Short answer. The undisposed remainder passes by legal (intestate) succession. Under Article 851, if you institute one heir but limit the gift to only a fraction of the inheritance, the rest of the estate goes to your legal heirs under the rules of intestacy. The same happens when several heirs are each given fractions that together do not cover the whole.
What the law says
If the testator has instituted only one heir, and the institution is limited to an aliquot part of the inheritance, legal succession takes place with respect to the remainder of the estate.
Civil Code, Article 851 — Sole Heir Instituted to Part of the Estate. Read the full provision →
What the law says
The same rule applies if the testator has instituted several heirs, each being limited to an aliquot part, and all the parts do not cover the whole inheritance.
Civil Code, Article 851 — Sole Heir Instituted to Part of the Estate. Read the full provision →
The leftover goes to your legal heirs
A will does not have to cover everything you own, and Article 851 explains what becomes of the part it leaves untouched. If the testator has instituted only one heir, and the institution is limited to an aliquot part of the inheritance, legal succession takes place with respect to the remainder of the estate. An aliquot part is a fraction — a half, a third, a quarter. So if you give your named heir only one-third and say nothing about the other two-thirds, that two-thirds is distributed by legal succession, the default rules that apply when there is no will covering the property. Your estate is not left ownerless just because the will stopped short.
The same happens with several partial heirs
The rule is not limited to a single heir. The article continues: The same rule applies if the testator has instituted several heirs, each being limited to an aliquot part, and all the parts do not cover the whole inheritance. So if you leave one-quarter to each of three heirs, that is only three-quarters of the estate; the missing quarter falls to legal succession. Whenever the fractions you actually assigned do not add up to the entire inheritance, the gap is filled by the intestacy rules rather than being shared out among the instituted heirs beyond the shares you gave them.
Why the gap is not just split among named heirs
It might seem simpler to enlarge the named heirs' shares to soak up the remainder, but Article 851 does not do that. It presumes that by fixing each heir to a specific fraction, you meant that fraction and no more. Stretching their shares to cover the leftover would give them more than you chose to give. Instead, the law treats the undisposed portion as property you did not dispose of at all, and sends it down the intestate line — to the heirs the law itself designates when a person dies without effectively willing part of the estate.
Who the legal heirs are
"Legal succession" means the estate's default heirs — those the law calls to inherit in the absence of a controlling will, generally the closest family members in the order the Civil Code lays down. They may or may not be the same people you named in your will. This is why leaving fractions that fall short of the whole can send part of your estate to relatives you did not specifically provide for. If you want a particular person to receive the remainder, the safer course is to draft the will so that your dispositions actually account for the entire estate.