of these three join the same two levels — the p and the d — and have exactly the same gap, to twelve figures. What differs is the dipole: 5.196152 against 4.500000, a ratio of 1.1547007, which is 2/√3.

Tested in Three events, and a ratio of two dipoles · the series on representation

“A localisation over more orbitals is a localisation over more angles.”

It is a larger group, and the difference matters. Two orbitals have one angle and a functional quadratic in it with no linear term; three have a three-dimensional group and eighteen dipole matrix elements, of which the off-diagonal ones in two different components are what make an interior maximum possible.

Tested in An interior maximum a third orbital allows · the series on hybrids

“Running more starts would settle it.”

It would find more descriptions and would not change which is commonest. On the one cage where quality varies, more starts move the reported answer away from the commonest and towards a rarer, better one — so more effort changes the answer rather than confirming it.

Tested in Fifty descriptions of one molecule · the series on multicentre

“The pairwise assembly's error grows with the basis, so a better calculation makes the fragment method worse.”

It grows as a fraction and falls in hartree — by a factor of 793 across the sweep, from 6.27 × 10⁻³ to 7.90 × 10⁻⁶ at the closest separation. Both are true because the correction itself falls by 3,631, and the fraction grows by exactly the ratio of those two shrinks, 4.579.

Tested in One basis size where it is worth doing · the series on basis

“Whether the growing fraction matters depends on the accuracy target, which is a matter of judgement.”

It is decidable once the target is stated. At one kilocalorie a mole the correction is above the line at one and two Gaussians a centre and below it at three and above; the assembly's error is above the line at one and below it at two and above. Two is the only basis where the first holds and the second does, at every separation.

Tested in One basis size where it is worth doing · the series on basis

“The one usable basis size is a robust finding.”

Its assembly error is 1.582 × 10⁻³ hartree against a line at 1.594 × 10⁻³ — under it by 0.74 per cent. Any convention that put chemical accuracy one per cent tighter would leave no usable basis size at the two closest separations at all.

Tested in One basis size where it is worth doing · the series on basis

“The rule of sixteen and the rule of eighteen are rules of the same kind.”

The sixteen-electron gap is between two d levels and closes to exactly zero by the time the coordination reaches six. Whatever a count of eighteen is about at six-coordinate, it is not this gap, because this gap does not exist there.

Tested in The ligand the rule was waiting for · the series on electron count

“The angle between a parent's principal axis and its daughter's is always a meaningful number.”

Not when the parent is a spherical top. Methane's three moments are equal, so every orthogonal triad diagonalises its tensor; the 54.74°, 45.00° and 35.26° a diagonaliser reports against CH₃D are the tetrahedral angles of whatever triad it happened to return.

Tested in Two moments about two different lines · the series on rotation

“Dropping the pairs that touch an end leaves a clean decay.”

It does on the two uniform chains, where the coefficient of determination rises from 0.95129 to 0.98986 and from 0.98954 to 0.99783. On the chain of two straight arms meeting at one angular ring it falls, from 0.84684 to 0.84376, so the scatter there is not at the ends.

Tested in An end effect with two signs · the series on aromaticity

“A diagnostic that works on one axis of a model works on the other.”

It works, in the sense that the transfer fails exactly where the broken solution has collapsed on both axes. It does not work as one rule: a target at a polarisation difference of 0.12546 removes 96.5 per cent of the low level's error and one at 0.13945 removes 39.2 per cent, a factor of 46 in the error at the same diagnostic value.

Tested in The half of the square a ring of four cannot show · the series on approximation

“A pair whose overlap vanishes has the spectrum of two free atoms.”

One of the three levels sits at the free-atom energy exactly, at every third-orbital energy — the antisymmetric combination, which has nothing of its own symmetry to mix with. The other two move by up to twelve electronvolts.

Tested in A bond order between atoms that do not interact · the series on overlap

“A quantity that changes sign on an interval has a zero on that interval.”

The solved give-back on a ring of six at U = 8 changes sign at V = 3.8955 and again at V = 4.1595. Only the first is a zero; at the second the numerator is −4.404 against a scale of 19.18, and it is the denominator that has vanished.

Tested in A sign change is not always a zero · the series on correlation

“The give-back is a fraction of the on-site saving, so it is a fraction.”

The on-site saving it divides by passes through zero at V = 4.1595, where the neighbour repulsion has restored exactly as much double occupancy as the on-site repulsion removed. On either side of that repulsion the denominator has opposite signs, so the same numerator is reported as a give-back and as a gain.

Tested in A sign change is not always a zero · the series on correlation

“V = U/2 is where the two interactions balance on this ring.”

It is the limit of both crossings and neither of them. At U = 6, 8, 10 and 12 the root sits at 0.4741, 0.4869, 0.4934 and 0.4968 of U and the pole at 0.5304, 0.5200, 0.5140 and 0.5103, so U/2 falls strictly between them at every repulsion, closing as 1/U.

Tested in A sign change is not always a zero · the series on correlation

“The gap above eight electrons is a gap between d(z²) and d(x²−y²).”

Only while a symmetry keeps them apart. In C₂ᵥ they share a representation and mix: the purity of the less pure frontier orbital falls to 0.9678 at twenty degrees and 0.7500 at forty-five, so at large fold there are no two named orbitals for the gap to be between.

Tested in The distortion that opens the gap · the series on electron count

“The response is largest on the ring that carries the heteroatom.”

For an interior position two adjacent rings share the peak to within a per cent, because the carbon furthest from the molecule's centre in an interior ring lies on a bond that ring shares with its neighbour. Only at the end does one ring own the perturbed carbon.

Tested in The reach is the molecule's · the series on delocalisation

“The two measured negatives are weak evidence for a systematic sign.”

They are the only evidence there is, and nothing computed here strengthens or weakens them. What is established is that this route cannot add to them: the model's residue on the one block both can reach disagrees in sign, so its twenty other blocks carry no information about sign at all.

Tested in The residue is below its own noise · the series on contour

“The orphan count steps somewhere along a distortion, and where it steps relative to the energy's minimum decides what a slightly distorted molecule counts as.”

On the tetrahedron-to-plane path it does not step anywhere along the distortion. It is zero at the tetrahedron and zero at every angle up to and including 89.99°, and one at 90° exactly. There is one step and it is at an endpoint.

Tested in A count that changes at one point · the series on hypervalency

“A filled-shell result is at least independent of the arbitrary parts of the model.”

It is independent of every energy — the six-electron value does not move by 10⁻⁹ as the third orbital is swept over twenty-five electron volts — and it is a pure function of the overlaps, which are the part of the model a basis choice fixes. It has traded one dependence for another.

Tested in A filled shell is not an empty statement · the series on overlap

“Excluding systems with a degenerate level at the Fermi energy excludes the systems where the contrast is a ratio of a quantity to itself.”

A ring of six at half filling has a Fermi-level gap of 2.000 against the ring of four's 3 × 10⁻¹⁷, so it passes the exclusion, and it still reads exactly one at five of the eight repulsions measured. The degeneracy that produces the tie is in the ion, and a one-electron shell gap does not determine it.

Tested in A ratio of exactly one is a tie · the series on photoelectron

“A ring of four's contrast stops being exactly one above U = 32 because its degenerate ground state leaves the eigensolver free.”

Its many-electron gap is 0.332 at U = 8 and 0.014 at U = 0.5, and the contrast is exactly one at both — widest and narrowest alike. It is not exactly one at U = 64 and 256, where the gap is 0.062 and 0.016, in between. The gap does not predict the tie; the cut simply stops landing inside a degenerate pair between 16 and 64, which is a change in the weight ordering.

Tested in A ratio of exactly one is a tie · the series on photoelectron

“Tilting the field separates the coincident events, so the count rises with the tilt.”

It rises from three to six, and then falls back to five at three angles and to four at forty-five degrees. The count is not monotone in the tilt and is six almost everywhere rather than at the end.

Tested in Four angles the shell chooses · the series on representation

“A field at a general angle couples every pair, so the events are generic.”

Every pair is coupled at a general angle, but four angles are not general: at arctan 1, arctan 2/√3, arctan √3 and arctan 2 two crossovers coincide exactly, and those angles are fixed by the ratios 3√6 : 3√6, 3√3 : 9/2, 9/2 : 3√3/2 and 3√3 : 3√3/2.

Tested in Four angles the shell chooses · the series on representation

“A large slope floor is a strong statement about the model.”

The largest floor of the four predictors is 28.5, and it is a rise of 0.880 eV divided by a run of 0.0308 — one per cent of that predictor's range. A floor built on a near-tie in the predictor is a small denominator rather than a steep relation.

Tested in The error bar that would be needed · the series on models

“The composite graph's third moment is the quantity that decides.”

It is the right predicate and the wrong coordinate. Split into the part inside a band and the part crossing the gap, the second sorts all nine combinations and the first sorts none — and against the total, the two families' crossovers sit a factor of twenty-seven apart, against about two when the crossing part is used.

Tested in The triangles that were never in the bands · the series on Bands in a solid

“If most of the family is degenerate, counting descriptions as a measure of ambiguity needs restating.”

Thirty-seven of forty-eight pairs are degenerate, and thirty-five of those find exactly one description — where degeneracy is trivially true and means nothing. Of the thirteen that find more than one, eleven are genuinely different. The measure survives on the family.

Tested in Counting was right except where it mattered · the series on multicentre

“The four coincidence angles are properties of the shell.”

They are properties of the shell's matrix elements and therefore of the two-state estimates built from them. In the exact spectrum the one avoided crossing's field runs 1.68, 1.73, 1.89 and 1.95 × 10⁻⁵ at 45°, 49.107°, 60° and 63.435°, which lies smoothly between its values at neighbouring tilts. Nothing marks them.

Tested in None of the six was a crossing · the series on representation

“An avoided crossing is where two levels exchange character.”

Here the gap at its minimum is between 1.39 and 2.74 per cent below its zero-field value. Two levels exchanging character would close the gap to a fraction of it; a dip of one and a half per cent is a perturbation, and the two-state vocabulary of a crossover does not describe it. The second crossing found at 90° does close, to 2.1 × 10⁻⁷ against 5.0 × 10⁻⁴, which is what a real one looks like.

Tested in None of the six was a crossing · the series on representation

“The criterion's value of 1.732128 for the two lone pairs agrees with √3 to five figures, which is suggestive.”

It is √3. The criterion equals 1/√(p fraction) in closed form, with the radial dipole integral cancelling between numerator and denominator, and the departure from √3 falls by a factor of 197 — from 1.7 × 10⁻³ to 8.5 × 10⁻⁶ — as the quadrature is refined from forty points a dimension to a hundred.

Tested in The five figures were an identity · the series on hybrids

“The two ends of the twist have counts because they are the symmetric arrangements.”

They have counts because their groups happen to be tabulated. So do the two angles 0.01° and 59.99°, which are not those arrangements: the finder reports D3h and Oh for them because the worst residual, 1.4251 × 10⁻² Å per degree of twist, is still under its threshold there. The countable stretch is 0.0702° at each end of sixty — 0.234 per cent of the path — and on every bit of it the group being counted in is not the group the molecule has.

Tested in The group nobody wrote a table for · the series on hypervalency

“The displacement is a perturbation-theory result and therefore approximate.”

First-order perturbation theory puts the level at the mean of the two site energies, and the exact diagonalisation sits 1.06 × 10⁻⁴ eV from that mean at a detuning of 0.05. The departure is quadratic — a slope of 1.974 on logarithms — so first order is the whole of the small-detuning behaviour.

Tested in A symmetry holds or it does not · the series on overlap

“A low-temperature feature helps a four-parameter fit by making it better determined, and that is what it does.”

It does that — the worst relative uncertainty falls from 37.3 per cent on an open chain of eight to 21.9–23.9 per cent on the frustrated rings — and it also rotates the free direction through a sign change, which no amount of better conditioning would do.

Tested in The sign a frustrated ring changes · the series on magnetism

“A composition with one local optimum is one with nothing to get wrong.”

It is at half filling on a bipartite net, where the two-colouring is the unique arrangement with every bond unlike. Everywhere else a single basin is a statement about a search of twenty starts, and the smallest compositions have one because there is almost nothing to exchange.

Tested in The composition that is hard is not the full one · the series on cohesion

“The orphan count is constant along the Bailar twist, because the twist keeps D3 throughout.”

The conclusion holds and the reason given for it does not carry the weight. The count is 2 at all fifteen answerable geometries — but four of those are not D3, they are D3h and Oh, and the count is 2 there too. So the constancy is not the group staying the same, since the group does not stay the same. It is that six σ combinations and four matched valence orbitals are the same at every one of them.

Tested in The count the table was hiding · the series on hypervalency

“Writing the missing table closes the gap in the count.”

It closes eleven of the seventeen uncounted geometries. Six remain — three at each end, from 0.070° to about 6° and about 54° to 59.930° — and they are refused because the finder's own residual check fires, not because anything is missing from the tables. Answered geometries go from 4 to 15, and the two gaps were never the same kind.

Tested in The count the table was hiding · the series on hypervalency

“The ring's contrast of exactly one at half filling was an artefact of its degeneracy, so the true limit is unknown.”

An open chain of six has no exactly degenerate neighbouring pair in its removal spectrum at any of the eight repulsions or any of the three fillings — zero, against up to thirty on the ring. Its half-filled contrast converges to 0.999999, with residuals the guard permits. The degeneracy invalidated the ring's argument and not the ring's number.

Tested in The number the tie got right · the series on photoelectron

“The contrast falling below two at half filling is what makes satellites indistinguishable there.”

On the chain it falls below two between two electrons and four — 2.905 at two, 1.642 at four — and half filling is not where the crossing happens. Half filling is where the contrast reaches its floor, which is a different statement about a different repulsion.

Tested in The number the tie got right · the series on photoelectron

“A finite upper end can be recovered by letting the two measurements differ in precision.”

It can, and it is useless as a bound: errors differing by ten per cent give ×14.14, by twenty-five per cent ×5.66, by half ×2.83, by a factor of two ×1.41. The ceiling exists only because of a precision ratio nobody has quoted, and it runs from 1.41 to unbounded as that ratio approaches one.

Tested in The other end of the bracket is not a number · the series on models

“Excluding the three functions the plane's mirror symmetry decouples is safe, since they genuinely do not couple.”

They do not couple, and excluding them leaves a subspace that is not closed under rotation — so the answer acquires a dependence on the field's direction that the whole problem does not have. The truncated crossing field runs 1.6435 to 2.2014 × 10⁻⁵ across ninety degrees; the whole shell's is 2.0876 × 10⁻⁵ everywhere.

Tested in The variation was the basis · the series on representation

“A diagnostic that collapses the scatter onto one curve is the rule a composite method needs.”

The one that collapses it needs the target's exact energy, which is the diagonalisation the composite exists to avoid. Of the seven candidates computable from mean fields alone, the best leaves a factor of 3.67 between two points at the same diagnostic value and the worst leaves 45.7.

Tested in The second number is the error, rearranged · the series on approximation

“The depolarisation ratio reads the distortion's component outside the totally symmetric species.”

It reads a direction as well as a magnitude. At equal magnitude in the non-symmetric plane the departure from three quarters runs from 8.18 × 10⁻⁶ to 1.62 × 10⁻⁵, a factor of 1.978, so 'the component' is two numbers rather than one.

Tested in One number was a direction too · the series on spectrum

“A quantity quadratic in the distortion and invariant under a three-fold axis must be isotropic on a two-dimensional representation, so the reading should be constant on the circle.”

That is true of a single band. The reading is the largest departure over four depolarised bands, which the distortion splits, and a maximum of functions related by the three-fold is not invariant. Its period is sixty degrees rather than one hundred and twenty, and the six directions the group relates agree to two parts in a million.

Tested in One number was a direction too · the series on spectrum

“So the convention does not matter and the discrepancy can be left.”

The margins move by up to a factor of eleven. The six-membered ring clears its measurement by 9.9 per cent under the convention the decided verdict used, by 265 per cent under the torsion count, and by a factor of 12.1 under the n − 2 count. A verdict is convention-free and a margin is not, and margins get quoted.

Tested in The count that was never written down · the series on strain

“So the defect does not matter to the comparison.”

It sets every scale in the problem and it sets them together. The zero-field s-p gap, the exact crossing's field and all eight estimates shrink with it, so the ratios between them tend to constants — 0.04000 for the crossing over the gap, 0.02356 for the depth of the dip, 0.9067 for the nearest estimate over the crossing. The defect matters and cancels.

Tested in Consistently wrong is not a limit · the series on representation

“The magnitude decaying with ring size shows the frustration washing out.”

Open chains decay the same way and have no frustration to wash out: 0.792, 0.440, 0.272 at five, seven and nine spins against the rings' 0.857, 0.463, 0.287. At each odd count the two agree to within 0.065, 0.023 and 0.015.

Tested in It was the count, not the frustration · the series on magnetism

“The five collapse cases differ in size and stiffness, so the collapse is tested across both.”

In the quantity the departure from the bulk amplitude actually depends on — the ring measured in its own alternations — they are 9.56, 4.89, 1.36, 7.33 and 9.77. That is a factor of 7.2, and three of them are rings of forty at three stiffnesses, which are three different ring sizes in those units.

Tested in Five rings that were five different sizes · the series on metal

“Matching the cases shows the collapse is exact to 1.6 per cent.”

A dimerised ring needs an even number of sites, so the target cannot be hit exactly: the matched set still differs by 1.09 per cent in n·δ∞. What is left is the same order as what the rounding leaves, so a tighter figure is not available on a lattice.

Tested in Five rings that were five different sizes · the series on metal

“A distortion mixing the two effects must pass through a direction in which the gap does not move to first order.”

It must, and there is more to say: the direction is the antisymmetric fold at every symmetric arrangement, and its blindness is exact rather than found. Folding one pair three degrees further and the other three less gives the same arrangement with the pairs swapped, so the two gaps agree bitwise, and an even function has a vanishing derivative at its centre.

Tested in The direction the gap cannot see · the series on electron count

“The depolarisation reading varies by a factor of two with the direction of the distortion, so it carries direction information.”

The largest departure over the four depolarised bands varies by 97.8 per cent. Their sum varies by 0.038 per cent — a factor of 2591 between the two departures — and a sum cannot be changed by reordering. The direction dependence is in the extremum.

Tested in The sum was flat all along · the series on spectrum

“Two of the four bands are nearly undeparted in the insensitive directions.”

They are exactly undeparted — zero to the last bit the arithmetic carries — in all six directions where one bond is stretched against another. Those distortions keep a mirror plane that fixes those bands' ratio at three quarters, so the zero is a selection rule and not a small number.

Tested in The sum was flat all along · the series on spectrum

“Then it is the third-order term instead, and the fitted exponent should have been 1.”

The cubic term is there — reversing a distortion changes the sum by 7.36 × 10⁻⁴ where the isotropy residual is 5.24 × 10⁻⁴, and only an odd power can do that — but it is not the whole residual either, since an exponent of 1.248 is 0.25 from the pure-cubic prediction and a single power fits to 7 per cent. Both terms are present and the cubic is the larger over the range swept.

Tested in The suspect that did not fit · the series on spectrum

“A cubic and a quartic term together account for it.”

They fit better — the worst misfit halves from 7.0 to 3.5 per cent — which is what a second parameter does whether or not it belongs. What says it does not is that the misfit keeps its shape: positive at both ends of the range and negative through the middle, two sign changes in nine points. A model with the right terms leaves residuals without a pattern.

Tested in The suspect that did not fit · the series on spectrum

“A coordinate set's count of totally symmetric vibrations is the molecule's count.”

Only when the set spans every vibration. Hydrogen peroxide's five coordinates have rank five against six vibrations and count three where the Cartesian displacements give four; both ferrocenes' sixty have rank forty-four against fifty-seven and count three where the molecule has four.

Tested in Five coordinates for six vibrations · the series on point group

“Seven candidates scoring between 3.67 and 45.7 is seven measurements of the same thing.”

Two of the seven have no comparable pair at all and cannot be scored. The other five are judged on 2, 4, 8, 10 and 12 comparisons — a sixfold spread — so the scores are not equally well founded and a ranking among them reads more into the numbers than they carry.

Tested in Five failures in five different places · the series on approximation

“A census of failures is a property of the systems rather than of where the line is drawn.”

At a threshold of zero every comparable pair counts as a failure and the census implicates every point in the square. The answer moves with the threshold, which is what a census that measures anything must do.

Tested in Five failures in five different places · the series on approximation

“The plateau is flat.”

It is a broad maximum whose position moves with the stiffness. At K = 1.1 the exponent falls monotonically outward from six; at K = 2.8 it rises to 0.697 at fifteen bonds and falls to 0.664 at thirty. Both movements are small against the finding, and neither is noise.

Tested in The rule of thumb was on the flat part · the series on Peierls distortion

“The quarter rule is conservative, so the stiff cases are being cut short and their reaches would rise.”

Right for half the series and wrong for the other half. Five stiffnesses — 1.1 through 1.8 — end on the noise floor inside a quarter of the chain, and their exponents do not move by a part in a billion when the window is opened to the midpoint. The rule was free for those and binding for K ≥ 2, and the split at two is sharp.

Tested in The window that was not a plateau · the series on Peierls distortion

“The capacity is not a function of the second ionisation energy, and the other candidate inputs remain open.”

It is not a function of any of them. Every one of the six has two atoms adjacent in it whose capacities differ several-fold: the cubic coefficient ×7.9, the hardness ×10.4, the electronegativity and the first ionisation energy ×24.2 each, the electron affinity ×43.9, the second ionisation energy ×192.6.

Tested in The worst of the six was the one we asked about · the series on electronegativity

“A minimum of a gap between adjacent levels of the whole spectrum finds the places where levels come closest.”

It finds adjacent ones. The only gap minimum between two levels the field couples is between the two lowest m = 0 levels, at 0.0196, and both |m| = 1 levels lie between them there — so in the sorted spectrum those two are not neighbours and the search never looks.

Tested in The crossing nothing couples · the series on representation

“A shortfall of ten per cent in αₑ means the curve's asymmetry is ten per cent too small.”

αₑ goes as −(1 + a₁), where the 1 is the harmonic spread's contribution of the opposite sign. So the cubic coefficient needs to be only 5.9 per cent steeper for HCl to raise αₑ by 10.7 per cent — a difference of two terms amplifies the change in the larger one.

Tested in The cubic a Morse curve guesses · the series on rotation

“The direction the gap cannot see is the one a complex is softest along without paying for it.”

On the symmetric line it is better than that: the repulsion falls by about a tenth of a per cent along the antisymmetric fold at every amplitude tried, so the distortion is paid to happen rather than merely free. Off the symmetric line it is worse: at fifteen and five degrees the same direction costs 1.30 per cent while the gap moves by 0.44.

Tested in A blindness that is inherited · the series on electron count

“One target is enough to establish that matching tightens the collapse.”

It tightens it at all three, by factors of 1.36, 1.65 and 2.10 — and the factor rises with the target. The single-target factor of 2.10 is the largest of the three rather than a typical one, and reporting it alone would overstate what matching generally buys.

Tested in Three points, and they all go down · the series on metal

“Three points make a curve of residual against target.”

They make a direction. The residual is 2.515, 2.070 and 1.621 per cent at targets of 5, 7 and 9.6 — monotone and still falling at the largest target that could be afforded, which is a finite-size effect that keeps shrinking rather than a limit approached.

Tested in Three points, and they all go down · the series on metal

“So the classification has genuine exceptions and the rule is approximate.”

The three are consistent with chance. The rule flags 19 pairs; four independent signs all fall the same way one time in eight, so 2.375 of those 19 are expected to look agreed for no reason, against 3 observed — a ratio of 1.26. And the tables are correlated, which makes agreement more likely still, so 2.375 is a lower bound. The boundary may be exact and the panel too small to show it.

Tested in Two numbers caught what one could not · the series on dipole

“The improvement is a smaller version of the same thing.”

It changes the kind of mistake as well as the number. The one-variable rule misses disputed pairs and over-flags agreed ones; the two-variable rule misses no disputed pair at all and over-flags three. As a screen for where a computed sign cannot be trusted, one direction of error is harmless and the other is not.

Tested in Two numbers caught what one could not · the series on dipole

“The chance calculation's 2.375 spurious agreements is a lower bound, and three observed sits comfortably above it.”

Measured near the boundary, the tables correlate at 0.325, and four signs with that correlation all agree 29.5 per cent of the time rather than 12.5. The expectation among the nineteen flagged pairs is 5.6, so the three exceptions sit below chance, not above it.

Tested in No panel of this kind can find an exception · the series on dipole

“A correlation between tables inferred from a model of their signs is only a model.”

It is a model, and it can be checked. The correlation from the tables' values and the one from their sign agreement agree to 0.003 near the boundary, and the fraction of pairs on which all four signs agree matches the prediction from pairwise agreement to within 0.02 in every window from nineteen pairs to sixty.

Tested in No panel of this kind can find an exception · the series on dipole

“The coupled minimum is a robust feature of the shell.”

It exists because the d level sits a fifth of the s–p gap above p, inside the quarter at which the f² coefficient of the separation, 2·54 − 27/δ, changes sign. At a quarter it is gone, and at the shell's own offset it is 0.436 per cent deep.

Tested in Two levels cannot make a minimum · the series on representation

“The inversion barrier is what decides how fast a pyramidal molecule turns inside out, so quoting it settles the question.”

The barrier enters through the action ∫√(2μ(V−E))dx, which is an area. Two wells with the same minima and barriers 12 per cent apart — 2020 and 2262 wavenumbers — give ground splittings of 1.3508 and 0.7935, a factor of 1.70. The height moves by a tenth and the answer by a factor.

Tested in A barrier is not what a splitting measures · the series on inversion

“The splittings are too large because the barrier used was too low.”

Raising it to 2329.9 reproduces the ground splitting exactly and leaves the excited one at 47.30 against 35.81, still 32 per cent out. And the barrier the excited splitting alone asks for is 2559.8 — 9.9 per cent higher again. One height cannot satisfy two lines.

Tested in A barrier is not what a splitting measures · the series on inversion

“There is exactly one basis size at which the three-body correction is worth computing and can be assembled pairwise.”

True at a kilocalorie a mole and at no other line tried without qualification. Over two decades the answer is basis 4, then nothing, then 3, then nothing, then 2, then nothing, then 1, then nothing. Four different sizes are the answer somewhere, and more of the range has no answer than has one.

Tested in The line was holding the answer up · the series on basis

“The published result is at least robust upward, since a looser line can only make the assembly adequate more often.”

It is robust upward by 72 per cent and fragile downward by 0.7. A looser line does make the assembly adequate more often and also stops the correction mattering, so the window closes at both ends — just very asymmetrically about the value that was chosen.

Tested in The line was holding the answer up · the series on basis

“The far end is limited by the noise floor.”

For the five softest it is: the reaches are 23, 30, 39, 54 and 74 bonds and each is where the excess falls below the relaxation's convergence. For the five stiffest all five reaches are exactly 77, which is a quarter of the 320-site chain — an arbitrary reading rule rather than a property of the profile.

Tested in The other window was a plateau too · the series on Peierls distortion

“A flat sweep means a robust quantity.”

It means one only where the sweep is doing something. The softest chain has exactly one far end inside its reach, so its perfectly flat curve is a statement about a single fit repeated eleven times, and its zero per cent carries no information at all.

Tested in The other window was a plateau too · the series on Peierls distortion

“The sign is a property of the ground state's total spin.”

It is right on fifteen of sixteen clusters at the coupling every earlier reading used, and wrong on a star of six, whose exponent is +28.893 because its pole lies at 47.03 cm⁻¹, three wavenumbers short of that coupling. It is right on all sixteen at once only between 20.1 and 47.0 cm⁻¹. The ground spin places the poles; it does not fix the sign.

Tested in The sign rule holds between two poles · the series on magnetism

“Which kinds of constant a spectrum determines is a property of the particular molecule.”

On the two available it is the same: every stretching constant is fixed on its own in both, no bending constant is fixed on its own in either, and the mixed stretch–bend kinds are partly determined in both. Two molecules is not a rule, and the pattern holding across a tetrahedral centre and a planar one is more than nothing.

Tested in The second molecule with a blind spot · the series on normal mode

“The quarter is a number about the n = 3 shell, found by bisecting a numerical search.”

It is 2(n² − 4)/(5(n² − 1)) for every shell, from the closed forms of two dipole matrix elements. At n = 3 that is exactly 27/108, which is the quarter the bisection found at 0.24999886; at n = 4 it is 8/25 and it rises to two fifths.

Tested in The quarter, generalised · the series on representation

“Correlated measurement errors cannot change which of this collection's claims is the most fragile.”

True for one common correlation, which is what the common-factor result proves: a common factor multiplies every price equally and the ordering slides rigidly. False in general — with a correlation per predictor the factors form a ratio, and three of the four adjacent pairs in the ranking can be swapped, the easiest at ρ = 0.154.

Tested in The ranking moved and the headline did not · the series on models

“A net that cannot be two-coloured behaves differently from one that can.”

Not at the smaller contrast. There the frustrated net's half filling has one basin and every start reaches it, exactly as the bipartite net's does. The difference appears only at the larger contrast, where the bipartite net loses four points of hit rate and the frustrated one loses thirty-six — so what the triangles cost is invisible until the site energies dominate the band term.

Tested in A net with no two-colouring · the series on cohesion

“Dropping one of the two runaway constants is the repair.”

It is a repair to the search. The converged field and the field moved ninety times its own norm project to the same point, agreeing to 5.8 × 10⁻¹³ in every one of fifty-five components — so the projection repairs the answer regardless of where the fit stopped, and dropping a constant is a choice about which coordinate to lose.

Tested in Adding data made it worse · the series on normal mode

“The clause about hydrogen repaired the bond lists.”

It repaired hydrogen peroxide, which is what it was written for. Across the twenty-three molecules the rule still gets seven wrong, and five of those come back with no bonds at all: both xenon fluorides and all three complex ions, every bond in which is longer than the cutoff.

Tested in No length separates them · the series on point group

“The reduced mass barely changes, which is why it does so little.”

It changes by 10.8 per cent, and the splitting's sensitivity to it is steeper than to the barrier — the mass exponent is exactly one below the barrier exponent, so 10.8 per cent of mass is worth more than 10.8 per cent of barrier. It does little because the change is small, not because the dependence is weak, and the same 10.8 per cent applied to the pyramid height would cost 2.5 times as much.

Tested in It was never the mass · the series on inversion

“A force field fitted exactly to every observed frequency, with its flat direction removed, is determined.”

Boron trifluoride's E′ block has three combinations of constants — r − rr, rt and t − tt — and two frequencies. Its exact fits form a closed conic on which the B–F stretch constant spans 3.94 to 12.11 millidyne per ångström, the stretch–stretch coupling changes sign and the stretch–bend coupling runs from −0.36 to 6.71, with all six frequencies fixed.

Tested in The residual was a loop · the series on normal mode

“A second isotopologue settles an underdetermined block.”

Boron-10 frequencies predicted by the first fit, fed back as data, agree with exactly two points of the boron-11 curve: the fitted field, with a stretch constant of 7.43, and a different one at 3.97 with a bend combination four times larger. Known to half a wavenumber they leave a stretch constant of 7.32 to 7.52 near the fitted field — a wider range than the four fits happened to agree to.

Tested in The residual was a loop · the series on normal mode

“The arrangement that binds best at half filling is the one that opens the widest gap at the Fermi level.”

True in eighteen of twenty composition–net–contrast cases. On the triangular net at half filling and a contrast of 4, the winner's gap is 4.947 and a runner-up's 5.088. On the square net at a contrast of 1 with six sites raised, the winner's gap is 0.806 and a runner-up's 0.898.

Tested in The gap follows the winner late · the series on cohesion

“A pairwise assembly of the counterpoise correction over-corrects.”

With the heavy centre inside it over-corrects on all twenty-four cells, with three centres and with four. With every centre alike it under-corrects on seventeen of twenty-four, and with the heavy centres outside on sixteen and eighteen. The overshoot is a property of one arrangement.

Tested in The overshoot was one arrangement · the series on basis

“Comparing the assembly's error with the accuracy line is a test of whether the assembly is adequate.”

Only if the error is taken by its size. With the heavy centres outside, one Gaussian a centre under-corrects by 5.17 kilocalories a mole, and a test on the signed error would call that adequate.

Tested in The overshoot was one arrangement · the series on basis

“Dunham's relations fix a potential's cubic and quartic, and the higher coefficients only refine the result.”

The Morse curve satisfies the relations exactly, and its own series cut after the quartic — with the same cubic and quartic — moves its αₑ by 0.7 per cent for CO, 2.8 for DCl, 4.9 for HCl and 6.3 for HF, which is between a sixth and a half of each Morse shortfall. Kept to the octic, every molecule is back within half a per cent.

Tested in Two coefficients are not a potential · the series on rotation

“A change of sign in the third direction's exponent is a pole.”

At two per cent every change of sign below 150 cm⁻¹ is a pole. At four per cent every doublet's lower separation is a zero of the same exponent instead, at a coupling moved by two per cent, because the third and fourth singular values have traded places. By sixteen per cent every cluster's upper separation is a zero.

Tested in Purity renames the poles · the series on magnetism

“The linear per-decade rise ρξ ln 10 predicts the growth wherever the run density is small.”

Where ρ is below 10⁻³ it does over the lengths computed: at x = 0.3 the rise over 10²⁰–10³⁰ is 0.97 of that over 10³–10⁶. At x = 0.75, where runs of six are most common at ρ = 0.011, it is 0.35 of it. What decides is ρ times the separation, not ρ alone.

Tested in The share that was read as a line · the series on defect

“The supply argument is the refutation without a margin in it, since it compares against bonds rather than against a convention.”

It compares a requirement against bonds and computes the requirement from the ceiling, which contains the gauche energy. The 250-fold acceleration needs seven rotors at 2.5 kilojoules a mole, five at 3.8 and four at 4.5, against a supply of four — so the argument holds at the bottom of the range and fails at the top.

Tested in The lever that was supposed to be smaller · the series on strain

“The fragility of a pairwise counterpoise assembly — one usable basis size, a window a hundredth wide — is a property of the method.”

It is a property of stopping at pairs. With the three-body increments added, every four-centre arrangement has a usable basis size at a kilocalorie a mole, two of them have two, and more than one size is usable over 23, 60 and 38 per cent of three decades of line where pairs gave 0, 0 and 3.

Tested in A repair that costs more than the whole · the series on basis

“Taking a many-body expansion one order further improves the assembly.”

It improves the usable range in every arrangement and the error on most cells. It makes the error larger on five cells of twenty-four with every centre alike and on four with the heavy centres outside, and it raises two of the eighteen window floors.

Tested in A repair that costs more than the whole · the series on basis

“The triples whose fragments are not consecutive can be dropped from a chain's assembly.”

With the heavy centres inside or outside, dropping them leaves the covered share exactly where every triple put it, 75 and 80 per cent. With every centre alike it falls to 52 per cent, below the 58 that pairs alone give, because the triples with a gap carry a median of 42 per cent of each cell's three-body increment there.

Tested in A repair that costs more than the whole · the series on basis

“Changing an overlap puts two first-order terms into the level's shift, which will add or nearly cancel.”

Both terms are there and both multiply the level's amplitude on C, which is zero, so the first-order shift vanishes without any cancellation — and so does every higher order, because the pair's two rows of H − αS stay proportional whatever their overlaps to C are.

Tested in A level no symmetry was protecting · the series on overlap

“A shorter run saturates sooner because it is commoner.”

That is one of two causes and they compound. Runs of four are sixteen times as common as runs of eight at half filling, which shortens the separation needed by the same factor. They are also less deeply bound: ξ is 1.438 against 2.556, so the coupling decays faster and the reach the chain buys per decade is smaller — 3.31 against 5.89. The first shortens the distance and the second lengthens the journey, and the distance wins by a factor of three in the decades required.

Tested in The commonest runs reach it in a grain · the series on defect

“The spreads are properties of the cages rather than of the search.”

A spread is a difference between the best and worst maxima found, so more starting points can only raise it. The ten-vertex cage's Boys spread is 7.17 × 10⁻⁴ at twenty starts and at sixty, and 1.07 × 10⁻³ at a hundred and twenty. Every spread quoted here is a lower bound, and the zeros are the only entries a longer search cannot move.

Tested in The cage is on both sides · the series on multicentre

“The count of third-row ions is the variable the model's error runs with.”

It is a coarse version of one. The sum of the two exponents correlates with the error at −0.986, matched by 4 of 720 orderings where the count is matched by 24, fits a line through all six at r² = 0.974, and orders the three pairs with one third-row ion, to which the count gives one value.

Tested in The sum of the exponents, not the softer ion · the series on contour

“The slope the sum gives is the slope a correction would need.”

A line through the three pairs outside the middle group predicts the middle group's order and a slope of −23.8 points per unit of exponent, where those three have −11.5 among themselves. The sum is the right direction and not the whole variable.

Tested in The sum of the exponents, not the softer ion · the series on contour

“The chain of six has no pair of removal lines of identical weight, so its contrast of one cannot be a tie.”

At the repulsions sampled it has none. At U = 28.916 its third and fourth strongest lines, at removal energies +0.664 and −1.666, carry weights equal to eight figures and the contrast is 1.00000004. The lines are not degenerate; their weights cross.

Tested in The limit of one is a parity · the series on photoelectron

“A half-filled chain's intensity contrast converges to one.”

At large repulsion the lines sit at 2cos(kπ/(n + 1)) in pairs at ±E whose weights agree as 1/U. With three up electrons on the chain of six, and one on the chain of two, the cut falls inside a pair and the contrast goes to one. With two on the chain of four it falls between pairs and the limit is 1.3125.

Tested in The limit of one is a parity · the series on photoelectron

“The plane's law can be written down, since it is the chain's with s replaced by πs².”

It is right where it matters and wrong where the objects touch. Measured in three nets of a quarter of a million sites, the share of runs of six with a twin inside s matches 1 − exp(−ρπs²) to within a point at s = 16 and 4.2 points at s = 8, but over-predicts by 4.9 points at s = 4 and 1.8 at s = 2 — because two runs of six cannot overlap and cannot share a row within seven sites, which a point process knows nothing about.

Tested in A plane spends the same reach on a disc · the series on defect

“The half-way separation of seven and a half sites is too small for the exponential coupling law to be trusted.”

It is inside the range the law was fitted over, which runs from two to fourteen sites, so it is interpolation rather than extrapolation. What it is not inside is the range where the hard core is negligible: at 7.5 sites the point-process law over-predicts close pairs by about four points, which moves the measured half-way separation to about 8.0 and the half-way length up by a tenth of a decade.

Tested in A plane spends the same reach on a disc · the series on defect

“A symmetry label resolves a distortion into vibrations.”

It resolves it into species, and 24 of the 29 internal coordinates of the five molecules with force fields fall in a species that appears more than once. Within that species the projection puts 0.761 of the share on one of the two occurrences, against 0.500 for a label that said nothing — so the label narrows the answer by about half the distance to a decision and settles six of the twenty-four.

Tested in Half the distance from a coin · the series on point group

“The largest molecule's labels are the least informative.”

Methane's mean is 0.746, above ammonia's 0.751 by nothing and below water's 0.876 — but it reaches that average without settling a single one of its ten distortions, where ammonia and boron trifluoride each settle three of six. The same mean is reached two quite different ways.

Tested in Half the distance from a coin · the series on point group

“The eighteen-electron rule's t2g set is a feature of octahedral symmetry.”

It survives the whole twist as a three-dimensional set, becoming A1 + E in the D3 interior and A1' + E' at the prism. A d6 centre therefore has the same closed shell at every geometry on the path, so the electron count cannot be what resists a Bailar twist.

Tested in The leftover changes sides · the series on hypervalency

“Contracting the model's p functions tests the range explanation.”

Not if every shell is contracted alike. At a factor of 1.1 every separation shrinks by between 9.7 and 10.2 per cent, the spread of the six errors narrows from 32.7 points to 30.1 — less than rescaling every distance by the mean change would give — and the root-mean-square error rises from 11.4 per cent to 14.4.

Tested in One contraction for two conditions · the series on contour

“Correcting the third-row shells' range would make the ionic model accurate.”

It removes the pattern and not the error. At the aligning contraction the spread of the six errors falls from 32.7 points to 5.2, and the root-mean-square error rises to 14.7 per cent, because every pair is brought to the second-row pairs' error of about −14 per cent rather than to zero.

Tested in One contraction for two conditions · the series on contour

“A family of orderings that brackets the measurement makes the improvement meaningless.”

The family is unbounded, so it brackets every number. The ordering that lands on the measured 0.7935 has exponents of +3.04 and −3.04, and the one that undoes the mass construction has α = γ = −4.63 and β = +8.26 — three and five times further out than any ordering proposed.

Tested in An ordering worth half a per cent · the series on inversion

“The ordering ambiguity is small because ammonia's mass varies only by a fifth.”

It is small because the ordering's potential carries ħ² and a reciprocal mass. Multiplying every nuclear mass by λ divides that potential by λ exactly, and the spread falls from 0.63 to 0.25 per cent between λ = 0.5 and λ = 4 while the mass construction's effect rises from 34 to 70 per cent.

Tested in An ordering worth half a per cent · the series on inversion

“The near-end and far-end plateaus were established on truncated profiles, so they may not survive the opened window.”

The near-end sweep never touched the truncation: its far end of sixty bonds lies inside every quarter-chain reach, and it returns identical exponents under either fraction. The far-end plateau survives inside the turn — 1.5 per cent at K = 2 and 2.0 at K = 2.2, under the 5.3 per cent published — and past the turn it is not a window choice at all.

Tested in The rate that turned back · the series on Peierls distortion

“So no isotopic data at all can resolve the block.”

A substitution that breaks the symmetry can. With one fluorine of mass 21 the two fields' frequencies differ by up to 0.45 cm⁻¹, and with one of mass 17 by 0.66 — ten thousand times more than any symmetric substitution, and less than the anharmonic corrections a real fit would carry.

Tested in The third mass that cannot help · the series on normal mode

“The census of failures was a negative result about the candidates.”

It was a negative result computed on a favourable sample. The scores it reported — a factor of three to five for the best candidates — were ten times better than the scores the same statistic gives on the interior of the same square. The conclusion survives and the numbers in it do not.

Tested in The arms were the kindest part of the square · the series on approximation

“The gap is blind to the antisymmetric fold because the two trans pairs are exchanged by a symmetry.”

True, and too narrow. With each ligand lifted separately, the gap's gradient at any symmetric fold has four parts equal to eleven figures, so it points along the one direction in which all four lift together. Every distortion whose lifts add to zero is blind: the antisymmetric fold, the tilt of either trans pair, and three up with one down.

Tested in Four lifts and one that matters · the series on electron count

“The D3 gap found by accident on the Bailar twist was likely to be the only one.”

It was not, and there is exactly one more. Eighteen paths sampled at fifteen geometries each give 270, of which 243 are named by a tabulated group, 25 are refused by the finder's tolerance and 2 are a finite group of order ten that no table here reaches. Two of 270 is a small answer and it is not zero.

Tested in The gap found on purpose · the series on hypervalency

“ND₃'s measured splitting is a sharp test of the bond-conserving reduced mass, because that mass changes with deuteration in a way no constant mass can.”

The mass does change differently — 23.7 per cent across the coordinate for ND₃ against 20.1 for NH₃ — but fitted to NH₃ and carried to ND₃ the two constructions predict isotope ratios of 19.26 and 19.00 in one well and 17.63 and 18.49 in another, against a measured 14.94. They differ by less than either misses, and which is nearer changes with the well.

Tested in Deuterium cannot tell the masses apart · the series on inversion

“One contraction of the third-row shells accounts for the pattern, since its two conditions agree to 2.2 per cent.”

Taken per ion, chloride needs 1.302, potassium 1.387 and calcium 1.441 — a spread of 10.7 per cent. The single factor's two conditions, 1.3765 and 1.3463, are within 0.03 and 0.11 per cent of the means of the per-ion factors of the ions each condition contains. They agreed because they averaged overlapping sets of ions.

Tested in Three contractions for one shell · the series on contour

“The (S + ½) band failed, so the upper separation has no simple spectral predictor.”

It has one that the band was a shadow of. The product of the upper separation with the first spin-raising excitation spreads by a factor of 1.50 across the doublets and 2.15 across the triplets, where the separation alone spreads by 3.59 and 5.70; its median is 359 and 351 cm⁻¹. For stars and complete bipartite graphs the raising gap happens to grow with the ground spin, which is why S + ½ looked like the variable.

Tested in The excitation that raises the spin · the series on magnetism

“A diagnostic that cannot be scored is best reported as a blank, since it has not failed.”

It has not, and it has not passed either, and the second half is what a blank cell does not convey. Scored by their nearest comparisons rather than only by comparisons inside a band, the two blanks give worst error ratios of 388 and 46 — the first and third worst of the seven. Neither was a success the band was hiding.

Tested in A blank is not a pass · the series on approximation

“A worst-case error ratio is a score.”

Not without the gap it was taken at. Ranked by worst ratio the best cheap candidate is polarisation × level shift at 11.4; ranked by how close its comparisons are, it is polarisation at a 14.0 per cent median gap, and that one's worst ratio is 45.7. The two orderings disagree at the top, so a table of ratios alone ranks them wrongly.

Tested in A blank is not a pass · the series on approximation

“Removing the band makes the scores harsher, so it is a stricter rule.”

It makes them harsher and it is not strictly stricter, because it also admits comparisons that mean nothing. The control — the quantity that is the error — goes from 1.16 under the band to 1.84 without it, and stays near one, which is the check that the rule is not simply inflating everything. Every cheap candidate exceeds 11.

Tested in A blank is not a pass · the series on approximation

“The four predictions can be tested against the measured splittings of the mixed molecules.”

Both are measured and neither is quoted here. What is computed is a prediction from a barrier fitted once to NH₃ — 2116.9 wavenumbers under the bond-conserving construction — and carried to the other three with only masses changed. The prediction for ND₃ is 0.0418 against a measured 0.0531, so the model is already a fifth low at the far end, and the two middle points inherit that.

Tested in The two that are not on the line · the series on inversion

“Making the two ends of the radical inequivalent is what shifts its spin off an even split.”

An overlap change does, at 2.0 per unit of overlap. A site-energy change that moves the level by a tenth of an electronvolt moves the spin by only 0.012, and a coupling between the ends — which moves the level by 0.54 eV at an overlap of 0.05 — moves the spin by nothing, because it leaves the ends equivalent.

Tested in The spin the count does not hold · the series on overlap

“The missing zero-point term cannot be assessed without a force field along the inversion path.”

Its value cannot; its required size can. The gap between the barrier each construction wants for NH₃ and for ND₃ is 97 wavenumbers under the bond-conserving mass and 109 under the rigid-triangle one. The transverse zero-point energies differ between the two molecules by 1783 wavenumbers, from eight measured bands, so the correction would have to be 5.4 or 6.1 per cent of that difference — an ordinary amount for five frequencies to change between a pyramid and a plane.

Tested in The ordering a manifold picks · the series on inversion

“The scatter within a separation class is an effect of where a pair sits relative to the ends.”

Over 702 pairs in 27 chains, whether a pair touches an end raises R² from 0.780 to 0.786. Counting the angular rings between a pair's two rings and under them raises it to 0.935, and adding the end term back changes nothing. Divided by those counts, no chain's worst spread exceeds 2.27, against measured spreads up to 5.95.

Tested in The scatter counts angular rings · the series on aromaticity

“A momentum measurement along a bond reads the bond length off the first zero of the profile.”

At π/R the profile per electron is the antibonding occupation times a number the atom fixes, exactly. Two hundredths of an electron in the antibonding orbital turn the zero into a minimum at 1.608 that reads the separation as 1.954 bohr instead of 1.997; past 0.104 there is no minimum at all; and the Heitler–London bond's occupation is above that threshold at every separation from 1.4 to 4 bohr.

Tested in The zero belongs to one determinant · the series on orbital

“A longer bond goes to the axial site of a bipyramid.”

At five sites it does, by 2.8 × 10⁻³ at a length ratio of 1.1 and 1.4 × 10⁻² at 1.5. At seven it does the reverse: a bond of 1.1 prefers the equator by 7.7 × 10⁻³ and one of 1.2 by 1.7 × 10⁻². What the long bond follows is the more crowded site, which is axial in the trigonal bipyramid and equatorial in the pentagonal one.

Tested in The long bond goes to the crowded site · the series on VSEPR

“The root of the give-back is where the structure beyond contact contributes nothing.”

That is true only when the interaction reaches one bond, where the numerator has one term and vanishes with it. With a second-neighbour term at half the nearest-neighbour strength the numerator vanishes at V = 5.5151, where the one-bond pair count is still enhanced by 0.185 and the two-bond count still depleted by 0.370. Neither separation is at its free value; the two terms are equal and opposite.

Tested in A cancellation between two separations · the series on correlation

“The action can replace the solver, since it does not underflow.”

It can replace it for a ratio and not for a value. Run on the four ammonia isotopologues the solver can still resolve, it is low by 5.64, 5.97, 6.26 and 6.51 per cent — and on ratios of those same four splittings it is out by 0.36, 0.67 and 0.94 per cent. The prefactor is the crude half of the formula and it is nearly common to two isotopologues of one molecule.

Tested in The same fraction at six times the barrier · the series on inversion

“The spin on the ends of the radical is independent of the middle orbital's energy.”

Only while the ends are uncoupled. With the coupling on, moving the middle orbital from 4.4 eV below the ends to 8.6 eV above them takes the spin on A from 0.177 to 0.456 — nearly back to an even split — where the uncoupled radical gives 0.236 at every energy.

Tested in Two changes that do not add · the series on overlap

“Mulliken and Löwdin spins agree here because the unpaired electron has no amplitude on the middle orbital and the ends do not overlap.”

That is one reason, and it is not the one that operates when all three site energies are equal. There the Wolfsberg–Helmholz Hamiltonian commutes with the overlap matrix, every orbital is an eigenvector of it, and the two partitions agree to 10⁻¹⁵ at every overlap and coupling. With the middle orbital at a different energy and both changes on, they part by up to 0.011 of the electron.

Tested in Two changes that do not add · the series on overlap

“Triples with a gap in them matter less the wider the gap.”

Nothing more than four separations away carries more than about a per cent. But the triples with a gap carry a median of 0.42 of the three-body increment by size at four fragments, 0.82 at five and 1.13 from six on, because two one-gap shapes seen from one side carry +20.8 and −11.0 per cent and a two-gap shape carries −16.5 — the gap triples do not fall away, they take back what the consecutive ones overshoot.

Tested in Length did not rescue the consecutive triples · the series on basis

“A heteronuclear bond's profile at π/R measures the orbital's polarity.”

It measures the mismatch between the two atomic contributions, which is a polarity plus a floor. At equal coefficients — no charge transfer at all — the depth is already 1.35 per cent of the peak for C–O, 1.63 for B–N and 10.2 for Li–F, and zero only when the two atoms are the same. The measure is monotone in the polarity in the chemical direction, and it has an offset rather than a zero.

Tested in A floor no charge transfer explains · the series on orbital

“IF₇'s equatorial ring puckers because five domains around an equator are crowded.”

Crowded, yes — the ring's 72° neighbours make the pucker the arrangement's softest motion by a factor of a hundred under 1/r. But with IF₇'s own bond lengths the flat ring is a minimum under 1/r, 1/r² and 1/r³, and its pucker is three times stiffer than with equal bonds. Crowding makes the pucker cheap; it does not make it happen.

Tested in The ring held flat by its long bonds · the series on VSEPR