Möbius values and prime adjunction — Exact Proof Explorer

Parity constructs an actual canonical code for the alternating unit at every natural exponent.

Constructive parity exclusivity makes the signed alternating-unit code unique.

Exponent zero has canonical positive-unit code two, not code one.

Actual unordered prime-factor uniqueness proves literal equality of factor counts; no canonical list is assumed.

The independent Möbius graph explicitly excludes the infinite-divisor boundary zero.

No canonical signed value is asserted for the excluded input zero.

A supplied actual prime-square divisor of a positive input constructs its canonical zero Möbius value.

Squarefreeness, a real prime-factor list and its actual length parity construct the nonzero Möbius value.

Finite prime-square search, actual prime factorization and parity construct the independently defined Möbius value for every positive natural.

For a squarefree input, every real factor list computes the same Möbius sign, independently of ordering or chosen witnesses.

Möbius values have literally unique canonical signed codes; square-divisor and squarefree branches are disjoint by proof.

Every positive natural has one actual, uniquely determined, factorization-defined canonical Möbius value.

The unit boundary has Möbius value positive one (signed code two), proved from its actual empty prime factorization.

A genuine divisor of a positive squarefree input is positive and squarefree, with no bound assumption on its prime-square witnesses.

No genuine prime has a squared prime divisor; both nonzero and nonunit boundaries are proved from primality.

Adjoining an actual prime not dividing a squarefree input preserves squarefreeness; Euclid cancellation excludes every possible squared prime divisor.

The beta extension theorem constructs a new actual prime list with one more occurrence and product n*p; no sorted or preselected factorization is supplied.

Canonical code two is positive one and code one is its genuine decoded additive inverse.

The alternating unit at the successor exponent is the canonical signed negation, proved by the two constructive parity cases.

Every independently defined Möbius value at an actual prime-square multiple is the canonical zero code.

For an actual prime not dividing n, adjoining that prime negates the genuine Möbius value, including all nonsquarefree zero cases.