CLOSEST-RESULTS AND SUBSUMPTION ASSESSMENT
Independent AI-assisted review, 2026-10-18
Scope: adjacent distinct edges in a uniformly selected nonempty, non-induced,
non-spanning-allowed subtree of labeled K_n, including singleton vertices.

Verdict
The finite strict lower bound is a plausible short-note contribution relative to
the primary statements inspected. It is NOT a new counting method or a new
moment inequality. No inspected prior theorem already states or automatically
supplies this finite lower bound. This is bounded evidence, not worldwide
priority certification. The final 13-page Luo-Xu-Tian journal text remains unread.

Closest primary statements inspected live
1. Chin, Gordon, MacPhee, Vincent (2018), Subtrees of graphs,
   DOI 10.1002/jgt.22359; author-posted full text:
   https://www.researchgate.net/publication/324905745_Subtrees_of_graphs
   Theorem 6.5, printed p.432: for fixed m-edge subtree S in K_n,
   the number of containing subtrees is
   (m+1) sum_{j=0}^{n-m-1} binom(n-m-1,j)(m+j+1)^(j-1).
   Its proof uses planted forest counts. Thus both the edge and adjacent-pair
   extension counts and their reduction to finite sums are prior work.
   Section 7, p.433, Conjecture 7.3 asks for the upper correlation bound.
   Its table gives finite examples, rather than a lower-bound theorem.
   Theorem 6.5 encodes the quantities needed but does not assert the required
   comparison between them; an exact enumeration formula is not itself a
   proof of the new inequality.

2. Luo, Xu, Tian, 20-page author manuscript:
   https://www.researchgate.net/publication/372825398_Random_subtrees_and_unimodal_sequences_in_graphs
   Theorem 3.3 gives a limiting ratio (m1+m2-1)/(m1*m2) for fixed
   subtrees intersecting in one vertex; adjacent edges yield 3/4.
   Theorem 3.4 gives that ratio exactly for uniform SPANNING trees.
   Theorem 3.5 proves the finite upper bound, ratio <=1.
   Conjecture 5.1, printed p.18, asks for the strict finite lower bound >3/4.
   They explicitly note that their upper-bound proof approach appears
   unsuitable near this limiting constant. Their exact spanning-tree result
   does not transfer unchanged to a mixture over proper vertex sets.
   Journal DOI 10.1016/j.disc.2023.113654 identifies the later version;
   none of this verifies that version's concluding statements.

3. Grimmett and Winkler, Negative association in uniform forests and connected
   graphs, author PDF:
   https://www.statslab.cam.ac.uk/~grg/papers/negass.pdf
   Theorem 1.4 states negative association for a uniform spanning tree.
   This is an upper dependence comparison, not a positive universal lower
   correlation ratio. Its model fixes the host vertex set and spans it.
   Uniform forests and uniform subtrees are also different sample spaces.
   Accordingly this theorem does not subsume the target.

4. Wagner and Wang, On the local and global means of subtree orders,
   DOI 10.1002/jgt.21869; publisher abstract inspected:
   https://onlinelibrary.wiley.com/doi/abs/10.1002/jgt.21869
   The reported results bound local vertex-conditioned mean order by twice
   global mean, and locate maximum local means at leaves or degree-two
   vertices, for a host TREE. These hypotheses and observables differ from
   adjacent-edge probabilities in a complete graph. The author PDF route
   returned a tool error; no claim to have checked its full theorem text.

Direct mathematical subsumption test (reviewer's deduction)
Let p_k be ANY distribution on 1,...,n; conditional on K=k choose a uniform
k-subset and uniform labeled tree on it. The manuscript's exact local counts
and Jensen apply to this broader mixture without using p_k proportional to
binom(n,k)k^(k-2). If mu=E(K-1) lies strictly between n/(n-2) and n-1,
then precisely the same algebra proves R>3/4. The sole additional input for
the uniform-all-subtrees model is the elementary mean-order bound, established
in the note by retaining order-four counts. Hence the proof is an application
of standard convexity plus classical counts, with a useful model-specific
choice of statistic and threshold factorization. Do not call Jensen itself new.

Conditioning on an edge size-biases p_k by k-1, immediately from the known
one-edge formula. Size-biasing alone is insufficient: a distribution concentrated
at K=2 has zero adjacent-pair probability. Even the spanning case K=n has
ratio exactly 3/4 and fails strictness. Thus neither an arbitrary mixture nor
negative dependence supplies the target without quantitative information on K.
The known limit 3/4 alone permits approach from either side and is likewise
insufficient. These observations identify the actual finite increment.

Assessment by category
- Method novelty: low. Classical enumeration, conditional symmetry, reciprocal
  convexity/Jensen, and one elementary count. The moment reformulation and
  factorization are the useful presentation mechanism, not a new general tool.
- Result novelty: provisional but concrete against the read author manuscript:
  it resolves its explicitly stated finite lower-bound conjecture, assuming
  the note's proof survives final verification. Counts/limit/upper bound are old.
- Significance: focused graph-enumeration/probability refinement. A concise note
  may be reasonable because a named explicit conjecture is settled elegantly;
  it is not evidence of broad impact or a breakthrough. Venue judgment requires
  a graph-theory editor/referee and the final-version/source-status check.

Coverage limitations
Targeted web discovery here used a tool whose engine is unspecified, not Google;
the pre-existing expanded assessment records separate Google coverage. Searches
included local/global means, size-bias terminology and spanning-tree negative
association. No broad final-PDF search was repeated. No new matching resolution
surfaced in these checks, which cannot exclude inaccessible or unindexed work.
No manuscript edit, public upload, contact, or submission was performed.
