Quadratic-reciprocity gate for the complementary orientation

Exact statement

Let $RS+3^g=2^K$ be the reduced $a=0<g$ equation, with $R$ the negative-character canonical odd factor and $S$ the positive-character factor. Of the twelve possible canonical pairs modulo 24, the equation and the two Jacobi identities leave exactly $(R,S;K,g)\equiv(5,1;1,1),(5,11;0,0),(7,1;0,0),(7,17;1,0),(13,1;0,1),(13,19;0,0)\pmod{(24,24;2,2)}$ .

StatusProved elementarily and independently reconstructed

External reviewNo documented external or specialist review of these FPRD results is recorded.

Context

Reducing modulo each odd factor gives 2^K congruent to 3^g. The supplementary laws for the Jacobi symbols of 2 and 3 then couple the two exponent parities to each factor's residue modulo 24.

Proof or evidence

The equation modulo 24 supplies one parity pair for every candidate residue pair; the two Jacobi identities eliminate six of the twelve pairs.

Complementary quadratic gate →

Verification notes

Independent Python and C++ implementations reconstruct the complete table and the exact residue cycle for odd parts of 1+3^b.

Limitations

The six rows are necessary conditions, not a classification of their integer solutions.
The theorem applies to the normalized a=0<g orientation, not arbitrary integer-complexity formulas.
No external review or novelty determination is recorded.

Open work

Use the six-row gate before any 3-adic lifting or odd-prime quotient in the complementary orientation.