The initial investigation lasted until the Portuguese closed the case and published their files.
Rather more than 3 months.
Allow me to make myself clearer.
The investigation coordinated by Goncalo Amaral into Madeleine's disappearance lasted all of three months: thereafter the investigation coordinated by Amaral became the investigation into the culpability of Madeleine's parents in her disappearance.
When Amaral was sacked the investigation was taken over and coordinated by Paulo Rebello a dyed in the wool Policia Judiciaria man. Funnily enough I have never heard of Rebello dreaming up any theories but there is plenty of proof that he relied on evidence not not speculation.
Rebelo and the Policia Judiciaria turned up
no evidence either against Kate and Gerry or Robert Murat. Which in my opinion shows the value of following evidence rather than following theories and it proves that much of not only the golden hours but the months and years of Madeleine's investigation have been tainted by Amaral's incompetence and Amaral's nonsensical theories.
I think the responsibility of ruining any chance of recovering a missing child is a pretty heavy burden for anyone to labour under, particularly for Amaral so it is no surprise he has spent his time in denial and so assiduously attempting to offload his incompetence and his blame.