[clarkstill]

Had time to review old papers.... 

If this follows from the imidization reactions: 

Make enolate, this attacks anhydride.  Result is first product shown by pgk in attached file.

The acetic anhydride reacts with the carboxylate by anhydride exchange: the new carboxylate formed in the first product is now a mixed anhydride with an acetate group, and the other piece of acetic anhydride exits as the acetate ion. 

An enolate is formed again between the three carbonyls - this displaces the acetate of the mixed anhydride to re-form the 5 membered ring. 

So now the product is a 1,3-cyclopentanedione with a 2-acetyl group and a 2-carboxyethyl group. 

This does the retro-Claisen as suggested by clarkstill. 

The second reaction is acid hydrolysis to the acid, which (like acetoacetic acid) readily decarboxylates.

Agreed

[pgk]

a). Indeed, enolate attacks take place therein and not hydride ones. But it was presented that way due to graphical text savings and simplicity reasons.
b). The retro-Claisen scenario is reasonable. But it demands a strong base; stronger than Et3N or AcO- or even, an enolate.
c). Contrary, decarboxylation via formic carbanion in equilibrium with the enolate that is formed between the three carbonyls, is possible because the equilibrium is pushed to the right via formic acid dehydration that is mediated by acetic anhydride (step 2).
d). Additionally, please note that the high ring strain of β-lactams explains among others, the antimicrobial activity of β-lactam antibiotics in a similar way as in step 3.

1). Recent Advances in the Retro-Claisen Reaction and Its Synthetic Applications, Current Organic Synthesis, 9(4), 488-512, (2012)
http://www.eurekaselect.com/100926/article
2). Base-catalyzed retro-Claisen condensation: a convenient esterification of alcohols via C–C bond cleavage of ketones to afford acylating sources, RSC Advances, 4(56), 29502-29508, (2014)
https://pubs.rsc.org/en/content/articlelanding/2014/ra/c4ra04618h
3). The catalytic decomposition of formic acid in acetic anhydride, Journal of the American Chemical Society, 45(2), 455-468, (1923) 
https://pubs.acs.org/doi/abs/10.1021/ja01655a022
4). Carbon-13 Kinetic Isotope Effects in the Decarbonylation of Liquid Formic Acid Assisted with Acetic Anhydride, Isotopes in Environmental and Health Studies, 4(3), 285-289, (1998)
https://www.tandfonline.com/doi/abs/10.1080/10256019808234061
5). Advances in the chemistry of β-lactam and its medicinal applications, Tetrahedron, 68(52), 10640-10664, (2012)
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3525065/

[rolnor]

Please, take a look at the attached file, below:

pgk, is this not a published mechanism?

[pgk]

No, as far as I know. But I have not exhaustively searched any prior art.

[pgk]

Is it more clear, now?

[rolnor]

Then pgk, I feel its more likely that the original scheme is wrong, it should be diethylmalonate.

[pgk]

Indeed, the reaction mechanism would be less obscure if starting with maleic andhydride. But the reaction can also go with succinic anhydride and see how:
Given that ethyl acetoacetate has pka ≈ 10.5 (water) and 14 (DMSO), it is expected that conjugate acid of the enolate that is formed between the three carbonyls, must have pka = 15 in water, at least.
On the other hand, the ionization free energy can be estimated by the equation:
ΔG = -2,303RTln(pka)
Which corresponds to formation energies of the particular enolate, about 20 kcal/mol at 25 oC and 27 kcal/mol at 100 oC, at least, and which is higher than 2.5 kcal/mol and 22 kcal/mol (ΔH and Ea of formic acid dehydrative - decarbonylation and catalyzed decomposition, respectively).
Also, please note that simply mixing formic acid and acetic anhydride, leads to the formation of the mixed formic-acetic anhydride.

1). Formic Acid, Ullmann’s Encyclopedia of industrial Chemistry
https://onlinelibrary.wiley.com/doi/pdf/10.1002/14356007.a12_013.pub3
2). On the Structure Sensitivity of Formic Acid Decomposition on Cu Catalysts, Topics in Catalysis, (17-18), Authors Manuscript, (2016)
https://www.osti.gov/pages/servlets/purl/1398775
3). Formylation of Amines, Journal of Organic Chemistry 1958 23(5), 727-729, (1958)
https://pubs.acs.org/doi/abs/10.1021/jo01099a023

[rolnor]

No, I mean that the nucleophile should be diethyl malonate.

[clarkstill]

I don't understand why we're all ignoring hollytara's answer, which is clearly correct. Without a clear and relevant literature precedent, I can see no reason to believe the reaction proceeds through an implausible intramolecular decarbonylative SN2 reaction to generate a cyclobutanone.

[pgk]

First of all:
We respect and we do not ignore hollytara's answer, which obviously seems to be the most reasonable scenario.
But it suffers of the following:
1). Formation of a mixed anhydride from acetic anhydride demands a stronger acid than acetic anhydride (e.g. formic, trifluorocetic, etc.).
Please note that succinic acid (pka1 = 4.2) is just a little stronger than acetic acid (pka = 4.7) and consequently, their anhydride exchange would be very slow, if happening at all.
2). Formation of the ending product (ethyl acetyl-succidinylene-acetate) is not thermodynamically favorable, as containing 4 electronegative atoms (carbonyl oxygens) that are very close to each other. Moreover, there is no possibility of stabilization via an electropositive bridge-like “β-enol-carbonyl” intramolecular H-bonding, such as in ethyl acetoacetate.
3). The following retro-Claisen step demands a strong base that does not exist in the reaction medium.

Besides, diethyl malonate nucleophile would also demand a strong base because the overall procedure would involve a decarboxylation step.

In situ preparation of mixed anhydrides containing the trifluoroacetyl moiety. Application to the esterification of cholesterol and phenol, ARKIVOC, (vi), 127-135, (2005)
 
http://www.arkat-usa.org/get-file/19336/

[pgk]

Concerning the proposed model.
1). The attacking enolate is not the one that is presented above, but it is the resulting enolate of three conjugated carbonyls groups. The said enolate has a pka ≈ 15-20 in water and quite higher in non-ionized organic medium, which corresponds to formation energy quite higher than 30 kcal/mol at higher temperatures in organic medium; getting very lose to ≈ 85 kcal/mol that is the C-C bond strength.
2). The leaving groups are gaseous CO and ionizable AcOH (ET3N salt) that both have a high entropic content and cannot be considered as poor leaving groups.
3). The so formed, highly strained cyclobutanone ring explains the formation of the final product due to the easy ring rearrangement.

[hollytara]

I based my proposed mechanism on experience: 

I have prepared N-substituted maleimides by a similar route:  Amine reacts with maleic anhydride to form amic acid, which is then re-cyclized with acetic anhydride as the reagent.  This reaction is almost identical - except that a C nucleophile from the 1,3-dicarbonyl compound replaces the N nucleophile. 

The alternative through the 4 membered ring goes against my chemical intuition - which tells me that 5 and 6 membered rings are favored by strain energy and 3 membered rings are favored by entropy/proximity but 4 membered rings are strained and not favored by proximity - so they are less favorable.  The mechanism through the 4 membered ring is possible but I would need more evidence to favor that over the simpler mechanism

[pgk]

Then, what about if the retro-Claisen retction occurs before the ring closure?
There is no steric hindrance to the nucleophilic attack; neither formation of an electronegativity crowdy molecule.