ADVERSARIAL PROOF ASSESSMENT -- 2026-10-18
Source: paper/main.tex; independently read from scratch; no earlier audits used.

Verdict: No fatal or nonfatal mathematical gap found in the stated finite inequality. The proof is complete for the explicitly defined model. Publication originality/current-priority clearance is a separate question and is not established by this proof review.

Model: Uniform over all nonempty labeled tree subgraphs of K_n, including n singleton trees; neither induced nor spanning. Cayley count binom(n,k) k^(k-2), with b_n(1)=n, and the conditional uniformity of vertex sets and spanning trees are correct. Edge events have positive probability, so all conditional probabilities and ratios used are defined.

Local counts: An edge and an adjacent pair have extension counts 2 k^(k-3) (k>=2) and 3 k^(k-4) (k>=3). Their conditional inclusion probabilities are 2(k-1)/(n(n-1)) and 3(k-1)(k-2)/(k n(n-1)(n-2)). Zero cases k=1,2 are correctly supplied separately by the simplified formulas. No spanning-tree probability has been accidentally substituted for the random-subtree probability.

Moment reduction: With mu=E(K-1)>0, averaging gives R_n=3n(n-1) E(K-3+2/K)/(4(n-2)mu^2). Jensen applies because K is positive and finite; E(1/K)>=1/EK=1/(mu+1), hence E(K-3+2/K)>=mu(mu-1)/(mu+1). The inequality direction is correct. Strict Jensen is unnecessary.

Mean estimate: For n>=4 all terms omitted from sum (k-3)b_n(k) are nonnegative. b_n(4)=16 binom(n,4). Thus N E(K-3)>=16 binom(n,4)-2n-binom(n,2)>=n(7n-27)/2>0, using (n-1)(n-2)>=6. At n=4 both displayed lower estimates are equal and positive (value 2); no endpoint is lost. Therefore mu>2>=n/(n-2). Proper subtrees have positive mass, so mu<n-1 strictly.

Final algebra: Expanding (n-2)(n-1-mu)(mu-n/(n-2)) gives exactly n(n-1)(mu-1)-(n-2)mu(mu+1). Both factors are strictly positive for n>=4. Since all divisors are positive, the Jensen lower bound for R_n is itself strictly greater than 3/4. For n=3, N=9, each edge occurs in 3 trees, the adjacent pair occurs in 1, and R_3=1. The n>=3 domain is correct for distinct adjacent edges.

Increment over asymptotics: Yes, the theorem adds an all-finite-order strict lower bound. A sequence converging to 3/4 need not lie above 3/4 at finite indices, so the limit alone does not imply this claim. Optimality of the constant follows from the cited ratio limit, rather than from the finite argument alone. This is a short resolution of the precise manuscript conjecture, not a new asymptotic constant or counting formula.

Fresh primary-source check: Read the author-uploaded manuscript through https://www.researchgate.net/publication/372825398_Random_subtrees_and_unimodal_sequences_in_graphs on 2026-10-18. Its introduction includes singleton orders and uniform tree subgraphs; Theorem 3.3 and the adjacent-edge specialization give the ratio limit 3/4; Conjecture 5.1 states the strict lower inequality. This confirms model and claimed increment against that manuscript. The final typeset article and possible subsequent resolutions were not independently cleared in this narrow proof review. The manuscript's explicit version caveat is appropriate.

Scope of independence: Separate AI-assisted adversarial proof review, not external human peer review. No manuscript changes or extensions made.
