First cubic collision of Hardy-Ramanujan number

The number of Hardy–Ramanujan (1729) is a significant in additive number theory because it marks the first time a positive integer can be written as a sum of two positive cubes in two different unordered methods. In this work, we show a complete and verifiable account of that fact. We begin with the classical setting, then replace anecdote with exact arithmetic which is a divisor-discriminant certificate detects all two-cube representations of the number, a finite lattice audit proves that no smaller positive integer has the same property and a restricted partition model records the same result as a coefficient statement in a generating function. We also introduce a collision-deficit graph that shows quantitatively where the cubic-sum map first loses injectivity. This number is not only historically memorable, but also the first collision value of the positive unordered cubic-sum outline.