Simply false — page 4 of 5

“The claims that can be reordered are reordered only by implausibly strong correlation.”

The easiest swap needs ρ = 0.154 in one predictor with the other independent. That is a weak correlation and an entirely ordinary one between two quantities measured in the same literature with the same instruments. It is the middle of the ranking that a per-predictor structure disturbs, at a cost nobody would call implausible.

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

“Displacing the collection's most fragile claim needs a correlation of about the same size.”

It needs ρ = 0.954 — near-perfect correlation — because the nearest challenger from a different predictor is nearly five times less fragile. A factor of six separates the two thresholds, so the ordering's top and its middle have quite different standing and quoting them together conceals that.

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

“Giving each donor the effective charge its own formal charge implies would move the computed π/σ ratios towards the fitted parameters, since a real ligand is an ion and the sweeps below used neutral atoms.”

Three of the five ratios move and all three move outwards. Chloride goes from 0.9456 to 1.4827 against a fitted 0.16, fluoride from 0.3052 to 0.3373 against 0.14, and cyanide from 0.5013 to 0.6562 against −0.10. Water's and ammonia's donors are formally neutral, so they do not move at all. Not one of the five moves nearer.

Tested in The correction that moves three of them backwards · the series on ligand field

“A donor's own contraction is a wider lever than the metal's, so somewhere inside it the fitted values must be reachable.”

It is wider — 2.45 units of effective charge against the metal's 1.40 — and every window still lies entirely above its target. The smallest π/σ ratio chloride's whole chemistry can produce is 0.2674 against a fitted 0.16, fluoride's is 0.2146 against 0.14, and water's is 0.2297 against 0.10: short by 1.67, 1.53 and 2.30 at the best each donor can do.

Tested in The correction that moves three of them backwards · the series on ligand field

“The donor sweep is the metal sweep again with the other end moved, so it can say nothing new.”

The metal's window moved the five ratios by 2.07, 2.09, 2.16, 2.17 and 2.18 — one factor, so no charge could reorder them. The donor's moves them by 1.57, 2.05, 3.15, 5.54 and 9.51, a six-fold difference between ligands. This correction reorders the series where the other could not, and reorders it into a different wrong order.

Tested in The correction that moves three of them backwards · the series on ligand field

“A charge outside the window is still a charge, so the search returning one means the repair exists in principle.”

A metal's effective charge can be pushed past Slater's window by supposing a core electron screens less. A donor's cannot be pushed past its own bare nucleus, which is where its window ends — 7.00 for chlorine, 7.30 for fluorine, 6.30 for oxygen. Chloride's answer of 9.28 would require 2.28 of chlorine's ten core electrons to stop screening altogether.

Tested in The correction that moves three of them backwards · the series on ligand field

“Half filling is the easiest composition to search because on a bipartite net the two-colouring is the unique arrangement with every bond unlike.”

A triangular net has no two-colouring and at the smaller contrast its half-filled composition has one basin reached by all two hundred starts — as uniquely easy as the square net's. And the arrangement it reaches makes thirty of forty-eight bonds unlike, against a counting ceiling of thirty-two, so it is not even the arrangement a count would pick.

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

“Removing the two-colouring should remove the recovery at half filling.”

All four sweeps have the same shape — easy at the sparse end, hardest between a quarter and a third full, easy again at half filling. Both nets at both contrasts have the half-filled composition no harder than their own hardest point, so the finding the explanation was for holds on a net the reasoning does not cover.

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

“On a frustrated net the arrangement with the most unlike bonds is the one the energy picks.”

At half filling and the larger contrast two arrangements attain the counting ceiling of thirty-two exactly and differ in binding by 1.8 × 10⁻², and a third arrangement making only thirty unlike bonds sits between them. So the count neither identifies the optimum nor orders the arrangements.

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

“Refitting a redundant force field with two isotopologues rather than one would reduce the spread the projection leaves.”

It produces a runaway the single fit does not have. With methane's bend–bend constant restored, a fit to CH₄ alone converges to a C–H constant of 5.4248 and a bend–bend of −0.0891; a fit to CH₄ and CD₄ together walks to −97596.4 and −97596.9 in two constants, equal to four figures and opposite in effect.

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

“More data constrains a fit, so a second isotopologue must at least not hurt.”

A redundancy is a combination of force constants that displaces no atom, so it is invisible at every mass. Moving methane's field ninety times its own norm along that direction — putting a constant at 289 mdyn per ångström — changes no frequency of CH₄ or of CD₄ by more than 1.67 × 10⁻⁶, and the two molecules give the same number.

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

“The residual left by that move is a small real curvature the extra data could grip.”

It is round-off. Making the move ten times smaller makes the residual ten times smaller, not a hundred times, on both isotopologues — which is how a difference of large numbers behaves and is not how a quadratic does.

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

“A bond list needs one distance cutoff and a clause for the awkward cases.”

No cutoff works at all. The longest bond in this collection is a platinum–chlorine bond at 2.3200 ångström and the shortest pair that is not a bond is water's two hydrogens at 1.5144 — so the two populations overlap by a factor of 1.53 and every cutoff is wrong about something.

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

“A rule from covalent radii would need its own special cases, since a metal is not a halogen.”

It needs none. Every bond drawn here has a length at most 1.1174 times the sum of its two covalent radii and every pair that is not a bond has one at least 1.2112 times it, so every tolerance from 1.1174 up to — but not including — 1.2112 classifies all twenty-three molecules correctly, including hydrogen peroxide and without the clause.

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

“The window is narrow enough that the choice of radii would decide the answer.”

The window is a factor of 1.0839 wide and the two edges are set by molecules at opposite ends of the collection — hydrogen peroxide's O–O bond above and bromochlorofluoromethane's chlorine–bromine contact below. Neither involves a transition metal, so the one genuinely ambiguous radius in the table decides neither edge.

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

“Phosphine does not invert because phosphorus is heavier than nitrogen.”

Substituting phosphine's reduced mass into ammonia's well and changing nothing else takes the splitting from 1.3508 to 0.8366 wavenumbers — a factor of 1.61. Its pyramid height alone costs 9,641 and its barrier alone 93,992. The mass is the smallest of the three contributions by four orders of magnitude, and it is the only one the usual account names.

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

“A heavy enough central atom would suppress the inversion by mass alone.”

It cannot, at any mass. The umbrella coordinate's reduced mass is 3mₗmₓ/(3mₗ + mₓ), with mₗ a ligand's mass and mₓ the central atom's, which is bounded above by three times the ligand mass — 3.0235 unified mass units for three hydrogens, against ammonia's 2.4866. An infinitely heavy centre lowers the splitting by a factor of 2.55, and the fall being explained is 5 × 10¹⁵.

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

“The three contributions multiply, so the whole is the product of the parts.”

In the action they would add. They do not: the three one-at-a-time costs are 0.4304, 8.5039 and 12.3896, summing to 21.32, against 36.24 for all three at once. The three reinforce each other by 70 per cent, because each of them moves where the turning points are.

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

“The ionic model's error on a separation has no identifiable pattern.”

It has a clean one. Counting how many of the pair's two ions have a third-row outermost p shell sorts the six errors into three groups that do not overlap: −13.6 and −14.9 per cent at zero, +0.6 to +6.7 at one, +17.9 at two. The narrowest gap between adjacent groups is 11.2 percentage points.

Tested in The error was the row, not the charge · the series on contour

“The charge is the likely culprit, since the model treats a double charge by squaring a Coulomb term.”

The charge separates nothing. The doubly-charged pairs run from −14.9 to +6.7 per cent, spanning 31.4 of the 32.7 points the whole set covers, and interleave with the singly-charged pairs throughout. Mg²⁺O²⁻ sits with Na⁺F⁻ and Ca²⁺O²⁻ with K⁺F⁻, in each case because the shells match and despite the charges differing.

Tested in The error was the row, not the charge · the series on contour

“Six points cannot distinguish two explanations.”

These six can, because the two labels cross: both charges appear at more than one shell count, so each label varies while the other is held. That is checked before either is tested — a set of pairs in which charge and shell moved together would make the comparison vacuous however clean the result looked.

Tested in The error was the row, not the charge · the series on contour

“So the model's error is a fixed offset that could be absorbed into the radii.”

It changes sign. The model puts two compact second-row ions about fourteen per cent too close together and two diffuse third-row ions eighteen per cent too far apart. No shift of any radius produces an error that reverses with the shell, because a radius shift is additive and this is not.

Tested in The error was the row, not the charge · the series on contour

“An acceptor ligand's π parameter comes out negative because its empty π* lies closer to the metal level than its filled π does.”

It does not lie closer. At a metal level of −4.75 eV, cyanide's filled π is 8.86 eV below and its π* only 7.01 eV above — but carbon monoxide's π is 12.16 eV below against 6.25 eV above, and dinitrogen's 12.23 against 7.05. The gaps favour the donor channel in all three, so if the two overlaps were equal every one of them would come out a π donor.

Tested in The channel that points at the metal · the series on ligand field

“The far atom of a diatomic ligand cancels part of the π* overlap, which is what separates the two channels.”

It contributes between 4.7 and 9.6 per cent of the near atom's overlap and it does cut the π* and add to the π. What decides the comparison is the near-atom coefficient: 0.394 in cyanide's filled π against 0.971 in its π*, a factor of 6.06 in the square. The far lobe modifies that by a tenth.

Tested in The channel that points at the metal · the series on ligand field

“A two-channel model lifts the refusal that kept ammonia out of the single-channel one.”

It lifts the arithmetic half — a difference of two positive terms can be zero. It does not reach ammonia, whose nitrogen has no π level to donate from and no π* to accept into, its other two p functions being in N–H bonds. The model computes a donor term of 1.39 × 10⁻² on a p orbital the molecule has already spent, and there is nothing to subtract from it.

Tested in The channel that points at the metal · the series on ligand field

“The chemical capacity might turn out to be a function of atomic size, since the quantities it was tested against were all inside the fit that produced it.”

The worst adjacent pair in the mean valence radius — lithium and iodine, neighbours in size — differs in capacity by a factor of 78.7, against a control floor of 4.95. That is sixteen times the floor, and the second-worst pairs are comparable, so the failure is not one outlier.

Tested in A size the fit was not made from · the series on electronegativity

“The verdict depends on which definition of size is used.”

The mean radius and the root-mean-square radius order a few of the atoms differently and give the same worst pair and the same ratio of 78.7. Their median adjacent ratios are 6.92 and 6.20. A single definition would have left that untested; two agreeing is what makes the answer about size.

Tested in A size the fit was not made from · the series on electronegativity

“Size and capacity are unrelated, since the pair test rejects size.”

Their rank correlation is −0.433, which is a respectable association — both quantities track the periodic table. The pair test asks something stronger and the correlation does not survive it, which is the same lesson a rank correlation of 0.857 taught three essays earlier when a control that could not be a mechanism matched it.

Tested in A size the fit was not made from · the series on electronegativity

“Once the redundancy is projected out, what four fits still disagree about is a search or rounding residual.”

Every one of the four fits reproduces all six frequencies, and a fit with a convergence criterion a million times tighter and eight restarts moves the stretch–stretch difference by seven ten-thousandths. The four sit within a few thousandths of a traced curve of exact fits; their 0.115 spread in the stretch constant is a distance along that curve.

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

“A large quantum defect gives a shell more room below its threshold, not less.”

That followed from the screening model's defects keeping the ratio 2 : ⅔ : ⅖ as they grow. Measured defects do not: the ratio of the p–d difference to the s–p difference, which each shell's offset approaches, is 0.128 for lithium and 1.70, 3.06, 2.68 and 2.29 for the heavier alkalis, against the model's one fifth.

Tested in Four alkalis the model cannot hold · the series on representation

“Where the gap fails to name the winner it is a small numerical disagreement between nearly equal arrangements.”

The frustrated net's two contenders, held fixed, exchange binding at a contrast of 3.790 and exchange gap at 4.849. Over that whole interval the one that binds better has the narrower gap, by up to 0.14 at a contrast of 4.

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

“The contrast at which a second basin appears at half filling is where the count of unlike bonds stops picking the winner.”

The count failed on the square net above 1.5272. A second half-filled basin appears on the square net at 1.237 and on the triangular net at 2.714, the first reached by five starts in two hundred. Neither is within a quarter of the count's threshold.

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

“The reduced mass of an umbrella coordinate is a defined quantity.”

Three constructions are defensible and none is derivable from a measurement. The apex against a rigid ligand plane gives 2.4866 unified mass units, the moving atom's own mass gives 14.0031, and holding the bonds at their measured length gives a function of position running from 2.4866 to 2.9874. The splittings they produce are 1.35075, 5.5 × 10⁻⁶ and 0.94420 wavenumbers.

Tested in The mass nobody chose · the series on inversion

“A position-dependent mass can be replaced by a suitable constant.”

It sits nowhere near the average. The two constants at the ends of its range give 1.35075 and 0.56237 wavenumbers; the varying mass gives 0.94420, which is 1.68 times the heavy constant and 1.43 times off the light one. A tunnelling integral is taken across the middle of the barrier, which is where this mass is lightest, so the answer is set by where the mass is rather than by its average.

Tested in The mass nobody chose · the series on inversion

“The usual construction is safe, since the honest one is a small correction.”

It changes the splitting by 43 per cent and moves it towards the measurement — from 1.7023 times the measured value to 1.1899. A correction that improves the agreement by that much is not a correction that can be neglected; it says the standard choice is the one in error.

Tested in The mass nobody chose · the series on inversion

“The choice does not affect a barrier fitted to the measurement, since the fit absorbs it.”

The barrier the same well infers from the same measured splitting is 2330 wavenumbers under the usual construction and 2117 under the bond-conserving one — a spread of 10 per cent, comparable to the whole disagreement between published values. And the third construction cannot reach the measurement at any barrier between 200 and 40,000, which is reported as a refusal.

Tested in The mass nobody chose · the series on inversion

“The mixing criterion is a tautology on the lone pairs and was discriminating on the double bond.”

The margins run the other way. The lone pairs clear the boundary by 1.732 and the carbonyl's σ–π pair by 2.455 — and ethene's σ–π by an infinite factor, since symmetry puts both centroids at the bond midpoint and the separation is 2.5 × 10⁻¹⁹. The case offered as the counterexample is the least marginal of the three.

Tested in The criterion that has never said no · the series on hybrids

“The σ–π analysis of the double bond found the criterion refusing to produce bent bonds.”

That analysis contains no refusal. Its two verdicts are that the σ–π and bent descriptions are one occupied space in two bases, unchanged to 10⁻¹⁵ at every mixing angle, and that the bent pair are sp⁵ hybrids at 50.768° — a quantitative picture rather than a rejected one. The refusal is attributed rather than found.

Tested in The criterion that has never said no · the series on hybrids

“So the criterion is vacuous and measures nothing.”

It refuses two of the five pairs available. The carbonyl's σ–n pair scores 0.111 and its π–n pair 0.739, both from the same three orbitals and the same dipole matrices as the σ–π pair that scores 2.455. The instrument discriminates; what was one-sided is the set of cases put to it.

Tested in The criterion that has never said no · the series on hybrids

“Sweeping the polarisation would find the boundary the double bond sits near.”

Its margin has a floor of 2.412 across s fractions from 0.05 to 0.99, and it rises to 39 at the top of that range. No physical polarisation of a carbonyl brings the pair within a factor of two of the boundary, so there is no boundary inside the range where molecules live.

Tested in The criterion that has never said no · the series on hybrids

“A fourth fragment narrows the window in which a pairwise counterpoise assembly is usable, because it leaves more terms out.”

Asked of three pairs of arrangements that differ by one added centre, the covered share of three decades of accuracy line goes from 44 to 58 per cent with every centre alike, from 27 to 54 with the heavy centres inside, and from 67 down to 60 with the heavy centres outside. Two widen and one narrows.

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

“Whether a fragment is over- or under-corrected follows from where it sits and what it is.”

An end fragment of the uniform chain, at two Gaussians a centre and 1.6 bohr, is under-corrected by 43 per cent with three centres and over-corrected by 18 per cent with four. Same position, same charge, opposite sign.

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

“Writing electronegativity theory on the exact piecewise-linear energy would repair the difficulties the fitted curves have.”

It removes them by removing the quantities. The hardness is a second derivative of straight segments, the capacity a third; neither exists. The chemical potential is a pair of one-sided slopes rather than a number, so there is nothing for equalisation to set equal. Four quantities go and one question stays.

Tested in Four quantities go and one question stays · the series on electronegativity

“The piecewise model at least predicts sensible charge transfer, since it has the right shape.”

On isolated atoms it predicts none, anywhere. Moving one whole electron costs the donor's ionisation energy less the acceptor's affinity, and the cheapest such pair among these seventeen atoms — potassium to chlorine — costs 0.729 electronvolts. That becomes favourable only at a separation of 19.75 ångström, longer than any bond.

Tested in Four quantities go and one question stays · the series on electronegativity

“Adding the ion pair's own attraction gives the model back a quantitative prediction.”

It gives it an integer one. At the measured bond lengths the attraction runs from 5.10 to 15.71 electronvolts and exceeds the transfer cost for five of the eight bonds with measured dipoles, so the prediction swings from no charge to a whole electron with nothing in between. Every measured dipole implies a fraction, from 0.058 to 0.842.

Tested in Four quantities go and one question stays · the series on electronegativity

“So the piecewise model is simply worse than the quadratic one.”

Rounded to the nearest integer its prediction agrees with six of the eight measured charges, with no parameters at all, and it gets the alkali halides and the heavy hydrogen halides on the right sides. It classifies better than a reader would expect and quantifies nothing, which is a different failure from being wrong.

Tested in Four quantities go and one question stays · the series on electronegativity

“A potential carrying the cubic and quartic coefficients the measured αₑ and ωₑxₑ imply will reproduce those measurements.”

Solved on four thousand points and read from its four lowest levels, the quartic with HCl's measured coefficients gives αₑ = 0.3215 against a measured 0.3072 and ωₑxₑ = 50.55 against 52.82. In all four molecules its αₑ lands above the measurement, where the Morse curve's landed below it.

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

“Where two potentials share their cubic and quartic, the difference in αₑ must come from the mean displacement.”

The quartic's mean displacement in HCl's ground state is only one per cent larger than the Morse curve's, and its first difference B₀ − B₁ is smaller, 0.2506 against 0.2802. Its αₑ is larger because its γₑ is +0.0355 against the Morse curve's −0.0014: the truncation shows up in the constant that was supposed to be a small correction.

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

“The identity between orbits and totally symmetric species was established on the molecules that happened to have bond lists, and might not hold for the ones that did not.”

Rebuilt on the radius rule, xenon tetrafluoride, the hexafluoridocobaltate ion and both tetrachloride ions join the census. The number of totally symmetric vibrations equals the symmetric orbits less the totally symmetric redundancies for all four, as for the fifteen before them.

Tested in The census a bond rule was hiding · the series on point group

“A bond list that gets every molecule's bonds right gives every molecule a complete set of internal coordinates.”

Xenon tetrafluoride and the tetrachloridoplatinate ion have the right four bonds each and coordinates spanning seven of their nine vibrations. The two they miss are exactly the out-of-plane A₂u and B₂u species, which stretches and in-plane bends cannot describe at a planar four-coordinate centre.

Tested in The census a bond rule was hiding · the series on point group

“The tolerance chosen in the middle of the radius rule's window is safe against the choice of radius table.”

A second single-bond tabulation gives a window from 1.171 to 1.262, bounded by the same two pairs as the first table's 1.117 to 1.211. The factor in use, 1.163, lies below the second window, where hydrogen peroxide's O–O bond is lost; a factor from 1.171 to 1.211 works under both.

Tested in The census a bond rule was hiding · the series on point group

“A compound whose purity is not known cannot know which side of its own sign pole it sits on.”

From half a per cent monomer to sixteen, the couplings at which the monomer fraction and the temperature-independent term are uncorrelated move by under 1.6 per cent for the two stars of spin three halves and two, under 3.1 for the two clusters of spin one, and under 11 for the three doublets. Where a sample sits relative to them is decided by its coupling, not its purity.

Tested in Purity renames the poles · the series on magnetism

“The share of runs in a resonant pair grows by a fixed amount per decade of chain length and does not saturate.”

Each decade adds −ln(1 − ρ) ξ ln 10 times (1 − ρ)^s, the share not yet resonant. For runs of six at x = 0.5 the rise is 1.74 points a decade over 10³–10⁶, 1.57 over 10⁸–10¹² and 1.20 over 10²⁰–10³⁰, and the share reaches 57.7 per cent by 10⁵⁰.

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

“Half of the runs of six are in resonant pairs at a chain of about 10³¹ sites.”

That is where a straight line through 10³ and 10¹² reaches a half. The exact share reaches it where s = ln 2 / −ln(1 − ρ), which at x = 0.5 is a chain of 10^40.7. At every concentration from 0.2 to 0.9 the straight line puts the half-way length too early.

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

“A plausible range in the gauche energy is a smaller lever than the factor of eleven the rotor conventions spanned.”

It is larger where it matters. The conventions moved the ceiling by 11.0-fold and moved no verdict. The gauche energy's reported range of 2.5 to 4.5 kilojoules a mole moves it by 8.7-fold and moves two of the nine verdicts, because the ceilings it moves are the ones sitting nearest a measurement.

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

“The five-membered refutation is safe, since a factor of eleven in the ceiling did not reach it.”

At 4.422 kilojoules a mole the ceiling under the convention behind the decided verdict reaches 250 and the 250-fold acceleration stops being refuted. That is inside the range the gauche energy is reported over. Only the 11,000-fold case is refuted at every energy under every convention.

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

“Whatever moves here is the spread between published determinations rather than the uncertainty in any one of them.”

The six-membered verdict changes at 3.630, and the standard compilation value is 3.8 ± 0.4. That is 0.42 standard deviations from the centre — well inside a single determination's own error bar — while a band of ±0.02 moves nothing, so the sweep is measuring instability rather than manufacturing it.

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

“The gap in the spread distribution might be the localisation criterion's coarseness rather than a property of the cages.”

Boys localisation, which scores descriptions by orbital centroids in space rather than by atomic populations, leaves a gap of 3.2 × 10⁷ on the same forty-eight pairs — seven decades, from numerical zero to a real spread with nothing between. Pipek–Mezey's is a factor of 210. Two independent criteria both bimodal is a property of the family.

Tested in A second criterion left a gap too · the series on multicentre

“So the two criteria are measuring the same thing and either would do.”

They classify six of the forty-eight pairs differently, and the disagreements go both ways — two are degenerate under Pipek–Mezey only and four under Boys only. A criterion that was simply the other with more resolution would disagree in one direction.

Tested in A second criterion left a gap too · the series on multicentre

“The family's most ambiguous cage is a fact about that cage.”

The eleven-vertex cage at twenty electrons has the largest Pipek–Mezey spread in the family, 5.3 × 10⁻², and under Boys its descriptions are identical to machine precision. The extreme case of one criterion sits on the floor of the other.

Tested in A second criterion left a gap too · the series on multicentre

“A higher-order assembly is a cheaper route to the correction a full calculation would give.”

Under a model that charges one cubic-cost diagonalisation per ghost set, the three-body assembly of four fragments costs 1.64 times the full calculation. It becomes the cheaper of the two only at eleven fragments. The pairwise assembly costs 0.375 times.

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

“The trio's level at the free-atom energy is exact because the two outer orbitals are equivalent.”

Raising the A–C overlap from 0.25 to 0.45 while B's stays at 0.25 leaves no symmetry relating A and B, and the level stays at −13.6 eV to 2.7 × 10⁻¹⁴ eV at every step and at seven energies of the third orbital from −24 to +2 eV.

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

“Any change that makes A and B inequivalent destroys the exact level.”

Changes that make A and B inequivalent to C do nothing. A site-energy difference moves it at 0.5 per electronvolt and a coupling between A and B at 10.20 eV per unit of overlap, and a fourth orbital that sees the pair in a different ratio from C moves it by 0.025 eV — while one that sees it in the same ratio, 1.4 to one, leaves it exact with no symmetry anywhere.

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

“A quantity that responds to the change the level ignores must be tracking the same physics.”

The filled-shell A–B bond order moves at 0.653 per unit of A–C overlap and at nothing per electronvolt of site energy — the exact mirror of the level. It is twice an entry of the inverse overlap matrix and knows the metric, not the energies.

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

“A residual that scales with a non-integer exponent has a non-analytic origin.”

The reversal average and difference split it exactly. The even half's anisotropy scales as the amplitude to 2.005, with a single power describing it to 0.6 per cent; the odd half scales as the amplitude to 0.981 below 0.04. The combined 1.248 lies between them, and the two halves cross at an amplitude of 0.084.

Tested in Two integers made one exponent · the series on spectrum

“The absolute value in |0.75 − ρ| could make the reading non-analytic wherever the ratio crosses three quarters.”

A depolarisation ratio 3γ²/(45ᾱ² + 4γ²) cannot exceed three quarters for any Raman tensor, so the quantity inside the absolute value never changes sign. Across every distortion swept, forward and reversed, the least departure is −1.1 × 10⁻¹⁶, which is rounding.

Tested in Two integers made one exponent · the series on spectrum

“The flatness of the sum over depolarised bands is what an experiment that cannot resolve the bands would measure.”

An unresolved degenerate pair is measured with its intensities added in each polarisation before the ratio is taken. That observable's anisotropy over the plane is 2.9 × 10⁻³ at an amplitude of 0.01, against 1.05 × 10⁻⁴ for the sum — 28 times larger — and it scales as the first power of the amplitude.

Tested in Two integers made one exponent · the series on spectrum

“The upper coupling at which a susceptibility fit's monomer fraction and temperature-independent term decouple scales as the reciprocal of the ground spin plus a half.”

True to a spread of 1.127 across the seven clusters it was found on. Of five clusters built afterwards, only the star of seven lands inside the predicted range: the star of eight is 8.5 per cent above it, K₂,₅ 12.0 per cent below, K₂,₆ 21.6 per cent below and K₃,₅ 0.7 per cent below. Across all twelve the spread is 1.559.

Tested in Five more clusters break the band · the series on magnetism

“The upper separation is where a cluster's susceptibility has become the Curie law of its ground multiplet across the measuring window.”

At the seven separations, χT at 300 K is 2.39, 2.01 and 2.37 times the ground Curie value for the doublets, 1.08 and 1.33 for the spin-one clusters, and 0.81 and 0.63 for the stars of spin three halves and two. A fixed tolerance would give one number.

Tested in Five more clusters break the band · the series on magnetism

“The energy of the first excited multiplet sets where the fit's nuisance parameters separate.”

At the twelve upper separations the first excitation runs from 40 cm⁻¹ on the star of eight to 345 cm⁻¹ on K₂,₃. The star of five and K₂,₃ share a band product of 114.7 and 114.8 cm⁻¹ with gaps of 57 and 345 cm⁻¹.

Tested in Five more clusters break the band · the series on magnetism

“A rule that flags fewer disputed-looking pairs at the same miss rate would bring the panel of tables needed within reach.”

No straight boundary in the plane that misses no disputed pair flags fewer than the published nineteen, over 61,353 boundaries searched. Letting disputed pairs through saves one or two flags each: at eight misses, half the disputed pairs, the rule flags eight and still needs 46 tables at the measured correlation, against 73.

Tested in The rule is not the lever · the series on dipole

“With a narrow enough rule, four tables could establish an exception.”

Four independent tables agree on a sign one time in eight, above the one in twenty a significant exception needs, so a rule flagging a single pair needs six tables even at zero correlation. At the measured correlation of 0.325 it needs fourteen, and at the top of the bootstrap range, 0.547, it needs fifty-eight.

Tested in The rule is not the lever · the series on dipole

“The number of tables needed is set mainly by how many pairs the rule flags.”

At the measured correlation, every rule from one flagged pair to all 153 spans 14 to 215 tables. At nineteen flagged pairs, the correlation's bootstrap range from 0.110 to 0.547 spans 16 to 1,903 — nine times as many tables.

Tested in The rule is not the lever · the series on dipole

“Reachability is a property of the run length.”

It is a line across run length and concentration. At three tenths of the sites low nothing beyond runs of three reaches half inside a cubic centimetre; at half, the line falls between four and five; at seven tenths, runs of seven do at 10^20.1 and runs of eight do not at 10^25.2. Three decades of ρ move the boundary by four sites of run length.

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

“The straight-line reading was a mistake specific to the case where it was found.”

It is early at every run length and by more the longer the run: 0.8 decades early at runs of four, 8.1 at runs of six, 31.2 at runs of eight. The instance the earlier essay corrected was among the mildest of them, because the error grows with how far the extrapolation has to run.

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

“A coordinate set built from every bond and every pair of bonds spans every vibration.”

Not for a square-planar centre. Xenon tetrafluoride and tetrachloridoplatinate have nine vibrations each and their stretches and in-plane bends reach rank seven, because the two out-of-plane species — the centre leaving the plane and the ligands puckering — change no angle in the plane to first order.

Tested in The two coordinates a square plane was missing · the series on point group

“One out-of-plane coordinate would close the gap, as it does for a planar three-coordinate centre.”

Two species are missing, so one coordinate can span at most one of them. The pair used here is the signed distance of the centre from each trans pair's axis: the centre leaving the plane raises both together and the ligands puckering raises one and lowers the other, so the pair spans both. The rank goes from seven to nine, which is two coordinates buying two vibrations.

Tested in The two coordinates a square plane was missing · the series on point group

“Adding coordinates risks the identity between orbits, redundancies and totally symmetric vibrations.”

It survives. The two coordinates form one new orbit, add no redundancy, and leave the count of totally symmetric vibrations at one — because the new orbit is not a symmetric one: the quarter-turn exchanges the two coordinates and their symmetric combination does not survive its own stabiliser.

Tested in The two coordinates a square plane was missing · the series on point group

“The incompleteness was there to be seen in the census.”

Neither molecule was in it. A xenon–fluorine bond is 1.94 ångström and a platinum–chlorine bond 2.32, against a default cutoff of 1.85, so both raise no bonds, no coordinates, and a refusal that the census catches and discards. Sixteen molecules were reported and the two with the defect were silently absent.

Tested in The two coordinates a square plane was missing · the series on point group

“Sweeping the temperature and sweeping the gauche energy are two independent checks on the same verdict.”

The ceiling is a function of g/RT alone, so a change in one is reproduced exactly by a change in the other. Evaluated at five matched pairs the two routes agree bitwise — a relative difference of exactly zero, not a small number — and the temperature range swept spans 1.2248 to 1.8054 in g/RT while the gauche range spans 1.0085 to 1.8153, which contains it.

Tested in Two sweeps and one lever · the series on strain

“So the temperature sweep was wasted work.”

Its subject was not the ceiling. It put an activation-enthalpy term on the measured rate ratio, and that term is a function of temperature and not of g/RT — at 5.308 kilojoules a mole the ratio moves 22 per cent between 298 and 340 kelvin while the ceiling's dependence is already accounted for. The half of that calculation that was new is the half the identity does not cover.

Tested in Two sweeps and one lever · the series on strain

“The temperature sweep and the gauche-energy sweep found two different places where the six-membered verdict gives way.”

They found one. The temperature sweep's 312.1 kelvin at 3.8 kilojoules a mole is u = 1.4644; the gauche sweep's 3.630 kilojoules a mole at 298.15 kelvin is u = 1.4643. Two sweeps, two units, four figures of agreement, and one crossing reported twice.

Tested in Two sweeps and one lever · the series on strain

“The identity is an approximation valid where the ceiling is large.”

It is exact and it is algebraic. g and T appear in ((1+2x)/2x)ʳ only inside x = exp(−g/RT), and nowhere else, so the ceiling is a function of one variable at every rotor count, every energy and every temperature. The agreement is to the last representable bit rather than to a tolerance.

Tested in Two sweeps and one lever · the series on strain

“The cages whose classification depends on the criterion are the ones with a large enough symmetry group.”

The automorphism group separates nothing. The icosahedron's graph has 120 automorphisms — the most in the family — and gives three disagreements out of twelve fillings and nine agreements. The octahedron's has 48 and gives none out of six. The eleven-vertex cage's has 4, the fewest, and gives one out of eleven.

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

“A disagreement is a property of a cage.”

It is a property of a cage and a filling. The icosahedron disagrees at fourteen, sixteen and eighteen electrons and agrees at the other nine fillings, with the same graph and the same symmetry throughout. Any explanation appealing to the cage alone predicts twelve disagreements or none.

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

“A criterion-dependent case is one the two criteria half-agree about.”

In all six, one criterion's relative spread is numerically zero — 4 × 10⁻¹⁴ or exactly nothing — while the other's runs from 7 × 10⁻⁴ to 5 × 10⁻². There is no intermediate case in the family where both see a spread and rank it differently.

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

“The ionic model's error runs with how diffuse the pair's softer ion is.”

An oxide's 2p exponent is 1.925 and a chloride's 3p exponent is 1.917, and the four pairs whose softer ion is one of them have errors from −14.9 to +17.9 per cent. The rank correlation of the error with the softer ion's exponent is −0.36, matched or beaten by 384 of the 720 orderings of six points.

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

“A continuous measure that orders the three middle pairs correctly has thereby shown it carries the pattern.”

Three points have six orders, so an uninformative measure lands on the right one a sixth of the time. Of seven measures tried, two order them — the exponent sum and the ratio of the mean radii — and the second fails the test on all six pairs at p = 0.175.

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

“The collection's fragility ranking is an ordering of its claims, with the most fragile at the top.”

It is a partial order. Five claims fall into chains of two, two and one — one chain per predictor — and only within a chain is the relative order independent of anybody's error correlations. Thirty of the hundred and twenty orderings respect the chains and ninety do not, so the ranking makes thirty times fewer statements than a list of five would.

Tested in Three chains and ninety orderings ruled out · the series on models

“The rigid part of the ranking is the top of it.”

The rigid parts are the chains, and one chain occupies ranks two and four with a claim from a different predictor at rank three. That intruder can be lifted above rank two at a correlation of 0.736 or dropped below rank four at 0.154, so a chain constrains its own members and nothing else — the rigid region is not an interval of the ranking.

Tested in Three chains and ninety orderings ruled out · the series on models

“The top pair being locked is a fact about how much more fragile the most fragile claim is.”

It is a fact about which chain both claims are in. They are the Hückel eigenvalue's discordance and its slope floor, they share the predictor and therefore the factor, and their ratio of 2.93 is untouched by any correlation at all. The headline would survive any numbers whatever, which is a stronger claim than a comfortable margin.

Tested in Three chains and ninety orderings ruled out · the series on models

“The chain's contrast approaching one from U = 32, moving away to U = 128 and returning is the shape of its approach to the limit.”

Every one of the six strongest lines is monotone in weight from U = 19. The turns in the contrast are exchanges of rank: an exact tie at the cut at U = 28.916, the third-ranked line changing identity at U = 17.91 and 32.40, and the strongest satellite changing identity at U = 156.6, where the contrast peaks at 1.0632.

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

“The resonant share's slow rise is a property of how defect couplings fall off.”

It is a property of dimension. The coupling law, the density and the level spacing are identical in the calculation for a chain and for a plane; the only change is that the number of sites within a separation s goes from s to πs². Half the runs of six are resonant at 10^40.8 sites in a chain, 10^3.93 in a plane and 10^3.06 in a solid.

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

“The label is worth nothing where the species repeats.”

Six of the twenty-four exceed 0.9, the best being ammonia's N–H stretch at 0.986 and boron trifluoride's B–F stretch at 0.949. Both are stretches on molecules whose two stretching vibrations of that species are far apart in frequency, which is the condition for the label to carry information.

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

“A distortion spread over a degenerate pair is a distortion the label failed to resolve.”

It is not an ambiguity at all — the partners of a degenerate set are indistinguishable. Summing the shares within each occurrence before comparing between them is what separates the two, and it matters: methane's T₂ has two occurrences of three modes each, so the difference is between a chance level of a half and one of a sixth, and reading the modes individually put methane at 0.522 where the honest figure is 0.746.

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

“The direction the sixteen-electron gap cannot see is a property of an antisymmetric fold.”

It is a property of four equal bonds. At a bond-length ratio of one the gap's slope along the antisymmetric fold is 2 × 10⁻¹⁶ per degree, which is the finite difference's own floor; at a ratio of 1.02 it is 2.15 × 10⁻³ and at 1.5 it is 2.68 × 10⁻². Two per cent of a bond length is enough to remove it.

Tested in One was a symmetry and one was not · the series on electron count

“So the repulsion's fall along that direction was equally fragile.”

It is not. The repulsion's curvature along the fold is −2.87 × 10⁻⁴ at equal bonds and −1.03 × 10⁻⁴ at double the length — negative throughout, shrinking by a factor of three rather than changing sign. And a four-degree excursion lowers the repulsion in both directions out to a ratio of 1.5.

Tested in One was a symmetry and one was not · the series on electron count

“Both results have the same status, since both were computed in the same sweep.”

One is a theorem and one is a measurement. Exchanging the two trans pairs is a symmetry of the arrangement when the bonds are equal, so the gap is an even function of the antisymmetric coordinate and its derivative vanishes identically — no computation could have made it otherwise. The repulsion's curvature is negative because of where the ligands are, and it is the sweep that says so.

Tested in One was a symmetry and one was not · the series on electron count

“The σ interaction's distance law would have to be known before any of this could be said.”

Not for the part that matters. Taking the σ strength as the inverse third, fifth or seventh power of the bond length changes the slope's size at a given mismatch by up to a half and does not move the ratio at which it vanishes, which is one under every law — because a symmetry fixes that and a parameter does not.

Tested in One was a symmetry and one was not · the series on electron count

“The angular shape of the depolarisation reading is a property of the molecule.”

It is the group's. Boron trifluoride and ammonia give circles with the same sixty-degree period — repeating to within 1.8 and 2.3 per cent — and a largest-over-smallest of 1.9779 and 2.0271, within two and a half per cent of each other. Normalised by its own maximum each curve lies on the other.

Tested in The shape was the group and the size was not · the series on spectrum

“So the exponent the residual scales with is the group's too.”

It is 1.2481 for the planar molecule and 0.8155 for the pyramid — on opposite sides of one, so the two do not even agree which candidate term dominates. The essay below's 1.248 was a property of boron trifluoride and was reported as a finding about the probe.

Tested in The shape was the group and the size was not · the series on spectrum

“The pyramid's residual is a power law with a different exponent.”

It is not a power law. A single power describes the planar molecule's to seven per cent and the pyramid's only to fifty-six; the two-term fit returns a quartic coefficient of −0.589, so there is no crossover amplitude to quote; and the residual falls from 1.17 × 10⁻² at an amplitude of 0.057 to 1.09 × 10⁻² at 0.08, which no sum of positive powers does.

Tested in The shape was the group and the size was not · the series on spectrum

“The two molecules differ by one scale factor, since one is a distorted version of the other.”

They differ by two. The departure from three quarters is 169 times larger for the pyramid at the same amplitude and the sum's residual 23 times, so the two quantities are set by different combinations of the force field. A single scale factor would have given the same ratio for both.

Tested in The shape was the group and the size was not · the series on spectrum

“The orphan count is what a symmetry matching produces, so a transition metal has one too.”

It has none, on fourteen of fifteen answerable arrangements. Nine valence orbitals against six ligand combinations means every combination finds a partner; the leftover is three spare metal orbitals at an octahedron, four at a trigonal bipyramid, five at a tetrahedron. Spare minus orphan is nine minus the ligand count at every arrangement, which is arithmetic rather than a finding.

Tested in The leftover changes sides · the series on hypervalency

“So a metal centre can always match whatever a ligand set spans.”

One arrangement in the census refuses. A planar hexagon's six sigma combinations span a B1u component in D6h, and no s, p or d function of a centre transforms as B1u there. It orphans one combination, and reaching it would need an f orbital.

Tested in The leftover changes sides · the series on hypervalency

“The count is constant along the Bailar twist because the twist keeps one point group.”

It passes through three — D3h at the prism, D3 in the interior, Oh at the octahedron — and the spare set is A1' + E', then A1 + E, then T2g. Three decompositions with no species and no dimension pattern in common, and the count is three in all of them, because nine minus six does not depend on what anything is called.

Tested in The leftover changes sides · the series on hypervalency

“Stiffening the repulsion about a common separation tests whether the overlap's range is what sorts the model's errors by shell.”

It cannot fail. Steepened about five bohr, the model's repulsion moves the second-row pairs out by 0.50 and 0.35 bohr and potassium chloride in by 0.33, as predicted. One exponential of one range for every pair, which carries no shell information at all, moves them out by 0.58 and 0.46 and in by 0.34. Under both, the shift falls steadily with each pair's own separation.

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

“The order of the three pairs with one third-row ion is part of the same range defect.”

Contracting the third-row shells moves all three and leaves their order — sodium chloride, potassium fluoride, calcium oxide — unchanged at every factor from 1.0 to 1.5, with their own spread falling only from 6.1 points to 5.2.

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

“The improvement a bond-conserving mass makes to ammonia's splitting could be an artefact of choosing the BenDaniel–Duke ordering.”

The five orderings in use — BenDaniel–Duke, Gora–Williams, Zhu–Kroemer, Li–Kuhn and Mustafa–Mazharimousavi — give 0.94036 to 0.94477 wavenumbers, a spread of 0.47 per cent. The mass construction moves the splitting by 43.1 per cent, ninety-two times as much.

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

“Each ordering has to be solved separately to know what it does.”

Every ordering is BenDaniel–Duke plus s·μ′²/μ³ + c·μ″/μ², so first-order theory needs two integrals over one doublet. On fifteen orderings with exponents between −1 and 0 it reproduces the exact change to within 0.005 per cent of the splitting.

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

“The slope floor's denominator can be priced the way its numerator is, once somebody does the arithmetic.”

It needs three prices, because the three runs are three kinds of object. The spin-only moment's run is a difference of two square roots of integers and no defensible variation moves it at all. The Hückel run is a graph eigenvalue difference, exact given the connectivity and moved only by changing the form of the model. The angle-strain run is computed through a force constant and a reference angle, both conventions.

Tested in A denominator needs three currencies · the series on models

“A denominator that carries no experimental error is a denominator that carries no uncertainty.”

Two of the three floors are cheaper to move through their computed run than through their measured rise. The angle strain's floor halves on an 11.6 per cent change in its run against a 33.8 per cent measurement error, a ratio of 2.91; the Hückel floor's ratio is 1.91. Only the spin-only floor is priced entirely by its measurement, and it is the one whose run cannot move.

Tested in A denominator needs three currencies · the series on models

“The largest slope floor in the collection rests on a run that is one per cent of that predictor's range.”

One per cent is what 0.0308 is as a fraction of the measured range of 3.10 electronvolts. As a fraction of the predictor's own range of 0.5858 it is 5.26 per cent, five times larger. Both numbers are true statements about different things, and the earlier report quoted one and named the other.

Tested in A denominator needs three currencies · the series on models

“An unsweepable denominator should be reported as an infinitely expensive one.”

An infinite price and an impossible one are different claims. √(n(n+2)) on a count of unpaired electrons has no parameter, no basis, no convention and no model form in it, so the spin-only run is not expensive to move — there is nothing there to move. It is reported as having no model price rather than a very large one.

Tested in A denominator needs three currencies · the series on models

“Opening the window to half the chain lets the stiff profiles show more of their tail, so every bond it adds is tail.”

On the four stiffest cases the local rate reaches a minimum and climbs — at 99 bonds for K = 2.2, 85, 75 and 69 for the stiffer three — and the bonds past it are the bulk reference being read. For K = 3.1 that is 88 of the 157 bonds the opened window supplies.

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

“The stiff exponents fall by up to a seventh when the window is opened because the quarter-chain window was truncating a decay whose rate was still falling.”

Fitted only up to each profile's turn, the four stiff exponents sit within 0.8 per cent of their quarter-chain values, and two of them are above those values rather than below. The fall to 0.5564 at K = 3.1 came from bonds past the turn, where the rate rises.

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

“K = 2.2 is resolved once the window reaches the midpoint: it runs 12.5 of its decay lengths, inside the spread of profiles that end on their own.”

Counted to its turn it runs 9.13, against 13.19 for the six profiles with no turn. Thirty-six of its 135 bonds lie past the turn. Six of the ten stiffnesses are measurements, not seven.

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

“A third isotopologue is the first data that can pick one of the two boron-10-consistent fields.”

Not if it keeps the symmetry. Boron of mass 9 or 13, all three fluorines of mass 17 or 21, or boron-10 with three fluorines of mass 21: each gives the two fields E′ frequencies that agree to 1.5 × 10⁻⁶ cm⁻¹. The sum of squared E′ frequencies is exactly linear in the two reciprocal masses, so two symmetric isotopologues already fix everything any third can say.

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

“Changing the mass of the atom that moves in the bend samples the block differently from changing the atom that moves in the stretch.”

It samples it through the same two functionals. Replacing all three fluorines leaves the same two fields as replacing the boron, to the same millionth of a wavenumber; the fluorine term and the boron term are the two coefficients of one plane, and there is no third.

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

“The two arms are a fair sample of the square, so a region where both parameters move behaves like something between them.”

It behaves worse than both. Every cheap candidate's worst error ratio between two interior systems exceeds its worst across the arms: mean-field double occupancy goes from 4.66 to 48.03, the mean-field energy from 3.67 to 43.36, and both are a factor of about eleven. The arms were not a fair sample of the square; they were the part of it where the diagnostics happen to work.

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

“Two of the seven candidates cannot be scored, since their two families never come within fifteen per cent.”

On a cross they cannot: twenty systems give at most a dozen comparable pairs. The grid gives 1,232, and 982 of them between two interior systems. Both previously unscoreable candidates are scored, and the highest occupied level comes out worst of all at a ratio of 627.

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

“A diagnostic with a real correlation to the error is a partial success.”

A grid can ask whether a candidate orders the systems as the error does, which a cross cannot. Three of the cheap candidates reach a rank correlation near 0.77, which is a real relation and useless: two systems it ranks adjacently can have errors a factor of forty-eight apart. The highest occupied level reaches 0.063 — it does not sort them at all — while the quantity that is the error reaches one exactly.

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

“A fold of three ligands is a compromise between the symmetric fold and the blind one, and should be partly blind.”

It is not blind at all. Its first-order change is exactly three quarters of the symmetric fold's per degree of lift, because each lifted ligand contributes one quarter whichever ligand it is. There is no partial blindness: a direction is blind exactly when its lifts sum to zero.

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

“Once unequal bond lengths remove the antisymmetric fold's blindness, the square plane has no blind direction left.”

Two survive. With one trans pair two per cent longer the antisymmetric fold's slope is 2.9 × 10⁻³ per degree, two and a half times the symmetric fold's, and the tilt of either pair stays at the difference quotient's floor at every ratio tried, up to thirty per cent — because swapping the two ligands of one pair is still a symmetry.

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

“A geometry with no answer is a geometry whose symmetry the finder could not establish.”

Three unrelated things produce no answer. The finder failing its tolerance accounts for 25 of the 27; a continuous group, which is declined deliberately rather than missed, accounts for the excluded path; and a finite group with no character table accounts for 2. Only the first is about how nearly symmetric a geometry is.

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

“Those cases were already distinguished, since the Bailar-twist analysis made exactly that distinction.”

It made it on one path and not in the census. The census reported an infinite group whenever the symmetry search produced no name, which covers both a linear arrangement and a group of order ten — so the pentagonal pyramid was reported as having an infinite group. It has C5v, of order 10.

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

“The remaining gap is in a path's interior, like the first one.”

Both occurrences are at t = 1.000, which is the pentagonal-pyramidal endpoint of two different paths. The D3 gap was a whole stratum of interior geometries; this one is a single arrangement that two paths happen to end at, and it is in the census in its own right.

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

“The bond-conserving mass is confirmed by reproducing ND₃'s ground splitting at the published barrier.”

It lands on 0.05308 against a measured 0.05310, within 0.04 per cent, in the same well where it puts NH₃ at 0.94420 against 0.79350 — nineteen per cent high. An agreement on one isotopologue that the other refutes is a coincidence of where one number fell.

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

“The better well should favour the better mass on the isotope test.”

In the well whose shape is fitted to both of NH₃'s lines, the bond-conserving mass widens the gap between the two isotopologues' barriers from 3.42 to 4.25 per cent and moves the predicted ratio from 17.63 to 18.49, away from the measurement.

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

“The middle group's order is a separate question from the range of the third-row shells.”

It is the same question asked one ion at a time. Sodium chloride, potassium fluoride and calcium oxide are ordered as the contractions their third-row ions need, 1.302, 1.387 and 1.441, and a contraction of a whole shell could not reach that order only because it gives all three ions one factor.

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

“The two cations share a shell, so one factor should serve both.”

Carried to calcium oxide, potassium's factor leaves its error at −12.21 per cent; carried to potassium chloride, calcium's leaves it at −15.38. Both land outside the band the two second-row pairs span, −13.56 to −14.86 per cent.

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

“Three factors fitted to three pairs is a tuning and cannot test anything.”

Potassium chloride contains two of the three ions and was fitted on nothing. At potassium's and chloride's factors its error is −13.81 per cent, inside the second-row band and 0.40 points from its middle. The potassium factor it would need on its own, 1.401, agrees with potassium fluoride's to one per cent.

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

“The slope floor is a bound on every model of the form measurement = f(predictor), established without fitting.”

It is a bound on every model of that form with the predictor written on one particular scale. Under thirteen strictly increasing reparameterisations the floor moves by a factor of 19.1 on the spin-only moment, 10.2 on the Hückel eigenvalue and 11,746 on the angle strain. Every reparameterised predictor makes the same predictions about order and a different claim about derivatives.

Tested in A floor on models written in one scale · the series on models

“The tie and discordance instruments are approximately scale-free, like most robust statistics.”

They are exactly scale-free. Across all thirteen maps on all three predictors the count of exact ties is identical, the count of discordant pairs is identical, and the share of the variation the ties leave unexplained agrees to the last bit. Both quantities are functions of the order of the predicted values alone, and a strictly increasing map is a bijection on order.

Tested in A floor on models written in one scale · the series on models

“The power family is an artificial stress test, so the movement it produces says nothing about a real modelling choice.”

The overlap family is not artificial. Hückel theory sets every overlap integral to zero; restoring a uniform overlap s gives eigenvalues x/(1 + sx), and every member of the family for s between 0 and 0.3 is a defensible calculation. Across it the discordance and both exact ties are unchanged and the slope floor moves from 28.5 to 36.4, a rise of 27 per cent.

Tested in A floor on models written in one scale · the series on models

“A floor that moves is still a floor, so the reported value is a safe lower bound.”

The reported value is the smallest of the thirteen on the Hückel eigenvalue and nowhere near the smallest on the other two. The angle strain's reported 3.38 sits between 0.005 and 55.9 across the family, and on two of the three predictors the steepest pair changes identity — so the quoted fragility of the floor moves as well as its value.

Tested in A floor on models written in one scale · the series on models

“At fixed size and ground spin, the upper separation is ordered by the cluster's first excitation.”

Across all twenty five-centre doublets the rank correlation between the separation and the reciprocal of the first excitation is 0.36, and across all twenty-one six-centre triplets 0.15 — both worse than counting bonds, at 0.43 and 0.27. Four doublets share a first excitation of exactly one coupling unit and separate at 118, 140, 307 and 413 cm⁻¹.

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

“Every spin cluster has two separations below 150 cm⁻¹, both poles.”

Every cluster with a ground spin has two poles, and eleven of the twenty five-centre doublets put the upper one between 152 and 413 cm⁻¹. The rule was a property of the twelve clusters it was read on, whose upper poles all happen to fall below 150.

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

“The two were unscoreable for the same reason.”

Their gaps say otherwise. The highest occupied level's nearest comparison on the other axis is 34 per cent away at its closest and 74 per cent at the median, and its rank correlation with the error is 0.063 — it separates the families and is unrelated to the error. Polarisation times double occupancy is 28 per cent away at its closest and ranks the systems at 0.769, the same as polarisation alone — it separates the families because it is a monotone function of one that does.

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

“The direction information disappears as soon as the line width reaches the smallest splitting.”

The stretching pair, split by 0.53 cm⁻¹, has merged into one peak at a width of one wavenumber and the probe still varies by 1.90 round the circle, against 1.96 resolved. It falls only as the width approaches the bending pair's 2.8 cm⁻¹ — 1.59 at three, 1.09 at five — because the largest departure was always the bending pair's.

Tested in What a linewidth leaves of the probe · the series on spectrum

“Once a pair shows a single peak, the spectrum reads its intensity-weighted blend.”

At a width of five wavenumbers each pair shows one peak, and the probe varies round the circle by 9.4 per cent — fourteen times the blend's 0.66. The ratio is read at the envelope's maximum, where the nearer component still weighs more. The excess over the blend falls as the square of splitting over width, to 1.5 times at ten wavenumbers and 1.1 at twenty.

Tested in What a linewidth leaves of the probe · the series on spectrum

“Better resolution would make the probe a usable structural measurement.”

Resolution is the easy half. The splitting a spectrum must beat is first order in the distortion, 141 cm⁻¹ per ångström, and the departure it must then measure is second order, about 0.02 per ångström squared. At a distortion of a hundredth of an ångström that is a line width under 1.4 cm⁻¹ and a depolarisation ratio known to two parts in a million.

Tested in What a linewidth leaves of the probe · the series on spectrum

“The three icosahedral disagreements at fourteen, sixteen and eighteen electrons are one degeneracy being filled rather than three separate coincidences.”

They are one calculation, which is stronger and has a different cause. All three hand the localisation the same nine orbitals, because the search selects by non-zero occupation and Hund's rule fills a five-fold shell singly before pairing. Their spreads are identical because they are the same arithmetic run three times, not because a degeneracy is being filled.

Tested in Six disagreements and three calculations · the series on multicentre

“A partly occupied degenerate shell is what makes the two criteria disagree.”

Twenty-one of the forty-eight fillings leave a shell partly occupied and three of those disagree; three of the six disagreeing rows have a closed shell. The condition neither implies the disagreement nor follows from it, so the explanation is refused in both directions.

Tested in Six disagreements and three calculations · the series on multicentre

“The family survey covers forty-eight cage-and-filling pairs.”

It covers forty-eight rows and thirty-three questions. Fifteen rows hand the localisation an orbital set another row has already handed it: four inputs on the six-vertex cage rather than six, six on the nine-vertex rather than nine, seven of ten, eight of eleven and eight of twelve.

Tested in Six disagreements and three calculations · the series on multicentre

“Counting the repeats is only a correction to a denominator.”

It changes the finding the earlier survey rested on. Four Boys-degenerate cases against two Pipek–Mezey ones, read as a trend with cage size, is two against one. The direction survives and the evidence is a third of what it was, which is the difference between a trend and two cases.

Tested in Six disagreements and three calculations · the series on multicentre

“The momentum picture has fewer nodes than the position one, since a transform smooths a function out.”

It has exactly as many. The position radial function carries a Laguerre polynomial of degree n − l − 1 and the momentum one a Gegenbauer polynomial of the same degree, because the substitution carrying one to the other is a rational map of degree one in the squared momentum. Checked on nine orbitals from 2s to 5d, and against the numerical transform on the 3s.

Tested in The nodes in the other variable · the series on orbital

“A position node at radius r corresponds to a momentum node near 1/r.”

Within one orbital the products of paired nodes are not equal under either ordering. The 4s gives 1.130, 1.653 and 1.607 pairing innermost with outermost and a different set pairing innermost with innermost; the 5s spreads from 1.144 to 2.082. A rule would collapse one of those sets to a single value on every orbital and neither does on any.

Tested in The nodes in the other variable · the series on orbital

“Nothing exact relates the two node sets.”

The product of all the position nodes times the product of all the momentum nodes is (n+l)! / ((2l+1)! · 2^(n−l−1)), to twelve digits on every orbital tested — and it contains no nuclear charge. Each position node contracts as 1/Z and each momentum node expands as Z, and across Z from 1 to 10 the product does not move.

Tested in The nodes in the other variable · the series on orbital

“A Compton experiment would show an orbital's momentum node as a zero in the measured profile.”

A Compton profile is an integral of a squared wavefunction from the measured momentum upwards, so it is strictly positive. Its derivative is minus the momentum density times the momentum, which vanishes exactly at the node — so the node appears as a horizontal tangent. On the 2s that tangent sits at q = 0.5 with the profile still at 24 per cent of its peak.

Tested in The nodes in the other variable · the series on orbital

“With mixed ligands the bond-conserving reduced mass is no longer a single function of one coordinate.”

It is. For the symmetric umbrella motion the three ligands slide outward together along fixed bonds, and their radial kinetic energy sums to a single function of the apex height. What the asymmetry adds is one extra term, not an extra variable: the horizontal centre of mass moves, and subtracting its motion takes half the sum of the squared mass differences, divided by the total mass, off the radial coefficient.

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

“A partly deuterated molecule's splitting is an interpolation between the two pure ones.”

Not even under this model. The steps in the logarithm are 1.175, 0.960 and 0.809, so the series is concave in the substitution count and both middle molecules sit off the line joining the ends. The reason is arithmetic rather than chemistry — a splitting falls with the square root of the mass and the mass grows linearly with substitutions — and the departure from a square-root law is what a measurement would test.

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

“The asymmetric term is a refinement too small to matter.”

It changes the predicted splitting by 0.507 per cent for NH₂D and 0.423 for NHD₂. The isotope test could not choose between two whole constructions of the mass because they differed by less than either missed the measurement by; this term is of that order, and it exists only for the molecules that test did not have.

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

“If a change leaves the radical's singly occupied level where it was, it leaves the unpaired electron where it was.”

Raising the A–C overlap from 0.25 to 0.45 leaves the level at −13.6 eV to rounding and moves the spin on A from 0.500 to 0.236 and on B to 0.764. The split is S(B–C)² : S(A–C)² exactly, at seven energies of C from −24 to +2 eV and at three Wolfsberg–Helmholz constants.

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

“A converged self-consistent field is the mean-field answer.”

On the ring of four with a staggered site energy, the conventional start converges above the lowest solution on nine systems of a hundred and forty-four — by 1.124 at a repulsion of six and a site energy of four, where the composite's whole error was 1.146. The unrestricted mean field is variational; a fixed point above its minimum is a stationary point the iteration found, not the method's answer.

Tested in The worst system in the square was the solver's · the series on approximation

“The polarisation fails as a diagnostic by a factor of forty-six on the cross.”

One of the two systems in that pair, a repulsion of eight and a site energy of six, is on the missed band: its error was 0.545 from the conventional start and is 0.040 from the lowest field. Recomputed, the polarisation's worst pair on the cross is 2.61, under the census threshold of three.

Tested in The worst system in the square was the solver's · the series on approximation

“Correcting the mean field rescues the diagnostics.”

It rescues one reading on one cross. On the grid the polarisation's worst pair falls from 48.0 to 14.7 and still fails by a factor of five; the highest occupied level does not move from 627; and the census still finds a failing pair for four candidates, none shared.

Tested in The worst system in the square was the solver's · the series on approximation

“A diagnostic that fails on its worst pair is no better than chance.”

Over the grid's several hundred comparable pairs, the median error ratio is 1.14 to 1.21 for every cheap candidate but one, where shuffling the candidate's values among the systems gives medians between 2.3 and 4.2. The exception is the highest occupied level, at 3.05, inside its shuffled range.

Tested in The worst system in the square was the solver's · the series on approximation

“A localisation is invariant to how the occupied orbitals are chosen, since any unitary mixing of them leaves the density alone.”

It is invariant to mixings of the occupied set with itself, which is what the basin search exploits. It is not invariant to a rotation inside a degenerate shell the occupied set only partly takes: that changes which combinations are occupied. On the icosahedron at ten electrons the best Pipek–Mezey functional moves from 2.5171 to 2.0852 across four re-orientations, a factor of 1.207.

Tested in The basis a diagonaliser happened to return · the series on multicentre

“The number of distinct localised descriptions is a property of a cage and a filling.”

On the icosahedron at twelve electrons it is 10, 8, 14, 11 and 13 from five orientations of one shell. Nothing about the cage or the filling differs between them. The count moves on three of the six affected inputs.

Tested in The basis a diagonaliser happened to return · the series on multicentre

“The degeneracy classification the family survey is built on is a fact about each cage.”

On three of the six affected inputs it is not. The six-vertex cage at four electrons gives Pipek–Mezey spreads of 0, 5.5 × 10⁻², 4.9 × 10⁻², 0 and 0 against a threshold of 3.1 × 10⁻⁴; the ten-vertex cage at sixteen flips under Boys instead; the icosahedron at twenty flips under Pipek–Mezey. Each is labelled both ways depending on the orientation an eigenvalue routine returned.

Tested in The basis a diagonaliser happened to return · the series on multicentre

“The effect is really the search's numerical noise under a different name.”

An input whose degenerate shells are wholly occupied or wholly empty is put through the same rotation and does not move at all — 0.1666666667 against 0.1666666667, identical to ten decimal places. The rotation is then a mixing of occupied orbitals with occupied orbitals, which is the invariance the method rests on, and the search reproduces it exactly.

Tested in The basis a diagonaliser happened to return · the series on multicentre

“A bonding orbital is extended along the bond, so its momentum distribution is extended along the bond too.”

The two anisotropies have opposite sign. In position ⟨z²⟩/⟨x²⟩ = 2.056 and in momentum ⟨p∥²⟩/⟨p⊥²⟩ = 0.600, on the same orbital at the same separation. The interference factor cos²(p∥R/2) cuts the momentum density off at π/R along the bond and does not touch it across, so the same orbital is flattened in the direction it is stretched in.

Tested in Oblate in the picture nobody draws · the series on orbital

“The interference fringes wash out of any measurement, since a Compton profile integrates over two momentum components.”

They wash out in one direction and survive intact in the other. The cosine depends only on the component along the bond, so measuring along the bond leaves it untouched and the profile is the atomic one times cos²(qR/2), with exact zeros at odd multiples of π/R. Measuring across the bond integrates over the cosine's own variable and the fringes become a Bessel weight with no zeros in it.

Tested in Oblate in the picture nobody draws · the series on orbital

“Reading a bond length off a momentum measurement would need a model of the electron density.”

The first zero of the profile along the bond is at π/R exactly, whatever the atomic function is. Across separations from 1.4 to 4.0 bohr the exponent the energy chooses contracts by a quarter and the overlap falls from 0.72 to 0.19, and π/q₁ returns the separation to twelve digits at every point.

Tested in Oblate in the picture nobody draws · the series on orbital

“The momentum oblateness is an artefact of writing the orbital as a sum of two atomic functions.”

Separate the centres and it goes. At two bohr the anisotropy is 0.600; at forty, with the overlap down to 2 × 10⁻¹⁵, it is 1.0000 — the fringes crowd together faster than the atomic density falls, their average becomes one half, and the half cancels between the two directions. A formula that kept a molecular signature at forty bohr would be reporting itself.

Tested in Oblate in the picture nobody draws · the series on orbital

“The ordering of the kinetic operator for a position-dependent mass is a choice that has to be made by hand.”

Not for a mass that comes from a constraint. The reduction is a one-dimensional Riemannian manifold with metric μ(x), which has a distinguished Laplacian; carried to the flat measure by ψ ↦ μ^(¼)ψ it is a von Roos ordering with square coefficient −7/32 and curvature coefficient 1/8. Fitted numerically over forty-one positions those come out to six decimals, with a residual of 2 × 10⁻⁸.

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

“The covariant operator is BenDaniel–Duke, since that is the ordering with no extra potential.”

It is Mustafa–Mazharimousavi. The fitted coefficients give α + γ = −1/2 and αγ = 1/16, whose roots are a double root at −1/4. BenDaniel–Duke is at (0, 0), a quarter of the von Roos plane away, and adds no potential at all — which is what makes it the one to reach for and not the one a manifold picks.

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

“Resolving the ordering removes the last unsettled choice in the one-dimensional model.”

It removes the last one inside the von Roos family. A constraint imposed as the limit of a stiff potential leaves the zero-point energy of the five degrees of freedom it froze, and that term is not in the family: it depends on the masses, where the two basis functions of the family depend only on the mass profile the constraint already fixes.

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

“A family of chains that differ in which single fusion is turned varies the position of one bend and nothing else.”

A fusion's entry is a direction, so turning one turns the chain and the next fusion turns it back. Turned at the first or last fusion that leaves one angular ring; turned at any of the six between, two adjacent ones. Built from those angular rings, the chains reproduce the family's spreads — 3.052 for the ends, 4.065 to 4.600 for the rest — to four figures.

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

“Whether two angular rings turn the same way or opposite ways — whether the chain's overall envelope bends — decides the scatter.”

At seventeen arrangements of two or three angular rings, turning them all one way instead of alternately changes the worst spread by at most 0.81 per cent, while the envelope goes from straight to bent by up to 180°.

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

“The more angular rings a chain has, the more its response scatters.”

Two angular rings at positions two and six scatter by 2.449, below every single angular ring (3.029 to 3.660). Two at one and three scatter by 5.244. A chain with all seven interior rings angular scatters by 1.367, less than the straight chain's 1.520.

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

“Changing the sign between the two atomic functions only moves where the momentum fringes fall.”

It changes the orbital's shape in momentum from oblate to prolate. The bonding combination's ratio of second moments along and across the bond is 0.600; the antibonding combination's, at the same separation and exponent, is 1.731. The antibonding orbital is prolate in position too, at 4.335, so the opposite senses in the two pictures belong to the bonding orbital alone.

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

“Filling both the bonding and the antibonding orbital cancels the interference, leaving two atoms.”

It inverts it. One electron in each gives a profile along the bond equal to the atom's times (1 − S cos qR)/(1 − S²): a dip to 0.683 of the atom at zero momentum and a rise to 1.867 at π/R, with contrast equal to the overlap. The dip costs 0.156 hartree of kinetic energy per electron at this separation, which is closed-shell repulsion.

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

“A single flag for an angular fusion in a chain of fused rings makes one angular ring.”

The flags are fusion directions, not turns. A single one among zeros turns the chain and the next zero turns it back, so rings five and six are both fused across meta bonds. The bend measured along the chain was a kink of two angular rings, and the heteroatom on the first of them reads 0.811 of the straight chain while on the second it reads 0.969.

Tested in An angular ring rescales what lies beyond it · the series on delocalisation

“A heteroatom's response passes through angular fusion unchanged, because the fitted decay length does not move.”

The decay length moves by at most eleven per cent, from 0.719 rings to 0.638 in the fully angular chain. The responses on the rings beyond angular fusions move by factors: the third ring keeps 0.453 of its straight-chain response and the rings beyond the fifth between 0.127 and 0.313, alternating. A length is a slope, and angular fusion changes the level.

Tested in An angular ring rescales what lies beyond it · the series on delocalisation

“The fifth an angular ring costs belongs to its two shared bonds.”

With the heteroatom on the kink's first angular ring, the summed response over that ring's two shared bonds rises from 0.0352 to 0.0434 and over its four outer bonds falls from 0.0815 to 0.0513. The loss is in the bonds that are not shared.

Tested in An angular ring rescales what lies beyond it · the series on delocalisation

“Changing one bond's length changes where it sits and leaves the arrangement alone.”

At a ratio of 1.5 neither seven-coordinate placement survives a nudge: both relax to a capped octahedron, C₃ᵥ, with the long bond as the cap and its three nearest neighbours at 69°.

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

“A longer-ranged interaction moves the root in a definite direction relative to the one-bond zero.”

It moves it to either side. At a reach of a quarter the root is 5.192 and the one-bond zero 5.072, so the root is later by 0.121; at a reach of a half the root is 5.515 and the zero 5.870, so the root is earlier by 0.355. The sign reverses between a third and a half.

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

“Adding a second-neighbour repulsion strengthens the interaction, so both crossings move earlier.”

Both move later. The root goes from 3.8955 with no reach to 5.5151 at half, and the pole from 4.1595 to 6.2212 — the second-neighbour term competes with the nearest-neighbour one for the same electrons, so a larger V is needed before the structure beyond contact stops paying for itself.

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

“The pair count at two bonds rises with the repulsion, as it does with a one-bond interaction.”

Only when nothing penalises it. With a one-bond interaction the count at two bonds rises from the start — the electrons pushed off their neighbours go somewhere. With a second-neighbour term it falls at weak coupling, reaching −0.024 at V = 2 for a reach of a half, and rises again once the state begins to order.

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

“The profile at π/R measures a molecule's antibonding occupation.”

It measures the electrons in sine combinations. In nitrogen the largest single contribution at π/R, 0.052 per electron of the molecule, comes from the 2p σ bond. Ordered by the valence profile's share at π/R the first-row diatomics run Li₂, C₂, B₂, N₂, O₂, F₂, which is not the order of their antibonding counts or their bond orders.

Tested in The zero is a parity, not a bond · the series on orbital

“A real first-row molecule can have a zero in its momentum profile along the bond.”

The 1s core pair fills a sine combination in every one of them, so no whole profile is zero at π/R — the smallest share, carbon's, is 0.113. Only lithium's valence profile is zero, because its valence shell is one s bond, and any s–p mixing in a σ bond gives it a value proportional to the p share.

Tested in The zero is a parity, not a bond · the series on orbital

“A main-group centre loses a usable valence orbital only when its ligands lie in one plane.”

A bare ring of four to eight ligands, folded off its plane to any polar angle from 60° to 120°, matches three of the centre's orbitals and orphans n − 3, one more than the formula — the same count as the flat ring. The unmatched orbital is totally symmetric off the plane and antisymmetric on it.

Tested in Folding the ring does not give the orbital back · the series on hypervalency

“Ligands spanning all three directions in space give a main-group centre four usable orbitals.”

A folded ring spans three directions and gives three. In C₄ᵥ to C₈ᵥ the centre's s and its p along the axis are both totally symmetric, and a single symmetry-distinct set of ligands supplies exactly one totally symmetric combination for the two. An apex supplies the second and restores four at every angle, 90° included.

Tested in Folding the ring does not give the orbital back · the series on hypervalency

“The pentagonal pyramid, the one arrangement the census could not count, sits where the counting formula is expected to fail.”

Reduced in C₅ᵥ, sulfur hexafluoride as a pentagonal pyramid spans 2A₁ + E₁ + E₂ against the centre's 2A₁ + E₁: four matched, two orphaned, which is n + L − 4. The census now counts twenty-seven arrangements and the formula is right for twenty-four, the three it misses still the three flat ones.

Tested in Folding the ring does not give the orbital back · the series on hypervalency

“A splitting the solver reports as a small number is a small splitting.”

Raising phosphine's barrier from 2,020 wavenumbers to its own 12,300, the reported splitting falls cleanly for three points and then stops falling: 7.8 × 10⁻¹⁰ at 5,000, 3.9 × 10⁻¹⁰ at 6,000, zero at 7,000 and 7.8 × 10⁻¹⁰ again at 8,000. A splitting that grows with the barrier it tunnels through is not a splitting, and every one of those values is below the floor the operator's norm sets.

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

“The floor is a fraction of the level, since each eigenvalue is bisected to the last bit of itself.”

It is a fraction of the operator's norm. A Sturm count is a recurrence over every grid point and its rounding is set by the largest thing in the matrix, which on a 2,400-point grid is tens of hartree where the levels are thousandths of one. The two differ by four orders of magnitude — 1.7 × 10⁻⁹ against 1.3 × 10⁻¹² wavenumbers for phosphine — and the spurious values above sit between them.

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

“A correction sized on ammonia says nothing about a molecule whose barrier is six times larger.”

It says almost exactly the same thing. The same fraction of each molecule's own transverse zero-point difference is 4.58 per cent of ammonia's barrier and 0.56 per cent of phosphine's — ten times less — while the action it acts on is 8.86 against 60.76, seven times more. The predicted isotope ratio moves by 21.4 per cent for ammonia and 16.1 for phosphine.

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

“Since the overlap asymmetry moves only the spin and the coupling moves only the level, a radical with both moves each by the separate amounts.”

True at the origin, where both mixed second derivatives vanish, and false beyond it. With the A–C overlap 0.2 above B's and an end-to-end overlap of 0.1, the level sits 0.20 eV below the sum of the two shifts and the spin on A at 0.218 against the 0.236 the asymmetry alone gives. The electron also reaches the middle orbital, which carries 0.012 of it.

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

“Keeping only the consecutive triples fails for a uniform chain because four fragments is too short a chain for the saving to show.”

At five, six, seven and eight fragments the consecutive assembly covers 45.6, 47.1, 48.8 and 48.1 per cent of three decades of accuracy line, against 60.0, 63.1, 65.8 and 67.9 for pairs alone. The shortfall is 5.2 points at four fragments and 19.8 at eight: it grows with length rather than closing.

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

“Adding every triple to a pairwise assembly is at least as good as adding only some of them.”

On the heavy-outside chain, every triple together covers 70.8, 56.2, 59.1 and 62.3 per cent from five to eight fragments, and the consecutive triples alone cover 77.4, 80.9, 84.4 and 87.9. What every triple leaves out reaches 1.05 kilocalories a mole at five and 2.70 at eight, so the complete order is the worse assembly by up to twenty-five points.

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

“The sign of the correction cannot be settled without frequencies at the planar geometry.”

It is settled by the direction of the gap. The deuterated molecule's fitted barrier is the lower one and its transverse zero point is also the lower one, so the fraction that closes the gap is positive for every band and under every construction of the mass — the transverse frequencies must stiffen as the molecule flattens, which is what an X–H bond does as its central atom goes planar.

Tested in The band that cannot supply what it changes · the series on inversion

“The bend is the obvious supplier, since a flattening pyramid changes its bends most.”

It is the worst. The degenerate bend carries a quarter of the isotope difference against the degenerate stretch's half, so it would have to stiffen by 22.25 per cent against the stretch's 10.92 — and its two components soften rather than stiffen as the molecule flattens, which turns its contribution into a debt. With the bend softening by eight per cent the stretches have to find 9.79 rather than 7.20.

Tested in The band that cannot supply what it changes · the series on inversion

“An underdetermined correction cannot fix a barrier.”

It fixes this one. The bare electronic barrier is the fitted value less the zero-point rise at the pyramid, which works out to the gap divided by one minus that band's isotope frequency ratio — and those ratios are 0.7252, 0.7422 and 0.7320, all just above 1/√2. Four choices whose required fractions span a factor of 2.04 imply barriers spanning 1.33 per cent: 1740.6, 1754.9, 1763.9 and 1750.1 wavenumbers.

Tested in The band that cannot supply what it changes · the series on inversion

“Counting which central orbitals a set of ligands can use needs the arrangement's point group and its character table.”

It needs the rank of the rows (1, x, y, z) over the ligand directions. On the 451 geometries where a table exists — the census, every folded, capped and flat ring, and every tabulated point on every path between arrangements — the rank and the table give the same count every time.

Tested in The count needed no table · the series on hypervalency

“A reduction by symmetry species gives the right count wherever the group is known.”

It gives an upper bound. Five ligands placed irregularly on a small circle of the sphere have no symmetry beyond C₁, so every function is one species and the table matches all four central orbitals. The rank is three: a combination of s and the p normal to the circle's plane reaches none of them.

Tested in The count needed no table · the series on hypervalency

“Some three-dimensional arrangement of a single ligand set might leave px and py short while matching pz.”

Two orbitals left short means a rank of two, and a rank of two means every ligand direction lies on one straight line. A line meets the sphere at no more than two points, so only a two-ligand molecule can do it — where nothing is orphaned. Every crown of four to ten alternating ligands tried reaches all four.

Tested in The count needed no table · the series on hypervalency

“The splitting's unequal steps down the isotopologue series are a property of the potential's shape.”

They are the shape of a square root. Fitting the logarithm of the splitting against the square root of the reduced mass leaves a worst residual of 0.0126 for the ground line and 0.0084 for the excited one; against the number of deuteriums the same fits leave 0.100 and 0.0676 — eight times more. A tunnelling exponent is linear in √μ and √μ is not linear in a count.

Tested in The steps were a square root of a mass · the series on inversion

“The excited doublet would show a different pattern, since it sits higher in the well.”

It shows the same pattern with a smaller slope. Its sensitivity to √μ is 0.8614 of the ground line's, and the ratio of the two step patterns is 0.8323, 0.8705 and 0.8935 at the three steps — inside four per cent of that slope ratio throughout. One factor, not a change of shape.

Tested in The steps were a square root of a mass · the series on inversion