Definition
A character-like structural signature is a proposed invariant χ_A of a communicative kind (TERM-040). It takes the same value, or nearly the same, on every realization of the kind: A194 §2 asks for “a suitable invariant profile χ” on which the spoken, translated and visual realizations agree. It must also keep the distinctions the theory cares about. A194 §7: “the character-like invariant preserves independently specified pragmatic and decision-relevant distinctions.”
The word “character” is reserved. A194 §2 first marked it: “Here χ is a proposed character-like invariant, not necessarily the trace of a group representation.” A203 §36 made it a rule: “The word character would be used literally when χ is a trace character of a specified representation. Otherwise, it should be described as a character-like structural signature. That terminological distinction prevents the proposal from making stronger representation-theoretic claims than its construction supports.” A214 §11 applies it to a learned signature Ẑ: when no genuine group representation is available, “Ẑ remains a character-like signature, without claiming the completeness theorems of ordinary character theory.” A218 Ch3.11 lists “Character-like signatures and their mathematical limitations”.
A203 §36 asks three things of it: stability under the designated transformations, sufficiency for a family of decisions, and power to discriminate designated contrasts. Whether any signature meets all three is a question, not part of the definition.
What it is not
- Not Propp’s character functions (THEORY-167). Those are dramatis personae of the wondertale. This is the group-theoretic sense of “character”, a trace.
- Not yet built. A194 named the construction as “The next formal challenge”: an explicit communicative character “even for a small toy model of teasing, reprimand, and promise-making”. No later turn constructs one.
- Not a factorization’s latent coordinates. Those are fixed only up to a gauge, an invertible change of coordinates, which is not a symmetry of the relational system (CLAIM-148). A signature must be a function of the data that does not depend on that choice.
- Not a genuine character when it is approximate. The characters of representations of a finite group are integer combinations of the irreducible characters, so for two of them ‖χ_A − χ_B‖² = Σ(m_i − n_i)² is a non-negative integer. A tolerance d_χ ≤ ε with ε < 1 then forces equality. An approximate signature must live in a continuous space: normalized characters, class functions in general, or traces of fitted operators. A194 §7’s distance “based on differences between character values” falls into this trap if χ is read literally.