Unified Formulas for Fermat Congruence Solutions via Cyclotomic Integers

We present unified closed-form formulas for the number of solutions to x^n + y^n = z^n (mod p) for n = 3 (cubic) and n = 4 (quartic). For the cubic case, Sol_3(p) = p^2 + (p-1) B(a,b) where B(a,b) = 2b sigma(b) - (a-b) sigma(a-b) - (a+b) sigma(a+b) and sigma is the mod-3 sign function (Legendre symbol mod 3), verified for all 2,556 primes p congruent to 1 (mod 3) up to 50,000. For the quartic case, an analogous formula via Gaussian integers is verified for all 609 primes p congruent to 1 (mod 4) up to 10,000. Complete proofs are provided via Jacobi sums in the Eisenstein integers Z[omega], including the explicit six-case determination of primary associates and a Hasse bound |B| <= 2 sqrt(p). Version 3 proves the universal congruence B_n(p) congruent to n-2 (mod n) for EVERY integer n >= 2 and every prime p congruent to 1 (mod n), replacing the version 2 result which covered prime n only. The proof is an orbit count in two lemmas. A cone identity gives Sol_n(p) = 1 + (p-1) N_n(p), where N_n(p) is the number of F_p-points of the projective Fermat curve, hence B_n(p) = N_n(p) - (p+1) and B_n(p) is an integer unconditionally. The mu_n action [x:y:z] -> [zeta x : y : z] on that curve has, for every nontrivial zeta, fixed locus exactly the n points with x = 0, so every other orbit is free of size n and n divides N_n(p); with p congruent to 1 (mod n) this gives B_n congruent to -2 congruent to n-2 (mod n). No Jacobi sums, Gauss sums or cyclotomic integers appear in this argument and the primality of n is never used. A corollary extends the congruence to all primes p via d = gcd(n, p-1). The Mobius-function obstruction reported in version 2, arising from Phi_n(1) = 1 when n has at least two distinct prime factors, is identified as an artefact of the Gauss-sum route rather than a property of the statement; the orbit proof covers n = 105 and n = 210 unchanged. Version 3 also corrects a sign error in version 2 Proposition 2.5, which stated J(chi,chi) = -pi and B = -2 Re(pi) while the same document's Theorem 2.7 asserted B = +2 Re(pi_primary). Direct evaluation of J(chi,chi) in Z[omega] shows it is itself primary in the normalisation c congruent to 2, d congruent to 0 (mod 3) and equals pi with no sign. An accompanying self-contained verification script (Python standard library only, exact integer arithmetic, no floating point in any certified quantity, no randomness, no external data) certifies every claim in eight groups: 20,254 checks, zero failures. The recorded run is included. We note that the Jacobi sum J(chi,chi) determining B(a,b) also equals the phase correlation function C(1) in p-level OAM quantum optical systems, connecting the Fermat solution count to measurement statistics in structured light experiments. The cubic formula further provides the arithmetic engine for the author's conformity gradient D(t) = W''/W on elliptic curve moduli (DOI: 10.5281/zenodo.19425788).