Constructed triangular convolution steps

Witness-defined non-strict order on natural numbers.

Witness-defined 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.

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.

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.

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

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

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.

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.