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2,2-Dimethoxypropane is the dimethyl acetal of acetone. Reaction with a diol under acid catalysis can form an acetonide and two methanol molecules.

An acetonide is an isopropylidene acetal formed from a diol and acetone or an acetone equivalent. Formation of its O–CMe2 bonds leaves the original stereogenic carbon–oxygen bonds intact.

The alcohol oxygen attacks sulfur of methanesulfonyl chloride, chloride leaves, and triethylamine removes the O–H proton. This gives an O-mesylate with retention of the carbon configuration: the original carbon–oxygen bond has not been displaced.

An accessible chair places the mesylate axial. triethylamine then removes the anti axial hydrogen on the carbon between the ketone and mesylate. Its C–H bond forms the new C=C bond while the C–O bond expels mesylate. The result is the conjugated cyclohexenone.

The neighboring carbon on the acetonide side has its relevant axial position occupied by its oxygen substituent, rather than an anti axial hydrogen. The elimination toward the ketone is therefore favored both by the trans-diaxial geometry and by conjugation of the resulting alkene with C=O. The acetonide remains intact. No inversion of the original alcohol carbon is required during mesylation, and it becomes planar during elimination.

Number the cyclopentenone carbonyl carbon $C_1$, its alkene carbons $C_2$ and $C_3$ with OR at $C_3$, and its saturated carbons $C_4$ and $C_5$. The ester alkene has alpha carbon $X$, bearing Me and CO2Et, and beta carbon $Y$, bearing the protected carbon substituent. lithium diisopropylamide generates the ordinary ketone enolate at the saturated alpha carbon $C_5$. The acetonide is retained.

The carbon end of this enolate attacks ester beta carbon Y. Its pi system shifts through X onto the ester oxygen, forming the first new bond $C_5-Y$ and an ester enolate at X. That enolate now attacks the original cyclopentenone beta carbon $C_3$ intramolecularly, giving the second new bond $X-C_3$. The original enone electrons move onto its ketone oxygen, and protonation of the final ketone enolate completes the product. The OR substituent is retained at the new bridgehead; it need not leave in either Michael addition.

The atom map provides a useful structural check. Between bridgeheads $C_3$ and $C_5$, the three paths are $C_3-C_2-C_1-C_5$, $C_3-C_4-C_5$, and $C_3-X-Y-C_5$. Their numbers of internal atoms are 2, 1 and 2, establishing the norbornane skeleton. Both original carbonyls survive and both initial alkenes are consumed. The two successive conjugate additions give this atom map without requiring a concerted cycloaddition.

Acid protonates one methoxy group of 2,2-Dimethoxypropane. methanol departure gives the oxocarbenium species $\mathrm{M}{\mathrm{e}}_2\mathrm{C}(\mathrm{OMe}{)}^{+}$, which is captured by one diol oxygen. After proton transfer, protonation and loss of the remaining methoxy group generate a second oxocarbenium center. Intramolecular attack by the other diol oxygen closes the five-membered ring; deprotonation regenerates acid. The net transacetalization exchanges two methoxy groups for the two diol oxygens, releasing two methanol molecules.

The two starred carbon centers retain their original configurations, and the two carbon side chains on the acetonide are trans. No C–O bond at either original stereocenter is broken; reaction takes place at the reagent-derived CMe${}_2$ acetal carbon. The apparent change in wedge/dash presentation comes from redrawing the backbone into the five-membered ring, not inversion at a starred center. In the planar pentagon used in the diagram, draw the alkynyl-bearing side chain behind and the $\mathrm{C}{\mathrm{H}}_2\mathrm{OPMB}$ side chain in front. The unstarred methyl-bearing center also retains its configuration. The acetal carbon itself is not stereogenic because its two methyl substituents are identical.