Independent grade of the E1033 whole-conclusion examination

Grade: pass for the complete frozen existence assertion and its negation of the displayed catalog lower bound under the supplied minimum convention. The accepted mathematical verdict is refutation-failed. The original report supplies the required attacks and argument; this grade does not replace an incomplete examination with a new proof.

The last three rulings do not void the complete mathematical examination: none is used as an unsupported premise of its verdict. No change to the frozen statement, proof, checker, inputs or report is required for the two mathematical passes. This grade edits no corpus standing and supplies no Lean warrant.

Subject and independence

Role: independent grader, distinct from the subject's author, collaborators and original examiner. Disclosed model: gpt-6-astra, also the original examiner's disclosed model. Grading date: 2026-10-15 UTC. The grader began in a fresh context without prior involvement in the subject or original review. Model identity alone establishes neither independence nor dependence. Project mathematics is credited to Plasma AI.

The graded report is wiki/research/graph_ramsey_interfaces/butterfly_degree_sum_certificate/evidence/verify/whole_conclusion/report.md, the unchanged whole-conclusion examination, examined in full. It records examination on 2026-10-14 and 2026-10-15 UTC. Its delivered package was frozen on 2026-10-15 at 00:10:12.871573 UTC. The mathematical subject was frozen on 2026-10-14 at 23:31:08.459536 UTC. The subject manifest maps these prepared extracts to the source paths:

The complete integer recount, control driver, control cases and deliberately false expectation were read and executed. The original thirteen-file delivery is preserved unchanged. Archive filename restoration changed no source bytes. The prepared extracts, whose omissions are marked [cut], delimit the mathematical subject. This grade does not certify excluded source-page text or its current correspondence with these snapshots.

The grading inputs also included the actual launched assignment, original operating manifest and accepted settings, steps and scripts, reviewer continuation, primary execution and isolation receipts, full retained command log, and terminal records. The complete supplied packet was assessed: repeated generated control artifacts were checked against their exact inputs and outputs; unused bundled skill assets received identity and role assessment, not mathematical reliance. The private coverage inventory distinguishes these depths rather than describing every file as a newly read mathematical source.

Both full verification contracts, docs/verification.md and campaign/guides/verification.md, were applied. Their admitted original instruction versions agree with the current versions. The relevant original and current corpus-law provisions also agree. Required portions of README.md, docs/anatomy.md, campaign/AGENTS.md, the research reading rules, docs/evidence.md, docs/math_authoring.md, docs/weaving.md and docs/approach_vetting.md supplied operating rules only. The enumerated runtime imports supplied execution infrastructure; source inspection stayed within tools/__init__.py and tools/core/harness.py.

The original operating identity receipt precedes the recorded operating-page reads. Its subject identity and empty subscription checks precede mathematical reading. The launched configuration specifies a fresh, blind, sealed review, no synchronization and no descendants. Recorded later checks and terminal records agree. The admitted terminal native settings retain ordinary native instructions, disable memories and web search, and agree with the accepted model and effort. The operator's pre-start native-settings check is part of the accepted operating record; the grader does not claim to have independently read an excluded pre-start configuration.

The retained command log is command and narrative evidence, not an exhaustive native conversation transcript. It records shell invocations and some writes, but does not retain every native edit operation or every command's output. Execution receipts, preserved sources, final package identities and terminal records therefore matter separately. Preliminary package receipts describe earlier report or keeper versions and are not treated as checks of the final delivered bytes. The final freeze identifies the delivered bytes; the fresh replay below checks those bytes directly.

Within that coverage, the original record shows the permitted extracts and operating reads, mapped executions, and an outbound terminal notice. No excluded mathematical reading, incoming advocacy, synchronization, outside search or delegation is identified. The report explicitly declares no such exposure. Empty subscriptions support this account but do not prove it alone. The observed iteration-end automatic persistence changed only the examiner's own continuation and plan; under the adapter this is engine behavior, not a reviewer commit or a verification gate. No content exposure requiring a materiality exception or a fresh original examiner was found.

The grader likewise checked empty subscriptions before mathematical reading, verified the admitted identities, and used only the frozen grading packet and allowed operating context. There was no incoming feedback, excluded source reading, web access, synchronization or delegation. Two procedural facts remain disclosed. One group of four and three groups of three brief, allowed read-only operations exceeded the commissioned concurrency cap of two; these were reported and subsequent operations obeyed the cap. They included no mathematical execution or excluded content. A discrepancy between operating time-limit reports was also disclosed, and work used the shorter limit. The private records retain the commands, outputs and corrections. These facts are not concealed by the pass on the original examiner's report, and no additional authority or exposure clearance is inferred from them.

Restatement and assignment audit

For each integer $n\ge10^{13}$, the assertion requires a finite simple undirected graph of exactly order $n$, more than $n^2/4$ unordered edges, and degree sum strictly less than $183n/125$ on every triangle. Degrees are ordinary vertex degrees and isolates are allowed. Under the supplied convention, $h(n)$ is the minimum over that graph class of the maximum triangle degree sum. The consequence is its strict upper bound and the failure of $h(n)\ge(2(\sqrt3-1)-o(1))n$; it is not an evaluation of the optimum.

The launched assignment satisfies all seven neutral assignment items:

The original assignment's own accepted operating settings govern its conduct; different time settings in the grading commission are not retroactively applied to it. Operating examples and installed capabilities supplied no extra reading or delegation authority. This grading is not offered as a second-cycle mathematical review or as an additional independent computational route.

Weakest steps and strongest attacks

Finite realization and graph conventions

The report tests the supplied construction rather than inferring a graph from rational densities. With $M=10000$, weights $(w_A,w_B,w_C,w_D,w_E)=(4273,2471,709,1274,1273)$, $q=4352$ and $r=3011$, its part $i$ is the tagged finite set $\{i\}\times[ w_i ]\times\mathbb Z/M\mathbb Z\times[s]$. All factors are positive integers, the weights sum to $M$, and total order is $N=M^2s$. This is the frozen labeling made explicitly disjoint.

Complete pairs are $AB,BC,AD,AE$. For $AC$, part names choose the difference $z_C-z_A$ once; membership in $[0,q-1]$ defines an unordered pair. A fixed $A$ endpoint has $q$ distinct allowed residues in $C$, each with $w_Cs$ vertices. A fixed $C$ endpoint has residues $z_C-u$ for the same $q$ distinct values of $u$, hence $qw_As$ neighbors. The $DE$ argument uses $r$. No within-part edges or repeated pair classes occur, so the graph is simple and undirected.

The report's strongest attack is substantive: $z_A=9999,z_C=0$ gives an allowed forward difference $1$, but reversed difference $9999$ is not in the interval. Requiring both orientations to pass the same test would break the calculation. The frozen rule instead names the parts, so reverse neighborhoods use the negated interval and retain the same cardinality. The attack fails against the actual rule. The phrase about equal degrees on each side of $DE$ means constancy within each side: the opposite-part counts are $3011\cdot1273s$ and $3011\cdot1274s$, respectively. The displayed formulas already make that distinction; the report does not silently replace equal raw degrees across unequal sides with a different construction.

Independent recalculation gives degree coefficients $d(v)=sD_i$ of

Counting each supported edge from one endpoint part gives

The report's six edge contributions reproduce this value without using its degree-sum identity. Independent parts rule out repeated-part triangles. The ten part triples leave exactly $ABC$ and $ADE$; both occur, for example with all residues zero and the other coordinates one. This verifies the maximum over actual triangles, not merely over a formal support graph:

The maximum is exactly $T_2N$. Neither a factor of two nor a residue correlation supplies a counterexample.

All orders and strict inequalities

Put $U=M^2=10^8$ and $n_0=10^{13}$. Integer division gives $n=sU+t$, $0\le t<U$, $s\ge100000$, and $N=sU>n-U$. Exactly $t$ isolates produce order $n$. Because $n>U$, the quantities being squared are positive, and

This defeats the attempted failure at the largest remainder $t=U-1$; at multiples of $U$ the first strict comparison also remains strict. The cleared numerator $4S(n_0-U)^2-M^3n_0^2=563113137168315440000000000000000$ is positive. Monotonicity on the positive domain, not sampled values of $n$, extends the cutoff inequality. Isolates preserve all old degrees and triangles, so every triangle still has sum $T_iN\le T_2N<(183/125)N\le(183/125)n$. There is no exceptional order or lost strict inequality in the report.

Minimum, asymptotic consequence and reconstruction boundary

The report checks the supplied definition's domain. For a triangle-free eligible graph, each edge $uv$ has disjoint endpoint neighborhoods, hence $d(u)+d(v)\le n$. Summing gives $\sum d(v)^2\le ne$; the squared-difference inequality $(\sum d(v))^2\le n\sum d(v)^2$ then gives $4e^2\le n^2e$, contradicting $e>n^2/4$. There are finitely many labeled graphs and the complete graph is eligible for $n>2$. Thus every relevant triangle maximum and the minimum over graphs exist. That minimum is exactly the largest threshold valid for every graph in the class.

The constructed graph therefore gives $h(n)<bn$ with $b=183/125$. Since $433^2=187489<187500=12\cdot125^2$, positive square roots give $b<c=2(\sqrt3-1)$. For $\delta=c-b>0$, any real error $\varepsilon(n)\to0$ eventually has $|\varepsilon(n)|<\delta/2$. The proposed bound would then give $h(n)/n\ge c-\varepsilon(n)>b$, a contradiction. Signed or oscillating errors do not avoid this. Equivalently, $\limsup h(n)/n\le b<c$ excludes the proposed lower asymptotic bound.

Reconstruction ruling: pass. The frozen argument already supplies the finite labels, edge rule, both directional counts, support classification, floor construction, isolates, monotonicity chain and fixed-gap consequence. The report expands those deductions; it does not supply a missing realization theorem or a new padding construction. Its triangle-existence argument is an explicit elementary check of the supplied minimum convention, as the report contract requires for definitional material. It imports no established external theorem. Moreover, the exhibited graph directly defeats the proposed universal degree-sum guarantee even before that guarantee is expressed as a minimum. Checking this definition's domain is not a repair of a failed central construction. No essential omitted lemma was identified. The broader request to estimate $h(n)$ receives an upper bound only; no optimality conclusion follows.

Computation, witnesses and reproduction

The original execution claim has primary support: recorded invocations, return codes, generated case inputs, outputs and final source identities. The final keeper receipts include the controls added before delivery; earlier runs alone are not used to establish final-source execution. The grader separately reran the exact preserved sources and relative asset layout on 2026-10-15 UTC. The checker and integer recount returned zero; the full control driver returned zero; its deliberately false specification returned one in both ordinary and optimized Python, identifying the incorrect expected rejection. Required failure labels were checked as well as exits, so an unrelated crash is not counted as a successful negative control.

The frozen checker accumulates normalized degrees and density together using exact fractions. The recount derives five integer degree coefficients and six unordered-edge counts separately, then clears denominators. This is a different route through the normalization and edge-counting interface and addresses conversion and factor-of-two errors. It is not a second independent proof of graph realization, and no such weight is assigned to it. The analytic arguments provide the universal graph and padding conclusions.

The exact rejection witnesses are retained in the controls: change $w_A$ to $4274$; change $q$ to $10001$; add $BD$ with numerator $1$; change $q$ to $4351$; use comparison bound $14639/10000$ or $3/2$; or use cutoff $10^{12}$ or $10^8$. Each makes both arithmetic programs exit one, in ordinary and optimized Python, at a named relevant obligation. The supplied input makes both exit zero. The lower-cutoff rejection is a failed sufficient certificate, not a disproof of graph existence at that cutoff. The control driver's own failing input incorrectly expects rejection of the supplied valid input.

Two contrary findings are retained without softening:

• The handshaking diagnostic is forced by the code's shared pair updates: each density contribution is added twice to the weighted degree sum. No well-formed parameter choice breaks that identity while these updates are unchanged. It has no independent checking weight. Direct finite counts and the separately derived recount supply the needed evidence.
• Negating the denominator and every part and pair numerator preserves all rational weights, densities and $M^2$. The frozen checker consequently returns zero, while the proposed negative integer construction parameters are invalid. The recount rejects them at the positive-integer denominator check. The supplied positive data are explicitly verified; the witness refutes general construction-domain validation, not the fixed theorem.

Both examinations executed mapped copies, not the canonical source-path command printed in the evidence description. The private execution record retains the exact mapping and actual commands. No path substitution is called verbatim canonical execution. The ordinary-clone commands below describe reproduction after adoption of the retained layout, rather than an action already performed at those public destinations:

export PYTHONDONTWRITEBYTECODE=1
export PYTHONPATH="$PWD"
export OMP_NUM_THREADS=1 OPENBLAS_NUM_THREADS=1 MKL_NUM_THREADS=1
export VECLIB_MAXIMUM_THREADS=1 NUMEXPR_NUM_THREADS=1
certificate_dir=wiki/research/graph_ramsey_interfaces/butterfly_degree_sum_certificate
review_dir="$certificate_dir/evidence/verify/whole_conclusion"
nice -n10 uv run --no-sync python \
  "$certificate_dir/evidence/assets/review_subject/evidence/main.py"
nice -n10 uv run --no-sync python "$review_dir/integer_recount.py"
nice -n10 uv run --no-sync python "$review_dir/adversarial_controls.py" \
  --scratch tmp/e1033_whole_conclusion_controls

The following two commands must each exit one, reporting the incorrect expectation for the supplied input:

nice -n10 uv run --no-sync python "$review_dir/adversarial_controls.py" \
  --controls "$review_dir/assets/wrong_expectation.json" \
  --scratch tmp/e1033_whole_conclusion_wrong_expectation
nice -n10 uv run --no-sync python -O "$review_dir/adversarial_controls.py" \
  --controls "$review_dir/assets/wrong_expectation.json" \
  --scratch tmp/e1033_whole_conclusion_wrong_expectation_optimized

These use the repository's existing local environment and installed tools package under pyproject.toml and uv.lock. They require no installation, source search or generated mathematical witness. The driver executes fresh checks and writes diagnostic outputs only to its explicit scratch path. Cached outputs and administrative identity checks are not mathematical gates. The integrator owns comparison with current canonical files, any required canonical replay and durable filing.

Complete contract and checklist grades

Every required report part is present in the unchanged report:

The Erdos checklist receives the following independent item-by-item grade:

All seven neutral canonical modes are also adjudicated:

All twelve named patterns receive a disposition:

The neighboring statistical-discipline requirement is inapplicable: there is no statistical inference or claim of independent samples. The evidence-coverage discipline is met by separating fixed arithmetic, general input validation, finite realization, infinite-order transfer and actual execution scope.

Premises, verdict and limits

The original premise finding at verdict filing on 2026-10-15 UTC was checked again for this grade on 2026-10-15 UTC against the entire supplied argument, scope and consequence text, catalog convention, code and inputs. No native L-claim or source-owned theorem is consumed as an established premise. There is therefore no tier, stale mark, pending dependency grade or external-source proof interface to certify. No ledger or additional literature was read. Finite counting, modular translation, integer division, ordered-field inequalities and the positive square root suffice. The catalog is the target and a verified definition, not an assumed lower-bound theorem.

The unchanged report meets the complete contract and supports refutation-failed for both mathematical scopes. Its independence is supported by the actual assignment and primary evidence, with the transcript coverage limit explicitly retained. The checker limitations remain contrary findings with exact witnesses and the limited weight specified above.

This grade covers the preserved subject and report only. It does not grade later repairs, current canonical correspondence, the optimum of $h(n)$, a stronger constant, historical novelty, a complete solution to the broader estimate request, third-party acceptance, or Lean work. Filing the unchanged report, this grade, keepers and exact inputs at their named durable paths is necessary before citing them as an ordinary-clone record; a private handoff alone is not that filing. No corpus tier or standing is changed here.