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In a two-level factorial design, a factor subset defines a sign column. The coefficient of this column is its response contrast divided by the number of runs when the columns are orthogonal. A singleton describes a main effect and a pair describes a two-factor interaction. On the full cube, distinct columns have zero inner product: summing over any coordinate in their symmetric difference cancels the terms.

Code each factor level by $x_j\in \{-1,1\}$. A nonconstant factorial contrast indexed by a nonempty factor set $A$ has sign column $g_A(x)={\prod }_{j\in A}{x}_j$. Flipping any coordinate in $A$ reverses that sign and pairs the treatment combinations, so exactly $2^{m-1}$ have each sign. Allocating the two signs to two experimental blocks therefore divides the full factorial design into equal experimental blocks. The contrast is confounded with experimental blocks because its column is constant within each experimental block: changing its coefficient and compensating by the two experimental block intercepts leaves every fitted value unchanged. It has no information separate from experimental block differences. Randomize the allocation of the two groups to days and the run order within days.

Write $A,B,C$ for temperature, pressure and catalyst, respectively, with low levels and catalyst one coded minus. In the supplied allocation, $ABC=-1$ on day one and $ABC=1$ on day two. Thus the three-factor interaction $ABC$ is confounded with days. Every other nonconstant factorial column is orthogonal to the day contrast, because its product with $ABC$ is a nonconstant sign column on the full cube and consequently sums to zero.

If all interactions are negligible, the fitted model contains an intercept, one day contrast and the three main effects. The corrected total statistical degrees of freedom partition asThe three residual statistical degrees of freedom are the unused two-factor interaction columns $AB,AC,BC$, pooled as error under the stated negligible-interaction assumption. There is no independent pure-error replication of treatment combinations.

For four days with two runs each, all three main effects can be estimated. One regular allocation pairs each treatment combination with its complete opposite:The block-generating columns can be $AB$ and $AC$; their product $BC$ is also constant in experimental blocks. Thus $AB,AC,BC$ span the three experimental block contrasts, while $A,B,C$ and $ABC$ vary within experimental blocks. The main effects remain mutually orthogonal and separate from the experimental block effects; with an additive main-effect model there is one residual degree of freedom, represented by $ABC$.

For the further request about $AC$, the distinction between regular confounding and unrestricted pairing is important. A regular four-block allocation cannot retain all three main effects and $AC$. Its three confounded columns form a rank-two defining subgroup. To preserve main effects, these columns must lie in $\{AB,AC,BC,ABC\}$. If the subgroup includes $ABC$ and a two-factor column, their product is a main-effect column, which is forbidden. Hence the only permissible nonidentity subgroup is $\{AB,AC,BC\}$, and it necessarily confounds $AC$. This proves impossibility in the usual regular factorial-blocking framework, rather than relying on a count of available statistical degrees of freedom.

The literal request does not explicitly restrict arrangements to regular experimental blocks. If arbitrary pairings are permitted, a nonregular construction can estimate $A,B,C,AC$ together, assuming the other interactions are omitted from the model. An explicit allocation isEvery treatment combination occurs exactly once. Fit one unrestricted intercept per day and the four coefficients ${\beta }_A,{\beta }_B,{\beta }_C,{\beta }_{AC}$. By estimability from within-block differences, subtracting the second run from the first in each experimental block removes all day effects and yields the coefficient matrixThus all four coefficients are uniquely determined from the four day differences. In this broader design class the answer is yes, under the reduced four-effect model. The resulting fit is saturated: four experimental block intercepts plus four treatment coefficients use all eight observations, leaving no residual estimate of error variance. Replication or external error information would be needed for inference. This does not contradict the impossibility for regular orthogonal confounding, and it does not claim simultaneous unbiased estimation when arbitrary additional interactions are retained.

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 $G_1,\dots ,G_k$ 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 henceTo 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 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 fractionIts 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 signsTheir product is $x_B{x}_C{x}_F$. The three nonconstant block contrasts therefore coincide with the treatment-column alias classesOnly $AF$ 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 setThere 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 isThe 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.

The resolution of a regular fraction is the shortest nonidentity word in its defining contrast subgroup. Two effects alias only when their word product belongs to that subgroup. Thus resolution VI separates all main effects and two-factor interactions from one another and from three-factor interactions, though some three-factor interactions alias in complementary pairs.