Full finite signed Möbius inversion

Witness-defined non-strict order on natural numbers.

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

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

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.

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

The natural number d divides n.

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.

Witness-defined strict order on natural numbers.

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.

An actual signed table has canonical one, code 2, at every 0<n<=N. Its value at zero is unrestricted, including when N=0. Existence with any prescribed zero entry and the divisor-sum identity are separately proved.

An actual signed table has code 2 at n=1 and code 0 at all other positive n<=N. Its zero entry is unrestricted and N=0 has an empty positive domain. Both convolution unit laws are theorems, not definition premises.

Actual multiplication of original canonical signed codes; opposite-sign products remain on the negative side of the balance.

At a positive divisor d, witness n=d*q, read the actual signed values F(d) and G(q), and multiply them. Zero and nondivisors contribute canonical zero without reading either input at zero. Source-table validity is separate.

An actual signed table M records the independently defined convolution entry at every inclusive index 0<=d<=l. The endpoint l is included and can differ from n; no sum or convolution identity is assumed.

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.

Require n>0, construct a real convolution-summand prefix through n, and take its actual signed fold over exactly S n entries. Zero is outside this sum's input domain; no value is assigned by a unit or inversion formula.

Three actual finite signed tables whose output H(n), at every 0<n<=N, is the independently defined convolution sum of F and G. All zero entries are unrestricted; only represented positive values, not table codes, are subsequently proved unique.

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.

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.

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 actual positive output G(n), for n<=N, is the separately defined signed divisor sum of F. Table validity is a separate prerequisite and both zero entries are unrestricted. No Möbius values or inversion conclusion appear in the graph.