Two Radical?

This problem does not require a prior understanding of all the reactions involved in the sequences.

The first reaction with cyclopentadiene should tell the reader that a diels alder reaction must occur. So we must start looking for a diene (note that a benzene ring cannot participate). A clue is provided in the structure of the by-product : an acetal . So the reagent DAIB and the solvent can induce a de-aromatisation through a tautomeric mechanism to form the required diene.

The structure of the cyclo-adduct is fairly simple to deduce and the next next should be obvious to the reader. The biradical mediated reaction must occur. Before moving on, we must make a key observation, which is illustrated below :

The numbers on the carbon atoms help understand how A will react to form B :

Now the structure of B is apparent to the reader. However let us redraw the compound for clarity, else we run the risk of creating a mess.

The next step is a variation of the popular ketone protection method which doesn't use an acid catalyst. This is done to avoid the decomposition of the acetal and provide regio-control with bulky TMS groups.

The following step is another use of $AIBN$ ; reduction. It breaks open the cyclopropane ring by providing hydrogen radicals.

The next reaction is a Triflate group protected enolate that is formed by $LiHDMS$ which is consistent with E 's chemical formula.

The next step is acidic hydrolysis. This implies that both acetals will decomposes to carbonyl groups. Knowing this we can compare the formulae to deduce that the second step is a coupling reaction resulting in the loss of the $OTf$ group.

Allyl Tributyl Tin and the copper salt give Michael addition away and L-selectride's use can be deduced from the formula of H . The boron atom is bound to large secondary butyl groups, and hence steric hindrance helps in regio-control :

The next few reactions see the loss of two hydrogen atoms. The $LDA$ and $TMSCl$ reagent is to generate an enolate, one hydrogen down two to go. The second set of reagents' application is deduce by understanding that another hydrogen atom must be lost and that the $TMS$ group must be removed. This implies a cyclo-condensation reaction must occur. This results in the formation of a six membered ring.

The set of reactions are simple and give the final product J that has four rings. Hence $\boxed{\alpha = 4}$

We now move to the by-product. Observe that it looses two molecules of carbon monoxide this implies that a fragmentation reaction must occur. However the formula of X is the same as its precursor, hence this must be an example of a tethered mechanism. The hint provided states that no carbonyl group is present in X providing a clue to the reader. Consider the adjacent $OMe$ group, if it were to fragment to give a methyl and enolic radical another carbonyl group would be formed which goes against IR data. Hence consider a hydrogen atom on the $OMe$ group, fragmentation results in radicals that do not tautomerise. This hydrogen radical attackes the carbonyl oxygen to generate compound X :

Y is the product of acid catalysed lactonization and esterification made evident by the formula and IR data. The mechanism is left as an exercise for the reader. $\boxed{\beta = 3}$

$\alpha + \beta = 7$