Exact residual growth for sparse-marker languages

Exact statement

For every strictly increasing $f:\mathbb N\to\mathbb N$ , the exact residuals of $L_f=\{a^n b a^{f(n)}:n\ge0\}$ are precisely the pre-marker families $A_n$ , singleton completions $R_r=\{a^r\}$ , and the empty residual. At horizon $N$ , $q_f(N)=N+2+|\{n:f(n)\le N-1\}|$ .

StatusProved and internally audited

External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The family is a sparse calibration between regularity's bounded quotient regime and unrestricted maximum growth. Its growth can exceed the nonregular floor arbitrarily slowly.

Definitions

$A_n=\{a^{m-n}ba^{f(m)}:m\ge n\}$ is the residual before the marker.
$R_r=\{a^r\}$ is the residual after the marker with $r$ symbols left.

Hypotheses and scope

The schedule f is strictly increasing on the natural numbers.
The alphabet is {a,b}, with exactly one marker b in each accepted word.

Proof or evidence

Prefixes before the marker reach A_n; legal post-marker prefixes reach singleton completions; malformed or overlong prefixes reach empty. At horizon N there are N+1 active singleton rows, one active A_n row for each f(n) at most N-1, and one common empty row.

Sparse-marker profile →

Verification notes

The M28 proof and independent state enumeration agree on the residual classification and formula. The challenge reports 486,493 checks. Hand-copied review-table errors were rejected and corrected without changing the theorem or repository computations.

Limitations

For arbitrary f, the language is not asserted to be context-free, automatic, PEG-recognizable, or otherwise finitely presented.
The scalar profile does not determine same-horizon access from any finite history; that stronger result is recorded separately as FPRD-T49.

Open work

Identify finite presentation classes that constrain the jump terms hidden by the scalar quotient sequence.

Notes

The family shows that the excess above the N+2 nonregular growth floor can be unbounded and arbitrarily slow. It is not asserted to be finitely presented or context-free for arbitrary f.