Cross-length reduction in stages

Exact statement

If $f_m(y)$ counts raw $m$ -digit sources over output $y$ , then the full fibre has size $f_{M-1}(y)+f_M(y)$ ; cross-length gluing adds at most one bit to the larger within-chart source-fibre budget.

StatusProved

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

Context

The fixed-length mirror quotient extends across adjacent length charts with one seam bit, but complete growth responses require infinitely many states and separated-flank inverse limits contain non-realizable ghosts.

Proof or evidence

The proof composes the fixed-length quotient with the adjacent-chart seam theorem.

Reverse-and-add cross-length atlas review and independent audit →

Verification notes

The mirror-profile charts, output-length bound, one-bit receipt, and raw-fibre sum were reconstructed. An independent implementation checked 1,208,130 admissible profiles, 310 overlap outputs, and a sharp collision in every base from two through twelve. Missing Stage 6 archival payloads remain separate at L1.

Limitations

This is an information budget, not a time or space lower bound for arbitrary algorithms.

Open work

Compare the budget with finite-horizon response quotients.