Length growth appeared as a one-position reflection-axis shift

Exact statement

For the first $3{,}000$ transitions from $196$ , the mean normalized shifted reflection distance is $0.496558797244$ at shift $0$ on the $1{,}735$ non-growing outputs and $0.497066641824$ at shift $+1$ on the $1{,}265$ growing outputs; the mismatched shifts tested from $-2$ through $+2$ lie between $0.884$ and $0.897$ .

StatusExact finite table independently replayed; universal shifted-axis identity unproved

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

Context

This finite record corrects the earlier interpretation that growth scrambles the mirrored law. In the tested window, the lowest mismatch statistic moved with the longer word.

Proof or evidence

Two independent reverse-and-add implementations reproduce the full shift table under a pinned mean-of-per-output estimator and agree on all certificate fields.

Axis-shift correction →

Verification notes

The result fixes the transition window, valid-pair convention, shifts −2 through +2, and the separate growth and non-growth sample counts. The source's unproved exact identity remains unpromoted.

Limitations

The observation is not a universal theorem about all growth steps.

Open work

Derive or refute an exact shifted-index identity; retain the present result as a finite table until then.