Does this alternative model require more data points for regression? I will preface this question by admitting my limited statistical background. If my technical jargon is off, please feel free to correct me

We're looking to improve our current linear, quadratic model (with interactions) for $Y$ that requires three predictors, $a$, $b$ and $c$:
$$ Y = f(a, b, c) = \beta_1a^2 + \beta_2b^2 + \beta_3c^2 + \beta_4ab ... + \beta_n$$
There is a (physics-oriented) suggestion to use an alternative approach which involves modelling parameter $d$ using a quadratic fit for $a$, $b$ and $c$, and use $d$ as a predictor for $Y$:
$$ d = g(a, b, c) = \gamma_1a^2 + \gamma_2b^2 + \gamma_3c^2 + \gamma_4ab + \gamma_5ac... + \gamma_n $$
$$ Y = h(a, b, d) = \xi_1a^2 + \xi_2b^2 + \xi_3d^2 + \xi_4ab + \xi_5ad... + \xi_n$$
Some additional notes


*

*Due to the unwanted expense of testing, it is desired to keep the dataset required for regression as low as possible.

*The data for $d$ is already available in the existing dataset

*In our very preliminary testing, the alternative model has a higher $R^2$ value over the existing approach.


There are two opinions within our group


*

*No additional data samples are required since the number of unknowns ($\xi_1, \xi_2...\xi_n$) has not changed

*More data samples are required to prevent overfitting ("preserve the quality of the regression") because the alternative model now has higher order terms ($a^4, a^2b, b^4, b^2a$)


The big question here is, which opinion (if any) is right, and why?
 A: Where did you get the gamma-parameters from? If you got those from the data, even indirectly, then that means that those still 'count' as extra parameters. So if you had had gotten completely different data from the very beginning, would the gamma-parameters be any different?
If the gamma-parameters are based on some prior theory that is independent of the measurements, you are doing the same quadratic regression, only you swap one variable for another. That shouldn't change things.  
edit: Ah, it took some time before I understood the question. So we have $a$, $b$ and $c$ as measured, independent variables and $d$ and $Y$ as dependent variables. Now the question is whether instead of using $a$, $b$ and $c$ to fit $Y$, we could fit $d$ first and then fit $Y$ using $a$, $b$ and $d$.
There are 2 ways to fit this problem. First you always have to fit $d$ using $a$, $b$ and $c$. This will give a fitted value $\widetilde{d}$ for each (training)-input. Now the question is, do we fit $Y$ with $a$, $b$ and $d$ or with $a$, $b$ and $\widetilde{d}$?
In the first option we just have n-degrees of freedom in fitting our model. Since we aren't basing our model on $\gamma$, but only on measured data, we don't get any extra possibility of overfitting. However the downside is that when we start making predictions with our model, we are going to replace $d$ with $\widetilde{d}$. Since there is the real possibility that our approximation $\widetilde{d}$ is not perfect, this is going to be an extra source of error. So when testing our model on the test-set, we need to use $\widetilde{d}$ and we expect that this will give a larger than usual difference between the error on the training-set and the test-set.
We could also fit $Y$ using $a$, $b$ and $\widetilde{d}$. This way the model can account for and possibly correct for any errors made when approximating $d$. However this is a source for overfitting. If the distribution of $a$'s and/or $b$'s is biased, this bias is going to affect the learning process twice. First in approximating $d$ and then in approximating $Y$. More importantly, the bias in $a$ and $b$ is going to be exactly the same both times. So here you do need to be more careful with overfitting and maybe apply some regularization or get more training examples.
