Reverse-and-add, carry sweep, and reflection distance

Exact statement

Fix $b\ge2$ , $A_b=\{0,\ldots,b-1\}$ , and $n\in\mathbb N_0$ . Write the canonical expansion $n=\sum_{i=0}^{L-1}a_i b^i$ least-significant first, with $L=1,a_0=0$ for zero and $a_{L-1}\ne0$ otherwise. Define $\operatorname{rev}_b(n)=\sum_i a_i b^{L-1-i}$ , $T_b(n)=n+\operatorname{rev}_b(n)$ , $c_i=a_i+a_{L-1-i}$ , $\gamma_0=0$ , $s_i=(c_i+\gamma_i)\bmod b\in A_b$ , and $\gamma_{i+1}=\lfloor(c_i+\gamma_i)/b\rfloor$ , retaining $\gamma_L$ . Reversal-created leading zeroes do not change the integer. For canonical digits $d_i$ of $m$ , $D_b(m)=\#\{i:d_i\ne d_{M-1-i}\}$ counts ordered positions.

StatusExact conventions reviewed; no theorem is embedded in this record

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

Context

Pairing mirrored digits separates the nonlocal reversal from directional carry propagation. The page uses 47+74=121 to show the symmetric column sums (11,11) and the one-bit sweep explicitly. Reflection distance records proximity to a palindrome without treating a long finite orbit as an infinite conclusion.

Definitions

rev_b(n) reverses the canonical base-b digit word; leading zeros created by reversal do not contribute to the resulting integer.
D_b(m) is the Hamming distance between the canonical base-b digit word of m and its reflection, counting ordered positions; D_b(m)=0 exactly for palindromes.

Proof or evidence

The public page fixes one indexing convention and works a decimal example column by column. The theorem that the recurrence computes the full sum belongs to FPRD-T144.

Definitions and conventions →

Verification notes

The tracked present edition and retained finite evidence were reconstructed and checked. The cited June packet, original experiment code, and raw runs remain an explicit custody gap. Leading-zero reversals such as 1200 -> 21 were checked directly.

Limitations

The definitions imply nothing about whether any decimal orbit avoids palindromes forever.

Open work

Find the smallest update-closed receipt that carries this one-step presentation across iteration.