Idol Theory

According to Index, the Idol Theory (Also called Idol Worship) is a form of basic knowledge that allows magicians to put the power of something greater like God or angels to good use.

Theory
Following this theory, if one were to build a duplicate of the cross when Christ was crucified, and put it on the roof of a church, they’ll get some Holy power from those actual Cross wielders. Although the power of the duplicate is not even 0.000000000001% of the original, and even the legendary strongest duplicate Holy Manger has only a small fraction of the power, if they were to get even 1% of the power of the original, their power could match even those of the twelve apostles. Kamijou Touma compares this to the Sun providing solar power for electricity as the power of the 'original' is almost limitless even with the power split across so many duplicates. One hypotheses related to this theory is that it can be switched around. In other words, not only will the original cause some effect on the ‘Idol’, the ‘Idol’ can also affect the original. This is a considered a hypotheses as there are some exceptions which cannot be explained. This hypothesis is used by people claiming that they’ll receive divine judgment if they treat the Bible badly. Index claims this isn't unreasonable as there are many instances in the Bible where several ‘Idols’ from different Greek religions that are against Christians are struck down and destroyed by lightning. The theory cannot be used, however, once the original 'image' is changed. As long as there’s a slight difference in the level of magic used between the idol and the original, the idol will lose it’s purpose.

Usage/Explanations
Here examples will be listed by appearance in the Light Novels if possible.

Saints
The power Saints have can be explained by Idol Thoery, since human are made in God's image humans are in a sense idols of God. Saints are idols that can bring out more power of God than 'normal' people.

Angel Fall
According to Motoharu, Angel Fall was activated by the over 3,000 'souvenirs' that Kamijou Touya brought back from his travels and placed around the house according to Feng Shui. Though each souvenir are only symbols of what they represent (for example, the 'byakko' dolls in the kitchen) and hold even less power than idols used by magicians, by gathering such a large number of them, their power increased by several magnitudes with the powers multiplying each other countless times. Motoharu also mentioned they are lucky that only 'Angel Fall' have been activated, since it could have easily been some other Grand Magic that could have frozen the entire planet or cause a spatial distortion that would devour the world. Even if one of the pieces were destroyed somehow the Grand Magic would just shift into something else other than Angel Fall, so if the ritual location need to be destroyed it need to be erased all at once.

Dai Nihon Enkai Yochi Zenzu
In Japan, the famous mapmaker Ino Tadakata used the Idol Theory in a reverse matter. Using it he plotted 47 ‘portals’ on Dai Nihon Enkai Yochi Zenzu, an intricate map of Japan created by Ino Tadakata, that originally didn’t exist on the Japanese islands. This caused these entrances that allow people to come and go as they please appear in modern Japan. In the entire history of magic, he is the only person who has been able to add something to the duplicate that the original didn't have while maintaining a perfect balance between the duplicate and the original. Out of the 47 'portals' created, only 23 of them have been identified. The special magic teleportation properties of the Dai Nihon Enkai Yochi Zenzu cannot be used completely freely as it is limited to specific times due to the fact it was mapped and measured to the location of the stars. Although the location of the 'portals' that Ino Tadataka set do not change, the development of the city changes everyday. Some 'portals' are located within certain rooms in certain condominiums or even in a large gold vault in a certain bank, making it impossible to use some for teleportation.

Other Examples

 * Volume 16: Acqua of the Back used the projection of the moon in the underground district of Academy City as a substitute to the real moon to activate his water magic.