Exact access for sparse-marker languages

Exact statement

For $L_f=\{a^n b a^{f(n)}:n\ge0\}$ , where $f:\mathbb N\to\mathbb N$ is strictly increasing, the first appearances of every active singleton row, pre-marker row, and empty row give exact formulas for the prefix/future rectangle (G_f(M,N)) and access radius (a_f(N)). A fixed scalar quotient history through horizon (N) can coexist with arbitrarily large access at that same horizon.

StatusProved and internally audited

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

Context

The scalar number of bounded residual rows does not record how far one must read before every row has appeared. This theorem calculates both quantities exactly for a sparse-marker family.

Definitions

$\nu_f(r)=\min\{n:f(n)\ge r\}$ is the first schedule index reaching remainder (r).
$c_f(r)=\nu_f(r)+1+f(\nu_f(r))-r$ is the first prefix length reaching the singleton row $\{a^r\}$ .
$e_f(N)=\min\{\nu_f(N),2\}$ is the first prefix length reaching the empty horizon-(N) row.

Hypotheses and scope

The schedule f is strictly increasing on the natural numbers.
The alphabet is {a,b}, and accepted words contain one marker b.

Proof or evidence

The residual classification from FPRD-T48 leaves only three row families. Exact first-arrival arguments for those families yield the rectangle and access formulas; the schedule f_H(0)=0, f_H(1)=H supplies the fixed-history separation.

Sparse-marker access →

Verification notes

The written proof, production tests, an import-independent challenge, mutation checks, and independently reconstructed state enumeration were reconciled. Hand-copied audit-table errors were corrected and do not occur in the theorem or code.

Limitations

This is an arbitrary-function calibration, not a finite-presentation theorem.
No context-free, automatic, PEG, or novelty classification is claimed for arbitrary f.

Open work

Classify which finite-presentation families constrain the jump term that controls access.

Notes

This proves that access is not determined by any finite initial segment of scalar growth. It does not claim that the pair classifies arbitrary presentations.