General finite signed Dirichlet inverses — Exact Proof Explorer

An actual unit-at-one lookup supplies its canonical signed unit code; no inverse property is hidden in the predicate.

A genuine lookup with a canonical signed unit value satisfies the two-case unit-at-one predicate.

The same actual delta table restricts to every smaller positive window, without changing its unrelated zeroth value.

One actual right-delta convolution supplies both inverse laws by the already proved finite commutativity theorem.

Swapping the two actual convolution identities makes the original table an inverse of its inverse.

The inverse graph entails actual valid input tables; it is never a vacuous equation between missing lookups.

Every pair of actual zero-window tables has a genuine delta witness and both empty positive-domain inverse identities, with no condition at one.

Construct the proper signed remainder, solve its unit-coefficient equation, append the actual new input value and preserve every earlier convolution; the target table is arbitrary.

Finite induction constructs an actual solution G*F=T for every target table and signed unit F(1), with an arbitrary prescribed G(0), including actual witnesses when N=0.

Construct an actual delta target and solve the genuine triangular convolution equation; both inverse laws and the independently prescribed zeroth value follow.

Either actual canonical unit value at one constructively supplies a finite Dirichlet inverse, with any requested value at zero.

The empty positive window has a genuinely constructed inverse for any prescribed zeroth value, without assuming any lookup or unit condition at one.

The exact empty-window-or-unit-at-one condition constructs actual inverse witnesses, preserving an arbitrary requested value at zero.

At the genuinely in-domain index one, the actual convolution is a signed product equal to one, forcing the original value to be +1 or -1; the nonempty-domain guard is essential.

Associativity and independently constructed delta identities force any two actual inverses to agree at every positive input; neither their codes nor their zero values are identified.

An actual inverse restricts with its actual delta witness to every smaller finite positive window, including zero.

Independently constructed inverse prefixes have identical represented positive values on their common smaller domain.

Taking an actual Dirichlet inverse twice recovers precisely the original positive represented values, with no encoding or zeroth-value uniqueness claim.

An actual finite signed arithmetic table has a Dirichlet inverse exactly when the positive window is empty or its actual value at one is +1 or -1.

On a nonempty positive domain the general constructive inverse criterion is precisely the actual signed unit-at-one condition.

Construct an actual inverse with any prescribed zeroth value and prove that every actual inverse agrees with it on all positive inputs, including the genuine empty-window boundary.