While the photographer is being urged to finish his initial photo shoot, his model is wearing large, dangling earrings. When he finally completes the shoot several moments later, the earrings are gone.
The photographer is taking multiple photos of the model with a still film camera, so the images instantly flashed on the screen would have to have been developed first.
Redding names the formula that Cinnamon wrote on the clear panel as Eosin Methylthionine Chloride. Chloride indicates that it contains chlorine, but none is present in the photograph of formula.
The climactic encoding of a secret message requires today's date, and a shot of a calendar indicates the action takes place on Sunday, August 23, 1967. But August 23, 1967 was a Wednesday, and the footer of the same calendar page says Sunday, September 23, 1962. (Nothing other than Sunday and 23 probably would have been legible on typical TVs of the time, however).
When Barney is using the transmitter to make the fake broadcast, the plate current and antenna meter pointers do not move from the zero indication. This would mean no transmission is being made.
When Redding is flipping through the telephone book, trying to locate page 823, one can see that the page numbers are not displayed on any of the upper right corners of the phone book until he reaches page 823.
The tape in the player that Mr. Phelps uses in this episode is running backwards; while it is possible to record a message backwards on another machine, it would be impractical since this machine already has a record function.
It is ludicrous that a former fashion model, now biochemist, working on a Top Secret government project, would want, or be allowed, to engage in a magazine photo shoot with the resultant public notoriety.
When Barney takes the clip out of the first gun and replaces it with blanks, he neglects to see if there is a live round in the chamber.