= Solution
Use a <circular orbit>, as in the stated <Roche lobe> approximation, with orbital angular speed $\Omega$ and negligible stellar spin. The <center of mass> distances are $a_1=aM_2/M$ and $a_2=aM_1/M$. Summing the two orbital <angular momenta> gives
$$
\boxed{J=(M_1a_1^2+M_2a_2^2)\Omega=\frac{M_1M_2}{M}a^2\Omega.}
$$
<Kepler's third law>, $\Omega^2a^3=GM$, makes this $J=M_1M_2\sqrt{Ga/M}$. In <conservative mass transfer>, both $M$ and $J$ are fixed, while $dM_2=-dM_1$. Thus
$$
\frac{d\ln M_2}{d\ln M_1}=-q,\qquad \frac{d\ln a}{d\ln M_1}=2(q-1).
$$
The <Roche-lobe radius response exponent> is consequently
$$
\zeta_L=\frac{d\ln R_L}{d\ln M_1}=2(q-1)+\frac13=2q-\frac53.
$$
During a dynamical mass-loss perturbation, the deep interior and <luminosity> do not have time to change. The supplied giant structure therefore has <stellar radius response exponent> $\zeta_*=-0.27$. Its fractional overfill changes by $d\ln(R/R_L)=(\zeta_*-\zeta_L)d\ln M_1$. The <binary mass ratio> uses donor mass divided by accretor mass. Since $d\ln M_1<0$, self-limiting <Roche-lobe overflow> requires $\zeta_*>\zeta_L$. Hence \b[the dynamical stability condition] is
$$
\boxed{q<q_{\mathrm{crit}}=\frac{5/3-0.27}{2}=0.698\overline3\simeq0.7.}
$$
At equality the linear restoring response vanishes. This <conservative mass-transfer critical mass ratio> uses the given radius exponent, rather than imposing the different fully convective $-1/3$ approximation.
The initially more massive component normally evolves first. If it first overflowed only after acquiring the given giant response, it would have $q>1$ and the mass loss would be dynamically unstable: its radius grows while its <Roche lobe> initially contracts. Starting overflow earlier can avoid this outcome. A <main sequence> or early post-main-sequence <donor star> can have a radiative envelope with a stabilizing contraction response. Transfer then reverses the <binary mass ratio> before the donor develops the giant structure. This is the route to an <Algol binary> through <Case A mass transfer> during core hydrogen burning, or <early case B mass transfer> after core hydrogen exhaustion but before a deep giant envelope develops.
For the initial <Roche-lobe overflow> to occur before the base of the giant branch, its lobe must be smaller than the base-of-branch stellar radius. Combining $R_L=0.462a(M_1/M)^{1/3}$ with <Kepler's third law> cancels the companion dependence:
$$
P_i<2\pi\left[\frac{R_{\mathrm{BGB}}^3}{0.462^3GM_1}\right]^{1/2}.
$$
Writing $m_1=M_1/M_\odot$, substitute the <mass-radius relation> to obtain \b[the pre-giant period limit]:
$$
\boxed{P_i<P_0m_1^2,\qquad P_0=2\pi\left[\frac{(1.68/0.462)^3R_\odot^3}{GM_\odot}\right]^{1/2}\simeq0.803\,\mathrm{days}.}
$$
We use the specified $0.8\,\mathrm{days}$ below. This is a <pre-giant Roche-lobe-filling period limit>, not a condition on the current stripped donor's radius.
At fixed total mass, <Kepler's third law> gives $P\propto a^{3/2}$, while fixed orbital <angular momentum> gives $a\propto(M_1M_2)^{-2}$. Therefore \b[the conservative period invariant] is
$$
\boxed{P(M_1M_2)^3=\text{constant}.}
$$
To find the smallest possible pre-giant period ratio, use solar-unit masses $m_1,m_2$ and fixed $m=m_1+m_2$. Let $K=P(m_1m_2)^3$. Along this <period-product invariant for conservative mass transfer>,
$$
F(m_1)=\frac{P}{m_1^2}=\frac K{m_1^5(m-m_1)^3}.
$$
The denominator has logarithmic derivative $5/m_1-3/(m-m_1)$, with a strictly negative derivative of its own. Its unique maximum, hence the unique minimum of $F$, occurs at
$$
\boxed{m_1=\frac58m,\qquad m_2=\frac38m.}
$$
Since $F$ diverges at either endpoint, this is the global <minimum period-to-donor-mass ratio for conservative evolution>.
For the quoted <Algol binary>, $K=3(1\times3)^3=81\,\mathrm{days}$ and $m=4$. Choosing the minimizing progenitor gives $m_{1,i}=2.5$, $m_{2,i}=1.5$ and
$$
P_i=\frac{81}{(2.5\times1.5)^3}=1.536\,\mathrm{days},\qquad F_{\min}=0.24576\,\mathrm{days}<0.8\,\mathrm{days}.
$$
Its pre-giant upper limit is $0.8(2.5)^2=5\,\mathrm{days}$, comfortably above this initial period. Subsequent <conservative mass transfer> of $1.5M_\odot$ produces the stated masses and period. Thus \b[the data permit the proposed early-overflow formation route].
For <RT Lac>, instead $K=5(0.5\times1.5)^3=2.109375\,\mathrm{days}$ and $m=2$. At its most favorable conservative progenitor, $m_{1,i}=1.25$, $m_{2,i}=0.75$ and
$$
P_i=2.56\,\mathrm{days},\qquad F_{\min}=1.6384\,\mathrm{days}>0.8\,\mathrm{days}.
$$
\b[No conservative progenitor meets the pre-giant overflow condition.] An initially more massive giant donor also fails the dynamical-stability condition derived above. Within this stable Algol-type formation model, the current RT Lac system therefore requires <nonconservative mass transfer>.
A plausible history is substantial envelope loss from the system, through a <stellar wind> or escaping overflow, possibly with a <common envelope> episode. Escaping matter also carries orbital <angular momentum>, so the total-mass and period-product invariants no longer constrain its initial orbit. A more massive progenitor can then shed its envelope and reach the present low donor mass without requiring the excluded conservative sequence. The supplied final data do not determine a unique mass-loss or angular-momentum-loss history.
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