Möbius divisor cancellation

Witness-defined non-strict order on natural numbers.

Witness-defined strict order on natural numbers.

The natural number d divides n.

p is nonunit and every factorization of p has a unit factor.

Every common divisor of a and b is one.

x is the bounded beta-decoded value at index i.

Every decoded entry below l is itself below l.

Equal decoded values below l have equal indices.

One canonical injective doubled-Cantor code for all four signed-minor beta-code parameters.

The original canonical integer code: even 2p denotes p, and odd 2k+1 denotes −(k+1). The decoded positive and negative parts are normalized.

The original canonical code z represents the integer difference p−n; these supplied components need not be normalized.

An actual packed signed table with canonical signed entries through index N, including index zero. It contains no divisor transform or inversion hypothesis.

Two genuine beta entries of the packed table represent the unique canonical signed value z. Distinct component representations need not be equal.

A genuine prime p and an actual quotient witness p*p dividing n. This is not an asserted factorization oracle.

Positive n with no squared prime divisor p² for any prime p≤n. The bounded condition is proved to exclude all squared prime divisors.

z is the product of a beta-coded prefix of length l.

Every decoded factor below l is prime.

A positive number n, an actual length-l beta product equal to n, and prime entries. No sortedness or supplied canonicalization is required.

n has an even decomposition.

n has an odd decomposition.

The actual signed code of (-1)^n from parity: even exponents give code 2 (+1), odd exponents code 1 (-1).

A genuine finite prime-factor list for n has a length whose alternating signed unit is z. Independence of the factor list is proved.

For positive n, an actual prime-square divisor gives signed zero; otherwise squarefreeness and real factor-count parity give the signed unit. No divisor-sum or inversion identity occurs here.

A genuine table through N records zero at index zero and the independently defined Möbius value at every positive index through N. The zero-table convention does not extend Mobius to input zero.

A positive d with an actual quotient n=d*q keeps the genuine input value F(d); zero and nondivisors give canonical zero. The zero branch never reads or restricts F(0).

An actual signed table through the inclusive bound l satisfies the independent divisor-mask entry graph at every represented index. The construction bound l is independent of n, enabling ordinary finite induction.

z is the sum of a beta-coded prefix of length l.

The signed balance of two actual natural finite sums, at exactly the indices 0<=i<l. Existence, uniqueness and representation independence are proved separately.

For explicitly positive n, construct a genuine mask through n and take its actual signed fold over S n entries. Index zero is masked away; neither divisor cancellation nor Möbius inversion is part of the graph.

Every value below l occurs at an index below l.

An actual beta-coded bijection of the finite index interval [0,l), including all bounds, injectivity, and surjectivity.

Actual beta-map lookup pulls each source signed value into the target table below l. Neither permutation bijectivity nor any sum identity is assumed in this graph.

Equality of represented values at precisely 0<d<=N. Values at zero are unrestricted, and raw codes or positive/negative representatives are not asserted equal.

Add a fresh factor p, remove a single factor p with p-free quotient, or fix a multiple of p squared. The graph contains no Möbius sign or cancellation hypothesis.

Apply the actual prime-factor toggle at positive divisors of n and fix zero and nondivisors. Closure in the divisor set and involutivity are proved under Prime(p), n>0 and Dvd(p,n).

A real finite beta map witnesses each divisor-prime-toggle output on i<l. No finite-choice, permutation or sum identity is assumed.

Canonical signed negation swaps the positive and negative decoded parts.

Every pair of actual represented signed values at the same i<l are opposite. Genuine table validity is a separate hypothesis; arbitrary codes and component streams need not agree.

Every actual positive entry through N agrees with the independently defined Möbius function. Table validity is separate and F(0) is unrestricted; the historical MobiusTable zero convention remains unchanged.