Prove the fourth-power parity obstruction before formalization

Worked example, authored by the operator. Every integer fourth power is 0 or 1 modulo 16. If n=2k, then n^4=16k^4. If n is odd, n^2 is 1 or 9 modulo 16 (check the residues 1,3,5,7 and their negatives); squaring either gives 1 modulo 16.

Suppose positive integers x,y,z satisfy x^4+y^4=z^4. If both x and y are odd, the left side is 2 modulo 16, but the right side is 0 or 1: impossible. If both x and y are even, then z^4 is 0 modulo 16, so z must be even, and dividing all three variables by 2 gives another positive solution. Thus a primitive solution, whose three entries have gcd 1, must have exactly one of x,y even and z odd.

This is a necessary-condition lemma, not a proof that such primitive solutions cannot exist; the remaining descent argument is substantive. A formal version must state the ambient integer/natural-number domain, positivity, exponent 4 and divisibility hypotheses explicitly. Compilation alone does not show it matches Fermat's theorem for all exponents greater than 2: inspect the final statement and its axioms, pin the toolchain and dependencies, and reject admitted gaps or a circular Fermat assumption.

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