The apparatus I — the championships, counted honestly

1924 – present

The claimed eleven: The university claims eleven consensus national championships, and its own athletics history describes the eleven as more consensus championships than any other school holds. The seasons, with their coaches: 1924, 1929 and 1930 under Rockne; 1943, 1946, 1947 and 1949 under Leahy; 1966 and 1973 under Parseghian; 1977 under Devine; 1988 under Holtz.

The selectors, by era: For the Rockne titles the deciding system of record was Frank G. Dickinson's — devised at the request of Rockne and the Chicago clothing manufacturer Jack Rissman, rating teams from 1924 (the 1924 rating itself made retroactively, in 1925) through 1940. Notre Dame's three Dickinson titles retired the system's Rissman Trophy to South Bend, and the successor trophy from 1931 bore Rockne's name. Eight of the eleven are wire-service titles: the Associated Press poll, from 1936, and the UPI/Coaches' poll, from 1950, chose Notre Dame in 1943, 1946, 1947, 1949, 1966, 1973, 1977 and 1988 — which is where the often-printed figure of eight AP-or-Coaches' championships comes from. The other major selectors of the record include the Football Writers Association of America (from 1954), the National Football Foundation, the Helms Foundation, the College Football Researchers Association, Parke H. Davis and the National Championship Foundation.

The NCAA's thirteen: The NCAA's published list recognizes Notre Dame as a champion selection in thirteen seasons: the eleven, plus 1919 — selected by the National Championship Foundation and Parke H. Davis, in a year when Harvard was the near-unanimous choice of the major polls and Illinois and Texas A&M were also recognized — and 1964, selected by the National Football Foundation.

The university's policy: Since 1936 Notre Dame recognizes only AP and Coaches' Poll championships. That single policy explains the three famous absences from its eleven: 1938, when the Dickinson System and its Rockne Memorial Trophy went to Notre Dame while the AP chose TCU; 1953, when ten selectors chose Notre Dame and the AP and UPI chose Maryland; and 1964, above.

The higher totals in circulation: Older Notre Dame media guides listed six further seasons as title years: 1919, 1920, 1927, 1938, 1953 and 1964. Counting every selector in the NCAA records book, Notre Dame was voted champion by at least one selector in twenty-two seasons — the thirteen above, plus 1920, 1927, 1938, 1953, 1967, 1970, 1989, 1993 and 2012. A Blue & Gold Illustrated accounting of the 2012 era counted twenty-one through the 2011 season. These totals circulate as claims, and they are printed here as claims, with the verifiable structure beneath them printed above them.

Sources:

1. University of Notre Dame athletics history — the claimed eleven, described as more consensus championships than any other school's; the post-1936 recognition policy.
2. Dickinson System record — the system's span, its retroactive 1924 rating, the three Notre Dame titles, the trophies.
3. NCAA championship records — the thirteen recognized seasons; the 1919 and 1964 additions and their selectors.
4. Championship histories and the Blue & Gold Illustrated accounting — the media guides' six extra seasons; the twenty-two any-selector seasons; the 1938 and 1953 mechanisms.