Distinct grade of the two-tree and star-sharpness review

Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 5

Recorded 2026-09-10. Role: distinct report grader, separate from the author and the whole-unit reviewer. All first-person readings, rederivations, findings and verdicts below belong to that historical grader. This is a native rendition of the original grade, not a fresh grading of the filed page or an independent review by this rendition's author.

The grader assessed the exact two-tree and star-sharpness subjects, then the original whole-unit review preserved substantively in two_tree_review.md. TTC/SS and page-line citations refer to the frozen assets. Source-context citations and all original report identities are explained in source reading and documentary corrections. The grade is PASS with corrections, relative to theorem_1 at its recorded standing; that source proof chain and PDF were not freshly rechecked.

All findings and historical integration conditions are retained. Section 6 preserves items 3 and 9's author-recorded filing instructions, followed by a separately attributed later standing reconciliation for the source digest and both result pages. Neither this rendition nor that decision supplies a native tier, changes catalog status, or supplies source-proof, execution or formal credit. The recorded rendition passed fidelity review and hand check before it was filed; those checks do not certify the label normalization here.

Documentary checklist labels name the grader's recorded checks on the exact assessed subjects; they do not introduce broader criteria. The negative findings are unchanged. No scan of the current pages is reported.

Independent grading of one reviewer report, plus a full independent re-review of the unit performed before the report was opened in detail. Read-only throughout. Nothing was modified, staged, stashed or committed in any repository; no git write command was run; the canonical repository was never touched. No code from the frozen material or repository was executed, no Lean was run, no network was used, no PDF was opened.

Grade of the report: PASS with corrections.

My own verdict on the unit: PASS with exact corrections.

The mathematics of both subject pages is correct, complete and correctly scoped. The report's mathematical work is also correct — every numeric and textual claim I spot-checked in it was exactly right. One of its three required corrections (D1) is not a defect and must be withdrawn: applying it would breach a written repository rule. Two integration conditions are missing from its list.

0. Input verification

The grader checked every allowed input against the assignment before reading it. The inputs, as they stood on 2026-09-10T11:12:29Z (the native baseline of the review checkout, at which every listed page carries the bytes the grader read), were:

../assets/reviewed_v1_two_tree_corollary.md.txt (exact reviewed subject)
../assets/reviewed_v1_star_sharpness.md.txt (exact reviewed subject)
library/extremal_graph_theory/adamczewski_2026_erdos548/theorem_1.md
library/extremal_graph_theory/adamczewski_2026_erdos548/_index.md
library/extremal_graph_theory/adamczewski_2026_erdos548/tree_ramsey_corollary.md
wiki/problems/ramsey_theory/E0547/_index.md
two plain-text captures of the source's forum thread, not retained in this repository: an earlier and a later capture (the SHA-256 values this record carried for them are not retained)
the assignment's input manifest (working storage; not retained)

Graded report: the unchanged historical review whose full native rendition is two_tree_review.md.

The isolated review checkout was a snapshot of the native baseline as it stood on 2026-09-10T11:12:29Z. The four native checkout files were byte-identical to the supplied copies, so the baseline is fixed by content. The report states the same facts; I confirmed them with read-only revision and short-history checks.

1. (a) My independent view, formed before reading the report in detail

1.1 What `two_tree_corollary.md` claims

Objects (lines 14–17). Two given trees T_1, T_2 on n_1, n_2 >= 2 vertices. R(T_1,T_2) is defined as the least positive integer N such that every red/blue coloring of E(K_N) contains a red copy of T_1 or a blue copy of T_2; copies are ordinary, not necessarily induced.

Conclusion (lines 21–27). R(T_1,T_2) <= n_1+n_2-3 if n_1 and n_2 are both odd, and <= n_1+n_2-2 otherwise.

Scope statements (lines 29–34). No ordering assumption on n_1,n_2; no large-order restriction; exact endpoint values R(K_2,T_2)=n_2 and R(T_1,K_2)=n_1; equality is claimed only for stars, not for every pair.

1.2 What `star_sharpness.md` claims

Objects (lines 14–15). S_n = K_{1,n-1}, the star on n vertices, n >= 2; two given stars.

Conclusion (lines 17–23). Equality: R(S_{n_1},S_{n_2}) = n_1+n_2-3 when both orders are odd, = n_1+n_2-2 otherwise, with the same definition of R restated at lines 25–28.

Scope (lines 30–31). Two-color stars only; no equality claim for arbitrary tree pairs or for a general number of colors.

1.3 Formulation check against E0547 and the source index

E0547.md.txt:16–20 states the diagonal question: for a tree T on n >= 2 vertices, R(T) <= 2n-2 — a bound on the Ramsey number of one given tree. Setting T_1 = T_2 = T, n_1 = n_2 = n, the subject's "both odd" branch becomes "n odd" and yields R(T) <= 2n-3 for odd n, <= 2n-2 otherwise. That reproduces E0547.md.txt:80–82, E0547.md.txt:98–100 and the q = 2 case of tree_ramsey_corollary.md.txt:29. _index.md.txt:54–59 states the same given-tree edge inequality the subject uses as its premise.

The bounded quantity is the Ramsey number of the two given trees. I found no place on either page where the statement slides to "some tree on n_i vertices" or "every tree on n_i vertices". theorem_1.md.txt:22–23 does carry an "average degree greater than t-2 forces every tree on t vertices" paraphrase; neither subject page uses that form. The premise is applied in its given-tree form once per color class.

1.4 My re-derivation of the upper bound from `theorem_1` as written

The premise (theorem_1.md.txt:15–20): tree T on t >= 2 vertices, finite simple G on n >= 1 vertices, no copy of T implies 2e(G) <= (t-2)n. The subject restates it at lines 40–48 with n renamed N. Faithful: no hypothesis dropped, no range widened, the non-induced convention carried across (theorem_1.md.txt:22 and two_tree_corollary.md:17).

All orders. N = n_1+n_2-2 >= 2. Suppose a coloring of K_N avoids both. G_1, G_2 are the spanning color graphs, each simple on N vertices, so 2e(G_1) <= (n_1-2)N and 2e(G_2) <= (n_2-2)N. The color classes partition E(K_N), so 2e(G_1)+2e(G_2) = N(N-1). Summing: N(N-1) <= (n_1+n_2-4)N. Here n_1+n_2 = N+2, so n_1+n_2-4 = N-2 and the bound is (N-2)N. Dividing by N > 0 gives N-1 <= N-2, false. Hence R(T_1,T_2) <= n_1+n_2-2. The page's arithmetic at lines 71 and 72 is exactly right.

Both odd. n_1,n_2 >= 3 odd, so N = n_1+n_2-3 is odd and >= 3. Each n_i-2 is odd, N is odd, so (n_i-2)N is odd; 2e(G_i) is even, so 2e(G_i) <= (n_i-2)N - 1. Summing gives N(N-1) <= (n_1+n_2-4)N - 2. Here n_1+n_2 = N+3, so n_1+n_2-4 = N-1 and the right side is (N-1)N - 2 = N(N-1) - 2. So 0 <= -2, false. Hence R(T_1,T_2) <= n_1+n_2-3. The parity strengthening and the exact edge arithmetic at lines 99–100 are correct, and the page correctly gets N-1 here where the previous case gave N-2. The margin is exactly 2, with no slack, and none is lost.

Smallest cases. n_1=n_2=2: N=2, N(N-1)=2 > 0 = (N-2)N. Contradiction holds with no special pleading. n_1=n_2=3: N=3, N(N-1)=6, (n_i-2)N = 3 strengthened to 2e(G_i) <= 2, sum <= 4 < 6. Holds. I also checked R(P_3,P_3) = 3 directly on K_3 by degrees: both color graphs would have to be 1-regular on 3 vertices, impossible.

Endpoints (lines 106–121). The only tree of order 2 is K_2. On K_{n_2}, either an edge is red (red K_2) or all edges are blue, and the all-blue K_{n_2} contains T_2 by any bijection of vertices — correct, including n_2 = 2. The avoiding coloring is all-blue K_{n_2-1}, correct including n_2 = 2 (a single vertex). Color interchange gives R(T_1,K_2) = n_1, and R(K_2,K_2) = 2. Line 121's remark that these fall in the second branch is right: n_1 = 2 is even, and n_1+n_2-2 = n_2 equals the computed value, so the bound is attained there.

1.5 My re-derivation of the star-sharpness constructions

Existence of d-regular graphs (lines 45–75). Hypotheses N >= 1, 0 <= d <= N-1, Nd even. Necessity (lines 46–48): a vertex has at most N-1 neighbors, and Nd is the degree sum = 2e(G), hence even. Both correct; the handshake condition is exactly Nd even.

Even d = 2k: circulant on Z/N with differences ±1,…,±k. From 2k <= N-1 we get k < N/2. No difference is 0 mod N (1 <= j <= k < N). Positive differences are pairwise distinct, as are negative ones. j ≡ -j' would need j+j' ≡ 0 mod N, impossible since 2 <= j+j' <= 2k <= N-1. So there are exactly 2k distinct neighbors; the difference set is symmetric, so the relation is an undirected simple graph. k = 0 gives the empty graph, covering N=1, d=0. Correct.

Odd d = 2k+1: Nd even with d odd forces N even. 2k+1 <= N-1 gives k <= N/2 - 1. The 2k circulant neighbors are distinct as before. The chord difference N/2 is nonzero mod N; N/2 != j since j <= N/2 - 1; and N/2 ≡ -j would force N/2+j ≡ 0, impossible since N/2+1 <= N/2+j <= N-1. The chord is an involution (x+N/2+N/2 = x+N ≡ x), so x+N/2 and x-N/2 are the same vertex and the chord contributes exactly one neighbor, giving degree 2k+1, not 2k+2. Correct, and (N,d) = (N,N-1) yields K_N in both parities, which the endpoint section later relies on.

Not both odd (lines 79–112). N = n_1+n_2-3 >= 1, d = n_1-2 >= 0, N-1-d = n_2-2 >= 0 so d <= N-1. Parity: n_1 even makes d even; n_1 odd forces n_2 even (branch hypothesis), making N = odd + even - 3 even. Exhaustive, so Nd is even in every case. Red degree n_1-2, blue degree N-1-d = n_2-2. Complement degrees correct. A graph contains an ordinary S_n iff some vertex has at least n-1 neighbors — correct in both directions, and the clause "irrespective of edges between them" is exactly what makes it valid for non-induced copies. n_1-2 < n_1-1 and n_2-2 < n_2-1, so both forbidden stars are absent. Line 107–108's restriction remark supplies the monotonicity that turns one avoiding coloring into R > N. Hence R >= n_1+n_2-2.

Both odd (lines 116–136). N = n_1+n_2-4 >= 2 and even (odd+odd−4), d = n_1-2 >= 1, N-1-d = n_2-3 >= 0. Nd is even because N is even — in this branch the handshake condition can never fail. Red degree n_1-2 < n_1-1; blue degree n_2-3 < n_2-1. Hence R >= n_1+n_2-3. The binding constraint is d <= N-1, i.e. n_2 >= 3, which "both odd" forces, with equality exactly at n_2 = 3.

Integrality closing (lines 138–140). R > n_1+n_2-3 gives R >= n_1+n_2-2; R > n_1+n_2-4 gives R >= n_1+n_2-3. Both meet the corollary's branches, so equality holds in each case. Correct.

Order-two endpoints (lines 144–158). n_1=2: N=n_2-1, d=0; red empty, blue K_{n_2-1}; no red S_2, and too few vertices for a blue S_{n_2}; R = n_2. n_2=2: N=n_1-1, d=n_1-2=N-1; red complete, blue empty; R = n_1. Nd = (n_1-1)(n_1-2) is a product of consecutive integers, hence even, consistent with the general parity argument. n_1=n_2=2: N=1, d=0, one vertex; R = 2. All correct.

Smallest parameters, checked independently.

n_1,n_2: 2, 2; branch: other; claimed R: 2; N,d: 1, 0; red deg: 0; blue deg: 0
n_1,n_2: 2, 3; branch: other; claimed R: 3; N,d: 2, 0; red deg: 0 < 1; blue deg: 1 < 2
n_1,n_2: 3, 2; branch: other; claimed R: 3; N,d: 2, 1; red deg: 1 < 2; blue deg: 0 < 1
n_1,n_2: 3, 3; branch: both odd; claimed R: 3; N,d: 2, 1; red deg: 1 < 2; blue deg: 0 < 2
n_1,n_2: 3, 4; branch: other; claimed R: 5; N,d: 4, 1; red deg: 1 < 2; blue deg: 2 < 3
n_1,n_2: 3, 5; branch: both odd; claimed R: 5; N,d: 4, 1; red deg: 1 < 2; blue deg: 2 < 4
n_1,n_2: 5, 5; branch: both odd; claimed R: 7; N,d: 6, 3; red deg: 3 < 4; blue deg: 2 < 4
n_1,n_2: 7, 3; branch: both odd; claimed R: 7; N,d: 6, 5; red deg: 5 < 6; blue deg: 0 < 2

External sanity check (mine, not the pages'). Writing m = n_1-1, n = n_2-1 for the two star degrees, the subject's formula reads m+n-1 when m,n are both even and m+n otherwise. That is the classical Burr–Roberts star Ramsey number, and "both n_i odd" is exactly "both degrees even". Every table row above agrees. This is my own cross-check; the pages do not rely on it and do not cite it.

1.6 My independent conclusion on the unit's mathematics

No mathematical error, and no formulation error. The statement is the Ramsey number of the two given trees throughout; the parity strengthening is justified from integrality alone; the constructions exist for every admissible pair with no exceptional case; the endpoints are handled rather than excluded; and the sharpness claim is confined to two-color stars.

One strictly-omitted routine step: under the page's own leastness definition of R, exhibiting one avoiding coloring on N vertices gives R > N only with monotonicity in the host order. star_sharpness.md:107–108 supplies it once; it is not restated at star_sharpness.md:134–136 (a different graph on a different host) and is absent from two_tree_corollary.md:113–116 and 118–120. The step is routine and the conclusions are true. I record it as a recommended one-clause correction, not a blocking defect.

2. (b) Verdict on every mathematical claim the report makes

I checked each "verified", "re-derived", "correct" and "confirmed" item.

All of the report's mathematical claims are correct. Specifically:

§1.3 — the theorem_1 restatement is verbatim-faithful. Correct.
§1.4 — the conflation check, including the observation that theorem_1's "every tree" paraphrase is not used. Correct.
§1.5 — the diagonal specialisation and its agreement with E0547.md.txt:80–82, :98–100 and tree_ramsey_corollary.md.txt:29. Correct.
§2.1 — the all-orders derivation, the n_i = 2 degenerate reading (2e(G_i) <= 0, colour class edgeless), the (N-2)N identity, and the n_1=n_2=2 boundary check. All correct.
§2.2 — the both-odd derivation, the integrality strengthening, the (N-1)N-2 identity and the explicit contrast with the (N-2)N of the previous case, and the n_1=n_2=3 check. All correct.
§2.3 — the endpoint upper bound, lower bound, colour interchange and second-branch remark. All correct.
§3.1 — the necessity direction, both parity cases of the circulant construction, the j+j' <= 2k <= N-1 crux, the involution argument showing the chord is counted once, and the spot checks (2,1), (4,3), (6,3), (N,N-1). All correct.
§3.2, §3.3 — the parameter arithmetic, the exhaustive parity split, the complement degrees, the star criterion, the monotonicity remark, and the integrality closing. All correct.
§3.4 — the three endpoint cases including the consecutive-integers parity observation for Nd = (n_1-1)(n_1-2). All correct.
§3.5 — every row of the six-row table. All correct; I reproduced them and added two more rows, which also agree.
§3.6 — the "same quantity, restricted family" analysis. Correct.
§4 — the premise inventory. Correct and complete. I found nothing consumed that the table omits.
§5 — every attribution claim. All correct. I verified: post 8731 is LouisD at 14:25 on 04 Sep 2026 in both snapshots (intermediate line 9, latest line 12); the earlier diagonal display gives 2n-2 for odd n and 2n-3 for even n while the later reverses these; the earlier off-diagonal display uses t_1,t_2 after the prose introduces n_1,n_2; the later adds n >= 2; both hashes match and are correctly assigned to later and earlier; the subject's displayed branches follow the later text. The report's added remark that the earlier version is the mathematically wrong one is also right, and correctly makes the version choice load-bearing.
§8 — W1, W2, W3 rederivations, including the d = N-1 equality boundary at n_1 = n_2 = 3. All correct.
§9 — the three attacks and why they fail. All correct, and the family-wide (rather than sample-point) treatment of the existence conditions is the right method.
§0 — the HEAD and byte-identity facts. Verified true.
§7 D1's nine column measurements — theorem_1.md.txt:30 = 82, tree_ramsey_corollary.md.txt:38 = 83, E0547.md.txt:23/83/98 = 86/89/86, _index.md.txt:70/76/78/80 = 87/84/82/86. All nine exactly right.

I found no incorrect mathematical claim anywhere in the report.

3. (b) Verdict on every defect the report lists

D1 — wikilink label beginning on the line after `|`

NOT A DEFECT. Withdraw.

Line citations are exact. two_tree_corollary.md:156, star_sharpness.md:36, :168, :184 do each end with a bare |. Four occurrences, correctly located.

The finding is wrong on the facts. The report's evidence base is the four native snapshots only. Its narrow claim — that none of those four files ends a line with a wikilink pipe — is true; I checked. But the report generalizes it to "the uniform native convention", and the worktree refutes that. Grepping the whole corpus for \[\[[^]]*|$ returns nine committed instances of exactly this pattern, in problem pages, library source indexes and library result pages:

wiki/problems/divisors/E0964/_index.md:71 and :75 and :90
library/divisors/eberhard_2025_ratios_consecutive_values_divisor_function/_index.md:87 and :104
library/distance_problems/petrov_2021_remark_sets_few_distances/theorem_1_1.md:25
library/group_theory/lettl_sun_2008_covers_abelian_groups_cosets/_index.md:38
library/group_theory/lettl_sun_2008_covers_abelian_groups_cosets/lettl_sun_2008_covers_abelian_groups_cosets.md:156
library/covering_systems/bispels_2025_further_investigation_covering_systems_odd_moduli/_index.md:28

These are tracked corpus pages in a repository whose gate runs wiki lint --path erdos (docs/tools.md:122). The corpus tolerates the pattern. It is rare — 9 instances against 9688 wikilink-bearing lines that instead run past 80 columns — but rarity is not prohibition, and the report offered no rule that forbids it. There is none in AGENTS.md, docs/anatomy.md, docs/math_authoring.md, docs/evidence.md, docs/verification.md or docs/tools.md.

The prescribed correction is not exact and not sufficient — it is harmful. I measured the report's replacement lines:

[[library/extremal_graph_theory/adamczewski_2026_erdos548/tree_ramsey_corollary|tree is 84 columns.
[[library/extremal_graph_theory/adamczewski_2026_erdos548/two_tree_corollary|two-tree is 85 columns.

Both breach the 80-column rule that is written down (docs/anatomy.md:368, docs/math_authoring.md:57). Applying D1 would replace a tolerated pattern with four new rule violations, and would contradict the report's own checklist item 10, which passes the subject at "max 80".

Worse, no compliant alternative exists for these two targets. The prefix [[library/extremal_graph_theory/adamczewski_2026_erdos548/two_tree_corollary| is 77 columns and the label's first word two-tree is 8 more — 85. The prefix [[library/.../tree_ramsey_corollary| is exactly 80 columns, so not a single label character can follow it within the limit. The author's pipe-at-end-of-line wrap is therefore the only form that satisfies the 80-column rule for these two links, and it is a form the corpus already carries. D1 must be dropped, and its correction must not be applied.

D2 — bare-date frontmatter `created`/`updated`

REAL DEFECT. Citation exact. Correction sufficient.

two_tree_corollary.md:6–7 and star_sharpness.md:6–7 read created: 2026-09-10 / updated: 2026-09-10.

The report checked nine files. I checked the whole repository: 4515 created: values across both wiki roots, every single one in the form YYYY-MM-DDTHH:MM:SSZ, and zero bare dates anywhere. The sibling pages in the same folder use created: 2026-09-05T04:10:15Z. The convention is unbroken, and the org AGENTS.md Consistency rules ("Match its patterns exactly", "When in doubt, emulate") make deviating from it a defect. No repository rule assigns created/updated to the wiki tool — AGENTS.md convention 5 (Generated views), docs/anatomy.md section "Authoring and checks" and docs/math_authoring.md:24 give the tool only name, the H1 and index link rows — so this is author-owned metadata and must be authored in the sibling form.

The report's correction (created: 2026-09-10T<HH:MM:SS>Z, updated: 2026-09-10T<HH:MM:SS>Z, with the real UTC time, declining to invent one) is sufficient. It leaves a placeholder rather than exact bytes, which is the right call: fabricating a timestamp would fabricate provenance. The successor must substitute the actual UTC time.

D3 — the two SHA-256 values name no artifact

REAL DEFECT. Citation exact. Proposed wording is exact, truthful and compliant.

two_tree_corollary.md:132–138 reads "The retained later thread text has SHA-256 11f8…" and "The retained earlier text containing that post has SHA-256 43e7…", with star_sharpness.md:166–169 referring back to "The two retained versions". The span 132–138 is exactly the two hash sentences and their two code-span lines; the citation is precise, and the following differences paragraph at 140–144 is correctly left untouched.

The wording does say the hashed object is thread text, so it is not meaningless. But it never says the captures live outside the corpus, and no such file exists in an ordinary clone, so a reader is sent looking for a corpus artifact that is not there. docs/evidence.md:44–46 asks web sources to record "the author or account, URL, date, and available version or commit"; author, URL and date are recorded, and the version is recorded as a hash that cannot be interpreted as written. I require the fix.

Checked against the corpus precedent. library/distance_problems/sallerk_2026_convex_nonagon_relations/sallerk_2026_convex_nonagon_relations.md:16–19 records a forum source as "a locally saved forum-thread copy" with its SHA-256 (not restated here) — the corpus already accepts hashing an out-of-corpus forum capture, provided the object is named.

Checked for the naming prohibition. The report's proposed replacement uses "retained as private contextual material and are not part of this corpus". That is the accepted generic form — the phrase "material from other repositories or private contextual material" appears in existing corpus review records (library/ramsey_theory/openai_2026_ten_advances_mathematics_theoretical_computer_science/chapter_9/evidence/verify/upper_bound_route_grade.md:32, library/discrepancy/kesten_1966_bounded_remainder/evidence/verify/anchored_necessity_source_reading.md:112 and :289). It names no repository, no path and no private location. historical input manifest:85 and :110 record the private repository path private repository locator; the proposed wording does not reproduce it, and neither does the subject as it stands.

Truthfulness. "Two plain-text captures of that thread page" matches the snapshots, which are plain-text renderings headed URL Source: / Markdown Content:. "The earlier capture, which also contains that post" matches earlier forum capture:9–21. Every replacement line is within 80 columns (longest 67). Endorsed as written. I note only that the existing corpus phrasing "a locally saved copy of that thread page" would be equally compliant and closer to precedent; either is acceptable.

O1–O5, I1–I5

O1 (implicit monotonicity at two_tree_corollary.md:113–116) — the observation is right, but I disagree with the disposition. Under the page's own leastness definition this is an omitted step, not merely an asymmetry, and it is also absent from star_sharpness.md:134–136, which the report does not mention. See defect 3 below.
O2 ("the last inequality" at star_sharpness.md:138) — accurate; presentational only. Agreed, no change required.
O3 (## Source and standing vs the siblings' ## Source and dependencies) — accurate. I confirmed the heading appears nowhere else in the corpus, and also that library result pages already use over forty distinct ## headings (## Source and dependencies 48 times, ## Source and scope 27, ## Source and verification 10, and so on). No rule fixes it. Recorded only. Agreed.
O4, O5 — accurate and immaterial. Agreed.
I1 (both pages must land together) — correct. I confirmed star_sharpness.md and two_tree_corollary.md are absent from the baseline worktree and that the other four targets exist.
I2 (wiki update supplies name) — correct.
I3 (E0547 has no managed incoming block) — correct. I confirmed ../../../../../wiki/problems/ramsey_theory/E0547/_index.md contains neither marker, while 117 problem pages do.
I4 (the source index's six-page review sentence) — correct as far as it goes; incomplete, see defect 5.
I5 (tree_ramsey_corollary's open sharpness remark; E0547's account) — correct.

4. (c) Defects the report missed

Numbered as in the summary. Items 1–3 concern the unit; 4–6 concern the report and the integration list.

1. (Report listed as D2.) Frontmatter created/updated bare dates, two_tree_corollary.md:6–7 and star_sharpness.md:6–7. Required correction: the sibling YYYY-MM-DDTHH:MM:SSZ form with the real UTC time, e.g. created: 2026-09-10T00:00:00Z replaced by the actual value. The report's evidence was nine files; the true base is 4515 values with zero exceptions.

2. (Report listed as D3.) The two SHA-256 values must name what is hashed and say it is outside the corpus. two_tree_corollary.md:132–138. Adopt the report's replacement verbatim.

3. (Report listed only in part, as observation O1.) The monotonicity step is omitted where it is needed, and the omission is in three places, not one. Required corrected wording:

two_tree_corollary.md:115 — after "This includes $n_2=2$ , when the host is the one-vertex graph." insert: "The same all-blue colouring on any smaller host also avoids both required copies."
two_tree_corollary.md:119–120 — after "is all red." insert: "and likewise on every smaller host."
star_sharpness.md:132 — replace "The same centre-degree criterion excludes both required stars." with "The same centre-degree criterion excludes both required stars, here and on every smaller vertex set."

Basis: both pages define R as the least N with the covering property (two_tree_corollary.md:15–17, star_sharpness.md:25–27), so a single avoiding coloring at N yields R > N only together with monotonicity in the host order. star_sharpness.md:107–108 supplies the clause once; the three sites above do not inherit it. I classify this as recommended rather than blocking, because each conclusion is true and the missing clause is one sentence in each place — but it is a real omission under the pages' own definition, and the report's "not a defect" disposition understates it.

4. Report defect — D1 is not a defect and its correction is harmful. Detailed in §3 above. The corpus carries nine committed instances of the pattern; the prescribed replacements are 84 and 85 columns; and for the tree_ramsey_corollary target the pipe already sits at column 80, so no compliant alternative exists. Required action: withdraw D1 and leave two_tree_corollary.md:156 and star_sharpness.md:36, :168, :184 unchanged. Root cause: the report established a corpus-wide convention from four files while it had the whole worktree available and used the worktree freely elsewhere (it grepped E0547.md for I3).

5. Missed integration condition — the source record's digest must introduce the two new pages, not merely have its review sentence qualified. The report's I4 addresses only _index.md.txt:115–117 ("All six written proof pages …"). Two further authored passages in the same file go stale: the "Results and method" account and the enumerated chain at _index.md.txt:68–82 ("The full written chain is:", six bullets). Once the folder holds eight pages, the digest must identify the two supplied two-color results, state that they are author-recorded and outside the six-page review, and keep them distinct from the source's own proof chain. Required wording is the successor's to draft; the condition is that the digest name them and their standing.

6. Missed integration condition — build_library_subjects.py. The documented workflow (AGENTS.md section "Checks" and section "Subject indexes", section "Incoming library links"; docs/anatomy.md:240–250) runs the incoming-library writer, then build_library_subjects.py, then wiki update/wiki lint; gate leg 3 (docs/tools.md:118–119) runs build_library_subjects.py --check. The report's integration list omits both the subject-index regeneration and the closing pre-commit run --all-files.

Non-defects I checked and cleared (so the successor does not chase them): no line in either subject exceeds 80 columns (max exactly 80, at two_tree_corollary.md:129 and :156 and star_sharpness.md:184 — the report's item 10 names only two of the three, which is a trivial imprecision, not a defect); both files end with a single newline; no trailing whitespace, tabs or CR anywhere; no clock time outside frontmatter; the only body dates are bare 2026-09-04 (two_tree_corollary.md:128, star_sharpness.md:164); no status word, no tier, no matches in the recorded project-attribution check, no private path, no other repository, and no matches in the recorded source-transfer check — I re-ran the scans independently and confirmed every one. The ## Source and standing heading, the absent frontmatter name, and the missing PDF locator for the premise are all acceptable: headings are unconstrained, name is tool-owned, and routing the premise through the native page at its recorded standing (rather than re-citing the PDF the page says it did not recheck) is the more honest choice.

5. (d) Completeness and honesty of the report

Reading and exposure: fully recorded, with line ranges. §11 tabulates all nine frozen material files with whole-file ranges, the seven worktree instruction files with whole-file ranges, the partial read of ../../../../../wiki/problems/ramsey_theory/E0547/_index.md (grep plus tail -5, disclosed as lines 96–100), the four worktree files hashed but not read as text, the read-only git commands, and an explicit list of what was deliberately not opened — author handoff, author reading receipt, author freeze manifest, the author difference, author subject mapping, prior source-triage directory, private coordination and operational directories, other reviewers' records, the PDF, Lean and the network. This matches the isolation contract.

Exposure disclosure: candid and specific. §11 discloses that historical input manifest interleaves locators with the author's own assessment strings, quotes two examples with line numbers, states that the private repository path at lines 85 and 110 and the prior triage directory at line 237 were seen but not opened, and judges the exposure immaterial while leaving the judgment to the grader. That is the correct handling under docs/verification.md:74–77, which requires immediate disclosure rather than silence. I agree the exposure is immaterial: none of those strings could have produced the report's derivations, and I reached the same mathematical conclusions without them.

Exposure ruling (materiality grader, Claude Fable 5.1, 2026-09-18): the commissioned frozen set carried standing and acceptance text — the subject's own author-recorded, awaiting-review wording at ../assets/reviewed_v1_two_tree_corollary.md.txt lines 146–153 and 158–159 and ../assets/reviewed_v1_star_sharpness.md.txt lines 172–181 and 187, the premise-chain acceptance at _index.md lines 102–107 and 115–117 and wiki/problems/ramsey_theory/E0547/_index.md line 7 and lines 22–24 and 69–70 at the baseline state of 2026-09-10T11:12:29Z, and the unretained input manifest's author role, scope and limitation strings — and the exposure is ruled immaterial by the content test, because none of that text states or implies a verdict on the two-page unit, the subject's own standing text withholds one, the premise acceptance was commissioned reading to be taken at recorded standing, and the review's and grade's derivations (review §2–§3; grade §1.4–§1.5) proceed from theorem_1.md lines 15–24 with E0547.md lines 80–82 and 98–100 used only as a consistency check after the derivation.

Checklist dispositions: complete. Forty-five rows across §6.1–§6.4, each with an explicit verdict, including "NOT APPLICABLE" with a stated reason. docs/verification.md:121–122 requires exactly this ("Give an explicit verdict for each item below, including why an item is inapplicable. Silence is not a verdict."). No item is left silent.

Contract parts: all present. Subject and independence (§0, §11), restatement (§1), checklist (§6), weakest steps — three, with composition (§8), strongest attack (§9), premises with reading depth (§10), verdict (§12).

Inherited coverage: correctly refused. The report states at line 6 that no earlier review coverage is inherited, and §4 independently confirms that the source record's six-page review cannot cover pages that did not exist when it was made. It does not lean on the prior triage, which it did not open.

Honesty: high. Every checkable factual claim in the report was true when I tested it — the nine column measurements, the HEAD value and commit subject, the byte-identity of the four snapshots, the hash assignments, the absence of a managed block on E0547, the absence of pipe-terminated wikilinks in the four snapshots, the grep results for the prohibited tokens, and all six rows of its parameter table. It does not overstate its premise reading depth (it says "claims checked" and names what it did not recheck). It does not claim to have run wiki lint and says so explicitly at D1. It flags its own uncertainty about the wiki tool possibly normalizing frontmatter (D2). The single failure is evidentiary reach, not candour: D1 asserts a corpus-wide convention on a four-file sample. That is a real methodological defect, because the resulting instruction would degrade the pages, but it is not a misrepresentation.

6. (e) Integration conditions I consider necessary

For the filed page:

Both pages land in the same change. two_tree_corollary.md:33 links star_sharpness, and star_sharpness.md:36 and :168 link two_tree_corollary. Neither exists in the baseline, so a partial landing leaves dangling links and fails wiki lint.
Apply defects 1–3 (frontmatter timestamps; the forum-capture wording at two_tree_corollary.md:132–138; the three monotonicity clauses). Do not apply the report's D1.
Source record. Amend ../../_index.md so that (a) the review sentence at its lines 115–117 names the six covered pages or states that the two supplied two-colour results are outside that review, and (b) the digest introduces the two new pages with their author-recorded standing, kept distinct from the six-page source proof chain enumerated at its lines 68–82.
tree_ramsey_corollary. Its lines 71–72 say the sharpness claims "require a separate lower-bound argument". Note there that the q = 2 case is now supplied, leaving general q open. Neither subject page is wrong as written; this keeps the sibling account current.
E0547 authored account. Per docs/anatomy.md:111–113, a source addition that changes a problem page's mathematical account needs an authored explanation and a result link there. Add the off-diagonal generalisation and the star sharpness of the diagonal bound to E0547's Progress and Known Results, without touching its frontmatter status.
Managed incoming navigation. ../../../../../wiki/problems/ramsey_theory/E0547/_index.md has no  block, so it is a newly linked destination the gate will not discover. Run uv run --no-sync python scripts/build_problem_library_links.py --problem E0547 and pass the same --problem E0547 to the gate.
Subject indexes and generated metadata. Then uv run --no-sync python scripts/build_library_subjects.py, then uv run --no-sync wiki update --path erdos (which supplies the tool-owned frontmatter name both pages currently lack) and uv run --no-sync wiki lint --path erdos, both clean.
Closing checks. uv run --no-sync pre-commit run --all-files, then uv run --no-sync erdos gate --path . --problem E0547, with every leg PASS or a visible SKIP. Lean stays skipped; neither page claims formal verification, and none is warranted.
Standing preserved at filing. Both pages must keep their author-recorded / awaiting-independent-review wording and their refusal to inherit the source record's six-page review. Filing this unit awards no tier and changes no problem status. If a later cycle promotes the unit, that requires a distinct grader's record under docs/verification.md:110–133, not this grade.

Documentary disposition of the standing instructions in items 3 and 9: a later filing decision accepts premise-relative reviewed standing for the two-page unit on the historical whole-unit mathematical review PASS and distinct grade PASS, relative to theorem_1 at its recorded standing. That reconciliation applies consistently to the source digest and both result pages, superseding their prescribed author-recorded wording. It does not claim a fresh source-proof/PDF check, inherit the prior six-page review, award a native tier, or reinterpret this historical grade as a review of the filed page. The original items 3 and 9 above, and the same historical recommendation in §4, finding 5, are preserved; this disposition belongs to the later filing decision.

7. Verdict

Report: PASS with corrections. Its mathematics is correct throughout, its restatement and formulation analysis are sound, its checklist coverage and reading disclosure meet the contract, and its honesty is high. It must withdraw D1, and its integration list is short by two items.

Unit: PASS with exact corrections. The mathematics of both pages is correct, the formulation is exact, the scope disclaimers are accurate, and the warrant boundaries are drawn honestly. Two documentary corrections are required (frontmatter timestamp form; forum-capture identification) and three one-clause monotonicity insertions are recommended.

Linked from (5)

Verification of the two-tree and star-sharpness unit Independent review of the two-tree and star-sharpness unit Source readings and corrections for the two-tree review Two-colour star sharpness Two-tree Ramsey corollary

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