Reverse-and-add factors through a symmetric fold and one carry bit

Exact statement

For each fixed $b\ge2$ and $n\in\mathbb N_0$ , the quantities in FPRD-D39 satisfy $\gamma_i\in\{0,1\}$ and $T_b(n)=\sum_{i=0}^{L-1}s_i b^i+\gamma_L b^L$ . After the symmetric fold supplies $c_0,\ldots,c_{L-1}$ in increasing digit index, the carry stage is a deterministic subsequential transducer over $C_b=\{0,\ldots,2b-2\}$ , with states $\{0,1\}$ , initial state 0, transition $\lfloor(c+\gamma)/b\rfloor$ , output $(c+\gamma)\bmod b$ , and terminal output 1 exactly when the final carry is 1. Within this folded one-step presentation, carry is the only orientation-dependent sequential state.

StatusProved on the public page; finite replay is corroboration, not proof

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

Context

Reversal is nonlocal in the ordinary digit order. Once the symmetric fold has supplied the column sums in increasing digit index, every addition column is local and only the carry travels directionally.

Hypotheses and scope

The base b is fixed, and the input uses its canonical finite base-b expansion.
The symmetric folded word is supplied before the carry transducer runs; no ordinary one-way transduction of the original digit stream is claimed.

Proof or evidence

The written induction maintains the partial-sum invariant column by column. Its base is γ_0=0; if γ_i∈{0,1}, then 0≤c_i≤2b−2 makes γ_{i+1}=⌊(c_i+γ_i)/b⌋ a bit. The identities c_i+γ_i=s_i+bγ_{i+1} then telescope to the full sum.

Folded carry theorem and proof →

Verification notes

A published checker corroborates 3,000,015 cases in bases 2 through 16 and inputs from 0 through 200,000. This finite replay is not the proof.

Limitations

This is a family indexed by a fixed base, not one finite transducer over unbounded bases.
The symmetric fold is supplied first; the theorem does not make reversal into an ordinary one-way transduction of the original digit stream.
The decomposition relocates the difficulty; it does not control a complete orbit.
No novelty claim is made.

Open work

Compare this elementary decomposition with prior reverse-and-add and finite-transducer literature before making any novelty claim.