Maximum residual diversity requires exponential prefix access

Exact statement

If a binary language realizes all $2^{V_N}$ horizon-N rows, its access radius is at least $V_N=2^{N+1}-1$ . The gamma-coded simultaneous maximizer realizes every row by radius at most $V_N+2\lfloor\log_2(N+1)\rfloor+1$ , and therefore has access $2^{N+1}+O(\log N)$ .

StatusProved and internally audited

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

Context

Residual diversity and the prefix depth needed to expose that diversity are different resources. Counting available prefixes gives an unavoidable access cost.

Definitions

$a_L(N)$ is the least $M$ such that prefixes of length at most $M$ already realize every row in $Q_N(L)$ .
$V_N=2^{N+1}-1$ is the number of binary words through length $N$ .

Hypotheses and scope

The language attains the full horizon-N row capacity.
The upper bound concerns the explicit gamma-coded maximizer from FPRD-T46.

Proof or evidence

There are only 2^{M+1}-1 binary prefixes through radius M. Covering 2^{V_N} distinct rows forces M at least V_N. Every declared gamma root has length V_N plus the gamma-header length, giving the upper bound.

Access bounds →

Verification notes

The M27 review corrected an earlier off-by-one: radius V_N-1 supplies only 2^{V_N}-1 prefixes. The proof, production code, independent challenge, and closure audit agree on the repaired lower bound.

Limitations

The root construction gives an upper bound, not the exact first appearance of every row; accidental earlier witnesses may occur.
The result is an unrestricted calibration, not a lower bound for every finite presentation.

Open work

Determine whether a simultaneous maximizer can attain the volume lower bound with smaller additive overhead.

Notes

The upper bound is witnessed by the declared roots. It is not an exact formula for the earliest accidental occurrence of every row.