The sampled 196 orbit stayed linearly far from reflection

Exact statement

For the outputs $T^k(196)$ , $1\le k\le4{,}000$ , the reflection distance has minimum $2$ , its minimum normalized value is $4/33$ at step $68$ , and its mean normalized value over the $3{,}771$ outputs longer than $100$ digits is $0.659803985542$ .

StatusExact finite window independently replayed by two implementations

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

Context

Reflection distance quantifies the familiar visual impression that the sampled orbit is not close to a palindrome.

Proof or evidence

A Python-integer implementation and an independent digit-list implementation agree on all 4,001 saved states, the state-sequence hash, the minimum distance and ratio, and the long-word mean.

Reflection-distance observation →

Verification notes

The replay pins outputs 1 through 4,000, defines D on ordered positions, and replaces the source's descriptive 0.66 slope with the exact finite mean used here.

Limitations

A positive distance over 4,000 steps does not imply a positive distance at every step.
The approximate 0.66 slope is descriptive, not a theorem or probability model.

Open work

Compare the same pinned statistic with convergent controls; do not extrapolate the finite lower bound to the infinite orbit.