General finite signed Dirichlet inverses

Witness-defined non-strict order on natural numbers.

Witness-defined strict order on natural numbers.

Exactly the canonical signed codes 2 (+1) and 1 (-1). Its equivalence with an actual signed multiplicative inverse and the affine-equation solver are separately proved, not assumed by this two-case graph.

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.

Pointwise equality of actual canonical signed lookups below l, not equality of table codes or their positive/negative components.

A genuine output signed table through l preserves the represented source values at i<l and records the prescribed signed value z at l. Existence is proved by recoding both beta streams.

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.

Actual addition of original canonical signed codes, witnessed by their decoders and balanced equality.

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

The natural number d divides n.

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.

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.

An actual lookup F(1) has canonical signed code 2 or 1. This direct disjunction contains ArithAt, not a SignedUnit or inverse subformula. Table validity, the finite window bound, and the inverse criterion are separate hypotheses or theorems.

Witness a real Kronecker delta table E and both actual convolution tables F*G=E and G*F=E on 0<n<=N. Their graphs include actual table validity; values at zero remain unrestricted. The unit-at-one criterion is not a definition premise, and its necessity requires N>0; zero-window inverse identities are separately proved for all actual input tables.