A SHARP LOWER BOUND FOR ADJACENT-EDGE CORRELATIONS IN RANDOM SUBTREES
Scott Schubr - pre-publication review package - October 18, 2026

START HERE
Read manuscript/main.pdf (or editable main.tex), then SOURCE_VERIFICATION.txt
and REVIEW_REQUEST.txt. This package is for assessment and the owner's human
review. It is not approved for publication, journal submission or public upload.

Claim
If T is uniform on all nonempty tree subgraphs of labeled K_n, including
singleton vertices, n>=3, then for distinct adjacent edges e,f,
  P(e in T | f in T) > (3/4) P(e in T).
The constant is optimal by the limit already obtained by Luo, Xu and Tian.

Contribution
The contribution is the finite strict inequality and its elementary proof,
relative to the conjecture in the authors' accessible manuscript. Cayley counts,
fixed-subtree extension formulas, size-bias ideas and Jensen are prior methods.
The note does not claim a new asymptotic constant, general correlation theory,
ratio monotonicity, or a result for arbitrary/weighted host graphs.

Review status
Two fresh separate AI-assisted reviews found no proof gap or inspected prior
theorem subsuming the finite inequality. Independent exact arithmetic and small
graph enumeration corroborate the identities. These are not human peer review
or formal proof-assistant verification. See review/ and verification/.

Final-source hold
Conjecture 5.1, printed p. 18, is verified in the20-page author manuscript. The final
13-page journal text remains unavailable through checked lawful public routes.
Do not claim final published conjecture numbering or complete novelty clearance.
The manuscript footnote and SOURCE_VERIFICATION.txt retain this limitation.

Contents
manuscript/  LaTeX source and compiled review PDF; proof does not depend on code.
review/      Proof, closest-results and publication-decision assessments.
verification/ Optional stdlib-only exact checker and fresh results/replay log.
SOURCE_VERIFICATION.txt  Exact source/version facts, links and remaining hold.
REVIEW_REQUEST.txt       Specific independent assessment requested by Scott.
PROVENANCE.json          Baseline, revision scope and proof identity.
MANIFEST.sha256.json     Per-file SHA256 and byte lengths (excludes itself).

Reproduce optional computation
Python 3 standard library only; run:
  python -B verification/independent_check.py
It checks exact rational counts through n=200 and independently enumerates
edge subsets for n=3, 4, 5. It rewrites independent_results.json in that folder.
Finite checks are corroboration, not the proof for arbitrary n.

Compile
Use a current LaTeX distribution with amsmath, amssymb, amsthm, geometry and
hyperref, or Tectonic: tectonic manuscript/main.tex (choose output directory
as appropriate). The source has no external data/font/image dependency.

Author review
Scott will obtain human review. Confirm authorship, AI disclosure, literature
attribution, final-source clearance and any eventual venue/license decisions
before public release. No license grant or publication authorization is implied.

Preserved baseline
D:/LUMINARY ODDITIES LLC/AI Research/subtree_short_paper_Riemann_A_20261018_201056
The baseline manuscript and original evidence remain intact. This package only
shortens the background/version presentation and labels the pre-publication
draft; its theorem and proof are unchanged.
