I'm currently reading Dameronia, Paul Combs' biography of Tadd Dameron. Combs describes Richard Carpenter as a "shadowy figure in the jazz world," who had some sort of long-term financial hold over Dameron. Combs avoids making accusations of illegality, but he is pretty clear that Carpenter made a business in the 1950s of buying musical rights from musicians with drug problems, who needed a little quick money. A Google search turned up some references to Carpenter from jazz writers who were more straightforward in their choice of words. For example, Chet Baker biographer James Gavin, in an interview on jerryjazzmusician.com (see page 2): "Chet wanted to kill him, literally. Carpenter is remembered as the greatest leech the jazz world has ever known." (Gavin's Chet Baker bio is Deep in a Dream: The Long Night of Chet Baker.)
Carpenter's hold on Dameron may or may not have had something to do with Dameron getting off more easily on a 1956 drug conviction because his previous arrest records had somehow disappeared from city files, according to Combs.
But let's get back to the question of who composed "Walkin'." It wasn't Carpenter. Miles recorded an extremely similar tune in March 1954 that he called "Weirdo," listing himself as the composer.
In April 1954 he recorded the same tune as "Walkin'," listing Carpenter as composer. The track was released on a 10" LP in 1954, and again in 1957 on the album "Walkin'."
In 1958 Miles recorded the same song under the name "Sid's Ahead," crediting himself this time. It was released as part of the album "Milestones."
Miles' later recordings of the song under the title "Walkin'" credited Carpenter; Miles frequently played the tune in live performances as late as 1965.
But Miles didn't compose "Walkin'," either. In 1950 Gene Ammons had recorded "Gravy," crediting Jimmy Mundy as composer. It's the same tune. Here's a link to listen to it.
I guess "Walkin'/Sid's Ahead" should be added to the list of tunes that Miles may not have written.
Combs, in Dameronia, offers his take: "...some maintain that ['Walkin''] was written by Lucky Thompson, and others cite Gene Ammons. The author is inclined to agree with the latter assignation but cannot prove it."
According to this 1964 Downbeat article, Richard Carpenter at one time managed Gene Ammons. According to James Gavin's biography of Chet Baker, Jimmy Mundy had been a client of Carpenter's too. From Gavin's book:
"After Mundy's death in 1983, Don Sickler, a trumpeter and music publisher who handled Dameron's catalogue, found the copyright certificate at the Library of Congress for a tune originally called 'Gravey.' Mundy was believed to be the composer, although some argued that it was Gene Ammons or Miles Davis. The title had been partly erased; 'Walkin'' was written over it and Carpenter's name inserted as composer." (This excerpt was quoted in a discussion of Carpenter at organissimo.org. If you're interested, there are some more details there.)
It's impossible to know exactly how that erasure and altered composer credit could have happened, but there would have been (and still is) a bit of royalty money at stake. Perhaps it was all legal, though.