= Solution
Code each two-level factor as a sign $x_j\in\{-1,1\}$. A <factorial contrast> indexed by a factor subset $A$ has sign column $\chi_A(x)=\prod_{j\in A}x_j$, with the empty subset giving the intercept. Construct a regular <fractional factorial design> by choosing $k$ independent words $G_1,\ldots,G_k$ and retaining the settings satisfying $\chi_{G_i}(x)=s_i$, where each $s_i\in\{-1,1\}$. In binary exponent notation these are $k$ independent linear equations over $\mathbb F_2$, leaving $m-k$ freely chosen coordinates and exactly $2^{m-k}$ settings. Dependence of the chosen words would fail to give the stated fraction size.
The words generated by multiplying the $G_i$ form the <defining contrast subgroup> $\mathcal H$, of order $2^k$. Multiplication cancels a repeated factor, so word multiplication is <symmetric difference> of subsets. On the fraction, $\chi_H=s_H$ is constant for every $H\in\mathcal H$, and hence
$$
\chi_{AH}=\chi_A\chi_H=s_H\chi_A.
$$
To see that these are all the aliases, average $\chi_A\chi_B=\chi_{AB}$ over the fraction. If $AB\notin\mathcal H$, this is a nonconstant character of its freely varying binary coordinates; pairing settings that reverse that character makes the average zero. If $AB\in\mathcal H$, the product is constant. Thus the <aliasing in a fractional factorial design> classes are exactly the cosets of $\mathcal H$. For a half fraction, $\mathcal H=\{I,G\}$ with $G\ne I$, and \b[every contrast column is aliased with exactly the other word $AG$], with a plus or minus sign according to the chosen fraction. This includes the defining word paired with the intercept.
For the six-factor experiment, take the 32-run half fraction
$$
I=ABCDEF,\qquad x_Ax_Bx_Cx_Dx_Ex_F=1.
$$
Its <resolution of a fractional factorial design> is VI. Every <main effect> is aliased with a five-factor interaction, and every <two-factor interaction> with a four-factor interaction. In particular, no two required low-order effects are aliased with one another. Under the stated negligible-higher-interaction assumption, all six <main effects> and the fourteen required <two-factor interactions> are estimable before introducing blocks.
To form four equal days, classify runs by the two signs
$$
u=x_Ax_F,\qquad v=x_Ax_Bx_C.
$$
Their product is $x_Bx_Cx_F$. The three nonconstant block contrasts therefore coincide with the treatment-column alias classes
$$
AF\equiv BCDE,\qquad ABC\equiv DEF,\qquad BCF\equiv ADE.
$$
Only $AF$ among the <main effects> and <two-factor interactions> is lost to <block confounding in a factorial design>; the other two block directions involve only negligible three-factor interactions.
Here is an explicit construction of every run without an ambiguous generator convention. For each day sign pair $(u,v)\in\{-1,1\}^2$, choose $(s,t,w)\in\{-1,1\}^3$ freely and set
$$
\boxed{(A,B,C,D,E,F)=(s,\ t,\ vst,\ w,\ uv sw,\ us).}
$$
There are eight distinct settings per day and 32 in total. Direct multiplication gives $ABCDEF=1$, $AF=u$ and $ABC=v$. Conversely those three equations solve $C,E,F$ uniquely from $A,B,D$, so there are no missing or repeated settings. Assign the four sign pairs randomly to the four days, and randomize the eight runs within each day.
Distinct nonaliased <factorial contrasts> are <orthogonal> over the fraction. None of the 20 required low-order contrast columns belongs to a block class, so all are also <orthogonal> to the three block-effect columns. Their joint <design matrix>, including an intercept, therefore has rank $1+3+6+14=24$. The <analysis of variance> partition is
$$
\boxed{\begin{array}{c|r}
\text{Source}&\text{Degrees of freedom}\\\hline
\text{Days}&3\\
\text{Main effects}&6\\
\text{Two-factor interactions except }AF&14\\
\text{Residual}&8\\\hline
\text{Corrected total}&31
\end{array}}
$$
The eight residual directions are the remaining unconfounded three-factor alias classes. They estimate error only under the negligible-higher-interaction assumption; they are not a separate replicated <pure error> stratum. This provides the requested comparisons within exactly four days of eight runs.
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