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A consistent proposed sequence is desilylation, five-membered hemiketal formation, retro hetero-Diels–Alder opening, intramolecular aldol addition and hemiacetal closure. The essential carbon rearrangement is loss of the original ketone–side-chain bond and formation of a bond between its two neighbouring carbons; it is not a simple Michael addition.

For an explicit atom map, label the starting six-membered ring $A-B=C-D-E-O_r-A$, with the ketone at $D$, methyl substituents at $A$ and $B$, the hydroxy group at $A$, and the side chain $E-X-T-\mathrm{Et}$. Here $X$ is the methylene and $T$ is the carbon bearing the protected alcohol. Let $O_D$ be the original ketone oxygen and $O_T$ the protected alcohol oxygen. These labels identify atoms, not new substituents.

First, Fluoride removes the silyl protecting group from $O_T$. The resulting alcohol attacks the protonated ketone at $D$, forming the five-membered hemiketal $D-E-X-T-O_T-D$. The original oxygen $O_D$ is now a hydroxy group; the original pyran ring remains intact at this stage.

Next, The original pyran ring can undergo a retro hetero-Diels-Alder reaction. The three electron-pair movements are $B=C\to C=D$, $D-E\to E=O_r$ and $O_r-A\to A=B$: break $D-E$ and $O_r-A$, and form the three indicated double bonds. The result is the tethered aldehyde/dienol esterThe temporary $D-O_T$ bond keeps both reacting ends within the same molecule. At $D$, both oxygen substituents remain attached, and the new $D-O_T$ tether prevents separation into independent fragments.

Then The $C=D$ enol attacks the tethered aldehyde at $E$. Formation of $C-E$, restoration of $D=O_D$, and proton transfer give the six-membered lactone ring $C-D-O_T-T-X-E-C$, with $E-O_{r}H$. This is an intramolecular aldol reaction, and it supplies the crucial new carbon–carbon bond without losing any carbon atom.

Finally, Keto-enol tautomerism at $A=B$ gives the methyl ketone at $A$ and introduces the hydrogen at $B$. The $E-O_{r}H$ alcohol then attacks that ketone, closing the five-membered hemiacetal ring $A-B-C-E-O_r-A$. The resulting fused rings have exactly the product's connectivity: $D$ is the lactone carbonyl, $O_T$ is its ring oxygen, and $O_r$ is the five-membered-ring oxygen.

The stereocentre at $T$ survives throughout this sequence. The configuration initially at $E$ is erased at the aldehyde stage, and those at $A$, $B$, $C$ and $E$ arise during the later cyclization and proton-transfer steps. A folded intramolecular aldol reaction followed by reversible hemiacetal formation is consistent with the drawn cis ring junction, but connectivity and orbital bookkeeping alone do not prove that it is the unique selected diastereomer. In particular, a claim of simple retention at $E$ would contradict this atom map.

A related pyranone-to-fused-lactone rearrangement under aqueous hydrogen fluoride is experimentally reported; that account also explicitly qualifies the relative stereochemistry as assigned on mechanistic grounds. The sequence above is a proposed explanation of the examination structure, rather than a mechanism claimed to have been experimentally established in that paper.^[ref]

The fluorite structure consists of a face-centered cubic lattice of calcium sites with all eight tetrahedral interstitial sites occupied by fluoride. In fractional coordinates of the conventional cubic unit cell, the calcium basis is $(0,0,0),(0,\frac{1}{2},\frac{1}{2}),(\frac{1}{2},0,\frac{1}{2}),(\frac{1}{2},\frac{1}{2},0)$, and fluoride occupies all combinations $(x,y,z)$ with $x,y,z\in \{\frac{1}{4},\frac{3}{4}\}$. The projection therefore superposes two fluoride sites at each quarter-coordinate position. Labels in the original diagram give fractional depth along $[001]$; periodic boundary sites are displayed on both edges.

Counting shared sites gives $8/8+6/2=4$ calcium sites and eight interior fluoride sites: four formula units per conventional cell.

Each calcium cation has eight fluoride neighbours at the vertices of a cube: cubic coordination. Each fluoride anion has four calcium neighbours in tetrahedral coordination. Both unlike-neighbour Euclidean distances are $\sqrt{3}a/4$, with $a$ the conventional cubic lattice parameter.

Apply the Goldschmidt packing principle at its limiting eight-coordinate geometry: the eight surrounding anions just touch one another and also touch the central cation. The anions form a cube of edge $a/2$, so $2r_a=a/2$ and $r_a+r_c=\sqrt{3}a/4$. Eliminating $a$ givesEquivalently, $r_c/r_a=0.732$. This is a limiting radius ratio rule, rather than a measured ionic-radius ratio. It does not assume that the calcium sublattice consists of mutually touching spheres: the calcium positions have face-centred cubic symmetry, but that symmetry alone does not specify which ions touch. A different construction with touching calcium spheres and fluoride just filling their tetrahedral interstitial sites would instead give $r_a/r_c=\sqrt{3/2}-1=0.225$; that is the opposite contact limit, not the eight-coordinate anion-contact ideal used here.

Using the contact model of (b), $r_a=a/4$ and $r_c=(\sqrt{3}-1)a/4$. Divide the volume of four cation spheres and eight anion spheres by $a^3$ to obtain the packing fraction:For one-radius cubic close-packed structure, face-diagonal contact gives $4r=\sqrt{2}a$, hence ${\eta }_{\mathrm{ccp}}=4(4\pi r^3/3)/a^3=\pi /(3\sqrt{2})\simeq 0.7405$. The limiting fluorite value is lower than the one-radius close-packed value. Its two sizes, stoichiometry and eight-to-four coordination impose a different geometry: the fluoride sites form a simple-cubic array, and the calcium sites are too far apart for calcium–calcium contact. A face-centred arrangement of one species is therefore not sufficient to infer a close-packed whole crystal.

A larger barium cation expands the fluorite structure and can widen the constrictions through which fluoride ions move. A plausible explanation is therefore a smaller ionic migration barrier and greater fluoride mobility. In a simple dilute-carrier description, ${\sigma }_{\mathrm{ion}}=n|q|\mu$, or $n{q}^{2}D/(k_{B}T)$ when an uncorrelated Nernst-Einstein relation applies. fluoride vacancies and interstitial defects, including those generated as Frenkel pairs, provide possible carriers.

More open migration pathways can raise the conductivity, but ion size by itself is not a proof: defect formation free energies set $n$, migration free energies set mobility, and correlated hopping can alter the simple relation. A controlled comparison would use crystals with comparable impurity content, thermal history and electrode conditions, and measure conductivity over temperature to separate plausible activation behaviour from extrinsic defect effects.