One decidable language realizes every finite residual table

Exact statement

Over the ordered binary alphabet, $q_L(N)\le 2^{2^{N+1}-1}$ for every language. One decidable language built from prefix-free Elias-gamma headers and table payloads attains this ceiling at every horizon simultaneously. For any language, simultaneous attainment is equivalent to density of its residual orbit in the full Boolean behavior shift and to equality of its bounded-observation completion with that full shift.

StatusProved and internally audited

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

Context

A horizon-N residual row is a Boolean table on all binary words of length at most N. The theorem gives the unrestricted capacity baseline and shows that decidability alone does not reduce it.

Definitions

$B_N=\{0,1\}^{\le N}$ has $V_N=2^{N+1}-1$ coordinates.
A root $r_{N,b}=\gamma(N+1)b$ stores a horizon and one complete Boolean table $b\in\{0,1\}^{V_N}$ .

Hypotheses and scope

The alphabet is the ordered binary alphabet and table coordinates use length-lexicographic order.
The capacity statement ranges over arbitrary languages, not finitely presented recognition systems.

Proof or evidence

There are at most 2^{V_N} Boolean tables. Gamma-coded roots are prefix-free, so their cones can independently realize every table at each horizon while preserving decidable membership. Realizing every finite ball pattern is exactly density in the product topology; completion then gives the full shift.

Maximum residual growth →

Verification notes

The M27 proof, production suite, and import-independent challenge verify the code, table ordering, malformed inputs, full row realization, and density equivalence. The challenge reports 10,614 checks and five mutation separators.

Limitations

Prefix codes, dense full-shift orbits, and the Boolean language carrier are classical ingredients; no novelty claim is made.
The theorem supplies no finite-presentation result and no classification of all growth profiles.

Open work

Classify attainable growth and access profiles inside named finite recognition-presentation classes.

Notes

This is an unrestricted-language capacity baseline. Full shifts, dense orbits, prefix-free coding, and gamma codes are classical ingredients; no finite-presentation or novelty claim is made.