Recorded 2026-09-10. Role: independent whole-unit reviewer, distinct from the author and collaborators who constructed the two exact subject pages. The reviewer reported a fresh context that had not built on the subject. First-person readings, deductions, checks and judgments below belong to that historical reviewer, not the author of this documentary rendition.

The exact reviewed subjects are the two-tree corollary and star sharpness assets. Every TTC/SS or result-page line citation below refers to those exact versions, not the later current pages. The source and context snapshot citations, input identities and reading limits are mapped in source reading and documentary corrections. The distinct grade accepts the report with corrections and withdraws D1; that original finding is preserved and annotated below.

This is the full substantive historical report, with native locators, private operational references replaced by role descriptions, and tables rendered as labeled lists. It is not a fresh review of the filed page. Its PASS is premise-relative to the exact named edge bound; its proof chain and PDF were not rechecked. No native tier, catalog-status change, mathematical execution or formal-verification credit follows. 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 reviewer'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, blind, read-only review of two author-recorded pages treated as one unit in the dependency order two_tree_corollary (consumer of theorem_1) then star_sharpness (consumer of two_tree_corollary). No earlier review coverage inherited. Nothing was modified, staged or committed; no code, Lean, PDF or network access was used.

Verdict: PASS with exact corrections.

The mathematics is correct on both pages. Every step re-derived independently from the exact text of theorem_1.md checks out, including the edge-count arithmetic, the both-odd parity refinement, the integrality step, the regular-graph existence proof, the complement degrees, and every boundary case down to the smallest parameters. The exact-formulation check passes: no conflation between "contains the given tree", "contains some tree on n vertices" and "contains every tree on n vertices" occurs anywhere. All remaining findings are documentary.

0. Input verification

The reviewer checked all nine allowed inputs against the assignment before reading them. 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 reviewer 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)

I additionally compared the four native files in my own worktree with the supplied copies and confirmed that the copies are byte-identical to the baseline repository files:

• ../../theorem_1.md (identical)
• ../../_index.md (identical)
• ../../tree_ramsey_corollary.md (identical)
• ../../../../../wiki/problems/ramsey_theory/E0547/_index.md (identical)

The isolated review checkout was a snapshot of the native baseline as it stood on 2026-09-10T11:12:29Z; that canonical revision object itself was not present in the review checkout. The four native files were byte-identical to the supplied copies, so the baseline is established by content despite the different snapshot commit. Recorded as a fact, not a finding.

1. (a) Restatement and exact-formulation check

1.1 two_tree_corollary.md — what it claims

Objects. Two trees T_1, T_2, given in advance (not quantified existentially, not quantified over all trees), with vertex counts n_1, n_2 >= 2 (lines 14–15).

Quantity bounded. R(T_1, T_2), defined on lines 15–17 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) subgraphs (line 17). This is the ordinary off-diagonal two-color Ramsey number of the two given trees.

Conclusion (lines 21–27).

• R(T_1,T_2) <= n_1 + n_2 - 3 when n_1 and n_2 are both odd;
• R(T_1,T_2) <= n_1 + n_2 - 2 otherwise.

Explicit scope statements (lines 29–34). No ordering assumption on n_1, n_2; no large-order restriction. Exact values at the order-two endpoints: R(K_2,T_2) = n_2 and R(T_1,K_2) = n_1. Equality in the displayed bound is established for stars only, not for every pair of trees.

Sub-results proved in the body.

• Lines 54–78: the n_1+n_2-2 bound for all admissible orders.
• Lines 80–104: the n_1+n_2-3 refinement when both orders are odd.
• Lines 106–121: the exact endpoint values, including R(K_2,K_2)=2, and the observation that these endpoints fall in the second branch because 2 is even.

1.2 star_sharpness.md — what it claims

Objects. S_n = K_{1,n-1}, the star on n vertices, for n >= 2 (line 14); two given stars S_{n_1}, S_{n_2} with n_1, n_2 >= 2 (line 15).

Quantity. The same quantity: R(S_{n_1},S_{n_2}) is defined on lines 25–27 by the identical phrase — least positive integer N such that every red/blue edge-coloring of K_N contains a red S_{n_1} or a blue S_{n_2}, with copies not required to be induced (line 27).

Conclusion (lines 17–23). Equality, not merely a bound:

• R(S_{n_1},S_{n_2}) = n_1 + n_2 - 3 when both n_1,n_2 are odd;
• = n_1 + n_2 - 2 otherwise.

Explicit scope (lines 27–31). No ordering of the two orders; no size restriction beyond n_i >= 2; "This is sharpness for two-colour stars only. It makes no equality claim for arbitrary pairs of trees or for a general number of colours."

Sub-results proved in the body.

• Lines 43–75: from-scratch construction of a simple d-regular graph on N vertices for every N >= 1, 0 <= d <= N-1 with Nd even, plus the necessity of those two conditions.
• Lines 77–112: the lower-bound coloring when the orders are not both odd.
• Lines 114–140: the lower-bound coloring when both orders are odd.
• Lines 142–158: the order-two endpoints, including n_1=n_2=2.

1.3 Match against theorem_1

theorem_1.md.txt lines 15–20 read: "Let T be a tree on t>=2 vertices and G a finite simple graph on n>=1 vertices. If G contains no copy of T, then 2e(G) <= (t-2)n." Line 22: "Copies are not required to be induced." Lines 23–24: "No relation between n and t is needed for the edge bound itself."

two_tree_corollary.md lines 40–52 restate this as: "for a finite simple graph G on N>=1 vertices and a tree T of order t>=2, absence of an ordinary copy of T implies 2e(G) <= (t-2)N", followed by "The premise imposes no relation between N and t."

This is an exact restatement, with n renamed to N and "no copy" spelled "absence of an ordinary copy" — which is precisely theorem_1's non-induced convention. Verdict: faithful. No hypothesis is dropped, no range is widened, and the non-induced convention is carried across unchanged.

1.4 The conflation check — the most important item

No conflation is present. Specifically:

• theorem_1 is a given-tree statement: for a fixed T, absence of a copy of that T bounds the edges. The corollary applies it in exactly that form, once to G_1 with T = T_1, t = n_1 and once to G_2 with T = T_2, t = n_2 (lines 56–63). It never uses a "some tree" or "every tree" reading.
• theorem_1 line 22–23 also carries the equivalent phrasing "average degree greater than t-2 forces every tree on t vertices." That "every tree" form is not used by either subject page. The corollary quotes only the per-tree inequality (lines 46–48) and states at lines 51–52 that no other theorem from the source chain is used as a separate premise. Correct: the "every tree" phrasing is a true consequence of the per-tree bound, but the corollary would not need it, and does not take it.
• The Ramsey quantity on both subject pages is the given-pair quantity: a red copy of the given T_1 or a blue copy of the given T_2. Not "some tree on n_1 vertices", not "every tree on n_1 vertices".
• The convention pinning is directionally correct in both places. The upper bound needs the Ramsey "copy" convention to be no stronger than theorem_1's (both non-induced — matched). The lower bound needs the constructions to exclude even non-induced copies, which is the harder requirement and is what the center-degree criterion at star_sharpness.md:103–106 actually establishes. If Ramsey copies were induced and theorem_1's were not, the upper-bound deduction would break; the pages pin both conventions explicitly, so it does not.

1.5 Match against the E0547 page and the source index

E0547.md.txt line 16–20 asks for the diagonal bound: "If T is a tree on n>=2 vertices then R(T) <= 2n-2." — again a given tree.

Specializing the corollary at T_1 = T_2 = T, n_1 = n_2 = n: "both odd" becomes "n odd", so the corollary yields R(T) <= 2n-3 for odd n and R(T) <= 2n-2 otherwise. This agrees exactly with:

• E0547.md.txt lines 80–82 ("For odd n>=3, parity improves the conclusion to R(T)<=2n-3");
• E0547.md.txt lines 98–100 (Known Results row);
• tree_ramsey_corollary.md.txt line 29 (R_2(T)<=2t-2, with R_2(T)<=2t-3 for odd t), which is the q=2 case of its R_q(T) <= q(t-2)+2 / q(t-2)+1 pair.

The _index.md.txt "Results and method" restatement at lines 54–59 is the same given-tree inequality 2e(G) <= (t-2)n for t >= 2, n >= 1. The subject's premise form matches it verbatim in content.

Both subject pages bear on #547 correctly. They neither restate #547's status nor claim to change it, and neither carries a status or tier word.

2. (b) Re-derivation of the upper-bound deduction

Re-derived from the exact text of theorem_1.md.txt as written, not from recollection of the literature.

2.1 The bound for all orders (two_tree_corollary.md:54–78)

Set N = n_1 + n_2 - 2. Since n_1, n_2 >= 2, N >= 2, so N >= 1 and the premise's host-order hypothesis holds (line 56 asserts N >= 2; correct). Both n_i >= 2, so the premise's t >= 2 hypothesis holds for each application.

Assume a red/blue coloring of K_N with no red T_1 and no blue T_2. G_1, G_2 are the spanning red and blue graphs — each a finite simple graph on N vertices, so the premise applies to each separately:

2e(G_1) <= (n_1 - 2)N,    2e(G_2) <= (n_2 - 2)N.

Checked. Both applications are legitimate. In particular the degenerate case n_i = 2 gives 2e(G_i) <= 0, i.e. the color class is edgeless — the correct conclusion, since a graph with no copy of K_2 has no edges.

Edge partition: E(G_1) ⊔ E(G_2) = E(K_N), so e(G_1) + e(G_2) = N(N-1)/2, hence

2e(G_1) + 2e(G_2) = N(N-1).

Arithmetic check of line 71. (n_1-2)N + (n_2-2)N = (n_1+n_2-4)N. Correct.

Arithmetic check of line 72. n_1 + n_2 = N + 2, so n_1 + n_2 - 4 = N - 2. Hence the bound is (N-2)N. Correct.

Therefore N(N-1) <= N(N-2). Since N > 0, division gives N - 1 <= N - 2, i.e. 0 <= -1. Contradiction confirmed. Hence every coloring of K_N contains a required copy, so R(T_1,T_2) <= N = n_1+n_2-2. Correct.

Boundary case n_1 = n_2 = 2. N = 2; N(N-1) = 2, (N-2)N = 0, so 2 <= 0 is false — the general argument covers the smallest case with no special pleading. Correct.

2.2 The both-odd refinement (two_tree_corollary.md:80–104)

n_1, n_2 odd and >= 2 implies >= 3 (line 82). Correct.

N = n_1 + n_2 - 3. Odd + odd = even, and even − 3 is odd, so N is odd (line 83). N >= 3 + 3 - 3 = 3 (line 83). Both correct, and N >= 1 so the premise applies.

Parity step (lines 86–91). n_i odd implies n_i - 2 odd. N odd. Odd × odd = odd, so (n_i - 2)N is an odd integer. Correct. 2e(G_i) is even, and an even integer cannot equal an odd one, so the premise's 2e(G_i) <= (n_i-2)N strengthens to

2e(G_i) <= (n_i - 2)N - 1,   i = 1, 2.

Integrality argument confirmed. This is a genuine strengthening and is correctly justified from parity alone, with no appeal to any further theorem.

Summation (lines 93–101). Adding gives N(N-1) <= (n_1+n_2-4)N - 2. Correct (the two -1s sum to -2).

Arithmetic check of line 100. Here n_1 + n_2 = N + 3, so n_1 + n_2 - 4 = N - 1, and the right side is (N-1)N - 2. Correct — and note this differs from the all-orders case, where the same expression equaled (N-2)N. The page gets both right.

So N(N-1) <= N(N-1) - 2, i.e. 0 <= -2. Impossible, as stated. Hence R(T_1,T_2) <= n_1+n_2-3 when both orders are odd. Correct.

Smallest both-odd case n_1 = n_2 = 3. N = 3; N(N-1) = 6; (n_i-2)N = 3, strengthened to 2e(G_i) <= 2, so the sum is at most 4 < 6. Contradiction holds. Correct.

2.3 The order-two endpoints (two_tree_corollary.md:106–121)

• "The only tree of order 2 is K_2" (line 108). Correct.
• Upper bound (lines 108–111): in any coloring of K_{n_2}, either some edge is red — a red K_2 = T_1 — or all edges are blue, and then any bijection from V(T_2) onto V(K_{n_2}) embeds T_2 in blue, because every pair is a blue edge. Correct, including n_2 = 2.
• Lower bound (lines 113–116): all-blue K_{n_2-1} has no red edge, and no blue T_2 because the host has n_2 - 1 < n_2 vertices. Including n_2 = 2, where the host is K_1. Correct.
• Color interchange (lines 118–121) gives R(T_1,K_2) = n_1, with the avoiding coloring all red on n_1 - 1 vertices, and R(K_2,K_2) = 2. Correct.
• "These endpoints are in the second branch … since 2 is even" (lines 120–121). Correct: n_1 = 2 is even so "both odd" fails, and the second branch reads n_1+n_2-2 = n_2, which is the exact value just computed. So the endpoints attain the stated bound. Correct.

Completeness observation (not a defect). Concluding R = n_2 from a single avoiding coloring on n_2 - 1 vertices uses the monotonicity of the Ramsey property in the host order (a coloring of K_M restricts to one of K_{M'} for M' <= M). star_sharpness.md:107–108 states this principle explicitly; two_tree_corollary.md:113–116 leaves it implicit. The step is routine and the conclusion is true; I record it as an omitted routine remark, not an error. See finding O1.

2.4 Independent cross-check of the resulting formula

Specializing to stars gives, in edge-count terms with m = n_1-1, n = n_2-1: n_1+n_2-3 when both n_i are odd, i.e. when both m,n are even; n_1+n_2-2 otherwise. Every worked case I computed by hand from the constructions (§3.5) agrees with the corollary's branches. No case was found in which the corollary's bound is violated or slack against the constructions.

3. (c) Re-derivation of the star-sharpness constructions

3.1 Existence of the required regular graphs (star_sharpness.md:43–75)

Hypotheses (line 45). N >= 1, 0 <= d <= N-1, Nd even.

Necessity (lines 46–48). A vertex has at most N-1 neighbors, so d <= N-1; and Nd = sum of degrees = 2e(G) is even by handshake. Both correct. The handshake constraint is correctly identified and the degree-parity condition is exactly Nd even.

Even case d = 2k (lines 50–63). Circulant on Z/N joining x to x ± 1, …, x ± k.

• 2k = d <= N-1 gives 2k <= N-1 < N, hence k < N/2. Correct.
• No difference is 0 mod N: each j with 1 <= j <= k < N satisfies j ≢ 0 and -j ≢ 0. Correct.
• Positive differences distinct; negative differences distinct: immediate since k < N. Correct.
• A positive j never coincides with a negative -j': j ≡ -j' would need j + j' ≡ 0 (mod N), but 1 <= j + j' <= 2k <= N-1 < N. Correct — this is the crux and the page's justification is exact.
• Symmetry: x ~ y iff x - y ≡ ±j, symmetric in x,y. Correct.
• Degree exactly 2k = d. Correct.
• k = 0 gives the empty graph, covering N = 1, d = 0 (line 63). Correct and needed — this is the n_1 = n_2 = 2 case.

Odd case d = 2k+1 (lines 65–75).

• Nd even with d odd forces N even. Correct.
• 2k+1 <= N-1 gives 2k <= N-2, so k <= N/2 - 1 < N/2. Correct.
• The 2k circulant neighbors are distinct as before (needs 2k < N; here 2k <= N-2 < N). Correct.
• The extra neighbor x + N/2: nonzero mod N since 1 <= N/2 <= N-1 for N >= 2. Correct.
• N/2 ≠ j for 1 <= j <= k <= N/2 - 1. Correct.
• N/2 ≢ -j: that would need N/2 + j ≡ 0 (mod N), i.e. j ≡ N/2, ruled out by j <= N/2 - 1. Correct (the page asserts this at lines 69–70; the derivation holds).
• Symmetry of the extra edge: x + N/2 + N/2 = x + N ≡ x, so the relation is an involution and the edges are undirected (lines 71–72). Correct — and note this also shows x + N/2 and x - N/2 are the same vertex, so the extra neighbor is counted exactly once, giving degree 2k+1 and not 2k+2. The page's count of 2k+1 is therefore right.
• Line 73–75: "This proves existence in every case with the stated conditions, without appealing to a regular-graph existence theorem." Confirmed — the construction is self-contained and covers every admissible (N,d). This matters for item (d): no unnamed external theorem is smuggled in here.

Spot checks. (N,d) = (2,1): k=0, neighbor x+1; single edge, 1-regular. (N,d) = (4,3): k=1, neighbors x±1, x+2; K_4, 3-regular. (N,d) = (6,3): k=1, neighbors x±1, x+3; 3-regular. (N,d) = (N,N-1) for N odd: k = (N-1)/2, all N-1 others. For N even: 2((N/2)-1)+1 = N-1 others. In every case the construction yields K_N when d = N-1, which is what star_sharpness.md:151–153 relies on. All correct.

3.2 Color graphs when the orders are not both odd (lines 77–112)

N = n_1+n_2-3, d = n_1-2.

• N >= 2+2-3 = 1. Correct (line 85).
• d = n_1 - 2 >= 0. Correct (line 85).
• N - 1 - d = (n_1+n_2-3) - 1 - (n_1-2) = n_2 - 2 >= 0, so d <= N-1. Correct (lines 87–91).
• Parity (lines 91–94): if n_1 even then d even, so Nd even. If n_1 odd then, since not both are odd, n_2 is even, and N = odd + even - 3 is even, so Nd even. Exhaustive and correct.
• Hence a d-regular simple H on N vertices exists by §3.1.

Complement degrees (lines 96–101). H red, complement blue. Red degree d = n_1 - 2 at every vertex; blue degree N - 1 - d = n_2 - 2. Correct.

Absence of the forbidden stars (lines 103–107). The criterion — a graph contains an ordinary S_n iff some vertex has at least n-1 neighbors — is correct in both directions and for n = 2. Red degree n_1-2 < n_1-1, blue degree n_2-2 < n_2-1. No red S_{n_1}, no blue S_{n_2}. Confirmed. The "irrespective of edges between them" clause is exactly what makes the criterion valid for non-induced copies, matching the statement's convention.

Monotonicity (lines 107–108). "Restricting this colouring to any smaller vertex set also avoids both required stars." Correct and necessary to pass from one counterexample at N to R > N.

Conclusion. R(S_{n_1},S_{n_2}) > N = n_1+n_2-3 (lines 110–112). Combined with integrality, R >= n_1+n_2-2, matching the corollary's second branch. Correct.

3.3 Color graphs when both orders are odd (lines 114–140)

N = n_1+n_2-4, d = n_1-2, with n_1,n_2 >= 3 odd.

• N >= 3+3-4 = 2, and odd+odd-4 is even, so N >= 2 is even. Correct (line 122).
• d = n_1-2 >= 1. Correct (line 122).
• N - 1 - d = (n_1+n_2-4) - 1 - (n_1-2) = n_2 - 3 >= 0. Correct (lines 124–126), so d <= N-1, hence 0 <= d <= N-1 (line 128).
• Nd even because N is even. Correct (line 128).
• Red degree n_1-2 < n_1-1; blue degree N-1-d = n_2-3 < n_2-1. Correct (lines 130–132) — note the blue degree here is n_2-3, one less than in §3.2, which is exactly what buys the extra vertex.
• Conclusion R > N = n_1+n_2-4 (lines 134–136), so by integrality R >= n_1+n_2-3, matching the corollary's first branch. Correct.

Integrality closing (lines 138–140). "In each parity case, the Ramsey number is an integer, so the last inequality gives the lower bound equal to the claimed upper bound." Both cases check out: 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. Correct. See finding O2 for a clarity note on "the last inequality" and on the placement of this both-cases paragraph inside a one-case heading.

3.4 The order-two endpoints (lines 142–158)

• n_1 = 2 (line 144): n_1 is even, so the first (not-both-odd) construction applies; N = n_2-1, d = 0. Red graph empty; blue graph K_{n_2-1}. No red S_2 = K_2 (no red edge). No blue S_{n_2}: the host has n_2-1 vertices, and the blue max degree n_2-2 < n_2-1. On n_2 vertices, any red edge gives a red S_2, otherwise the all-blue K_{n_2} has a vertex of degree n_2-1 and contains S_{n_2}. Hence R(S_2,S_{n_2}) = n_2. Correct, and it agrees with the second branch n_1+n_2-2 = n_2.
• n_2 = 2 (line 151): again even, so the first construction; N = n_1-1, d = n_1-2 = N-1. Nd = (n_1-1)(n_1-2) is a product of consecutive integers, hence even — consistent with the page's general parity argument. The red graph is K_N (§3.1 confirms d = N-1 yields the complete graph) and the blue graph is empty. Red max degree n_1-2 < n_1-1; no blue edge. On n_1 vertices, either a blue edge gives S_2 or the all-red K_{n_1} contains S_{n_1}. Hence R(S_{n_1},S_2) = n_1. Correct, agreeing with n_1+n_2-2 = n_1.
• n_1 = n_2 = 2 (lines 156–158): N = 1, d = 0; the avoiding graph is one vertex with no edges; every coloring of K_2 has its single edge red or blue, giving S_2 in that color. R = 2. Correct, and n_1+n_2-2 = 2.

3.5 Independent numeric checks at the smallest parameters

• n_1,n_2: 2, 2; branch: other; claimed R: 2; construction: N=1, d=0; red deg: 0; blue deg: 0; verdict: ok

• n_1,n_2: 2, 3; branch: other; claimed R: 3; construction: N=2, d=0; red deg: 0; blue deg: 1 < 2; verdict: ok

• n_1,n_2: 3, 2; branch: other; claimed R: 3; construction: N=2, d=1; red deg: 1 < 2; blue deg: 0; verdict: ok

• n_1,n_2: 3, 3; branch: both odd; claimed R: 3; construction: N=2, d=1; red deg: 1 < 2; blue deg: 0 < 2; verdict: ok

• n_1,n_2: 3, 4; branch: other; claimed R: 5; construction: N=4, d=1; red deg: 1 < 2; blue deg: 2 < 3; verdict: ok

• n_1,n_2: 5, 5; branch: both odd; claimed R: 7; construction: N=6, d=3; red deg: 3 < 4; blue deg: 2 < 4; verdict: ok

In each row the constructed graph exists under §3.1's conditions, both forbidden stars are absent, and the resulting lower bound coincides with the corollary's upper bound. No gap between the two bounds in any case checked.

3.6 Which quantity is shown sharp — explicit answer

Sharpness is shown for exactly the quantity the corollary bounds, not for a different one. star_sharpness.md:25–27 reproduces the corollary's definition of R word-for-word in substance (least positive N; every red/blue coloring of K_N; a red copy of the first graph or a blue copy of the second; copies not required to be induced). The lower-bound constructions attack that same quantity.

The restriction is in the family, not the quantity: equality is established only for pairs of stars, and only for two colors. Both pages say so — two_tree_corollary.md:31–34 ("Equality … is established for stars … not for every pair of trees") and star_sharpness.md:30–31 ("This is sharpness for two-colour stars only. It makes no equality claim for arbitrary pairs of trees or for a general number of colours"). These scope sentences are accurate and are exactly the anti-overreach statements this checklist looks for.

4. (d) Premise audit — is theorem_1 the only external theorem?

Yes. Inventory of everything the two proofs consume:

• Ingredient: Sharp tree-free edge bound (theorem_1); Where used: two_tree_corollary.md:40–48, applied at 56–63 and (implicitly, same form) at 82–91; External?: Yes — the single external premise

• Ingredient: Edge partition of K_N into two color classes; Where used: two_tree_corollary.md:65, 98; External?: elementary, no theorem

• Ingredient: e(K_N) = N(N-1)/2; Where used: two_tree_corollary.md:69, 97; External?: elementary

• Ingredient: Parity of an integer product; even ≠ odd; Where used: two_tree_corollary.md:86–91; External?: elementary

• Ingredient: Integrality of R; Where used: star_sharpness.md:138–139; External?: elementary

• Ingredient: Circulant d-regular graph existence; Where used: star_sharpness.md:45–75; External?: proved in place, explicitly "without appealing to a regular-graph existence theorem" (73–75)

• Ingredient: Handshake lemma (necessity direction only); Where used: star_sharpness.md:46–48; External?: elementary; used only to justify the hypotheses, not the construction

• Ingredient: Complement degree N-1-d; Where used: star_sharpness.md:98–101, 130; External?: elementary

• Ingredient: Star containment ⟺ a vertex of degree >= n-1; Where used: star_sharpness.md:103–106; External?: proved in place, both directions

• Ingredient: Monotonicity under vertex restriction; Where used: star_sharpness.md:107–108; External?: proved in place

• Ingredient: Two-tree corollary; Where used: star_sharpness.md:35–41; External?: internal to the unit, correctly attributed

Nothing else is used as a premise. In particular:

• The general-q account is not used. two_tree_corollary.md:155–159 and star_sharpness.md:183–187 reference tree_ramsey_corollary only to delimit scope ("treats one tree in q colours"; "leaves general-q star sharpness separate"). Neither page derives anything from it. The two-color arguments are written out independently from theorem_1, and I verified they do not silently reuse the q-color computation.
• The forum post is not used. two_tree_corollary.md:129–130 states "The comment is attribution and context, not the proof used here." star_sharpness.md:165–169 states "It supplies no lower-bound construction … Neither forum version is a proof premise. The regular graphs and all degree and parity checks above are supplied here." Both true: the post contains no proof of anything (see §5).
• The PDF is not used. Both pages say the canonical PDF was not rechecked (two_tree_corollary.md:148–150, star_sharpness.md:176–177) and neither cites a PDF page or theorem number as a step. The premise enters through the native theorem_1 page at its recorded standing.
• No prior review is inherited. two_tree_corollary.md:152–153 and star_sharpness.md:179–181 both state that the source record's review of six written pages did not assess these supplied arguments. Consistent with _index.md.txt:115–117 ("All six written proof pages … have passed independent mathematical review for this corpus"), which does not and cannot cover pages that did not exist at that time. Warrant boundary correctly drawn.

5. Attribution against the two forum snapshots

Fewer than 20 words are quoted from the snapshots in total.

Claim: post 8731 is by LouisD and dated 2026-09-04. Both snapshots head the post **LouisD** — 14:25 on 04 Sep 2026 (#post-8731, depth 0) (intermediate line 9; latest line 12). Accurate. The date is carried in bare form and the clock time is correctly not reproduced on either page.

Claim (two_tree_corollary.md:127–130): the post "states the diagonal and off-diagonal bounds and star sharpness in a comment beginning 'Assuming #548.'" Latest snapshot lines 13–24 contain a diagonal display, a "tight for stars" sentence, an off-diagonal display, and a second tightness sentence, all under the opening words Assuming #548. Accurate.

Claim (star_sharpness.md:162–166): the post "asserts sharpness for stars after the tree bounds, under its opening 'Assuming #548.' It supplies no lower-bound construction." Accurate. Both tightness assertions follow their respective bounds, and the post contains no construction, no degree argument, and no proof of any kind.

Claim (two_tree_corollary.md:132–138): the two retained texts have the two SHA-256 values stated there (later and earlier). These match later forum capture and earlier forum capture respectively. Hashes correct and correctly assigned to later/earlier. But see finding D3 on what the hashed artifact is.

Claim (two_tree_corollary.md:140–144): "The earlier diagonal display reverses the later parity branches and the earlier off-diagonal display uses t_1,t_2 after introducing n_1,n_2. The later text adds the explicit diagonal condition n>=2 and uses n_1,n_2 consistently … The statement above follows the later text."

Checked term by term:

• Earlier diagonal (intermediate lines 10–13): 2n-2 for n odd, 2n-3 for n even. Later diagonal (latest lines 13–16): 2n-3 for n odd, 2n-2 for n even. The earlier version does reverse the later branches. Accurate — and materially so: the earlier version is the mathematically wrong one, since parity buys the improvement when n is odd. Recording which version is followed is therefore genuinely load-bearing provenance, not decoration.
• Earlier off-diagonal (intermediate lines 16–20): introduces n_1, n_2 in prose then writes t_1+t_2-3 / t_1+t_2-2 in the display. Accurate.
• Later text adds for all $n$-vertex trees $T$ with $n\geq 2$ (latest line 13). Accurate — "adds the explicit diagonal condition n>=2".
• Later text uses n_1,n_2 throughout the off-diagonal display (latest lines 19–23). Accurate.
• The subject's displayed off-diagonal bound matches the later text's branches exactly. Accurate: "The statement above follows the later text."

Observations, not defects. (i) The later snapshot also contains a fifth post (JakeMallen, latest lines 9–10) and a Posts: 5 header versus Posts: 4; the subject's difference sentence is scoped to post 8731's displays and does not claim to be an exhaustive thread diff, so it is not inaccurate. (ii) The later version also promotes the inline $…$ displays to $$…$$; unmentioned and immaterial. (iii) The subject writes the two-word fragment as “Assuming #548.” where the source reads Assuming #548,; the quoted words are exact and the period inside the quotation marks is ordinary American typographic practice. I record it and do not require a change.

URL fidelity. Both pages cite https://www.erdosproblems.com/forum/thread/547#post-8731, matching the snapshots' URL Source: https://www.erdosproblems.com/forum/thread/547. Note that the pre-existing native pages cite the same discussion as https://www.erdosproblems.com/forum/discuss/547 (tree_ramsey_corollary.md.txt:69, E0547.md.txt:30). That is a pre-existing inconsistency in the baseline corpus, not a subject defect; the subject's URL is the one the snapshots record.

6. (e) Corpus checklist — explicit disposition of every item

Line numbers are given for each disposition. "TTC" = two_tree_corollary.md, "SS" = star_sharpness.md.

6.1 Reviewer-instruction checklist

• Item number: 1; Item: Standing wording: author-recorded / awaiting independent review; TTC: 146–147; SS: 172–173; Disposition: MET. TTC: "The argument here is author-recorded and awaits independent review of its whole statement and proof." SS: "This result is author-recorded and awaits independent review of its whole statement and proof, together with the supplied upper bound." Matches docs/evidence.md:82–83.

• Item number: 2; Item: Warrant boundaries: premise standing not inflated; TTC: 148–153; SS: 174–181; Disposition: MET. Both record that only the premise's statement/result page was read, that the proof chain and PDF were not rechecked, that the premise is used at the standing in the source record, and that the six-page review is not transferred.

• Item number: 3; Item: Warrant boundaries: no formal-verification credit; TTC: 158–159; SS: 187; Disposition: MET. "No formal verification or mathematical program was run …" on both.

• Item number: 4; Item: Warrant boundaries: general-q scope preserved; TTC: 155–159; SS: 183–187; Disposition: MET. Both correctly say only that general-q star sharpness is untouched, which is true; neither overclaims that the whole boundary is unchanged.

• Item number: 5; Item: Dates in bare form (body); TTC: 128 (2026-09-04); SS: 164 (2026-09-04); Disposition: MET. The only body dates are bare YYYY-MM-DD.

• Item number: 6; Item: No clock-time labels outside frontmatter; TTC: —; SS: —; Disposition: MET. Grep for [0-9]{2}:[0-9]{2} returns nothing in either file. The forum posts' 14:25 is correctly not reproduced.

• Item number: 7; Item: No status or tier claims; TTC: —; SS: —; Disposition: MET. Grep for the expressions status, tier, proved, disproved, solved, decidable, verifiable, refutation-failed, refuted-as-stated, independently reviewed, and independently accepted returns only SS:93, "The construction just proved therefore gives …", which is ordinary mathematical usage, not a standing label.

• Item number: 8; Item: Wiki link targets exist in the worktree; TTC: 41, 151, 156, 161; SS: 39, 179, 184, 189; Disposition: MET for all external targets: theorem_1.md, _index.md, tree_ramsey_corollary.md, E0547.md all exist.

• Item number: 9; Item: Wiki link targets — intra-unit; TTC: 33 (star_sharpness); SS: 36, 168 (two_tree_corollary); Disposition: MET conditionally. These two targets do not exist in the baseline worktree because they are the unit's own pages. They resolve iff both pages land together. Recorded as an integration condition (I1), not a defect.

• Item number: 10; Item: Lines at most 80 columns; TTC: max 80 at 129; SS: max 80 at 184; Disposition: MET. No line exceeds 80 in either file.

• Item number: 11; Item: No project-attribution matches; TTC: —; SS: —; Disposition: MET. The recorded scan of the exact TTC/SS subjects found no matches in that check or in the private workflow-record vocabulary.

• Item number: 12; Item: No private paths; TTC: —; SS: —; Disposition: MET. A scan for private filesystem and agent-configuration paths returns nothing.

• Item number: 13; Item: No other repositories; TTC: —; SS: —; Disposition: MET. Grep for repositor|hub\.com returns nothing. Contrast _index.md.txt:91–104, where such links legitimately appear on the source record.

• Item number: 14; Item: No source-transfer matches; TTC: —; SS: —; Disposition: MET. The recorded import scan of the exact TTC/SS subjects found no matches. Neither page uses the accepted generic wording either, because it has no occasion to.

6.2 docs/verification.md audit checklist (lines 149–173)

• Item: Quantifiers and scope; Disposition: MET. Every hypothesis is explicit and every range is stated: n_i >= 2 (TTC:14, SS:15), no ordering assumption and no large-order restriction (TTC:29, SS:27–28), the parity split (TTC:24–25, SS:20–21), the endpoint cases (TTC:106–121, SS:142–158). No "almost all", eventual, or limit statement is involved.

• Item: Circularity; Disposition: MET. No circularity. theorem_1's proof (a permutation/marked-cut count, theorem_1.md.txt:26–40) is independent of Ramsey theory, so applying it to color classes is not circular. Within the unit the dependency is strictly one-way: SS consumes TTC, TTC consumes theorem_1; TTC does not consume SS. TTC:31–34 forward-references SS for equality only, not for the upper bound.

• Item: Model and convention changes; Disposition: MET. The single convention that must transfer is "copy" = non-induced subgraph. It is pinned identically in theorem_1.md.txt:22, TTC:17 and 43–44, and SS:27 and 103–106. No relaxed, averaged or abstract substitute is introduced.

• Item: Finite and statistical overreach; Disposition: NOT APPLICABLE. No finite computation, sample or heuristic is used anywhere. The parameter checks in §3.5 are my own cross-checks, not part of the pages' warrant.

• Item: Uniformity; Disposition: MET. There are no constants, error terms, limits or infinite families with parameter-dependent bounds. The bounds hold for every admissible (n_1,n_2) with no threshold, and the pages say so.

• Item: Extremal conclusions; Disposition: MET. This is the item that matters most for SS. Sharpness is checked in the proposition's own units (§3.6), the required existence is proved rather than assumed (SS:45–75), and boundedness is not at issue. The attained values at the endpoints are computed exactly, not asserted.

• Item: Consequences and composition; Disposition: MET. Each "hence" was checked separately: TTC:76 (division by N>0), TTC:104 ("This is impossible"), TTC:116 and 118 (endpoint equalities), SS:106–108 (criterion + restriction ⟹ R > N), SS:132–136 (same in the both-odd case), SS:138–140 (integrality ⟹ lower bound meets upper bound). All compose correctly. The one routine step left implicit is at TTC:113–116 (see O1).

• Item: Computation; Disposition: NOT APPLICABLE. No program, certificate or numerical enclosure. Both pages state that none was run (TTC:158–159, SS:187), which is the correct disclosure.

• Item: Reproduction; Disposition: NOT APPLICABLE. No rerun command or coverage claim is made, so there is nothing to reproduce.

• Item: Source and verdict fidelity; Disposition: MET for the mathematics; one documentary correction. The theorem_1 restatement is verbatim-faithful (§1.3). The forum characterizations are accurate (§5) and are not strengthened. The residual issue is that the two SHA-256 values are attributed to an unnamed artifact (D3).

6.3 docs/evidence.md requirements

• Requirement: Exact hypotheses, quantifiers, ranges, conclusion (line 15); Disposition: MET. TTC:14–34, SS:14–31.

• Requirement: Endpoint exceptions and convention changes explained (line 18); Disposition: MET. TTC:106–121, SS:142–158; the n_i = 2 cases are handled rather than excluded.

• Requirement: External dependency explicit, without implying its proof is included (lines 22–24); Disposition: MET. TTC:40–52 and 148–150; SS:37–41 and 174–177.

• Requirement: Sketch/scope labeled at actual scope (lines 26–27); Disposition: MET. Both pages give complete arguments and label them author-recorded, not reviewed.

• Requirement: Web-source provenance: author, URL, date, version (lines 44–46); Disposition: PARTIALLY MET. Author, URL and date are recorded on both pages; the version is identified by two hashes whose artifact is not named. See D3.

• Requirement: Living verification record: standing, exact mathematics checked, source versions, external premises, limitations (lines 81–86); Disposition: MET. TTC:146–159, SS:172–187. Each element is present.

• Requirement: No dated agent activity logs (line 88); Disposition: MET. Neither page contains a log, a pass count or a chronology.

• Requirement: Hashes pin identity, not truth (lines 68–69); Disposition: MET. TTC:143–144 says explicitly "These are retained textual differences, not proof warrants", and SS:169 says "Neither forum version is a proof premise."

• Requirement: Premise identified by version, exact statement, specialization used (lines 114–117); Disposition: MET. TTC:40–52 gives the exact statement and states the specialization ("used below for each spanning colour graph separately").

• Requirement: Required evidence resolves from an ordinary clone (lines 154–156; AGENTS.md convention 8 (Self-contained corpus)); Disposition: MET for evidence — there is none. See D3 for the hashed forum captures, which the pages correctly say are not warrants.

6.4 docs/anatomy.md and docs/math_authoring.md page mechanics

• Requirement: Result page carries title: (anatomy:163); Disposition: MET. TTC:2, SS:2.

• Requirement: One-sentence desc (anatomy:163); Disposition: MET. TTC:3–5, SS:3–5. Both are single sentences and both accurately describe the page.

• Requirement: Precise statement (anatomy:163); Disposition: MET. TTC:12–34, SS:12–31.

• Requirement: Proof or proof pointer (anatomy:163); Disposition: MET. TTC:36–121, SS:33–158. Full proofs, not pointers.

• Requirement: Results it depends on (anatomy:164); Disposition: MET. TTC:38–52 and 149–153; SS:35–41 and 174–179.

• Requirement: "Bears on" list (anatomy:164–165); Disposition: MET. TTC:161, SS:189, both linking problems/ramsey_theory/E0547. Listing only #547 is right: neither page concerns #548's own statement or the multicolor #557.

• Requirement: Descriptive result-page name when the paper gives none (anatomy:160–162); Disposition: MET. two_tree_corollary and star_sharpness follow the sibling precedent tree_ramsey_corollary, and both pages state they are supplied results, not numbered results in the exposition (TTC:125, SS via TTC).

• Requirement: Prose wrapped at 80 columns (anatomy:368, math_authoring:57); Disposition: MET. See item 10 above.

• Requirement: Display math in $$ blocks on their own lines (anatomy:369, math_authoring:57–58); Disposition: MET. Every display in both files opens and closes with $$ alone on its line (TTC:21/27, 46/48, 60/63, 67/74, 89/91, 95/102; SS:17/23, 52/54, 81/83, 87/89, 99/101, 110/112, 118/120, 124/126, 134/136).

• Requirement: Headings on one line (math_authoring:58); Disposition: MET. All H2/H3 headings are single-line.

• Requirement: No indented display equations in list items; no list continuation that is only a code-span path (math_authoring:58–60); Disposition: NOT APPLICABLE. Neither page uses lists.

• Requirement: Tool owns name, the H1 and index link rows; author desc and body below *** (anatomy:360–361, math_authoring:24); Disposition: MET. Neither page hand-writes a name, an H1 or a link row. Both open the body with *** at line 10. The absent name: is the tool's to supply on wiki update; see I2.

• Requirement: Trailing newline (org AGENTS.md, Consistency 7); Disposition: MET. Both files end with a newline; no trailing whitespace, no tabs, no CRLF.

7. Findings

Defects requiring correction (documentary)

D1 — wikilink broken immediately after the pipe, against the uniform native convention. Four occurrences.

Documentary disposition: this finding and all four proposed replacements below were withdrawn by the distinct grader, §3, preserved in two_tree_grade.md. They remain here as the original reviewer's finding, not a current correction request. The four pipe-ended links are retained. The grader's contrary evidence and width measurements are not incorporated into the original reviewer's reasoning.

two_tree_corollary.md:156, star_sharpness.md:36, :168, :184 each end the line with a bare |, so the link label begins with a newline.

Evidence that this deviates: across the four native snapshots, no line ends with a wikilink pipe (the only |-terminated lines are the YAML desc: | keys). Where a wikilink token is too long, the native corpus consistently exceeds 80 columns rather than breaking after the pipe — theorem_1.md.txt:30 is 82 columns, tree_ramsey_corollary.md.txt:38 is 83, E0547.md.txt:23 is 86, :83 is 89, :98 is 86, and _index.md.txt:70/76/78/80 are 87/84/82/86. Every one of those breaks inside the label, after at least one label word. The convention therefore resolves the 80-column conflict in favor of the long line.

I could not run wiki lint (execution is outside my remit), so I do not assert that the link fails to resolve; the target is intact and the likely effect is a label with leading whitespace. The finding is the convention deviation, which is uniform and easily fixed.

Required corrections (replace the two-line span in each case):

• two_tree_corollary.md:155–157

  textCopy

  The existing
  [[extremal_graph_theory/adamczewski_2026_erdos548/tree_ramsey_corollary|tree
  Ramsey corollary]] treats one tree in $q$ colors. Its separate boundary

• star_sharpness.md:35–37

  textCopy

  The upper bound is the
  [[extremal_graph_theory/adamczewski_2026_erdos548/two_tree_corollary|two-tree
  corollary]], applied to these two stars. That supplied result uses the

• star_sharpness.md:167–169

  textCopy

  identified in the
  [[extremal_graph_theory/adamczewski_2026_erdos548/two_tree_corollary|two-tree
  corollary]]. Neither forum version is a proof premise. The regular

• star_sharpness.md:183–185

  textCopy

  The existing
  [[extremal_graph_theory/adamczewski_2026_erdos548/tree_ramsey_corollary|tree
  Ramsey corollary]] treats general $q$ and leaves general-$q$ star

D2 — frontmatter created/updated use bare dates, not the sibling timestamp form.

two_tree_corollary.md:6–7 and star_sharpness.md:6–7 read created: 2026-09-10 / updated: 2026-09-10. Every page I read uses the full UTC form: theorem_1.md.txt:7–8, _index.md.txt:6–7, tree_ramsey_corollary.md.txt:7–8, E0547.md.txt:8–9, docs/anatomy.md:11–12, docs/evidence.md:7–8, docs/verification.md:7–8, docs/math_authoring.md:7–8, docs/tools.md:6–7 — nine for nine, e.g. created: 2026-09-05T04:10:15Z.

No written rule mandates the format, and the wiki tool may normalize it; the basis for the correction is the org AGENTS.md Consistency rule ("Read the surrounding code first. Match its patterns exactly", "When in doubt, emulate"), against an unbroken convention.

Required correction, both files, lines 6–7: use the same YYYY-MM-DDTHH:MM:SSZ form as every sibling page, with the actual UTC timestamp — i.e. created: 2026-09-10T<HH:MM:SS>Z and updated: 2026-09-10T<HH:MM:SS>Z. I do not supply a clock value, since inventing one would fabricate provenance.

D3 — two SHA-256 values are attributed to an artifact that is never named and cannot be resolved from an ordinary clone.

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…". A reader cannot tell what object is hashed (a captured page? a file? which file?) or where "retained" means. No such file exists in the corpus; star_sharpness.md:166–169 then refers back to "The two retained versions", inheriting the same gap.

The hashes themselves are correct and their later/earlier assignment is correct (§5), and the material difference between the two versions is genuinely load-bearing, so the right fix is to name the artifact rather than remove the hashes. docs/evidence.md:44–46 requires recording "the author or account, URL, date, and available version or commit" for web sources; the version is recorded but is uninterpretable as written.

Required correction, replacing two_tree_corollary.md:132–138:

Two plain-text captures of that thread page are retained as private
contextual material and are not part of this corpus. The later
capture has SHA-256

<removed: the later capture's digest; the bytes are not held here>.

The earlier capture, which also contains that post, has SHA-256

<removed: the earlier capture's digest; the bytes are not held here>.

All lines are within 80 columns and the wording uses the accepted generic form ("private contextual material") rather than a private path or a repository name.

Observations (no change required)

O1 — implicit monotonicity at two_tree_corollary.md:113–116. Concluding R(K_2,T_2) = n_2 from one avoiding coloring on n_2-1 vertices uses monotonicity of the Ramsey property in the host order. star_sharpness.md:107–108 states this explicitly; TTC leaves it implicit. The step is routine and the conclusion is correct, so this is not a defect; adding a clause such as "and to any smaller host, by restriction" would make the two pages symmetric.

O2 — "the last inequality" at star_sharpness.md:138. The paragraph at 138–140 concludes both parity cases but sits inside the H3 "Colour graphs when both orders are odd" and refers to "the last inequality" in the singular. The opening "In each parity case" forces the correct distributive reading, so the mathematics is unambiguous; a reader may need a moment. Promoting 138–140 to its own short paragraph outside the both-odd H3, or writing "each case's last inequality", would remove the friction.

O3 — section heading differs from the siblings. Both subject pages use ## Source and standing; theorem_1.md.txt:62 and tree_ramsey_corollary.md.txt:60 use ## Source and dependencies. The subject's heading is arguably more accurate for a page whose main burden is standing, and no rule fixes the heading. Recorded only.

O4 — desc slightly under-describes TTC. two_tree_corollary.md:3–5 says the page "Derives the two-colour Ramsey upper bound"; the page also establishes the exact endpoint values R(K_2,T_2)=n_2 and R(T_1,K_2)=n_1 (lines 106–121). One sentence cannot carry everything, and the desc is not misleading.

O5 — quoted fragment punctuation. “Assuming #548.” at TTC:130 and SS:165 where the source has Assuming #548,. Standard typographic practice; the quoted words are exact.

Integration conditions for the parent (outside the subject text)

I1 — the two pages must land together. two_tree_corollary.md:33 links star_sharpness and star_sharpness.md:36,168 link two_tree_corollary. Neither target exists in the baseline. Landing one without the other leaves a dangling wiki link and will fail wiki lint.

I2 — wiki update must run before the gate. Neither subject page carries the tool-owned frontmatter name: that every sibling has (theorem_1.md.txt:2, tree_ramsey_corollary.md.txt:2). This is correct authoring — docs/anatomy.md:360 assigns name to the tool — but wiki update --check, gate leg 6, will not pass until wiki update --path erdos has run.

I3 — E0547 is a newly linked destination with no managed block. Both pages carry **Bears on.** [[problems/ramsey_theory/E0547/_index|#547]], and ../../../../../wiki/problems/ramsey_theory/E0547/_index.md contains no <!-- BEGIN problem library links --> block in the baseline (I grepped it; the file ends at the Known Results row). Per AGENTS.md section "Incoming library links" and docs/tools.md:154–167, the incoming-library writer must be run with an explicit --problem E0547 and the same selection passed to erdos gate; the gate does not discover newly linked destinations on its own.

I4 — the source record's review sentence becomes ambiguous. _index.md.txt:115–117 says "All six written proof pages, including the Ramsey corollary, have passed independent mathematical review for this corpus." Once these two land, the folder holds eight written pages. The sentence remains literally true and both subject pages defensively disclaim inheritance (TTC:152–153, SS:179–181), so this is not a subject defect — but the source record should be amended to name the six covered pages, or to state that the two supplied two-color results are outside that review, so no reader infers eight-page coverage.

I5 — tree_ramsey_corollary's open sharpness remark is now partly answered. tree_ramsey_corollary.md.txt:71–72 says "the comments' additional claims of sharpness for stars require a separate lower-bound argument." star_sharpness supplies that argument for two colors. Both subject pages correctly limit themselves to saying the general-q boundary is unchanged, so nothing on them is wrong. The parent may wish to note on tree_ramsey_corollary that the q = 2 case is now supplied. Also, E0547.md currently says nothing about star sharpness; per docs/anatomy.md:111–113 a source addition that changes a problem page's mathematical account needs an authored explanation there.

8. Weakest steps, independently rederived

Per docs/verification.md:124–127, the three weakest steps and how they compose:

W1 — the integrality strengthening at two_tree_corollary.md:86–91. This is the only place where the conclusion improves on a direct summation, and it is where an off-by-one would hide. Rederived: n_i odd ⟹ n_i-2 odd; N = n_1+n_2-3 with both n_i odd ⟹ N odd; odd × odd = odd; an even integer 2e(G_i) bounded above by an odd integer is bounded by that integer minus one. Composition: the two strengthenings contribute -2 in total, and because n_1+n_2-4 = N-1 here (not N-2 as in the all-orders case), the right-hand side lands exactly on N(N-1)-2, one step below the left-hand side's N(N-1). The margin is exactly 2, so the argument has no slack to lose — and it does not lose any. Survives.

W2 — existence of the d-regular graph at star_sharpness.md:65–75 (odd d). This is the step that a construction proof most often gets wrong, because the perfect-matching chord x ↦ x+N/2 can collide with a circulant difference or be double-counted. Rederived: k <= N/2 - 1 strictly, so N/2 ∉ {1,…,k}; and N/2 ≡ -j would force j ≡ N/2, again excluded; and x+N/2 = x-N/2 in Z/N, so the chord contributes exactly one neighbor, giving 2k+1 and not 2k+2. Composition: the graph is used only through its degree, so the degree count is the whole interface, and it is exact. Survives.

W3 — the choice N = n_1+n_2-4 in the both-odd case (star_sharpness.md:116–128). The lower-bound host must be exactly one below the claimed Ramsey number, and here the claimed number is n_1+n_2-3, one smaller than in the other branch — so the construction has one fewer vertex to work with while still hiding both stars. Rederived: red degree n_1-2 is unchanged, but blue degree drops to N-1-d = n_2-3. The binding constraint is n_2-3 >= 0, which holds because both orders odd forces n_2 >= 3; and at n_2 = 3 the blue graph is 0-regular, which is still admissible. Composition: n_2-3 < n_2-1 with room to spare, so the blue star is excluded comfortably; the tightness is entirely on the existence side (d <= N-1), which holds with equality exactly when n_2 = 3. Survives, including at that equality boundary (n_1 = n_2 = 3: N = 2, d = 1 = N-1, red K_2, blue empty).

9. Strongest attack and why it failed

The strongest attack I could mount was on the both-odd branch's consistency between the two pages: if the upper bound n_1+n_2-3 were correct but the lower-bound construction only reached n_1+n_2-4 vertices for a different reason than claimed — for instance if the required d-regular graph failed to exist for some odd pair — then the equality claim in star_sharpness would be false while both pages looked locally fine.

I pursued this by checking the existence conditions across the whole both-odd family rather than at sample points. With N = n_1+n_2-4 and d = n_1-2: N is always even (odd+odd−4), so Nd is always even regardless of d's parity — the handshake condition can never fail in this branch. And d <= N-1 reduces to n_2 >= 3, which is implied by "n_2 odd and >= 2". So the construction exists for every admissible both-odd pair, with no exceptional case. The attack fails.

I also attacked the not-both-odd branch the same way: there Nd even is not automatic, and the page's two-case argument (n_1 even ⟹ d even; n_1 odd ⟹ n_2 even ⟹ N even) is exactly what is needed. I looked for a pair escaping both cases and found none — the disjunction "n_1 even or n_1 odd" is exhaustive, and the second case genuinely uses the branch hypothesis. The attack fails.

A third attempt targeted the n_i = 2 degenerate inputs, where theorem_1 degenerates to 2e(G) <= 0 and stars degenerate to single edges. Every degenerate case is handled explicitly and correctly on both pages (TTC:106–121, SS:142–158), including n_1 = n_2 = 2 where the host is a single vertex. The attack fails.

10. Premises consumed

• External: the sharp tree-free edge bound, theorem_1.md at the review baseline, statement at lines 15–24. Reading depth: claims checked — I read the full page (1–72) for the interface, and I did not recheck its proof chain, the linked rooted_word_bound and marked_cut_count pages, or the canonical PDF. Specialization used: applied once per color class, with t = n_1 and t = n_2, on hosts of order N >= 1. Interface: the inequality 2e(G) <= (t-2)N under absence of a non-induced copy. I take it at its recorded standing per the assignment and do not re-prove it.
• Internal to the unit: star_sharpness consumes two_tree_corollary for the upper bound only (SS:35–41). Acceptance order is therefore theorem_1 (given) → two_tree_corollary → star_sharpness, which is acyclic and matches the assignment's dependency order.
• Assumptions I made: none beyond the premise's recorded standing.
• Not consumed: the general-q corollary, the forum posts, the PDF, the pinned Lean source, and the source record's six-page review.

11. (f) Actual reading and exposure

Frozen-material files opened (allowed list only)

• File: ../assets/reviewed_v1_two_tree_corollary.md.txt; Lines read: 1–161 (whole file)

• File: ../assets/reviewed_v1_star_sharpness.md.txt; Lines read: 1–189 (whole file)

• File: theorem_1.md.txt; Lines read: 1–72 (whole)

• File: _index.md.txt; Lines read: 1–140 (whole)

• File: tree_ramsey_corollary.md.txt; Lines read: 1–81 (whole)

• File: E0547.md.txt; Lines read: 1–100 (whole)

• File: earlier forum capture; Lines read: 1–30 (whole)

• File: later forum capture; Lines read: 1–33 (whole)

• File: historical input manifest; Lines read: 1–281 (whole)

Additionally I ran shasum -a 256, wc -l, awk width scans and grep pattern scans over the two subject files and the four native snapshots; no other frozen material bytes were read.

Worktree files opened (read-only)

• File: AGENTS.md; Lines read: 1–146 (whole)

• File: docs/anatomy.md; Lines read: 1–377 (whole)

• File: docs/evidence.md; Lines read: 1–226 (whole)

• File: docs/verification.md; Lines read: 1–215 (whole)

• File: docs/math_authoring.md; Lines read: 1–74 (whole)

• File: docs/tools.md; Lines read: 1–191 (whole)

• File: organization-wide AGENTS.md; Lines read: 1–480 (whole; read in three passes — a 1–~30 preview, then 24–323, then 324–480)

• File: ../../../../../wiki/problems/ramsey_theory/E0547/_index.md; Lines read: grep for problem library links (no match) and tail -5, i.e. lines 96–100

I hashed but did not read as text (their content reached me only through the byte-identical snapshots above): ../../theorem_1.md, _index.md, tree_ramsey_corollary.md, and ../../../../../wiki/problems/ramsey_theory/E0547/_index.md.

I also ran width scans over the four native snapshot files to establish the corpus line-wrapping convention (finding D1), and listed the directory ../../ and the frozen two-page subject directory (filenames only, no content).

Read-only git commands

Read-only revision, object-presence and short-history checks confirmed the isolated baseline facts in §0. The native baseline object was absent from that review checkout. No write command of any kind was run. Nothing was modified, staged, stashed, checked out, restored or committed. The canonical repository was never touched.

Deliberately not opened

The author handoff, author reading receipt, author freeze manifest, author difference and author subject mapping; the prior source-triage directory whose locator appears in the historical input manifest; private coordination and operational directories; any other reviewer's or grader's records; the canonical PDF; any Lean source; and the network. No code from the frozen material or the repository was executed and no mathematical program was run.

Exposure disclosure

historical input manifest is a single JSON object in which the locators are interleaved with the author's own role, scope and limitation strings (for example, characterizations such as "Only external mathematical premise" at line 11 and "not an independent proof assessment" at line 241). Reading the file for locators necessarily exposed those strings. Per the assignment I took only the locators and disregarded every assessment phrase; none of them entered my mathematical reasoning, and each conclusion above rests on bytes I re-derived from the subject and native snapshots themselves. The same file names a private repository path at lines 85 and 110 and the prior triage directory at line 237; I opened neither. I judge this exposure immaterial to the verdict but record it so the grader can weigh it.

On 2026-09-18 a separate materiality grader (Claude Fable 5.1) re-read, as they stood on 2026-09-10T11:12:29Z, the standing paragraphs of the frozen subject (../assets/reviewed_v1_two_tree_corollary.md.txt lines 146-153, ../assets/reviewed_v1_star_sharpness.md.txt lines 172-181), the premise acceptance sentences in the commissioned context (_index.md lines 115-119, wiki/problems/ramsey_theory/E0547/_index.md lines 69-70) and the manifest strings quoted above, and ruled the exposure immaterial under the content test: none of that text states or implies the verdict on the two-tree bound or star sharpness, and the report's reasoning does not lean on it.

Independence

I am distinct from the author of the two subject pages and from any collaborator who constructed them. This context was fresh at the start of the review and had not previously built on the subject. No sibling verdict, prior triage report, advocacy, or reconciliation material reached me.

12. Verdict

PASS with exact corrections.

• two_tree_corollary.md: mathematics correct, formulation correct. Documentary corrections D1 (one occurrence, line 156), D2 (lines 6–7), D3 (lines 132–138).
• star_sharpness.md: mathematics correct, formulation correct. Documentary corrections D1 (three occurrences, lines 36, 168, 184), D2 (lines 6–7).

No mathematical or formulation defect was found in either page, so neither page fails. The unit's standing wording, warrant boundaries and scope disclaimers are accurate and, in several places, notably careful — in particular the refusal to inherit the source record's six-page review, the restriction of sharpness to two-color stars, and the preservation of the general-q boundary.