INDEPENDENT COMPUTATION AND PROOF REVIEW
Date: 2026-10-18
Source: SUBTREE_PROOF_NOTE.txt in bounded_feasibility_and_new_search_Riemann_A_20261018_195244.
No existing verification code was opened before or during this independent implementation.

VERDICT: No mathematical gap identified in the specified finite-n lower bound.

Independent route and results:
1. Enumerated edge subsets for every vertex subset, retaining precisely connected graphs with k-1 edges. This is independent of Prufer generation. Full enumeration for n=3,4,5 gives total counts 9,38,250 and edge-pair ratios 1,152/169,5500/6561.
2. Compared moment-based probabilities against direct fixed-forest extension sums: edge count sum binom(n-2,k-2)*2*k^(k-3), with k=2 case 1; adjacent-pair count sum binom(n-3,k-3)*3*k^(k-4), with k=3 case 1. These are consistent with contraction and weighted Cayley. Exact rational agreement n=3..200.
3. Verified exact factor identity and strict mean interval for n=4..200 and strict ratio >3/4 for n=3..200. Computation supports but does not supply the all-n proof.

Analytic audit:
For fixed K=k, vertex inclusion times uniform-tree forest probability gives the two displayed conditional probabilities. The adjacent-pair expression has value zero at k=1 and k=2, so there is no undefined singleton term. The distribution has mu>0.
The Jensen step is correctly applied to 1/K. Substitution gives mu(mu-1)/(mu+1). For n>=4 the established interval mu>n/(n-2)>=1 ensures this lower expression is useful. The displayed factorization is exact. Both factors are strictly positive: the lower mean follows from monotone likelihood ratio and the n=4 calculation; the upper mean follows from positive probability of smaller trees. No strict Jensen assertion is needed.
The likelihood-ratio comparison on common support is sound. Normalize b_(n+1) restricted to 1..n; it is the old law tilted by a strictly increasing positive weight. The covariance identity raises its mean. The missing atom n+1 has positive mass and lies above all common-support values, so adding it raises the mean further. The n=3 exception is handled exactly.
The proof in fact establishes a useful sufficient condition for any exchangeable mixture of uniform complete-graph spanning trees over vertex subsets: whenever mu is in the given interval, the same adjacent-edge lower bound holds. This is an explanatory observation, not an asserted new literature result.

Publication-value assessment:
Resolving an explicitly stated specialist conjecture can warrant a concise specialist note even when all tools are standard. Simplicity alone does not eliminate mathematical value. Here the contribution is narrow and clearly describable: the strict finite-size complement to the known limit and upper inequality for complete graphs. It is not a general correlation theory, nor an RH-related advance. Suitability is conditional on a current literature check confirming the conjecture has not already been resolved and on accurate attribution. A full general-interest research article is not justified by this proof alone. A short note, possibly bundled with a genuinely useful extension or explanatory criterion, is proportionate; acceptance cannot be predicted.
The known methods closest to this argument are Cayley forest extension/contracted component counts, Jensen convexity, and monotone likelihood-ratio stochastic ordering. These are standard techniques, so any originality claim should attach only to their application to Conjecture 5.1, not the ingredients.

Literature-access limit:
I opened the full primary Luo-Xu-Tian manuscript through the web tool and confirmed the model, known upper inequality and limiting constant from its introduction. This review did not conduct an independent exhaustive forward-citation or Google discovery search. The root originality reviewer must supply that current search; no worldwide-priority claim is warranted by this review.
Primary source: https://www.researchgate.net/profile/Kexiang-Xu/publication/372825398_Random_subtrees_and_unimodal_sequences_in_graphs/links/64c9a5916814a775a9f4beea/Random-subtrees-and-unimodal-sequences-in-graphs.pdf

Artifacts: independent_check.py; independent_results.json. Python execution passed with exit code 0.
No manuscript, publication package or external contact was created.
