Simply false — page 3 of 5

“An acceptor's π* level cannot be brought into the arithmetic because it is a molecular orbital rather than an atomic one.”

It is measured directly. Electron transmission spectroscopy captures a slow electron in the π* and reads the energy: 1.50 eV for carbon monoxide, 2.30 for dinitrogen, and 2.26 for hydrogen cyanide, which stands in for cyanide's.

Tested in A denominator that fails both ways · the series on ligand field

“Since the denominator over-predicts for donors, the overlap must be roughly constant across the series.”

It must shrink down the halides to cancel part of an over-prediction and grow from cyanide to carbon monoxide to make up an under-prediction. Two statements about an overlap, in opposite directions, neither of which comes from computing one.

Tested in A denominator that fails both ways · the series on ligand field

“One number separates disputed bonds from agreed ones with nothing in between.”

On thirty-one bonds it does. On all one hundred and fifty-three pairs the four tables cover, the largest disputed difference is O–Br at 0.1628 and the smallest agreed one is Na–K at 0.0398, and twenty-seven agreed pairs lie below the largest disputed one. The gap was a property of which thirty-one bonds were chosen.

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

“Measuring the difference against how much the four tables disagree would restore the separation, since a dispute needs the difference to be small compared with the disagreement.”

The steepest disputed pair sits at 0.5950 of the way to the diagonal and the shallowest agreed one at 0.3330, so the two groups overlap on that measure too. What is exact is only the arithmetic bound: four numbers straddling zero cannot have a mean above their range, so every disputed pair must fall below the diagonal, and they all do.

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

“The disagreement is about one awkward element.”

Fifteen of the eighteen elements appear in at least one disputed pair; only fluorine, phosphorus and potassium never do. Hydrogen, which the interhalogen series was designed to exclude, appears in three of sixteen — fewer than carbon, sulphur, iodine or beryllium.

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

“The arrangements that break the counting formula are the expensive ones, which is why the formula is reliable in practice.”

The three failures sit 4.20, 6.29 and 9.80 per cent above their own repulsion minimum. The arrangements the formula gets right run from exactly zero to 26.47 per cent, so the most expensive arrangement in the census is one it survives.

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

“An arrangement decides whether the formula holds for it.”

A square plane of four ligands breaks it on methane and holds on xenon tetrafluoride, at an identical 4.197 per cent excess. Two lone pairs take the orbital perpendicular to the plane, and the orphan then has no spare orbital to sit beside.

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

“A centre cannot have an unused valence orbital and an unmatched ligand combination at the same time.”

It has both in exactly three arrangements, and they are exactly the three where the formula fails. A planar arrangement of four or more ligands cannot reach the p orbital perpendicular to its plane, so that orbital is spare while a combination goes unmatched.

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

“An amplitude that a scaling collapse divides out is a number that has to be measured.”

All five of the amplitudes the collapse divided by are reproduced to better than one part in three million by a sum of n/2 square roots — 0.23894451 against 0.23894449, 0.12316658 against 0.12316659, and so on — with no matrix diagonalised anywhere.

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

“A more detailed version of a working expression fits better.”

Split by principal axis, the coefficient the expression needs spans a factor of 2.33 among the positive cases — 0.4944 to 1.1499 — against the 1.36 the molecule-averaged version spans. Three times as many parameters, twice the spread.

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

“Zero-point motion makes a molecule's moments of inertia larger.”

It makes eleven of these twelve principal moments larger and one smaller. Water's smallest moment falls by 0.46 per cent, so its coefficient is −0.1499 and no positive constant can describe it.

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

“The expression fails per axis because the moment enters it wrongly.”

Plotted against their own moments the twelve coefficients lie on no curve: two axes of boron trifluoride whose moments differ by exactly two need 0.5863 and 0.8274, while boron trifluoride's smaller axis and ammonia's larger one differ in moment by a factor of eighteen and need 0.5863 and 1.0112.

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

“The assembly's overshoot is a small-basis artefact that a better basis removes.”

From one Gaussian a centre to six, at a separation of two bohr, the overshoot goes 5.25, 7.10, 15.69, 23.16, 32.27, 37.98 per cent. It grows at all four separations tested, by factors of 4.6, 7.2, 9.1 and 13.0.

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

“Six Gaussians a centre is enough to see where the overshoot is heading.”

Plotted against the reciprocal of the basis size, where a limit would show as a flattening, none of the four curves flattens: every one is still rising at six functions and rising faster in those coordinates than in the basis size itself.

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

“A structural response that stops falling has stopped.”

On a chain of twelve fused rings the response flattens at 4.48 × 10⁻⁵ after five rings when each molecule's couplings are measured from its own mean bond order, and falls to 7.89 × 10⁻⁹ at the twelfth ring when both are measured from the same one. The molecule is identical in the two runs.

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

“A floor in a self-consistent calculation is the convergence tolerance.”

The floor is 2.2 × 10⁻⁵, 4.5 × 10⁻⁵ and 7.0 × 10⁻⁵ at feedback strengths of 0.2, 0.4 and 0.6 — proportional to a model parameter — while the fixed point's own residual is 7.3 × 10⁻¹⁴ at all three. It scales with the physics and not with the arithmetic.

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

“A reach set by a gap goes as the reciprocal of the gap.”

Across a factor of five in the gap the structural reach goes as the gap to the power −0.688 and the magnetic reach to −0.680, both fitted over profiles spanning three to ten decades. Neither is −1, and the two agree with each other far better than either agrees with the argument.

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

“Starting a local search from the arrangement with the most unlike bonds finds the best one.”

It does on sixteen sites at both contrasts and on thirty-six at a contrast of one. On thirty-six at a contrast of four it arrives at 2.22569848 with 28 unlike bonds and is beaten by a random start reaching 2.23046805 with 24 — fewer unlike bonds, more binding.

Tested in Twelve basins where there were two · the series on cohesion

“An exhaustive search is available for any net worth studying.”

Twelve raised sites among thirty-six is 1,251,677,700 arrangements. One steepest ascent on the same net used 1,484 diagonalisations. The exhaustive answer is not expensive there; it does not exist.

Tested in Twelve basins where there were two · the series on cohesion

“A shell with three levels in one subshell and two in another has four crossovers.”

It has three. The m = 0 half has two coupled pairs and the m = ±1 half has one, because it is a two-level ladder — and the m = ±1 subshell occurs twice over, at m = +1 and at m = −1, which gives one field rather than two.

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

“The events of one subshell fall together, so a shell's crossovers separate by subshell.”

They interleave. The three fields are 9.5965 × 10⁻⁶ (m = 0), 1.1081 × 10⁻⁵ (m = ±1) and 3.5087 × 10⁻⁵ (m = 0), so the m = ±1 event sits between the m = 0 pair's two.

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

“Every pair of levels in a subshell can be coupled by a field.”

The s and the d of the m = 0 subshell have a dipole of exactly zero between them — a field changes the angular momentum by one — and no matrix element joins the m = 0 subshell to the m = ±1 one at all, because a field along z leaves m alone.

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

“A defect band's width grows without limit as the chain grows.”

The largest coupling any chain can produce is the splitting between two runs one site apart, which is 0.0877β for runs of four and 0.0158β for runs of eight. No chain, however long, contains a closer pair, so the width has a ceiling that no length can raise.

Tested in A band that is a hundred and seventy decades of nothing · the series on defect

“As runs get longer their levels crowd together, so their bands must eventually merge.”

The spacing between the levels of runs of L and L + 1 and the sum of their widest half widths both fall as the cube of L + 1. Their ratio has a minimum of 1.6187 at L = 5 and rises at both ends, so two defect bands never touch.

Tested in A band that is a hundred and seventy decades of nothing · the series on defect

“A defect band can broaden until it overlaps the host band and closes the gap.”

At a barrier of four the highest defect level of a run of twelve sits 0.0581β below the barrier band and its band is at most 0.0026β across — a margin of 22.7 of its own half widths, which is larger than the 8.7 a run of four has, not smaller.

Tested in A band that is a hundred and seventy decades of nothing · the series on defect

“A localisation criterion that has no interior maximum for two orbitals has none for three.”

The Boys functional over the whole rotation group of a carbonyl's σ, π and oxygen lone pair peaks at 1.106642 Ų against 1.079600 for the best two-orbital mixing and 0.966555 for the canonical set — an interior point 0.80 radians from any relabelling.

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

“What the third orbital adds is the third orbital's own localisation.”

The lone pair is barely moved: the maximum takes 8.44 degrees of it into the two bonds, and what improves is the bonds. Removing the lone pair from the space costs 2.50 per cent of the functional, which is the amount no pair of orbitals can find.

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

“The rabbit-ear description of a carbonyl's lone pairs is the same phenomenon as a bent bond.”

It is the two-orbital criterion applied to two orbitals, and it wins outright: the two lone pairs' centroids are 0.31652 Å apart and twice the dipole between them is 0.54826, a ratio of 1.732128. A double bond's mixing is a competition; two equivalent lone pairs' is not.

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

“A description a search reaches rarely is a worse description.”

On a nine-vertex cage at eight electrons the fifty descriptions span 2.20 × 10⁻⁵ of the functional's value, and on a twelve-vertex cage at twenty electrons the five span 1.68 × 10⁻⁶. The rare ones are not worse; they are the same answer reached from different starts.

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

“Where the descriptions do differ, the one a search usually returns is the best.”

On the twelve-vertex cage at twelve electrons the descriptions differ by up to 3.9 per cent, and the most-reached one is 0.34 per cent below the best. Two rarer descriptions beat it on the functional the search is maximising.

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

“A sixteen-electron complex has a gap, and adding a ligand is a separate question from whether it does.”

The gap is (2 − kf)eσ with k axial ligands at a fraction f of the equatorial strength, exactly, at every point. It is 1eσ at a full square pyramid and exactly zero at an octahedron — so the gap and the coordination are one quantity, not two.

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

“Distorting a square plane costs the gap its exactness, so adding a ligand will too.”

The derivative of the gap with respect to eπ is zero at every point of both axial approaches, to 10⁻⁹. Bending all four ligands loses it at the first degree. The fourfold axis is what protects it, and an approach along that axis keeps it.

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

“As the gap closes the complex becomes less stable, which is why it stops at sixteen.”

The stabilisation of the sixteen filled electrons rises the whole way — 2.0, 3.0, 4.0, 5.0, 6.0 eσ as two axial donors come in from nothing to full strength — while the gap falls from 2 to 0. The two move in opposite directions along the same path.

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

“Comparing a molecule's moment of inertia with its deuterated form's compares the same quantity twice.”

The principal axes are eigenvectors of a tensor built from the masses, so changing one mass turns them. One deuterium turns water's frame by 21.122°, and the daughter's moment about the parent's own smallest axis is 0.87431 u Ų against the 0.72911 it has about its own — a difference of 19.9 per cent, none of which is physics.

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

“An isotopic substitution always changes a rotational constant, which is why the method works.”

¹¹BF₃ → ¹⁰BF₃ changes every rigid moment by exactly zero, to machine precision, because the boron is at the centre of mass. The substitution reports no coordinate at all, which is Kraitchman's known blind spot arriving as an identity rather than as a caution.

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

“A substitution that leaves the rigid moment alone leaves the spectrum alone.”

It does not. ¹⁰BF₃'s rotational constants differ from ¹¹BF₃'s by 310 kHz on two axes and 58 kHz on the third, entirely from zero-point motion; and CH₃D's constant about its own threefold axis differs from methane's by 4,038 MHz on a moment that is identical to twelve figures.

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

“The scatter within a separation class is an end effect, so it collapses against how far the pair sits from the nearer end.”

The end pair's response relative to the deepest pair at the same separation is 0.6579 in a straight chain of nine and 1.1610 in a zigzag of nine, at four fusions' separation. One variable cannot take two signs.

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

“An end correction measured on one molecule can be carried to the next.”

At two fusions' separation the ratio runs 0.820, 0.762, 0.789, 0.839 for straight chains of five, seven, nine and eleven rings, and 1.512, 1.184, 1.277, 1.244 for zigzags. Neither converges and the parity of the ring count moves it as much as the length does.

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

“The decay length of a fused system's response is a property of the system.”

It is a property of which pairs are fitted. On a straight chain of nine it is 1.9005 fusions over all pairs, 2.0690 over the interior ones and 1.718 over the end pairs alone — a spread of a fifth, with no measurement between them to prefer.

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

“A four-site ring is a fair miniature of a correlated system for questions about symmetry breaking.”

Its half-filled shell is degenerate to 3 × 10⁻¹⁷, so its symmetric solution is unstable at every repulsion however small and its broken solution survives down to 0.25 with a polarisation of 0.516. A chain of four breaks only above 1.67 and a ring of six only above 2.36.

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

“The two ways of moving away from a reference are symmetric, so one measurement covers both.”

The broken solution exists BELOW a critical site-energy modulation and ABOVE a critical repulsion. On the site-energy axis the failure comes from turning the modulation up; on the repulsion axis it comes from turning the repulsion down, and the worst case there is 1,538 per cent worse than the calculation it was meant to correct.

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

“A composite correction is at worst no better than the cheap calculation it corrects.”

On a chain of four with a correction built at a repulsion of six, transferring it to a repulsion of a half gives an error of 0.425 where the mean field alone was wrong by 0.026 — sixteen times worse, and the practice's own assumption forbids it.

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

“Two orbitals with no overlap and no resonance integral are not bonded to each other, so their bond order is zero.”

It runs from +0.031 to −0.954 as a third orbital is swept past them, and to −0.999899 when that orbital is put 80 eV below. A bond order is computed from the occupied orbitals, and the third orbital decides which combinations are occupied.

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

“The three regimes a third orbital creates survive the pair's own interaction vanishing.”

They collapse to one point. What the pair is bonded by, on the difference-of-two-calculations definition, is exactly zero at every third-orbital energy — there is no interaction to remove, so the quantity the regimes were measured in has no variation left.

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

“If the pair contributes nothing, the system is not bound.”

The trio is bound at every third-orbital energy tested, most strongly by 5.33 eV. Every electronvolt of it is A–C and B–C, and none of it is A–B.

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

“A chain has to be angular along its length to change how the response scatters.”

One turned fusion is enough. A straight chain of nine rings scatters within a separation class by a factor of 1.520 at worst; a chain turned at its first fusion, which has one angular ring, scatters by 3.052, and the chains turned at an interior fusion, which have two adjacent angular rings, by 4.065 to 4.600.

Tested in One integer, and everything it changes · the series on aromaticity

“The worst place to turn a fusion is the middle, where it divides the molecule most evenly.”

The worst is at the third fusion of eight, at 4.600, and the centre positions give 4.065. The trend along the chain is 3.052, 4.304, 4.600, 4.065 and then its mirror — though the first member has one angular ring and the rest two, so the trend mixes a count with a position.

Tested in One integer, and everything it changes · the series on aromaticity

“Two pairs at the same separation and the same depth in one molecule respond alike.”

In a chain bent at its second fusion, the pair touching the left end and the pair touching the right differ by a factor of 2.58 at two fusions' separation — 0.331 against 0.855 of the deepest pair's response. Both are at depth zero.

Tested in One integer, and everything it changes · the series on aromaticity

“The scatter is worst at the largest separation, where the response is smallest and a fit is most fragile.”

Only for the straight chain, which is the one case that rises monotonically. The chain bent at its second fusion peaks at 4.304 at four fusions but the one bent at its third peaks at 4.600 at three, in the middle of the range where a fit carries the most points.

Tested in One integer, and everything it changes · the series on aromaticity

“The dimension of a net decides how its coupled bands behave.”

A chain and a cubic structure are one- and three-dimensional and both give a gap exponent of −2; a chain reaching to its second neighbours and a triangular net are one- and two-dimensional and both give −1. Each group contains both, so no dimension sorts them.

Tested in Seven points that looked like a switch · the series on Bands in a solid

“The coordination decides it.”

Coordination six occurs in both groups — a cubic structure at −2.0277 and a triangular net at −1.1054 — and coordination four occurs in both, a square net at −2.0118 and a chain of reach two at −1.0644.

Tested in Seven points that looked like a switch · the series on Bands in a solid

“A census whose points fall on two lines has found a switch.”

Turning the third moment on continuously slides the exponent from −2.0118 to −1.0835, and it is half-way at a skewness of 0.0300. The smallest non-zero skewness any of the seven nets has is 0.7500, twenty-five times larger. The two lines are where the nets sit, not where the behaviour changes.

Tested in Seven points that looked like a switch · the series on Bands in a solid

“The size of an effect and the power it follows change together.”

Along the same family the quotient falls by a factor of 27.6, smoothly and monotonically, with no feature at the skewness where the exponent turns over. Watching the amplitude alone would never locate the crossover.

Tested in Seven points that looked like a switch · the series on Bands in a solid

“Distorting a square-planar complex closes the gap above eight electrons.”

Folding two of the four ligands to the same side raises it, from 2.000000 to 2.093826 eσ at twenty degrees. It falls only past twenty-five, and reaches 1eσ at forty-five. The four-ligand path and the axial approach both close it monotonically; this one does not.

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

“A gap that is getting larger is a gap that is getting safer.”

The derivative with respect to eπ is −0.00490 at two degrees, −0.13698 at ten and −0.63921 at twenty — where the gap is at its largest. Every point at which the gap exceeds the plane's is a point at which its exactness is already gone.

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

“Any tilt of two ligands out of the plane is a distortion of it.”

Tilting them so that they stay trans to each other changes no level at any angle, to nine figures — because two perpendicular linear pairs always span a plane, so the result is the same square plane in a different orientation. The levels are 0, 0, 0, 1, 3 at zero degrees and at ninety.

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

“A decay measured from the end of a molecule might be the end's rather than the molecule's.”

Moved to the fifth ring of twelve, the response decays at 0.687 rings in one direction and 0.707 in the other, against 0.719 measured from the end. The three agree to within three per cent, so one number describes the reach of every case.

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

“Where the heteroatom sits does not matter, then.”

The peak response is 3.763 × 10⁻² on the end ring and 1.942 × 10⁻² on every interior one — a factor of 1.94 — and it is shared between two rings rather than one. The end-ring measurements report a decay that is the molecule's and an amplitude that is a choice of position.

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

“The two directions differ because one has more molecule beyond it.”

The step ratios run 4.00, 4.86, 3.78 on the short side and 3.96, 4.80, 4.08, 3.47 on the long one. They track each other until the short side runs out of rings, and the departure is the boundary rather than the amount of molecule.

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

“A larger block would settle whether the non-additivity has a systematic sign.”

A model with no fitted length in it supplies twenty-one blocks, nine negative and twelve positive, with a root-mean-square of 0.1118 Å. The measured residues are 0.031 and 0.056 Å. The model's scatter is twice the larger measurement, so its split in sign is its own error sorted, not a verdict.

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

“A model that reproduces the measured separations can be trusted on their differences.”

It reproduces six measured separations to a mean of 0.242 Å and a worst of 0.563 Å. The residue is a fourfold alternating difference of four such numbers, and the largest measured residue is 0.056 Å — four times below the model's error on any one term. On the one block it can check, it returns +0.115 Å against a measured −0.031.

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

“The survey covers every block the four cations and four anions make.”

The overlap rule refuses two of the sixteen separations — magnesium and calcium against sulfide, a compact p against a diffuse one at three bohr — and says so rather than returning a number. Those two appear in fifteen of the thirty-six blocks, including the alkaline-earth chalcogenide block, which is one of the two the measurements supply.

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

“A harmonic zero-point correction to a moment of inertia is the leading term, and anharmonicity is a refinement on it.”

A rotational constant averages 1/r², whose expansion carries +2⟨Δr⟩/rₑ from the anharmonicity and −3⟨Δr²⟩/rₑ² from the harmonic spread. In all four molecules the anharmonic term is the larger, by 1.94 to 2.58 times, and it is exactly zero in any symmetric well.

Tested in The term a harmonic field cannot produce · the series on rotation

“The harmonic and anharmonic corrections to a moment both make it larger.”

They have opposite signs. For hydrogen chloride the anharmonic term is +2.4008 per cent and the harmonic term −1.1335, and the exact answer, +1.3289, is a difference between them rather than a sum.

Tested in The term a harmonic field cannot produce · the series on rotation

“A more anharmonic well is one in which the anharmonic term matters more.”

The ratio runs the other way: carbon monoxide, the most nearly harmonic of the four at ωₑxₑ/ωₑ = 0.61 per cent, has the largest ratio at 2.575, and hydrogen fluoride at 2.17 per cent has the smallest at 1.941. Both terms grow with the anharmonicity and the mean square grows faster.

Tested in The term a harmonic field cannot produce · the series on rotation

“The moment of inertia of a vibrating diatomic is a single number.”

One ground-state wavefunction gives two. μ⟨r⟩² sits 2.415 per cent above the equilibrium moment for hydrogen chloride and μ/⟨1/r²⟩ sits 1.329 per cent above it — nearly a factor of two apart, and the second is the one a spectrum reports.

Tested in The term a harmonic field cannot produce · the series on rotation

“A molecule sitting slightly off a square plane might have either count.”

It has the tetrahedron's. At 89.99° a ligand sits 1.7 × 10⁻⁴ of a bond length from where the plane would put it — two orders below the 0.06 tolerance the symmetry finder uses and far below anything diffraction resolves — and the count there is zero, as it is at 54.74°.

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

“The count is defined at every geometry, whether or not it is useful.”

Within about two degrees of the tetrahedron the symmetry finder refuses: the tetrahedral operations still map the molecule onto itself inside its tolerance but cannot be refined to the precision it demands, so no group is returned and no count exists. Four of the twenty geometries computed have no count at all.

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

“A filled shell's bond orders vanish, because the sum over occupied orbitals is a sum over a complete set.”

The sum over a complete set in a non-orthogonal basis is the inverse overlap matrix, not the identity. Between two orbitals whose overlap is exactly zero the filled-shell bond order comes out at 0.142857 — one seventh — by diagonalisation and by inverting a three-by-three matrix, two routes that share nothing.

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

“The bond order between two orbitals says something about those two orbitals.”

On one geometry with one Hamiltonian it is 0.00000 at zero electrons, +0.36940 at two, −0.63060 at four and +0.14286 at six. Nothing about the pair changes between those rows.

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

“A bond order between non-interacting orbitals is a small artefact.”

At four electrons with the third orbital eighty electron volts below the pair it is −0.999899, which is a full antibond to four figures; at two electrons with the third orbital at zero it is 1.03059, which is more than a full bond.

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

“The contrast limit at half filling is a measurement of how distinguishable satellites become.”

On a ring of six it is exactly 1.000000 from U = 64 upward, and at every one of those repulsions the third and fourth strongest lines agree in weight to 6 × 10⁻¹⁶ and in energy to 8 × 10⁻¹⁵. The contrast is a line divided by its own degenerate partner.

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

“The coincidences are an artefact of deciding when two computed fields count as one.”

Two degrees either side of each coincidence the count is back to six, so each is a point rather than a stretch. A tolerance of one, which calls every crossover the same as every other, reports one event at every angle — that is what a threshold artefact looks like and it is not what is measured.

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

“Propagating uncertainties would remove one of the three discordant pairs.”

The three need errors of 51.6, 43.0 and 16.0 per cent of their set's measured range to be reversed — 59, 49 and 18 kJ/mol on strain energies spanning 115. They are the three most robust refutations in the collection.

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

“A claim's fragility can only be assessed once the uncertainties are known.”

The error that would destroy a claim is computable from the claim alone: a difference Δ between two measurements is reversed at about Δ/√2 on each. Expressed as a fraction of the set's own range it is comparable across four predictors in four different units, with nothing quoted.

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

“The π overlap must shrink from fluoride to chloride, because the energy denominator over-predicts the trend and something has to cancel it.”

Computed at the measured bond lengths with Slater radial functions, chloride's π overlap squared is 2.919 × 10⁻² and fluoride's is 8.317 × 10⁻³ — 3.5 times larger, not smaller. The two factors push the same way and their product over-predicts a fitted ratio of 1.14 by about five.

Tested in The overlap the model is not proportional to · the series on ligand field

“The angular overlap model's eπ/eσ is the ratio of the squared overlaps, as its derivation says.”

The computed ratio is 0.3052 for fluoride and 0.9456 for chloride; the fitted values are 0.14 and 0.16. The computation over-predicts by 2.2 and 5.9, and the two columns are not even monotone together.

Tested in The overlap the model is not proportional to · the series on ligand field

“A ligand with no π interaction is one with no π overlap.”

Ammonia's fitted eπ is exactly zero and its computed Sπ²/Sσ² is 0.3546 — larger than fluoride's. The zero records that nitrogen has no p orbital to spare, which a bare p function placed on the donor atom cannot know.

Tested in The overlap the model is not proportional to · the series on ligand field

“Whatever eπ is, a computed overlap is the right kind of quantity for it.”

The fitted value runs +0.14 for fluoride, 0 for ammonia and −0.10 for cyanide. A ratio of two squared overlaps is positive by construction, so no computation of that kind can produce the column at all.

Tested in The overlap the model is not proportional to · the series on ligand field

“A band's own third moment decides how it responds to being coupled to another band.”

Two triangular bands, whose own third moment is 7.296, give an exponent of −1.1054 coupled along their own bonds and −2.1040 coupled along a matching. The bands are identical in both rows and the answers are a whole power apart.

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

“Two bands with no odd cycles between them keep the constant.”

Two square bands coupled along a triangular set of bonds have an intra-band third moment of exactly zero and give −1.0732, with a quotient spread of 128 per cent. The triangles are entirely in the perturbation.

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

“A weaker coupling is a weaker version of the same perturbation.”

A matching has one coupling bond per site where a square net has four, and the difference is not a factor. A matching has no site joined to two others across the gap, so it closes no triangle at all, and its gap-crossing third moment is 4 × 10⁻¹³ rather than a quarter of anything.

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

“Two couplings — the closest and the typical — describe the distribution between them.”

The closest is the extreme: 0.125 per cent of runs of six are coupled at least that strongly. The typical is the mean of a geometric distribution, at 250 sites against a median of 177, so 95.8 per cent of pairs are coupled more strongly than the coupling quoted as typical.

Tested in A count rather than an average · the series on defect

“How much of a defect band is a resonant pair is a property of the disorder.”

It is a property of the chain length, and it rises logarithmically. Runs of six go from 1.26 per cent of runs in a resonant pair at a thousand sites to 16.1 per cent at a million million — about 1.7 percentage points per decade, slowing slowly as the share grows.

Tested in A count rather than an average · the series on defect

“The share cannot be predicted without running the chains.”

It rises by ρξ ln 10 per decade, the run density times the coupling's decay length. For runs of four to eight that predicts 5.172, 3.093, 1.802, 1.027 and 0.575 points against measured 4.518, 2.878, 1.735, 1.007 and 0.569.

Tested in A count rather than an average · the series on defect

“Longer defects couple over shorter distances, which is why they are more isolated.”

They couple over longer distances: the decay length rises from 1.4376 sites for runs of four to 2.5563 for runs of eight, a slope of 0.2801 sites per site. What makes them isolated is the density, which halves with every extra site, and the density wins.

Tested in A count rather than an average · the series on defect

“The measure therefore survives everywhere.”

It fails at the extreme. The nine-vertex cage at eight electrons finds 44 descriptions — three times the next case — and their spread is 2.2 × 10⁻⁵, which is one answer. By count it is the most ambiguous system in the family; by spread it is not ambiguous at all.

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

“The two readings rank the family similarly.”

The case at the top of the count ranking has the third-smallest spread of any case with more than one description. The case at the top of the spread ranking, an eleven-vertex cage at twenty electrons, finds two descriptions and does not appear in the count ranking's top three at all.

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

“A large spread needs many descriptions to be spread over.”

Ten of the thirteen cases with more than one description find exactly two, and those two include the largest spread in the family at 5.3 × 10⁻². Two descriptions that differ by five per cent of the functional are more ambiguity than forty-four that differ by two thousandths of a per cent.

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

“One of the two-state estimates agreeing with the exact crossing to one per cent shows the picture works near that angle.”

At a general tilt the six estimates span a factor of four to twenty-one while the exact crossing moves by a factor of 1.34 across the entire ninety degrees. Scattering six numbers over that range guarantees one lands near the target. At 60° the nearest is 1.5 per cent away and at 20° the nearest is 37 per cent away, with no angle in between behaving differently.

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

“The depolarisation ratio's departure from three quarters measures the distortion.”

Adding any amount of the totally symmetric breathing distortion to any other changes the departure by at most 0.35 per cent, and breathing alone gives exactly zero at every size tried. The ratio reads the distortion's component outside the totally symmetric species, not the distortion.

Tested in The distortion the ratio cannot see · the series on spectrum

“Distortions the ratio can see separately combine into one it can see.”

Each of the three single-bond stretches of boron trifluoride moves the ratio by 1.07 × 10⁻⁵. Their sum moves it by zero — not a small number, exactly zero, at a displacement of 0.02 and at five times that.

Tested in The distortion the ratio cannot see · the series on spectrum

“A blind direction of a second-order probe is a small effect that a larger distortion would expose.”

The breathing distortion gives exactly zero at displacements from 0.005 to 0.1, a factor of twenty. A small effect grows as the square of the displacement, as the visible distortions do; this one does not grow because it is an identity of the point group.

Tested in The distortion the ratio cannot see · the series on spectrum

“The criterion tests whether the two lone pairs are better described as mixed rabbit ears or as a σ and a π lone pair.”

It equals 1/√(p fraction), which exceeds one for every hybrid with any s character at all. Over nine hybrids from 90 per cent s to 10 per cent s it returns 3.162, 2.236, 2.000, 1.732, 1.581, 1.414, 1.291, 1.155 and 1.054 — mixed wins nine times out of nine. It has no input that could make it say otherwise.

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

“A criterion with a closed form can be checked at one molecule.”

At 50 per cent s character the correct form 1/√(p) and the wrong form 1/√(s) are both √2, and the integrals return 1.414234. A single carbonyl at that hybridisation would confirm both. The sweep over nine hybrids is what separates them, and away from that one point the wrong form misses by at least 8.9 per cent.

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

“A geometry with no orphan count is one the symmetry finder could not resolve.”

Two different failures are being run together. In two bands, 0.070° to about 6° and about 54° to 59.930°, the finder refuses — its own check that every operation maps the molecule onto itself to better than 10⁻³ Å fires. Across the eleven interior angles sampled it does not refuse: it succeeds, reports order 6, and produces no count. The geometry is exact and the arithmetic is absent.

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

“A level placed exactly by a symmetry is at least approximately placed when the symmetry is nearly there.”

Detuning one of the two equivalent orbitals by 0.05 eV moves the level by 0.025106 eV — 0.502120 of the detuning. There is no regime in which most of the exactness survives; the ratio approaches one half as the detuning falls, not zero.

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

“Breaking the symmetry changes the level and leaves everything else alone.”

The two bond orders that were equal by symmetry separate at once, to 0.525795 and 0.518996 at a detuning of 0.05, and the pair's own antibond deepens from −0.630602 to −0.632329. Every quantity moves first order.

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

“The free combination is ρ·χ_TIP raised to some power, and the power is a property of the model.”

The power is +0.412 for an open chain, +0.225, +0.278 and +0.285 for rings of six, eight and ten, and −0.857, −0.463 and −0.287 for rings of five, seven and nine. The sign is the ring's parity.

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

“Being a ring rather than a chain is what changes the answer.”

Rings of six, eight and ten give positive exponents, like the open chain. Only the odd ones — where the antiferromagnetic couplings cannot all be satisfied — go negative, so it is the frustration and not the boundary condition.

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

“The exponent is a fixed number worth quoting for a frustrated ring.”

It decays with size: 0.857 at five spins, 0.463 at seven, 0.287 at nine. An odd ring's frustration is a finite-size property and the exponent follows it down, so a value quoted without the ring size is not a value.

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

“The metal's contraction could account for the discrepancy between the computed and fitted π/σ ratios.”

Across the entire range Slater's rules allow chromium — 4.60 at the metal to 6.00 at chromium(VI) — the computed ratio changes by a factor of 2.07 to 2.18 depending on the ligand. The discrepancy runs from 2.2-fold at fluoride to 5.9-fold at chloride. The window is smaller than the gap for three of five ligands and irrelevant to the other two.

Tested in A contraction that cannot reach three of them · the series on ligand field

“A different effective charge per ligand would work, since a real 3d contracts as the ligand field strengthens.”

Water needs 9.49 and fluoride 7.20 — two different charges, neither inside 4.60 to 6.00, and 2.30 apart. Chromium has 24 protons and eighteen core electrons; an effective charge of 9.49 on a 3d electron would need nine of the eighteen to stop screening. Chloride needs more than 16, which is where the overlap rule stops agreeing with its own verifier, so the model cannot even evaluate its own answer there.

Tested in A contraction that cannot reach three of them · the series on ligand field

“The discrepancy is a matter of size that a better radial function could close.”

Ammonia's fitted e_π is exactly 0.00 and cyanide's is −0.10. The model's ratio is S_π² over S_σ², which is strictly positive for every radial function, every bond length and every basis. Two of the five ligands are outside the range of values the model can produce at all.

Tested in A contraction that cannot reach three of them · the series on ligand field

“Contracting the metal could be tuned to trade one ligand against another.”

Every ligand's computed ratio falls with contraction — by 2.17, 2.18, 2.16, 2.07 and 2.09 for chloride, fluoride, water, ammonia and cyanide. They move together and by nearly the same factor, so the ordering across the series is fixed and one charge cannot be chosen to improve one ligand at another's expense.

Tested in A contraction that cannot reach three of them · the series on ligand field

“The hardest composition to optimise is the one with the most arrangements.”

On thirty-six sites at a contrast of one the half-filled composition has 9.08 × 10⁹ arrangements, one local optimum and a hundred per cent hit rate; nine raised sites has 9.4 × 10⁷ arrangements, six local optima and a hit rate of ten per cent. The larger space is the easier problem.

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

“A search over arrangements gets steadily harder as the composition rises towards half.”

It is not monotone in either direction. On sixteen sites at a contrast of one the hit rate runs 100, 100, 100, 33.5, 21.0, 67.0, 100, 100 per cent from one raised site to eight.

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

“Difficulty is a property of the problem, so the two contrasts should agree about which composition is hardest.”

On thirty-six sites the hardest composition is nine raised at a contrast of one and twelve at a contrast of four, and the second has twelve distinct local optima against the first's six. Changing the contrast moves the hard composition by a third of the way across the range.

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

“The nineteen tables were a principled set and D3 was excluded for a reason.”

C3v is also of order six and was present; D2, D4, D5d and D6h were present. D3 needs three classes and three irreducible representations, its class sizes sum to 6, its squared dimensions sum to 6, and every pair of rows and columns is orthogonal to a part in 10⁹. It was absent because nothing had asked for it.

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

“Reading the contrast at a large repulsion gives its limit.”

The chain's half-filled contrast is 1.0316 at U = 32 and 1.0617 at U = 128 — it moves away from one over a factor of four in the repulsion before turning back. A reading at U = 32 is nearer the limit than a reading at U = 128, and neither is the limit.

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

“The guard refuses an extrapolation when the number is wrong.”

It refuses both of the ring's degenerate fillings and permits all three of the chain's. The half-filled ring, which it refuses, gives 1.000000; the half-filled chain, which it permits, gives 0.999999. The guard is a statement about the evidence, not about the answer.

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

“Recomputing every price under perfect correlation gives the other extreme, so the true value lies between two numbers.”

At perfect correlation between measurements of equal precision the difference of the two has standard error zero, so no measurement error overturns any claim and the price is unbounded. The bracket runs from the independent price to infinity, which is not a bracket.

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

“Correlated errors would turn each price from a point into a bracket around the ordering.”

The multiplying factor is √2/√(1 + r² − 2ρr), which depends on the correlation and the precision ratio and on nothing about the claim. Every one of the five priced claims is multiplied by the same number, so the ordering by fragility is identical at correlations of 0, 0.25, 0.5, 0.75, 0.9, 0.99 and 1.

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

“The most fragile claim might not be the most fragile once errors are correlated.”

Overtaking would require a factor below one, and every factor is at least one — the smallest is exactly 1 at zero correlation and equal precision. Raising the correlation on one claim alone moves it further from the next, never towards it. The required correlation is reported as non-existent rather than as a value outside [0, 1].

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

“The one avoided crossing's field varies by a third across the tilt.”

In the complete shell it does not vary at all. Thirteen directions spread over the sphere give the same field to eleven decimal places, because the zero-field Hamiltonian depends only on l and is therefore invariant under every rotation, while a uniform field's coupling rotates with the field. The whole spectrum is independent of direction, exactly.

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

“The second crossing at ninety degrees is a feature of the field lying along x.”

It is the truncation's. The five-function basis retains two d functions and a field along x leaves them degenerate, so a gap closes to nothing. The whole shell's field lifts the entire d quintet at every direction, and the nine-level count is one at ninety degrees as it is everywhere else.

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

“The product of the polarisation and the correlation-energy change collapses the scatter better than either alone.”

The change alone scores 1.157, which is the floor this statistic can reach. Multiplied by the polarisation it scores 1.910 — the multiplication moves it off the floor rather than towards it.

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

“The chemical capacity is essentially a statement about the second ionisation energy.”

Fitted directly, the second ionisation energy accounts for 2.9 per cent of the capacity's variation across fourteen atoms. The best of five two-parameter forms accounts for 43.8, and needs a logarithm the proposal did not include.

Tested in A correlation is not an account · the series on electronegativity

“The capacity's relation to the second ionisation energy is monotone enough to be a rule.”

Nitrogen and potassium differ by 2.02 eV in the second ionisation energy, less than almost any other pair, and by a factor of 193 in capacity — 31.54 against 0.164. No function of one variable gives two nearly equal inputs two answers two hundredfold apart.

Tested in A correlation is not an account · the series on electronegativity

“The direction dependence appears when the molecule is pushed hard.”

The departure divided by the square of the amplitude varies by 2.0 per cent across a factor of eight in amplitude, and the anisotropy itself moves only from 1.989 to 1.915. Both extremes are second order and so is the ratio between them.

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

“The two extremes are an arbitrary pair of directions on the circle.”

The minimum is at one bond stretched against another and the maximum at two bonds against the third — the two kinds of distortion the point group distinguishes within the plane. Every direction related to the minimum by the three-fold rotation and the mirror planes reads the same to two parts in a million.

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

“The verdicts on the gem-dimethyl effect depend on a rotor count nobody had settled.”

Nine calculations — three measured accelerations against three conventions — and the verdict column never changes. The five-membered ring's 250-fold and 11,000-fold accelerations exceed the ceiling under every convention; the six-membered ring's 10-fold falls under it under every convention. The unsettled choice is not load-bearing for any of those verdicts.

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

“The two conventions differ by an offset, so one is the other shifted.”

They order the ring sizes oppositely. n − 2 gives the five-ring 3 rotors and the six-ring 4; the decided verdict used 4 for the five-ring and 2 for the six. One rises with ring size and the other falls, so no constant reconciles them and at most one can be a count of anything.

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

“The tail exponent settles near two thirds as the gap closes.”

The four cases that produce the flattening have between 3.8 and 6.2 coherence lengths of visible tail. Raising the floor from 10⁻⁸ to 10⁻⁵ moves their fitted powers from 0.21 to 0.58, from −8.2 to 0.30, from −37.8 to −6.0 and from −196 to −36.9. A number that moves by tens when a threshold moves is not a measurement.

Tested in The exponent was the floor · the series on Peierls distortion

“The drift in the fitted power comes from fitting over a fixed window of bonds.”

Refitting each case over a window scaled to its own coherence length gives 0.47, 0.51, 0.54, 0.57, 0.61, 0.67, 0.68 — the same drift. The window's position in bonds is not what decides it.

Tested in The exponent was the floor · the series on Peierls distortion

“The drift is a subleading term the two-parameter fit is absorbing.”

Fitting rate = 1/λ + p/b + c/b² makes it worse, not better: p then runs 0.42 to 0.75 and c changes sign midway through the range.

Tested in The exponent was the floor · the series on Peierls distortion

“A chain of 320 sites is too short to hold the tail.”

The centre of the chain is bulk to better than 10⁻⁸ at every stiffness here: the excess forty bonds before the centre is 8 × 10⁻⁵ at K = 3.1 and the centre itself is −3 × 10⁻⁸. The limit is the arithmetic's resolution, not the chain's length, and a longer chain would not move it.

Tested in The exponent was the floor · the series on Peierls distortion

“The two-state picture is a limit this problem is far from, and closing the l degeneracy would bring the two counts together.”

Swept from a defect of 0.02 to 0.0002 the exact count is one throughout and the estimated count is eight throughout, with the estimates spanning a factor of 7.08 at both ends. Nothing approaches anything.

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

“The nearest estimate agreeing to nine per cent shows the two-state picture is roughly right.”

It is the nearest of eight scattered over a factor of seven, and its ratio to the exact field converges to 0.9067 rather than to one. An estimate that settles at a fixed wrong value is a different quantity computed accurately, not the right quantity computed approximately.

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

“The convergence of these ratios is a numerical artefact of the sweep's smallest defect.”

They move over the sweep and then stop moving: the crossing's fraction of the gap runs 0.04097, 0.04048, 0.04024, 0.04009, 0.04004, 0.04001, 0.04000 as the defect falls by a factor of a hundred. A quantity constant by construction would not have moved at the top of the range.

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

“The exponent's sign is negative because the ring is frustrated.”

Open chains of five, seven and nine spins are not frustrated — a chain is bipartite and every antiferromagnetic bond can be satisfied — and their exponents are −0.792, −0.440 and −0.272. Three unfrustrated systems carry the sign the account reserves for frustrated ones.

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

“Removing the frustration would change the sign.”

Weakening a ring of five's closing bond from full strength to absent removes the frustration continuously and leaves the count alone. The exponent moves by 0.065 across the whole path — from −0.857 to −0.792 — and crosses nothing. At zero strength the system reproduces the open chain of five to a part in 10⁹ in every singular value.

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

“A bend in a fused chain is a boundary of the same kind as an end, so the reach measured on a straight chain is a property of straight chains.”

Putting one kink — two adjacent angular fusions — in the middle of a chain of twelve changes the response on the heteroatom's own ring by 0.12, 0.09 and 0.52 per cent when the heteroatom is three rings before, three rings after or on the end, and by 18.9 per cent only when it sits on the bent ring itself. An end halves it.

Tested in A bend is not an end · the series on delocalisation

“The bend changes how far the response reaches.”

The fitted decay lengths on the bent chain and the straight one agree to within 10.7 per cent everywhere, and the straight chain's own two directions already differ by up to 7 per cent. The bend moves the range by less than the direction of measurement does.

Tested in A bend is not an end · the series on delocalisation

“Which side of the bend the heteroatom sits on matters.”

Three rings before the bend the ratio is 0.9991 and three rings after it is 0.9859; one ring before, 0.9948, and one ring after, 0.9685. The effect does not change sign or size across the bend, which is what a boundary would do.

Tested in A bend is not an end · the series on delocalisation

“A bend is therefore structurally irrelevant to a fused system's response.”

A ring-current response changes by a factor of four with a single bend. The same structural change costs this response a fifth of its amplitude on one ring. Two responses, one structural change, and a difference of twenty in sensitivity.

Tested in A bend is not an end · the series on delocalisation

“The 3.41 per cent residual is what the collapse is worth.”

Re-run on five cases matched at n·δ∞ = 9.6 — rings of 40, 56, 78, 108 and 150 — the worst spread is 1.62 per cent, a factor of 2.10 tighter. The residual was finite size.

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

“The square plane is the natural place to look for the blind direction.”

The gap's gradient there is zero in both directions — 4.9 × 10⁻⁸ against 2.9 × 10⁻³ ten degrees out — so every direction is flat to first order and none is distinguished. A null direction computed there is undefined, and the calculation reports that rather than an angle.

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

“The blind direction is a fixed coordinate of the four-ligand problem.”

It is at 45° to the two fold axes only on the symmetric line. At ten degrees and zero it is at 90°; at fifteen and five it is at 59.52°; at twenty and ten it is at −72.76°. The direction is a property of where the molecule sits.

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

“A molecule distorting along the blind direction keeps its gap.”

It keeps it to first order only. Three degrees along the direction from a ten-degree symmetric fold moves the gap by 0.55 per cent, and from fifteen degrees by 1.04 — upward in every case, so the symmetric line is a valley floor in the gap and a molecule leaving it pays quadratically.

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

“The symmetry argument that predicted an isotropic reading was wrong.”

It was right about each band and applied to the wrong observable. The sum over bands is the quantity the argument governs, and it is constant on the circle to four parts in ten thousand — which is confirmation rather than consistency.

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

“Following one band instead of the maximum removes the anisotropy.”

Ordering the four bands by size at each direction and following the k-th gives curves that vary by a factor of two or more, because an ordering crosses between bands where they cross. An ordering is not a band, and the distinction is the whole mechanism.

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

“The residual in the sum's isotropy is the fourth-order term in the distortion.”

A fourth-order origin makes the relative residual scale as the square of the amplitude. Measured across a sixteen-fold range, the residual scales as amplitude to the power 1.248, and a single power describes it to within 7 per cent over that whole range. The predicted exponent of 2 is 0.75 away from the measurement, which is five times the scatter.

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

“The residual is the solver's own precision showing through.”

A precision limit does not move with the amplitude. This one falls by a factor of 32 as the amplitude falls by a factor of 16, from 3.31 × 10⁻³ to 1.05 × 10⁻⁴. It is a term in the expansion, whatever order it is.

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

“A ring with a twofold axis through two of its bonds would test whether an orbit of torsions can be reversed.”

A twofold axis through a bond's midpoint carries the torsion about that bond onto itself end for end, and being proper it cannot change a dihedral's handedness. Benzene's ring torsion is fixed by four of its group's twenty-four operations, and the two that reverse it are the mirror bisecting the bond and the plane of the molecule.

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

“Torsions complete any coordinate set that stretches and bends leave short.”

They complete ethene, benzene and hydrogen peroxide. Ferrocene's iron is 2.064 ångström from every carbon, beyond the 1.85 at which bonds are listed, so it has no coordinates at all: torsions take the rank from 44 to 48 of 57, and only bonds to the iron reach 57.

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

“Completing a coordinate set brings the old orbit formula closer to the answer.”

It moves it further away. Benzene goes from −4 to −22 and each ferrocene from −11 to −66 against true counts of 2 and 4, and ethene, which the formula had right, comes out wrong.

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

“The cheap diagnostics all fail because of one pathological corner of the square.”

The five testable candidates fail on five distinct pairs involving ten distinct systems, and no system appears in more than one failure. There is no small set whose removal leaves the diagnostics working.

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

“The near-end cut is a second window with a second exponent hiding in it, so the quantity carries two arbitrary choices.”

From six bonds outward the exponent moves by between 0.5 and 9.6 per cent of itself, against a range of 0.41 to 0.66 across the stiffnesses — a sixty per cent effect. The quantity carries one arbitrary choice, and sweeping it is what establishes that.

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

“Since the choice does not matter, the first few bonds could have been kept.”

Fitting from two bonds gives 0.403, 0.429, 0.452, 0.478, 0.499, 0.515, 0.526, 0.535, 0.536 and 0.534 — every one lower than its value at six, and compressed into a range of 0.133 against 0.247. The rule of thumb is avoiding a real systematic error, not a hypothetical one.

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

“Every stiffness can be fitted from the same range of starts.”

A fit needs at least six local rates and the soft chains run out of visible tail early: K = 1.1 admits no start beyond fifteen bonds while K = 1.4 and above reach thirty. Comparing across stiffnesses at a fixed large start would be comparing fits with different numbers of points behind them.

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

“The three windows in this fit are all plateaus, like the two that have been swept.”

The third is not. Sweeping the read fraction from a tenth to a half moves the fitted exponent by 13.3 per cent at K = 3.1, 9.9 at 2.8 and 5.3 at 2.5. The two windows already swept move their exponents by under ten per cent across their whole range and mostly by far less; this one moves more than either, on the cases quoted at the upper end of the range.

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

“Opening the window fixes the affected cases.”

It fixes one of five. A profile that ends on its own runs 13.26 of its own decay lengths, the same to within 3 per cent across every case where the quarter rule was not binding. At half the chain K = 2.5, 2.8 and 3.1 have run 9.2, 6.5 and 5.2 lengths, so they are still cut — and K = 2.2, which reaches 12.5, runs only 9.1 once it is counted to the bond where its local rate turns back, so it is cut too. Their exponents are not mis-measured; they are unmeasured, and no fraction of a 320-site chain measures them.

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

“The exponents move in whichever direction the extra points happen to push them.”

Every one of the five affected exponents falls. The published upper end of the range, 0.6586 at K = 2.8, becomes 0.5932. A window that truncates a decay whose local rate is still falling systematically overestimates the exponent, so the direction is forced and the published range's top was the window rather than the chain.

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

“The second ionisation energy is one candidate among several that the capacity might have been a function of.”

It is the worst of the six by a factor of twenty-four over the best, and by thirty-nine times the floor that fourteen atoms allow. Its rank correlation with the capacity is −0.5033, so the relation a rising straight line was fitted to is a falling one.

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

“A pair test can find the quantity the capacity really is.”

The capacity comes from a cubic fitted through the dication, the cation, the atom and the anion, so every input the model can offer is either one of that cubic's three coefficients or one of the three energies it was fitted through. The three coefficients rank ahead of the three data with no interleaving, which is the test behaving rather than a discovery.

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

“Frustration decides the sign of the free direction's exponent.”

Four frustrated clusters with even spin counts — a tetrahedron, a square with one diagonal, an octahedron and a triangular prism — give +0.2084, +0.2136, +0.1924 and +0.2048. The frustration account predicts a negative exponent for every one of them, and none is negative.

Tested in The frustrated cluster with an even count · the series on magnetism

“A design of chains and rings alone could have settled this.”

Every one of its eleven cases had frustration and odd parity aligned or both absent, because an odd ring is frustrated, an even ring is not, and a chain never is. A frustrated system with an even count needs a graph that is neither a chain nor a ring, and its solver could not build one.

Tested in The frustrated cluster with an even count · the series on magnetism

“The sign follows the ground multiplicity, which is the mechanism the parity rule appeals to.”

A tetrahedron's ground state is two singlets at one energy, so its degeneracy is two where every other even cluster's is one — and its exponent, +0.2084, sits with theirs rather than with the odd clusters'. What the sign follows is the ground state's total spin: zero for every even cluster here and one half for every odd one.

Tested in The frustrated cluster with an even count · the series on magnetism

“So the exponent is a function of the spin count and nothing else.”

Topology moves its magnitude by up to 38 per cent at a fixed count. A trigonal bipyramid of five gives −1.1809 against a ring of five's −0.8573, and the four four-spin clusters span 0.2084 to 0.2871. What no topology moves is the sign.

Tested in The frustrated cluster with an even count · the series on magnetism

“The crossing's field converges to four hundredths of the zero-field s–p gap.”

With the defect removed the problem has no λ in it, and its minimum sits at 0.0399865255. The swept sequence passes 0.04000 between the two smallest defects and is still falling; a line through those two reaches 0.0399869 at zero.

Tested in The crossing nothing couples · the series on representation

“The limit is most likely a ratio of the shell's own angular integrals, like the four coincidence angles.”

It is the field at which two eigenvalues have equal slopes — one the lowest root of a cubic, the other 11/10 − √(1/100 + 81f²/4) — with the squared dipoles 54, 27 and 81/4 as coefficients. That is an algebraic number, and not a ratio of two integrals.

Tested in The crossing nothing couples · the series on representation

“The whole shell has one avoided crossing, at the minimum of its s–p gap.”

The two levels at that minimum are the lowest m = 0 state, 92.5 per cent 3s, and the lowest |m| = 1 state, 74.3 per cent 3pₓ. The field commutes with the angular momentum about its own axis, the z matrix has no element between those blocks, and the minimum is where their induced dipoles are equal, −3.93340 bohr each. Nothing avoids anything.

Tested in The crossing nothing couples · the series on representation

“A Morse curve built from a molecule's measured ωₑ and ωₑxₑ reproduces its vibration–rotation constant.”

Averaged exactly over its solved states it gives αₑ of 0.27747, 0.10240, 0.01674 and 0.68164 cm⁻¹ for HCl, DCl, CO and HF, against measured 0.3072, 0.1133, 0.0175 and 0.798 — ratios of 0.903, 0.904, 0.957 and 0.854.

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

“The shortfall is a numerical error in the average.”

The average agrees with Pekeris's closed form for the same curve to 0.04 per cent or better in all four, and six hundred grid points instead of four hundred move HCl's value by under half a per cent. The formula and the solver describe the same curve; the curve is what misses.

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

“The mismatch could belong to either molecule's data rather than to the potential.”

HCl and DCl share one Born–Oppenheimer potential, and their measured constants imply the same cubic Dunham coefficient, −2.3646 and −2.3644. Their Morse curves imply the same one too, −2.2330 and −2.2334, and miss by the same fraction. Two masses, one miss.

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

“That the two models agree on the symmetric line is a coincidence worth measuring.”

It cannot be otherwise. Exchanging the two trans pairs maps the arrangement to itself, and the repulsion is a function of the arrangement alone — so it inherits the same parity and the same vanishing first derivative. The two null directions agree to the last representable bit at every amplitude, and no computation could have made them differ.

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

“So a blind direction found in one model can be expected to be blind in another.”

Only when a symmetry produces it. Off the symmetric line the gap's null direction and the repulsion's have alignments of 0.3425, 0.0242 and 0.8026 — the second is all but perpendicular — and there is no geometry off the line where they agree.

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

“The square plane is the natural place to compare the two models.”

Neither has a gradient there. Both fold coordinates change sign under reflection through the plane and both models are even in each of them, so every direction is null in both and the comparison has no content. The comparison is declined rather than reported as an angle.

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

“The 1.62 per cent left after matching may be the even-site rounding rather than a departure from scaling.”

At a target of five the rounding leaves 6.48 per cent un-matched and the collapse residual is 2.51 — the collapse is tighter than the mismatch it is supposed to be limited by. The rounding bounds the residual loosely at one end and not at all at the other, and it cannot be what is being measured.

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

“The collapse can be run at any target.”

Below about one, two stiffnesses round onto the same ring — at a target of 0.5 the sizes are 2, 4, 4, 6, 8 — and two identical rings agree with each other perfectly. A target that produced them would report a tighter collapse for a reason that has nothing to do with scaling, and the right response is to report the collision rather than collapse the two into one case.

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

“Six misclassified pairs is the floor for a rule of this kind, because the difference is what decides a sign.”

Three is reachable, using the same data. Adding the inter-table dispute as a second variable and taking the best straight boundary in the plane brings the count from 6 to 3, and the exact one-dimensional threshold is used at each slope so the comparison is between two optima rather than between an optimum and a guess.

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

“The right two-variable rule separates them completely.”

No straight line in the plane does. Of 401 slopes swept from −2 to 2, none puts every disputed pair below the line and every agreed pair above it. The best possible is 3 wrong of 153, so the exceptions are not an artefact of projecting onto the wrong axis.

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

“Enough tables would eventually show whether the rule has a real exception.”

Independent tables would need ten. At the measured correlation the requirement is seventy-three, a bootstrap over the pairs puts the correlation anywhere from 0.11 to 0.55, and at a correlation of a half not even four hundred tables bring the chance of a spurious agreement below five per cent.

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

“Four tables are four independent opinions.”

Over the nineteen pairs with the smallest differences, Allred–Rochow and Allen correlate at 0.92, Pauling correlates with both at about 0.7, and Mulliken runs slightly against both. The participation ratio of that correlation matrix counts 2.02 effective tables.

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

“The eight two-state estimates scatter around the shell's avoided crossing.”

They are projections. With the field along its own axis the same formula gives three estimates, along x three different ones, and at the tilt eight, none equal to an axial one — while the exact spectrum is the same in all three directions. The eight described the axes the functions were written along.

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

“The nearest estimate, two per cent from the coupled minimum, was estimating that minimum.”

Moving the d level moves the estimate up in proportion and the minimum down: 0.0327 at a small offset, 0.0196 at the shell's own fifth, nothing past a quarter. The two curves cross at an offset of 0.2013, and at an offset of 0.001 they differ by a factor of 340.

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

“A two-state estimate locates where its pair of levels comes closest.”

Two coupled levels separate as √(Δ² + 4f²d²), which never falls. The p₁–d₁ pair and the s–p₀ pair on its own have no minimum at any field, and the one minimum between coupled levels needs its block's third level.

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

“The two-parameter well is simply too crude, and any error is spread across everything it computes.”

It is not spread. The zero-point energy comes out at 586.81 wavenumbers and the harmonic frequency at the minimum at 1226.6, both within a few per cent of what a fitted potential gives; four states sit below the barrier in both. Only the splittings are wrong, and they are wrong by 70 and 91 per cent.

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

“The difference between the two wells is where their barriers are tallest.”

At the ground state's own energy the two barriers are 0.2591 and 0.2607 ångström wide at half height — a difference of six parts in a thousand. Their actions are 5.6872 and 6.1229, and e raised to that difference is 1.546, which is nine tenths of the factor of 1.70 in the splitting. The area differs where the width does not.

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

“The accuracy line is a convention, so moving it rescales the verdicts without changing their shape.”

It changes which size is the answer and whether there is one. The window in which the published answer holds runs from 0.9926 to 1.7224 times a kilocalorie a mole; below 0.9926 no basis size is usable at every separation. A convention seven tenths of a per cent stricter erases the finding.

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

“The four answers are a scatter, so the sweep is reading noise in a marginal comparison.”

They are a staircase. The usable size falls 4 → 3 → 2 → 1 in order, no size repeats, and each stretch occupies a comparable width on a logarithmic axis — roughly a fourfold loosening of the line per basis size. That is the structure a threshold cutting across quantities that fall geometrically with basis size produces.

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

“The far end of the fitting window is the one that matters, because it runs into the noise floor.”

Inside the ends each profile actually reaches, the exponent moves by 0.00, 0.12, 0.23, 0.72, 0.87, 0.50, 1.24, 2.89, 3.52 and 3.02 per cent of itself. The worst is 5.3 per cent of the published value, against the near end's 9.6. Both windows are plateaus and the near one is slightly the worse.

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

“The sign of the exponent is a property of the spin count.”

A star of four spins and the complete bipartite graph on two and four are even and unfrustrated, and their unequal sublattices give each a ground spin of one. Their exponents are −6.841 and −3.812, where the parity rule predicts positive. Across sixteen clusters there is no coupling from 2 to 1000 cm⁻¹ at which the parity of the count predicts every sign.

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

“The exponent that has been followed is the free direction's.”

It is the third singular direction's, fixed to between 1.0 and 16.1 per cent. The free direction is the fourth, fixed to between 11.5 and 121 per cent. The two lie in one plane and their exponents are negative reciprocals to within three per cent, so every sign reported for the third is the opposite sign for the free one.

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

“The capacity's ordering across the atoms is a property of the quantity rather than of the fit.”

A fifth point changes the verdict on six of fifteen atoms. Carbon, nitrogen, fluorine, silicon and chlorine go from a finite capacity to an unbounded one and beryllium goes the other way, so a third of the set is reclassified by one more measurement in a fit that was already exact through its four.

Tested in Six of fifteen change verdict · the series on electronegativity

“Whatever a fifth point does, it will do it evenly across the set.”

The five it unbounds are exactly the five largest finite capacities the cubic reports — nitrogen at 31.5, chlorine at 6.28, silicon at 4.72, fluorine at 3.98 and carbon at 3.20. The seven that stay bounded keep their order exactly, and five of those seven still move by more than a fifth.

Tested in Six of fifteen change verdict · the series on electronegativity

“Beryllium is one of the three atoms with the largest capacities, two of which are unbounded.”

Under five points beryllium's capacity is 0.0122 of an electron, the smallest in the whole set. It crosses from one end of the ordering to the other on the addition of a single measured energy.

Tested in Six of fifteen change verdict · the series on electronegativity

“A larger molecule has a larger blind spot in its force field, since it has more constants and the same kind of redundancy.”

Methane has ten flat directions in fifty-five constants and boron trifluoride seven in twenty-eight — eighteen per cent against twenty-five. One redundancy each, and the smaller molecule loses the larger share, because a three-angle centre spreads its redundancy over a smaller field.

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

“The number of unmeasurable constants is the number of redundancies.”

Each molecule has exactly one redundant coordinate and the flat spaces are ten-dimensional and seven-dimensional. A redundancy is one linear relation among coordinates; what it costs in constant space is a relation among every product of coordinates that involves it, which is many.

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

“Every coordinate of a planar molecule is caught by its angle-sum redundancy.”

The out-of-plane coordinate is not. Its own constant is fixed on its own and so are all three of its couplings to the stretches, because the redundancy is the statement that three angles at a planar centre sum to a fixed value and a wag is not one of those angles.

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

“The shell's own offset of one fifth is a property of the n = 3 shell too.”

It is one fifth at every shell. As the quantum defect goes to zero it tends to λ/(l + ½), so the three levels' shifts are in the ratio 2 : ⅔ : ⅖ whatever n is — an s–p gap of 4λ/3 and a p–d gap of 4λ/15 — and their ratio carries no n at all.

Tested in The quarter, generalised · the series on representation

“A shell with a smaller threshold would have its minimum only marginally.”

The second shell's threshold is exactly zero and not small. Its m = 0 levels are s and p and stop there, so there is no third level for the push that makes a minimum, and the numerator of the ratio is absent rather than tiny. No quantum defect produces a minimum there.

Tested in The quarter, generalised · the series on representation

“The third shell's minimum is a special case, found because that shell was the one available.”

It is the first case. The m = 0 levels of a shell number n and a minimum needs three, so n = 3 is the smallest shell that can have one — and every shell above it has one with more room, from twenty per cent of the threshold at n = 3 to nearly half at n = 10.

Tested in The quarter, generalised · the series on representation

“Two published inversion barriers that differ by ten per cent are close enough to be treated as the same number.”

The splitting goes as the barrier to the power −3.56, so ten per cent on the barrier is thirty-six on the splitting. Between 2020 and 2262 wavenumbers — the published value and the one a fit to both measured splittings returns — the computed ground splitting runs from 1.3508 to 0.7935, a factor of 1.70.

Tested in The exponent that runs both ways · the series on inversion

“That sensitivity makes the barrier a badly determined quantity.”

It makes it a well determined one. The same exponent divides in the other direction: a splitting known only to a factor of two fixes the barrier to nineteen per cent, and ammonia's splitting is known to five figures. The exponent is a problem for prediction and a gift for inference, and the two are usually cited the wrong way round.

Tested in The exponent that runs both ways · the series on inversion

“The seventy per cent over-prediction of ammonia's splitting is an error in the reduced mass.”

The same potential solved for ND₃, whose only difference is a reduced mass 1.70 times larger, over-predicts by 1.61 against ammonia's 1.70. An error that survives a seventy per cent change of mass and moves by five per cent is not the mass's.

Tested in The exponent that runs both ways · the series on inversion

“The splitting is a power law in the barrier, with exponent −3.56.”

The local slope is −2.5325 at 1212 wavenumbers, −3.5628 at 2020, −4.5692 at 3030 and −6.1504 at 5050. It more than doubles across the swept range, and a single straight line through the sweep leaves a worst residual of 0.90 in the logarithm — a factor of 2.5, which is larger than the whole seventy per cent discrepancy in ammonia's splitting.

Tested in The exponent that runs both ways · the series on inversion

“The barrier, the reduced mass and the pyramid height are three independent sensitivities.”

Two rescalings of the equation fix two of them. Multiplying the mass by λ and dividing the coupling by λ divides every eigenvalue and leaves every eigenfunction, so the mass exponent is exactly one below the barrier's; substituting x = αy leaves the barrier invariant and forces the geometry exponent to be exactly twice the mass's. Checked at four barriers, the two hold to six decimal places.

Tested in The exponent that runs both ways · the series on inversion

“So the ordering by fragility is not worth quoting at all.”

Its top is safe for a reason no numbers can disturb. The two most fragile claims both depend on the Hückel eigenvalue, so any correlation in that predictor multiplies both prices by the same factor and their ratio stays 2.933 at every value. No per-predictor structure reorders them, whatever it is.

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