This is not really answering your question. But it's worth pointing out that periodicity of Clifford algebras (closely tied to Bott periodicity) already gives you the "periodicity" in the explicit lower bound. If you want a sphere with $8a-1$ independent vector fields, you can take an irreducible representation $M(8a)$ of the real Clifford algebra $C(8a)$; the Clifford module structure provides that many vector fields on the unit sphere in $M$. The algebras $C(8a)$ are all Morita equivalent, and since $\dim_{\mathbb{R}}C(8(a+1)) = 16^2\\, \dim_{\mathbb{R}}C(8a)$, you must have $\dim M(8(a+1))=16\\, \dim M(8a)$, so we learn that $\rho(16^a-1)\geq 8a-1$.

As for the upper bound, this reduces to something about the real K-theory of truncated projective spaces, which Adams calculated, and which clearly has something to do with Bott periodicity.

But I would agree that this all sounds like a big coincidence. I wish I knew something better to say about it.

This is not really answering your question. But it's worth pointing out that periodicity of Clifford algebras (closely tied to Bott periodicity) already gives you the "periodicity" in the explicit lower bound. If you want a sphere with $8a-1$ independent vector fields, you can take an irreducible representation $M(8a)$ of the real Clifford algebra $C(8a)$; the Clifford module structure provides that many vector fields on the unit sphere in $M$. The algebras $C(8a)$ are all Morita equivalent, and since $\dim_{\mathbb{R}}C(8(a+1)) = 16^2\, \dim_{\mathbb{R}}C(8a)$, you must have $\dim M(8(a+1))=16\, \dim M(8a)$, so we learn that $\rho(16^a-1)\geq 8a-1$.

As for the upper bound, this reduces to something about the real K-theory of truncated projective spaces, which Adams calculated, and which clearly has something to do with Bott periodicity.

But I would agree that this all sounds like a big coincidence. I wish I knew something better to say about it.

This is not really answering your question. But it's worth pointing out that periodicity of Clifford algebras (closely tied to Bott periodicity) already gives you the "periodicity" in the explicit lower bound. If you want a sphere with $8a-1$ independent vector fields, you can take an irreducible representation $M(8a)$ of the real Clifford algebra $C(8a)$; the Clifford module structure provides that many vector fields on the unit sphere in $M$. The algebras $C(8a)$ are all Morita equivalent, and since $\dim_{\mathbb{R}}C(8(a+1)) = 16^2\\, \dim_{\mathbb{R}}C(8a)$, you must have $\dim M(8(a+1))=16\\, \dim M(8)$, so we learn that $\rho(16^a-1)\geq 8a-1$.

As for the upper bound, this reduces to something about the real K-theory of truncated projective spaces, which Adams calculated, and which clearly has something to do with Bott periodicity.

But I would agree that this all sounds like a big coincidence. I wish I knew something better to say about it.

This is not really answering your question. But it's worth pointing out that periodicity of Clifford algebras (closely tied to Bott periodicity) already gives you the "periodicity" in the explicit lower bound. If you want a sphere with $8a-1$ independent vector fields, you can take an irreducible representation $M(8a)$ of the real Clifford algebra $C(8a)$; the Clifford module structure provides that many vector fields on the unit sphere in $M$. The algebras $C(8a)$ are all Morita equivalent, and since $\dim_{\mathbb{R}}C(8(a+1)) = 16^2\\, \dim_{\mathbb{R}}C(8a)$, you must have $\dim M(8(a+1))=16\\, \dim M(8a)$, so we learn that $\rho(16^a-1)\geq 8a-1$.

As for the upper bound, this reduces to something about the real K-theory of truncated projective spaces, which Adams calculated, and which clearly has something to do with Bott periodicity.

But I would agree that this all sounds like a big coincidence. I wish I knew something better to say about it.