Grade

---

Judgment and accepted weight

Grade: pass, for the complete frozen cubic assertion examined in the settled report. The accepted mathematical verdict is refutation-failed. The report satisfies the whole-claim contract in docs/verification.md and its campaign adapter, including the mathematical work beyond the numerical checker.

The accepted weight is one complete independent whole-claim examination: the strict family inequality, the exact gamma value, the upper bound for every monic complex cubic, and both directions of the equality classification. All eight proof sections and the actual local deformation used from the proof of Eremenko–Hayman Lemma 5 are covered. This meets the tier-1 report contract at its stated external-source and arithmetic boundaries. It supplies neither a second independent review cycle nor a kernel proof, and it accepts no unconditional general Problem 824 conclusion. No corpus standing is edited by this private grading delivery.

The four failed endpoint-equality assertions in the auxiliary diagnostic have zero weight as successful calibration checks. The diagnostic remains a failed program. Its successful finite root comparisons and rejection tests retain their stated, limited weight. The required rectangle inequalities are established in the report's written argument independently of those failed assertions. No missing required mathematical coverage remains on this account.

The reviewer's initial import of the live runtime is ruled immaterial to independence, for the content-based reasons below. The source's false unnormalized escape sentence remains a useful adverse finding at its precise scope; it is not a refutation of the cubic statement.

Subject, identities, and reading depth

Grading date: 2026-10-15 UTC. The exact settled report is preserved as review/report.md, with its derivations, reproduction instructions, independent diagnostic, and exact diagnostic inputs. The settled report was completed on 2026-10-15 at 16:53 UTC. Its mathematical subject was frozen on that date at 14:29:00 UTC, and the supplied neutral extraction was prepared at 15:09:22 UTC. These are different events; the grade does not substitute the report's settlement for the mathematical freeze. The private identity record separately verifies the grading-packet identity, settled-report identity, original mathematical identity, and operating sources.

The preserved mathematical subject is:

The reviewer report, all three independent derivations, reproduction account, complete diagnostic code, and exact inputs were read and assessed in full. The mandatory verification standard and campaign adapter were read in full, along with the required corpus, evidence, authoring, and weaving instructions. The frozen operating snapshots matched the permitted checkout pages before reading. The original reviewer manifest and delivered copies agree, including the mathematical files and runtime support; the retained reviewer delivery is unchanged.

For runtime support, the checker/parser, package import surfaces, constants, and relevant wrappers were read. The twenty supplied possible live-import modules were compared byte for byte with their frozen counterparts, and their import-time statements were inspected statically. This is not a full audit of their unrelated repository-command implementations. The project specification and lockfile were retained and identity checked; no environment was installed from them. No PDF, original external book, source-tree checkout, live claim ledger, or linked research page was opened.

The historical audit examined the actual launched commission, seed and input mapping, visible incoming instructions, all recorded command batches, returned text establishing the essential complete readings, environment and exposure records, mathematical execution records, communication records, and settlement paths. Earlier startup records were checked as exact prefixes of the final records. Repeated headless-log text and ancillary inventories were used only for scoped corroboration, not represented as a second full textual reading. An opaque native compaction payload was not decoded; the visible inputs and actions before and after it remain available. The report and mathematical keepers were already written before that compaction and were unchanged during its subsequent final checks.

Original project mathematics is credited to Plasma AI. Eremenko and Hayman retain authorship of the external lemniscate results. Reviewer and grader are distinct fresh roles; both disclosed model gpt-6-astra. The grader is also distinct from the author, preparer, and commissioner, and did not participate in constructing the subject or its review.

Eligibility and exposure rulings

Reviewer eligibility passes. The launched assignment charged refutation of the entire statement and supplied the neutral mathematical extraction, execution support, and operating rules. It did not supply author plans, prior verdicts, or mathematical advocacy. The actual incoming step commissions match the original seed after the ordinary worktree-path substitution. The initial preflight contained only a request for an acknowledgment. The visible record contains no substantive incoming reconciliation or encouragement.

Empty subscription tables support the wiring facts at startup and later boundaries. More decisively, the command history shows the commissioned reads and the reviewer's own continuation, derivations, code, and outputs; no inbox/feed read, synchronization, source-tree extraction, web search, subagent, or additional mathematical execution appears. Outbound disclosures are not incoming mathematical steering. A supplied record of the coordinator reading one of those disclosures is the coordinator's action, not a reviewer inbox read. The automatic settlement touched only the reviewer's own continuation and delivery paths. A completion label is not used as mathematical or independence evidence.

Initial live-package import: immaterial exposure. Before mathematical execution, an identity-only Python invocation on standard input selected the checkout's live tools package ahead of the intended frozen import path. The recorded command and output establish what happened; the report accurately discloses it. The reviewer immediately reported the exposure and repeated the probe with safe-path mode. Both mathematical processes subsequently printed the correct frozen-package identity.

The ruling does not rest merely on the later correction. All twenty supplied candidate implementation files are byte-identical to permitted frozen runtime files. Inspection of their import surfaces finds definitions, constants, regular expressions, and package initialization, with no invocation reading mathematical corpus content. The identity probe displayed package/interpreter locations and versions, not a theorem, proof, prior verdict, or advice. No mathematics in the report could only have come from those excluded imports, and neither the mathematical attack direction nor the refutation charge followed them. The operating correction followed the import-path error alone. On this content test the exposure does not require a fresh reviewer. It must remain disclosed together with this ruling when the report is filed.

Native configuration and automatic activity. The preserved final native configuration retains the entire original prefix and adds only the permitted project-trust leaf. No agent command edited it. The recorded automatic setup and final private settlement are operating activity, not independent proof warrants. The settlement path inspection found no mathematical corpus edit.

Grader's own isolation. The grader used only the individually commissioned packet and operating inputs, plus its own new continuation and outputs. The historical report and transcripts are expressly allowed grading evidence; they are not a second blind review of the mathematics. Own boundary and resumed subscription records are empty. No excluded exposure has been identified in this grading. An administrative file-write tool failure and its recovery are preserved privately; neither introduced mathematical content or another check. No own exposure is being excused through the reviewer ruling above.

A final custody comparison found the supplied packet and static operating pages unchanged, but two retry controls in the grader's own operating configuration differed from the accepted launch snapshot after resumption. The difference and the effective resumed time allowance were disclosed to the operator; no configuration edit, further launch, or additional mathematical execution was made by the grader. Operator acceptance of that operating change remains an intake obligation. This grade of the reviewed report does not self-approve the grader's operating configuration.

Exact restatement and clause coverage

Let $\ell(p)$ denote Euclidean arclength of the geometric set $\{z\in\mathbb C:|p(z)|=1\}$. For every real $0<a\le 2^{-1/3}$, with the nonnegative real square root $c=\sqrt{1-4a^6}$, the statement is

$$\ell(z^3+3a^2z-c)<L_0, \qquad L_0=\ell(z^3-1)=\frac{2\pi\Gamma(1/3)}{\Gamma(2/3)^2}.$$

For every monic complex polynomial of degree exactly three, $\ell(p)\le L_0$, and equality holds exactly for

$$p(z)=(z-z_0)^3-w, \qquad z_0\in\mathbb C,\qquad |w|=1.$$

The report preserves these quantifiers, strictness, monicity, complex domain, and geometric length convention. It includes the endpoint $c=0$, excludes $a=0$ from the strict assertion, and treats the latter as an auxiliary equality case. It does not replace geometric length by a ramification multiplicity. The family bound, constant evaluation, global bound, and equality equivalence all receive pass as coverage rulings within the single whole-contract grade; none is being substituted for the whole statement.

Contract and mathematical coverage

The eight proof components have the following substantive rulings:

Strongest attack: the actual deformation

The report correctly refuses the invalid inference that existence of one maximizer with critical values on the unit circle classifies all maximizers. Its first derivation examines the stronger local fact used in the frozen proof. For a maximizing monic cubic $p$, take an off-circle critical value $v$ and a disc separated from the other critical value and the unit circle. The perturbation $\phi_\lambda(z)=z+\lambda\Phi(z)$ gives the coefficient

$$\mu_{\phi_\lambda\circ p}(z) =\frac{\overline{p'(z)}}{p'(z)} \frac{\lambda\Phi_{\bar z}(p(z))}{1+\lambda\Phi_z(p(z))}$$

almost everywhere. Properness gives compact support, and small complex parameters keep its norm below one. Through the explicitly identified normalized Beltrami theorem, the straightening has $\psi_\lambda(z)=z+o(1)$ at infinity. Consequently $\phi_\lambda\circ p\circ\psi_\lambda^{-1}$ is an entire monic cubic. The normalization is essential: arbitrary scaling would change the optimization problem. Local topological degree preserves the two simple critical points and moves only $v$.

The support stays away from the original lemniscate. The report checks both $E(p_\lambda)=\psi_\lambda(E(p))$ and conformality on a common neighborhood. Analytic parameter dependence and local uniform boundedness justify integrating $|\psi_\lambda'|$ as a subharmonic function of the parameter. Global maximality then forces this length to be constant near zero. These are the essential deductions from the source construction, not an appeal to Lemma 5's statement alone.

The continuation objection is also answered in the retained report. Along an arc avoiding the other value $w$, depressed coefficients satisfy $b=(t+w)/2$ and $|u|^3=|w-t|/4$. Their boundedness and source coefficient continuity make the realizable-maximizer set relatively closed as well as locally open. They also supply the terminal limit. A second move, if needed, ends with two distinct unit critical values, contradicting the proved strict family bound. This is sufficient without assuming a global continuous choice of cubic roots or completion of infinitely many local moves. The grader finds this attack answered in the settled report; no new proof repair is supplied by the grade.

Uniform integral warrant and failed stopping-rule attack

The small-parameter derivation checks

$$x=\frac{a^3(y^2-4)}{c+\sqrt{1-a^6y^2}}, \qquad |d\sigma|=\frac{dy}{\sqrt{1-a^6y^2}},$$

and accounts for the head, the window about $y=2$, both regular intervals, and the entire omitted tail. In particular, the tail bound already includes both tails, whereas conjugation doubles the head, window, and regular terms. The written rational overestimate proves the conservative tail constant without a numerical sample. Negative upper bounds for $I(a)$ on full balls combine with $a>0$ to give strictness arbitrarily close to zero; a positive lower cutoff on $a$ has not been substituted.

The signed rectangle rule was checked directly against its code. If an actual integrand lies in $[v_-,v_+]$ and its nonnegative width in $[d_-,d_+]$, an upper scalar bound is $v_+d_+$ for $v_+\ge0$ and $v_+d_-$ otherwise. The lower rule uses $v_-d_-$ for $v_-\ge0$ and $v_-d_+$ otherwise. Ball products and sums enclose those scalar bounds; the appropriate final endpoint supplies the inequality. This argument remains valid with outward rounding of every constructor radius.

A finite enclosure accepted at an integration depth limit therefore remains a rectangle bound; it need not meet the requested subdivision tolerance. Nonfinite exhaustion raises an error. An unresolved parameter-cover box at its threshold records failure instead of acceptance. The program's floating subdivision heuristic changes efficiency, not the enclosing inequality. These facts supply the required stopping-rule warrant independently of the four failed ideal-endpoint predicates. The report does not claim that a returned lower-bound ball is wholly a scalar lower bound: its conservative stated control uses a scalar below the lower endpoint.

Premises and source boundaries

The grader independently derived the premise census from the frozen statement, proof, and consequences on 2026-10-15. There is no consumed native L-theorem premise. The self-reference identifies L144 and definitions supplied in the extraction. The normalization-sketch link is provenance for algebra that is actually reproduced. No live standing read was authorized, necessary, or performed. There is no pending native grade whose prospective standing is being substituted for a current one.

Ordinary reliance on these identified source theorems is permitted by the verification standard; it does not require an infinite reconstruction of their literature. The grade does not strengthen the actual reading depth. In particular, the local Lemma 5 deductions have been checked through the stated Beltrami boundary, rather than pretending that the unread book proofs were checked. Lemma 6, the source's general linear-in-degree bound, and its quadratic extremizer result are not premises of the cubic argument.

Adverse source finding. The supplied Lemma 4 proof says that length tends to zero when the entire unnormalized root vector tends to infinity. The report's witness is valid: for $p_R(z)=(z-R)^3$, all three roots escape while $E(p_R)$ is a unit circle of length $2\pi$. This refutes that sentence. It does not refute coefficient continuity or existence modulo translations; the preceding source argument concerns large root diameter. The grade retains the witness and the report's partial source-proof qualification, and does not claim a repaired or fully verified proof of Lemma 4. The cubic proof consumes the identified theorem interface, not the false unnormalized asymptotic.

Reproduction and exact diagnostic weight

The grader reconstructed the individually permitted delivery files with their original relative hierarchy inside private scratch and used the child-local installed interpreter. The only mathematical path changes are that relocation and the explicit import path to the preserved runtime. The retained identity wrapper and both mathematical programs are unchanged. The inherited virtual environment and project-selection variables were cleared; safe-path mode, explicit interpreter selection, one-thread settings, and the correct runtime were verified in the executing processes. CPython 3.14.6 and python-flint 0.9.0 match the recorded mathematical environment. The supplied project/lock files were not used for installation.

Exactly two mathematical reruns were made after reading the code and stating input sizes and time limits. The full run had a 900-second limit; the finite diagnostic had a 60-second limit. Both completed. Administrative identity and historical-record parsing did not execute additional mathematical code. Reproduction details preserve the command substitutions, input scope, outcomes, and the distinction between raw private receipts and portable mathematical inputs.

The actual discrepant endpoints are absent from both the original and repeated diagnostic output. Neither their numerical sizes nor the cause of each failure has been measured here. The documented possibility of outward radius rounding explains why exact ideal equalities are unjustified expectations; it is not reported as a measured diagnosis. The independent scalar rectangle argument already establishes the required inequalities for outward enclosures, so no missing required warrant is being excused as an optional probe.

The two root-generation routes share the same arithmetic library. They are structurally different for the suspected Cardano branch/normalization error, not independent implementations of ball arithmetic. The review is a single claim, not a batch or commissioned second cycle. Its finite alternative route has its actual scope; repeated executions by the grader add no new independent review cycle. Exact failing inputs and predicates remain in the unchanged code and JSON, including the indeterminate integrand, $\beta=0$ window, unresolved $[0,2^{-41}]$ cover, singular $2i$, and wrong origin factor. Neither program reads cached mathematical success.

Checklist adjudication

The following rulings assess the report's explicit verdicts against the whole contract, rather than accepting the presence of table rows as evidence.

Canonical failure modes

Named patterns

Erdos-specific checklist

The neighboring discipline concerning correlated statistical samples is inapplicable. The evidence-coverage discipline is satisfied by the explicit division between certified integral bounds and the written reductions and equality proof; no computation is asked to fill an unproved bridge.

Retained evidence and remaining obligations

The unchanged reviewer report, three derivations, diagnostic, exact JSON, mathematical extraction, runtime support, API capture, and executed wrapper are retained together under the relative paths above. The source-sentence counterexample and failed calibration predicates remain useful keepers. Private primary history, identity comparisons, reconstruction map, exact commands, raw outcomes, and grading-administration scripts are retained separately. They are not public mathematical inputs or substitutes for reruns.

The coordinator owns durable filing of the grade and review at wiki/theory/polynomials/L144_cubic_lemniscate_length/evidence/verify/, with all required exact inputs accessible from an ordinary clone. Filing must keep the exposure disclosure and this materiality ruling in the record and owning standing account, preserve the diagnostic's nonzero outcome, and respect the source-text publication boundary. Relocating the package requires corresponding path maintenance, not changes to mathematical bytes or retrospective claims about what was examined. Private handoff alone does not perform that filing.

There is no outstanding required mathematical coverage obligation for this frozen report after this grade. The source/PDF/book/library limits above remain limits of what is accepted; they are not silently declared completed. Measuring or correcting the four auxiliary endpoint tests would require a separately retained diagnostic if the coordinator wants that additional coverage. It is not a condition of this pass. Changes to the frozen statement or substantive proof/code require the standing treatment prescribed by the verification standard; this grade covers no later repair. The coordinator owns intake and filing, including disposition of the disclosed resumed operating change. The campaign owner inherits any unfinished administrative obligation if that work expires.