The 5"x8" record might refer to one of two (closely related) entities.
The first is a tool that would have been used in the office managing naturalisation and probably so-named by the staff to distinguish it from the big book from which the certificates were constructed.
This is the description of items held in the Warren County, NY Archival Collection
Bound, pre-printed receipt books of varying sizes (av. size 5 x 8 ) containing the stub portion to the Certificate of Naturalization and including the following information: certificate number, name, age, date of admission, declaration and/or petition number, alien registration number, family status (i.e. name, age, place of residence of spouse and of minor children; court and filing information. Overall size of original volume with Certificate of Naturalization 14 1/2" x 8 1/4". Receipt books discontinued c. 1927, replaced by form N-550 (10" x 3"). OCA Schedule # 21030.
The second possibility is that it is a copy made from one of those books, called a dexigraph .
The National Archives–Northeast Region (Boston) has a unique set of records that brings together naturalization documents created in various New England courts. These records consist of photostatic copies, called "dexigraphs," made in the 1930s, of naturalization proceedings in all courts—federal, state, county, and local—in five of the New England states (Maine, Massachusetts, New Hampshire, Rhode Island, and Vermont) between 1790 and September 26, 1906.
Sadly in your case of a petition in a NY court, if there is a dexigraph, it is not in the Boston office of the National Archives.
However they are indexed elsewhere as M1674, Index (Soundex) to Naturalization Petitions Filed in Federal, State, and Local Courts in New York, New York, Including New York, Kings, Queens, and Richmond Counties, 1792–1906. 294 rolls.
Thanks to @efgen for the last para who reminded me of the good genealogical practice of reading to the end of a document.