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Counterexample · negative result

FPRD-CPA-C05

The factorial derivative algebra is not finitely generated

Exact statement

Finite generation of the derivative Hadamard algebra D(f) is not necessary for a positive commutative-control presentation; the failure already occurs over a unary alphabet and also over the displayed two-letter alphabet.

StatusProved by an explicit factorial automaton; internally audited
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The most tempting reduction asks whether the output's derivative algebra is finitely generated. The counterexample shows that auxiliary commutative observables may finitely present an infinitely generated visible residual algebra.

Definitions

  • Use controls X,YX,Y with identical aa- and bb-transitions X↦XYX\mapsto XY and Y↦Y+1Y\mapsto Y+1, and output values F(X)=F(Y)=1F(X)=F(Y)=1.
  • The recognised controls are f(w)=∣w∣!f(w)=|w|! and g(w)=∣w∣+1g(w)=|w|+1.

Hypotheses and scope

  • The alphabet is nonempty and every letter induces the same transition; the displayed source-facing presentation uses two letters.

Proof or evidence

The recurrences give δf=f⊙g\delta f=f\odot g and δg=g+1\delta g=g+1, so both controls are commutative. Algebraic independence identifies Q[f,g]\mathbb Q[f,g] with Q[x,y]\mathbb Q[x,y]. The derivatives correspond to xPr(y)xP_r(y), whose degree-one component is xQ[y]x\mathbb Q[y], an infinite-dimensional space; a finitely generated graded algebra cannot have that degree-one component.

Factorial automaton counterexample →

Verification notes

The FPRD proof audit rechecked the factorial recurrence, the algebraic-independence limit argument, and the homogeneous-component grading obstruction. An independent checker confirmed 507 unary and binary derivative identities and the rising-factorial triangular basis through degree twelve.

Limitations

  • The counterexample refutes only necessity of finite generation of D(f).
  • It does not refute existence of some other finite commutative over-algebra; here Q[f,g] is such an over-algebra.

Open work

Retain the written grading proof as the general justification and the bounded checker only as corroboration.