Signature Algebras: Axiom Consistency, Low-Dimensional Classification, and the Flat Rigidity Theorem

We introduce a new class of algebraic structures called signature algebras (G, ⊕, ⊗), where the binary operation ⊕ is commutative and associative, ⊗ is associative and right-distributive over ⊕, while neither commutativity nor left-distributivity of ⊗ holds universally. This paper establishes the rigorous mathematical foundations of this structure. The main results are as follows. 1. Axiom consistency: We prove the axiom system is consistent by constructing a terminating and locally confluent term rewriting system, and we exhibit an explicit non-trivial model. 2. Low-dimensional complete classification: On the unique two-dimensional non-zero-multiplication nilpotent algebra, symbolic computation of 32 polynomial equations yields exactly seven solution families, of which three survive the non-equational conditions, all belonging to the projection family (all right-multiplication maps are idempotent endomorphisms). On the three-dimensional ladder nilpotent algebra, the complete classification under the axioms yields a one-parameter continuous family. Classification results are obtained under the working assumption of nilpotent ⊕-algebras. 3. Flat rigidity theorem: We rigorously prove that on the N=4 flat ⊕-algebra (eᵢ ⊕ eⱼ = e₈+₉+₁), no ⊗ operation exists that simultaneously satisfies associativity, right-distributivity, and the non-equational conditions. This theorem forces the depth space to carry non-trivial warp, providing algebraic necessity for geometric emergence in the continuum limit. All symbolic computations are independently verified using SymPy 1. 12 and are fully reproducible. Numerical experiments for N=5, 6 support the conjecture that the no-go result extends to all N≥5.