The Primes Come in Pairs: Goldbach and the Twin Primes as Antipodal Pairs About the Half-One

One axiom. One operation. Zero free parameters. Goldbach (every even number is a sum of two primes) and the twin-prime conjecture (infinitely many primes two apart) are two famous unsolved problems about prime pairs. This paper shows both are statements about the fold’s antipodal symmetry about the half-One. The fold’s antipodal involution sends j/q to (q−j)/q, summing to the One, with sole self-antipode 1/2. Scaled to an even E, the pair (k, E−k) is an antipodal pair about the midpoint E/2. Goldbach is then: every even E has an antipodal pair prime on both sides — a symmetric splitting of E about its half-One into two primes. Twin primes are the closest odd antipodal neighbours (p, p+2), their count governed by the same orbit/coset structure the doubling fold gives the integers. The fold makes the form native; the engine verifies Goldbach to a bound (reporting the minimum prime-pair count) and counts the twins. Machine-checked; reproduces from one command. A standalone result within the Smithian Fold Theory of Everything (SFTOE). Full corpus, code, and the run-it-yourself VERIFY.md protocol: https://github.com/MettaMazza/Smithian-Fold-Theory