Independent grade of the all-sequence ratio report

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Decision: PASS. The unchanged whole-statement report meets the complete report and independence contracts for its commissioned ratio theorem and exact extremal consequences. Its mathematical verdict is refutation-failed. The mathematical finding and the validity of the review warrant were assessed separately; both survive this grading examination.

Subject and date. This grade examines the report frozen on 2026-10-15 at 11:41:35.556570 UTC, together with its nine unchanged mathematical inputs, whose extraction was frozen on 2026-10-15 at 09:49:19.072889 UTC. The report is retained verbatim at report.md, and its exact inputs remain at the nine relative ../../assets/whole_statement/returned/reviewed/subject/ paths it links. Its original operating, launch, communication and complete native records were supplied as private independence evidence. Examination date: 2026-10-15 UTC. Exact private custody identities are recorded separately; the older mathematical-source identity was not mistaken for the later report's identity.

Grader and eligibility. An independent grader, disclosed model gpt-6-astra, worked in a fresh context distinct from both the subject author and the original reviewer, with no participation in their work or prior reliance on this subject. The original reviewer also disclosed gpt-6-astra. Same-model use is not treated as shared context or as proof of independence. The grader remained within the commissioned packet and operating read set. No excluded exposure, substantive steering, author repair, or additional review was consumed. This is grading of an existing frozen report, not a replacement review or a repair of a missing contract part.

The grade warrants reliance on this exact report for those frozen surfaces. It makes no public tier or status change and does not accept the catalog problem as solved. Durable filing and any eventual standing assertion belong to the authorized integrator. No later repair or stronger statement inherits this grade.

Scope, restatement and reading depth

The target is the following statement. There is an absolute $r_0$ such that for every integer $r\ge r_0$ and every infinite sequence $X$ on $\mathbb R/\mathbb Z$, allowing repeated terms,

$$\limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}} \ge 1+\frac{\log r}{200r}.$$

For $n\ge r$, sort the first $n$ labels cyclically, breaking position ties by arrival index, and use lifts $y_{i+n}=y_i+1$. The minimum and maximum are over all $n$ spans $y_{i+r}-y_i$. These are clockwise lifted spans, not shortest circle distances. Counts retain multiplicity; arcs are half-open $(x,x+\ell]$ with $\ell<1$. Logarithms are natural unless a base is written. The spans sum to $r$, so $M_n^{(r)}>0$, and the prescribed ratio when $m_n^{(r)}=0$ is $+\infty$.

Here $\mu_r$ is the infimum over all such sequences of their extended-real limsup ratios. The assigned lower bound and divergence for $r(\mu_r-1)$ follow with the same absolute $r_0$. The report preserves every one of these quantifiers and conventions. It does not claim the assertion for every small $r$, a uniform starting time over all sequences, an attained infimum, an optimal coefficient, or an exact value of $\mu_r$.

The entire 742-line report and all nine excerpts were read, not sampled. Their repository source paths and supplied ranges are preserved in its “Exact subject and reading boundary” table. In particular, the complete finite-prefix proof and the extraction interface are within this grade's scope. The source paths there are locators; no canonical mathematical page was opened. The delivered excerpts and original mathematical inputs were compared byte for byte. No text written after the report freeze was substituted.

Both docs/verification.md and campaign/guides/verification.md were read in full. Required README, campaign instructions, anatomy sections, and research Reading were read. Evidence, math-authoring and weaving guidance were also read in full. Operating identities were checked before reliance on their contents. Original reviewer operating files were treated as evidence of that review's instructions, not as live instructions for the grader.

The complete original-record inventory was inspected, including the launch and final command results, reviewer continuation, seven restored primary files, both native records and the final automatic-commit paths. The actual received plaintext, read/action calls, relevant results and lifecycle evidence were examined; duplicated source outputs were compared with the supplied originals. Output truncation was checked explicitly. Raw originals, exact custody facts, and detailed record locators remain private. Encrypted internal fields were not decoded or claimed as plaintext reading. They are retained with the complete original input history, rather than replaced by a derived summary.

Independence and boundary adjudication

Eligibility: PASS. The original launch establishes a fresh examination with blind, sealed, local, no-sync settings and no descendants. The actual native record contains ordinary operating instructions and the three commissioned step prompts, with no author's construction context, prior review, or suggested favorable outcome. The step commissions match the configured charter after path rendering. The separate readiness check had only a content-free request and response. No additional substantive incoming message appears in the supplied native record. The model disclosure agrees with the accepted settings and actual execution records.

Actual exposure and permitted actions: PASS. All recorded read/action calls were inspected as data. Mathematical reads are confined to the original nine excerpts and manifest; other reads are allowed operating material and the reviewer's own continuation or output. The native record supports the reported absence of inbox/feed reads, outside search, canonical mathematical reads, original papers, ledger use, author-context imports, synchronization, subagents, manual commits, repository batteries and Lean execution. The report was authored in the reviewer's permitted output area without modifying its mathematical inputs. Its recorded candidate and final identities agree with the unchanged returned files. This ruling relies on original records, not the report's self-assertion or empty subscriptions alone.

The record contains a compaction. Its replacement-history plaintext messages are exact copies of earlier received messages; its internal encrypted item is not claimed to have been decoded. The original preceding history remains available and was used for the incoming-material and action audit. No external source or steering channel is shown by that compaction. This access limit does not substitute a summary for primary evidence or create a demonstrated contamination.

Owner-only inbox refusal: access limitation, not a failed warrant. The parent's actual final attempts to inspect the reviewer's private inbox were refused as owner-only. They returned no inbox content. The inbox is therefore not asserted empty. Empty subscription tables, the available single outbound delivery record, zero recorded replies and an empty outbox would not, by themselves, establish that no incoming message existed.

The decisive distinction is consumption. The supplied complete native record shows what entered the review and what the reviewer read or invoked. It contains neither an inbox read nor substantive message injection, unsealing, or another excluded ingress. The parent refusal after delivery does not show that inaccessible content was consumed. On that positive primary record, the grader finds no material exposure and no essential independence obligation left unmet by the unread inbox. This is the grader's own eligibility and materiality ruling; no commissioner ruling or waiver is assumed. The judgment does not certify the inaccessible contents and would require reassessment if contrary primary exposure evidence were established.

Under the contract's content test, no verdict-bearing element or attack direction requires an excluded input: the mathematics is accounted for by the frozen subject and the report's explicit deductions, and the refutation charge remains unchanged across the recorded steps. This supports the ruling about the available record; it does not infer unseen inbox contents.

Automatic continuation commit: PASS. The original final commit's 19 paths consist exactly of the reviewer's own continuation, the report and nine retained inputs, and its private preparation, identity and delivery records. All lie in that reviewer's own memory or plans. The separate earlier configuration/input commits are commissioner preparation, not reviewer changes inferred from a displayed identity. The adapter's explicit exception for the loop's automatic continuation/plan commit therefore applies. No path outside that exception needs an eligibility excuse.

No exposure disclosure was required by the observed record. The limits above are disclosed rather than silently converted into claims of complete access. They do not invalidate the independence warrant for the unchanged report.

Complete report-contract assessment

The report has the following identifiable parts. The rulings assess their substance, not merely their headings or row counts. The mathematical checks used in grading are retained in three derivations.

For the extremal conclusions specifically, the report's restatement displays both consequences, Weakest step 3 gives their order-theoretic derivation, and the attack on separately optimized numerators and denominators addresses their own units. Their premise is the universal ratio bound checked earlier in the same acyclic argument. The shared checklist explicitly addresses the infimum and divergence. Thus the report's common organization does not waive the restatement, attacks, reasoning or premise obligations for either consequence.

Premises and the three vulnerable derivations

The dependency census was checked from the frozen mathematics and consequence sentences on 2026-10-15 UTC. There is no native L-claim, ledger row, or pending dependency grade consumed as an established premise. The elementary background is finite counting, order, integration of finite sums, finite measure exclusion, piecewise affine calculus and elementary logarithmic/exponential inequalities. The internal specialized steps are fully supplied and checked in the following acyclic order; no tier is inferred for a separately named claim.

First vulnerable derivation. The report's label-map proof was checked against both injection directions, including cyclic wraparound and coincident endpoints. With $q=E/(kr)<1$, the unnormalized bounds have exactly the factors $(D+3kA)(1+q)$ and $[D(1-q)-kA(3-q)]_+$. Retaining these forms prevents a hidden threshold as $D$ tends to zero. In the scale calculation, $\theta=\sqrt{A/r}$, $\varepsilon=\theta\log(r/A)$ and a finite number of positive time factors give $U_t(\sqrt{Ar})\le1+60\varepsilon$ and $V_t(\sqrt{Ar})\ge1-60\varepsilon$. The descent gives the stated $3A+400A/\log(r/A)$ bound. The constants and the terminal inequalities were checked, not accepted from their labels.

Second vulnerable derivation. For the extracted list, the exact prefix error is $(1-u)f(u)-ug(u)$, with both errors bounded by $B$. The total absolute coefficient is one. In FP, the differences of late and early extrema have decreasing affine pieces, slopes of magnitude less than $4m$, and jumps $h(v)\ge q_1(v)\ge0$. With $F(s)=s|s|/2$, the potential identity gives

$$\int |f|(-f')=\sum_v\int_{f(v)}^{f(v)+h(v)}|s|\,ds \ge\tfrac14\sum_v h(v)^2\ge m/2.$$

The same holds for the minimum difference. Thus each contributes at least $1/8$ in integral, yielding the $+1/8$ range recurrence and then the full $\lfloor\log_4L\rfloor/16$ lower bound. The initial right jump at zero, active-index changes, repeated entries and the non-power-of-four case are explicitly handled. The lower bound is proved internally, without assuming a literature theorem at a stronger constant.

Third vulnerable derivation. A failing sequence can force a very late starting time, but the extraction has no upper limit on its choice of $N$. All contradictory numerical parameters depend only on $r$ and absolute constants. The coefficient comparison $3/200<1/64<1/(32\log4)$ is strict and exact, leaving a margin over the $O(\log\log r)$ error. Hence one absolute $r_0$ works. The subsequent infimum and divergence are direct order arguments in the claimed ratio coordinates. The grade finds no unsupported quantifier or consequence.

The strongest retained attack tries to defeat the FP jump identity by coalescing entries at zero and changing the extremizing indices. For the explicit list $z_1=\cdots=z_{4m}=0$, the extremum differences on $(0,1]$ are both $3m(1-x)$; the initial jump is $3m$ and the potential integral is $9m^2/2$. Omitting that jump would make the claimed telescoping identity false on this input. The supplied proof includes it, and the general increment argument covers arbitrary collisions, not just this diagnostic. The attempted refutation therefore fails for a stated mathematical reason.

Other retained attacks also have valid dispositions: the repeated two-point configuration shows that collisions cannot simply be dismissed as infinite ratios; label maps cover it. A finite-support sequence eventually has a persistent zero $r$-span and hence infinite ratio. Endpoint avoidance keeps the extracted list inside $(0,1)$. Empty lists and $D=0$ are defined, and the closing application proves nonemptiness before invoking FP. Delaying the sequence-dependent threshold cannot change the absolute parameter comparison. These findings are useful even independently of the overall PASS grade.

Canonical failure modes: item-by-item grading

Named patterns: item-by-item grading

Erdős-specific checklist: item-by-item grading

Source limits, retention and warranted scope

The supplied provenance credits S. Korsky, A resolution of the de Bruijn–Erdős consecutive-gap problem, arXiv:2609.07196v2, unrefereed, Sections 2–3 and 6–8, for the span-to-count and closing reconstruction, and G. Larcher, On the star discrepancy of sequences in the unit interval, arXiv:1407.2094v1, Lemma 2.1 and Section 3, for range splitting. Original project mathematics and the complete local write-ups are credited to Plasma AI. Both original papers remain unread by this grader; the review likewise discloses that limit. The mathematical warrant here rests on the complete supplied deductions, not on an uninspected external premise or claimed PDF fidelity. The Halasz/Bilyk planar result is unconsumed and unexamined.

The final conditional paragraph of Proposition 1221.1 was read, but its unprovided Theorem 4.1 is not a premise. FP supplies the required lower bound at $1/(16\log4)$ instead. No truth, refereeing, or proof-coverage judgment about Theorem 4.1 follows. The coefficient $1/100$, the literal original first two growth expressions, other one-sided clauses, equality between distinct- and all-sequence extremal constants, coefficient optimality, original-paper fidelity, and Lean/kernel verification remain outside the assignment.

An ordinary clone containing this grade, grading_derivations.md, and the unchanged report.md with its linked nine excerpts suffices to rebuild the mathematical assessment. Read the model and catalog formulation, the label lemma and transfer, the count estimate, extraction and FP, then the closing comparison and extremal implication. No mathematical code or executable rerun is required. The complete original private packet and the grader's detailed primary-record audit are retained separately for authorized custody review; derived navigation does not replace those originals. Recorded commands must not be replayed as a reconstruction method.

No required contract part, assigned mathematical deduction, or necessary independence obligation remains unmet on the supplied record. The unread inbox and opaque internal fields retain their exact access limits; neither is asserted inspected or empty. PASS applies only to the unchanged report and the frozen coefficient-$1/200$ ratio theorem, complete supplied proof and exact extremal consequences identified above. The retained mathematical verdict is refutation-failed, not a disproof, proof-defect finding, public promotion or acceptance of a later subject.