Arithmetic cube identity for Weil heights ($h(P+Q+R)-h(P+Q)-h(P+R)-h(Q+R)+h(P)+h(Q)+h(R)-h(O)=O(1)$)

For a line bundle on an elliptic curve over a number field, the geometric cube identity remains exact as a line-bundle identity. The Weil height machine translates it into the displayed bounded-error identity for a chosen logarithmic height, uniformly on the rational triples. This distinction is essential: arbitrary naive heights do not obey an exact quadratic identity. Dividing the identity for $2^{n}P,2^{n}Q,2^{n}R$ by $4^n$ produces an exact identity for the canonical height of an elliptic curve.