One-step decimal palindromes have two column-sum classes

Exact statement

A positive decimal input has a palindromic one-step reverse-and-add output exactly in one of two cases: every symmetric column sum is at most $9$ , or every sum lies in $\{0,11\}$ and the outer sum is $11$ . On $1\le n\le1{,}000{,}000$ , the two classes contain $151{,}250$ and $808$ inputs.

StatusGeneral classification proved; complete bounded census independently reproduced

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

Context

Vaughn Suite's 2003 note states the general decimal carry classification and gives separate counting formulas for the two cases. The finite FPRD census independently checks the first million inputs.

Hypotheses and scope

The structural classification is for every positive integer in base ten, using its canonical decimal expansion.
The displayed counts are restricted to 1≤n≤1,000,000.

Proof or evidence

The published checker enumerates the complete interval, directly tests the output palindrome, classifies the column sums, and finds 151,250 carry-free plus 808 saturated cases with zero uncovered inputs.

Classification proof and bounded census →

Verification notes

The general criterion was reconstructed from the carry recurrence. A new public checker exhausts the complete interval and recovers 151,250 carry-free and 808 saturated cases with no uncovered palindrome-producing input.

Limitations

Only the decimal classification is asserted; other bases have different boundary behavior, notably base two.
The finite counts do not predict whether later iterates of a particular orbit become palindromic.

Open work

Translate the decimal carry proof to general bases, treating the exceptional base-two growth pattern separately.