Non-growing reverse-and-add outputs obey a mirrored ±1 law

Exact statement

Under FPRD-T144, if $\gamma_L=0$ (equivalently, the step does not increase digit length), then for $j=L-1-i$ , $s_i-s_j\equiv\gamma_i-\gamma_j\pmod b$ and the integer $\gamma_i-\gamma_j$ lies in $\{-1,0,1\}$ . This is a residue statement, not an absolute-distance bound; its residue-set corollary is nonrestrictive in bases 2 and 3.

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

Mirrored columns have the same raw sum. Their output digits can differ only because their incoming carry bits differ.

Hypotheses and scope

The output has the same digit length as the input; the comparison uses the unshifted reflection axis.

Proof or evidence

For j=L−1−i, symmetry gives c_i=c_j, so s_i−s_j≡γ_i−γ_j (mod b). T144 makes both carries bits. The page also proves that γ_L=0 is equivalent to no length growth, including the canonical zero singleton, and works 150+051=201 as a sharp modular example.

Mirrored carry law and proof →

Verification notes

A published checker corroborates the result with no violation among 10,888,899 ordered mirrored pairs in the stated finite domain. The proof, not the finite replay, establishes the universal claim.

Limitations

The congruence is not an ordinary absolute-distance bound: digits 0 and b−1 differ by −1 modulo b.
The residue set gives no exclusion in bases 2 and 3.
The law permits equality and therefore does not exclude palindromes.

Open work

Audit prior art, especially the documented observation that a palindromic output requires a palindromic carry pattern.