When Vallejo Police Chief Jack K Stiltz questioned whether the author of the codes and letter writer on July 31st 1969 was the same person as the killer, Zodiac's response was immediate in the 'Debut of Letter' the following day, supplying more details of the first two crimes at Lake Herman Road and Blue Rock Springs to the police chief.
All Zodiac had to do was use the same weapon at Lake Herman Road, Blue Rock Springs and Presidio Heights, and ballistics and shell casing analysis would have confirmed that the three attacks were inextricably linked or at the very least left little doubt. This would have saved Zodiac the effort of proving he was the killer in all four attacks, even employing his gun at Lake Berryessa. But Zodiac changed his weapon every time, casting doubt on all the crimes being linked, however everything Zodiac did and said the rest of the time was ultimately designed to prove he was the killer in all four attacks.
So what can you take from this. Zodiac was prepared to take risks, but was not prepared to extend this risk for weeks or months between his crimes, albeit hiding the weapon away from his home residence would seem to be foolproof if the gun is not registered or possesses identifying features.
The overarching belief is that Zodiac disposed of each weapon immediately after each crime. The quicker you can separate yourself from the 'smoking gun' the better. But we still have the conflict of a killer quite prepared to take innumerable risks in the commission of his crimes, but wastes not a second in the disposing a weapon he could easily conceal. This is the dichotomy of the Zodiac Killer, somebody of whom logic simply doesn't apply.