With unrestricted positive cell gains at DC, mean responses from one fixed connected passive gap model can match any positive matrix

We tested whether averaging responses can create apparent direction in a passive electrical circuit. At direct current (DC), each condition used a fixed positive symmetric transfer matrix with separate positive source and readout gains. We proved that n gain conditions can reproduce any strictly positive n by n response matrix as their equal arithmetic mean. The circuit stays fixed. If each gain changes by at most a factor K and every entry uses the same condition weights, a reciprocal reference exists within a factor (K+1)^2/(4K). The scalar bound is sharp; we do not claim a sharp network bound. A matched geometric mean preserves the reciprocal loop condition for arbitrary positive gains, even when the symmetric circuit changes. A three-condition test on a 296-cell Cook C. elegans gap model used twofold gains. Its arithmetic mean had loop product ratio 1.728 and minimum reciprocal change factor 1.09545. Its matched geometric mean had factor one. Unequal sampling broke both pooling rules. A full 296-cell construction reproduced a chosen positive target with maximum relative error 1.94e-15, but used unrestricted gain ranges near 1e31. Exact rational checks, independent solver checks, and 100 seeded software designs passed. All functional responses are synthetic. We did not fit animal recordings or identify the cause of an empirical loop signal. The package preserves proofs, code, anatomical provenance, complete numerical witnesses, controls, figures, and repeat instructions.

electrical synapses · reciprocal transfer · connectome · response averaging · cell gains · Caenorhabditis elegans · geometric mean · identifiability