# Canonical statement and exact scope Frozen record OWR-14299905-027, record_id 30006553, UnsolvedMath v1.6.0, commit c6e7f41b4eb6ce876feb83e7eb626be1b243fd7e. Statement SHA-256: 554c75aed814816aad160884bb858d7d4b37b556a6b74eea34d411c251d28000. No upstream proof/result file was supplied. The broad frozen question asks for a variant of graph removal under the normalized maximum-degree edit metric. The primary report already establishes several variants. The exact source statement addressed here is its subsequent Conjecture 8, pp.60--61: local H-removability is equivalent to uniformly o(n^(k-a(H))) copies through every a(H)-set of roots independent in the copy. Definition 4 fixes a(H) through endomorphisms. These pages and the metric definition on p58 were read in text and visually inspected. The complete counterexample refutes that precise criterion. It gives a general obstruction for non-well-covered cores, a six-vertex wheel example, and C9 blow-ups with exact normalized removal distance ceil(m/2)/(6m+3), while the largest independent-four-root count is only m^4. The universal broad question about all possible variants is not closed. The neighboring frozen record OWR-14299905-028 is not separately claimed, edited, or counted. The report's adjacent odd-cycle assertion is incompatible with the formula as printed; a missing qualification is possible. We do not claim to refute an unavailable unpublished intended theorem. The independence requirement cannot be silently dropped or changed to roots independent in the host graph. # Detailed self-audit Accepted proof: complete_counterexample.md and the corresponding English TeX. Outside review is not a required gate; no outside review or formal proof assistant verification occurred. Both flags remain false. 1. **Distance convention.** Symmetric-difference edits reduce to deletions by intersecting with G because non-induced H-freeness is monotone. Costs use Delta(D)/n, not total deleted edges/n^2 or a pointwise matrix norm. 2. **Copies and multiplicity.** Projecting every embedding into a core gives an automorphism. Each copy therefore uses one vertex of every part, with no folded embeddings or extra non-induced copies. Labeled counts differ by exactly |Aut(H)|, a harmless fixed factor. 3. **Parameter.** Every independent set of a core is admissible in Definition4. Odd cycles have only their 2l automorphisms. The wheel's hub cannot map to a rim vertex because its neighborhood is bipartite; the rim then maps automorphically to C5. Hence a(C9)=4 and a(K1 join C5)=2. 4. **Quantifiers.** The count bound is uniform over every root set S. Repeated part indices contribute zero. A maximum independent index set cannot contain a smaller maximal independent set B. At least one singleton part is omitted, forcing the lost power of m. This is maximal versus maximum, not a switch of graph model or a finite-size exception. 5. **Local lower bound.** B is independent, so no deleted edge has two singleton endpoints. Count singleton-bulk and bulk-bulk edges separately; the factors m^(r-1), m^(r-2), b Delta and rm Delta/2 follow directly. A union bound suffices even though cycles may be hit more than once. 6. **Sharp C9 degree.** Every surviving length-three gadget can be combined with the other two because internal parts are disjoint. Therefore one gadget must lose all terminal-to-terminal paths. Nonempty terminal-neighbor sets force deletion of their full complete bipartite join. The four degree terms a,c,m-a,m-c force ceil(m/2), attained by the explicit balanced cut. The m=1 endpoint is valid. No minimization over total edge count is used. 7. **Root convention negative control.** The three singleton C9 vertices plus one bulk vertex lie in m^5 cycles but are not independent in any of them. This defeats an attempted misreading that drops independence. The example has a dense three-root witness, so an all-smaller-root formulation is not disproved or proved by this note. 8. **Every fixed q.** For qa root counts vanish. These are permitted different sequences for different proposed q, refuting a universal fixed-q characterization. 9. **Scope.** The adjacent reported odd-cycle result may have an omitted qualification. Only the literal displayed criterion is refuted. The unpublished manuscript was not obtained. Broad-record closure is false. 10. **Computation.** Integer endomorphism enumeration, bounded independent direct cycle enumeration, eight SAT boundary tests (m=1..4), lift counts through m=6, and explicit deletion witnesses all pass. Normal and Python -O modes agree after removing run time and optimization metadata. SAT is extra finite corroboration, not the proof of the all-m theorem. Conclusion: complete scoped counterexample to the exact printed conjecture, with a general structural obstruction and exact C9 distance. Accepted by the originating research workflow. Publication does not certify priority. # Bounded literature and novelty review, 27 October 2026 The main primary source is the report itself (source_audit.md). It states the metric, parameter, and fixed-cardinality independent-root conjecture, and attributes graph work to Fox and Wei. The constructions in this note use the standard graph-homomorphism projection of a blow-up and an elementary edge covering count; these techniques are not claimed as new. Fan Wei's public research list, https://sites.google.com/view/fan-wei/home, was freshly read and archived as fan-wei-home.html with transport receipt. It lists Local graph removal lemma (Jacob Fox, Fan Wei) as forthcoming. No full unpublished manuscript was obtained. Consequently an intended corrected formulation could exist; the claim is specifically about the printed formula. The related paper by Viet Hung Le, Ruilin Shi and Fan Wei, A Complete Dichotomy for L_infinity Permutation Removal, arXiv:2610.28271v1 (20 October 2026), was checked through its primary abstract and introductory HTML. Its bottleneck earth-mover metric for permutations is different from the maximum deleted degree for graphs. Its result is not transferred to this problem, and its formal-verification statement is not attributed to our work. Public abstract bytes and transport receipt are retained. Targeted searches included local removal lemma/C9, local removal/well-covered, Fox/Wei/independent/homomorphism/removal, local removal/pendant, and local removal/vertex-transitive. No identical source-connected wheel or C9 counterexample was found in these bounded searches. Many search returns were irrelevant; absence of a result is not proof of novelty. The exact constructions and general core obstruction have not received independent review. No absolute priority, exhaustive literature coverage, or correction of an unpublished author-intended theorem is asserted. The source-connected claim is nevertheless mathematically complete and can be published as an unrefereed note: literal Conjecture8 fails, with explicit root counts and a sharp C9 deletion calculation. The broad frozen question remains partially addressed. Nothing is counted as a new general local removal lemma.