Grade of the consecutive-gap conjunction examination

---

PASS. The original whole-statement examination meets the report contract and independence requirements for the complete frozen L165 conjunction and its implication for the supplied mean-normalized modified Statement of Problem 1221. Its mathematical verdict, refutation-failed, is accepted at precisely that scope. Each assertion and contract ruling is recorded below. No defect requires a partial disposition or a hold at the recorded dependency reading.

Original project work and this independent assessment are credited to Plasma AI. Grading date: 2026-10-16 UTC.

Subject and independence

The subject is the original eight-file examination delivery, frozen on 2026-10-15 at 22:29:05.281749 UTC. Its underlying mathematical subject was frozen on 2026-10-15 at 21:10:27.616818 UTC and examined that day. The originals, including every marked omission, were read in full. This grade assesses those originals, rather than a later rendering or the current L165 card.

Their mathematical identifiers are wiki/theory/analysis/L165_consecutive_gap_growth_conjunction/_index.md, wiki/theory/analysis/L165_consecutive_gap_growth_conjunction/_proof.md, wiki/theory/analysis/L163_all_sequence_span_ratio/_index.md, wiki/theory/analysis/L164_all_sequence_span_deviations/_index.md, and the selected formulation sections of wiki/problems/analysis/E1221/_index.md. The preserved extracts specify the examined text; these identifiers do not extend coverage to omitted sections or subsequent text.

Grader. Independent fresh-context grader, assigned model gpt-6-astra with maximum reasoning effort, distinct from both the subject author and the original reviewer. This grading context neither constructed nor previously built on the subject. The commission charged the grader to find a mathematical, contract or eligibility defect. Reading the original report and the reviewer's own continuation was expressly part of grading that report; neither was accepted as an instruction or as evidence of correctness by itself.

Reviewer eligibility: PASS. The original assignment was a distinct, fresh review under a refutation charge. Its input contained the frozen mathematical extracts and the permitted operating material, without author preparation or an earlier assessment. The supplied original primary records support this separation. The preliminary exchange contained an operating acknowledgement and no mathematics. All 30 recorded examination calls were matched to their returns; the displayed returns, supplied instructions and later continuations were examined for their actual content. The continuations carry the reviewer's own work. No incoming mathematical steering, excluded verdict, surrounding mathematical search, other working context, or external source acquisition is shown. The recorded reading and writing remain within the original assignment.

All five mathematical extracts were displayed completely before the substantive assessment. The final report and keeper were also displayed completely. Initial truncation affected operating reading; subsequent reads supplied the missing required portions. Merely issuing a read command or computing an identity was not counted as seeing the file. The final dependency file was assembled from a fully displayed earlier selection and a fresh, displayed equality comparison of the complete selected interfaces; this distinction is retained in the provenance assessment. Its bytes agree with the final observation.

The original empty subscription observations support the recorded separation; they do not establish that an uninspected communication store was empty. The narrow settings corroborate the declared model and isolation route but are not a substitute for the actual-read audit. No excluded exposure is identified in the supplied record, so there is no exposure requiring an immateriality waiver. The ruling concerns the supplied complete primary record of this examination; it does not purport to observe unrecorded conduct or inaccessible internal reasoning.

Grading isolation: PASS. The grader read the supplied frozen packet and its original provenance, the identified operating rules, and the permitted current dependency selection. No current L165 standing, surrounding mathematical material, prior grade, author composition or outside source was read. The input identities matched before use and the frozen originals remain unchanged. Empty subscription observations support this reading record. No excluded exposure occurred in this grading. The separate operational trust comparison assigned to the commissioning authority is not certified here.

Exact restatement and assertion rulings

Let $\mathcal A$ be all infinite sequences in $\mathbb R/\mathbb Z$, including repetitions, and let $\mathcal D$ be its subfamily of sequences with distinct terms. For positive integers $n\ge r$, retain all first-$n$ labels, order them by position with arrival index breaking ties, and extend ordered lifts by $y_{i+n}=y_i+1$. Define $M_n^{(r)}$ and $m_n^{(r)}$ as the maximum and minimum of $y_{i+r}-y_i$ over all $n$ starting labels. These are forward sums of $r$ consecutive gaps on a circle of circumference one. If the minimum is zero, the ratio is $+\infty$; the maximum is positive.

The exact quantifier order is $\exists c\in\mathbb R_{>0}\ \exists r_0\in\mathbb Z_{>0}
\forall r\in\mathbb Z_{\ge r_0}\ \forall X\in\mathcal A$. For this one pair of absolute constants the three separate assertions are

$$\limsup_n(nM_n^{(r)}-r)\ge c\sqrt{\log r},\qquad \limsup_n(r-nm_n^{(r)})\ge c\sqrt{\log r},\qquad \limsup_n\frac{M_n^{(r)}}{m_n^{(r)}} \ge1+\frac{\log r}{200r}.$$

The logarithm is natural. Each limit holds $r$ fixed and uses integer $n\ge r$. No common subsequence or uniform onset time in $n$ is asserted. For either $\mathcal F=\mathcal A$ or $\mathcal F=\mathcal D$, the quantities are

$$\Lambda_r^{\mathcal F}=\inf_{X\in\mathcal F}\limsup_n nM_n^{(r)},\qquad \lambda_r^{\mathcal F}=\sup_{X\in\mathcal F}\liminf_n nm_n^{(r)},\qquad \mu_r^{\mathcal F}=\inf_{X\in\mathcal F}\limsup_n \frac{M_n^{(r)}}{m_n^{(r)}}.$$

Optimization follows each sequence limit. The sequence may depend on $r$; $c$ and $r_0$ may not. For every integer $r\ge r_0$ the claimed bounds are

$$\Lambda_r^{\mathcal F}-r\ge c\sqrt{\log r},\qquad r-\lambda_r^{\mathcal F}\ge c\sqrt{\log r},\qquad r(\mu_r^{\mathcal F}-1)\ge\frac{\log r}{200}.$$

Each expression tends to $+\infty$ through the positive integers, for each family. In particular the $\mathcal A$ conclusions answer the supplied mean-normalized modified Statement. This is the original report's scope.

These are rulings on assertions already made in the original delivery, not new claim identifiers or a replacement examination. Outside-scope historical, stronger-constant, optimality, attainment, family-equality and kernel assertions receive no mathematical warrant from this grade.

Assessment of the weakest steps and strongest attacks

The grader reproduced the indispensable keeper reasoning below to test the existing examination. None of these calculations supplies a new premise or repairs the frozen proof.

Objects, boundary cases and the geometric attack

For the keeper's $g_i=y_{i+1}-y_i$, periodicity gives $g_i\ge0$ and $\sum_{i=1}^n g_i=1$. In $\sum_i\sum_{j=0}^{r-1}g_{i+j}$ each gap occurs once for each of the $r$ values of $j$. Therefore the sum of spans is exactly $r$, and $0\le m_n^{(r)}\le r/n\le M_n^{(r)}\le1$. This establishes the definitions without relying on an unproved catalog assertion. Rotation permutes the spans; coincident labels are retained. At $n=r$ the lifted span is one even though its endpoint residues agree. Shortest circle distance would give a different observable and fails this check.

For the constant repeated sequence and $n>r$, one gap is one and the others are zero. Exactly $r$ spans contain the unit gap. The keeper's values $M=1$, $m=0$, deviations $n-r,r$, and infinite ratio follow exactly. The bounded second deviation creates no contradiction: for fixed real $c>0$, $c\sqrt{\log r}\le r$ eventually.

The dyadic sequence in keeper Section 2 is a realizable attempt to defeat the universal claim. Between a grid of $N=2^k$ points and the next complete grid, every gap is either $1/N$ or $1/(2N)$. Hence each span lies between $r/(2N)$ and $r/N$. For $N\le n<2N$ this gives $nm\ge r/2$, $nM<2r$ and $M/m\le2$. Along complete grids all deviations vanish and the ratio is one. After the first $N/2$ midpoint insertions, the two contiguous runs have $N$ short gaps and $N/2$ long gaps. When $N\ge2r$, each run contains $r$ gaps, giving $M=r/N$, $m=r/(2N)$ at $n=3N/2$. The exact deviations are $r/2$ and $r/4$, and the ratio is two. The reported liminf counterexample and failed limsup attack are therefore valid. This same witness establishes the claimed finite outer extrema, without asserting their exact values.

Tail identities, uniformity and outer optimization

At fixed $r$, let $u_n=nM_n^{(r)}$ and $v_n=nm_n^{(r)}$. The second sequence lies in $[0,r]$. On each tail,

$$\sup_{n\ge K}(u_n-r)=\sup_{n\ge K}u_n-r,\qquad \sup_{n\ge K}(r-v_n)=r-\inf_{n\ge K}v_n.$$

The first identity remains valid at an infinite supremum; the second uses only bounded real quantities. Passing to the limits of the tails proves exactly the two identities used in the report. In particular the lower-span conclusion concerns $\liminf v_n$. There is no replacement by a limsup and no indeterminate subtraction of infinities.

Set $a=c\sqrt{\log r}$ and $b=\log r/200$. For each $X$, the consumed interfaces give $U_X\ge r+a$, $V_X\le r-a$, and $Q_X\ge1+b/r$, where $U_X,V_X,Q_X$ are the three sequence limits. Taking the indicated outer infimum, supremum and infimum retains these inequalities. A strict violation by an outer extremum would force some member to violate the same common bound. This holds even when an extremum is unattained or the selected sequence changes with $r$. Restriction to $\mathcal D$ has the directions printed in the proof; subtracting its supremum from $r$ reverses that inequality once. No quotient of separately optimized numerator and denominator appears.

The strongest attack in the original report targets precisely an illicit exchange of these operations. For the keeper's $f_k(n)=0$ when $n<k$ and $f_k(n)=a>0$ otherwise, each individual limsup is $a$, while for every finite $n$ the pointwise infimum over $k$ is zero. Thus $\inf_k\limsup_n f_k(n)=a$ and $\limsup_n\inf_k f_k(n)=0$. For $h_k=r-f_k$ with $0<a\le r$, the corresponding values are $r-a$ and $r$. This exact diagnostic exposes the failure mechanism; it is not asserted to satisfy the geometric constraints. It cannot defeat the frozen proof, which applies a bound to each infinite sequence before optimizing. The geometric dyadic attack supplies an actual circle-sequence test alongside this scalar diagnostic. Together they are substantive attacks, not agreement with a displayed answer.

Conjunction, divergence and the formulation bridge

The two deviation clauses share the absolute $c$ supplied by L164. L163's threshold and L164's threshold can be replaced by their maximum. This operation does not intersect witnessing subsequences. Alternating scalar excursions, as specified in the keeper, indeed have positive separate limsups and a zero pointwise minimum; that rejected inference is absent here. The logarithmic ratio improvement is supplied by L163. The square-root deviations alone yield only corresponding square-root-scale ratio information through the mean inequality, so the report correctly identifies the ratio premise as essential.

For every real $T>0$, all integers $r$ sufficiently larger than $r_0$, $\exp((T/c)^2)$ and $\exp(200T)$ satisfy $a>T$ and $b>T$. This verifies each stated divergence, with no monotonicity assertion and no exchange of the $n$ and $r$ limits. It also checks the distinction between a real target and the integer eventual range.

The catalog extract's modified Statement and Notes specify circumference one, forward spans through $r$ gaps, extrema over cyclic starting labels and outer extrema over all infinite sequences after the sequence limits. The Notes resolve the displayed maximum/minimum subscripts and the phrase about consecutive points at exactly this convention. The algebraic identities $r(\Lambda_r/r-1)=\Lambda_r-r$ and $r(1-\lambda_r/r)=r-\lambda_r$ give the stated normalized coordinates. The ratio coordinate is already $r(\mu_r-1)$. Thus the claimed implication has the correct domain, operation order and units. The supplied text's historical explanation is not needed to establish that correspondence.

Premises and current reading

The consumed list is derived from the frozen proof and every consequence: exactly L163 and L164. L163 supplies the ratio clause for every sequence, and L164 supplies both separate deviation clauses with one absolute positive constant. The other deductions are the elementary identities and order arguments assessed above. The catalog is the specified formulation target, not an additional native theorem or external-source premise.

The original final reading was on 2026-10-15 at 22:26:35.670409 UTC. The grader's permitted current reading was on 2026-10-16 at 03:50:41.816039 UTC. It was restricted to each premise's id, status, tier, standing, depends_on, complete statement, Conventions and scope, and exact generated row in wiki/standing.md.

All selected fields also agree with the original final observation. No stale mark, changed premise, missing field needed by the application or pending tier was returned. This discharges the current-standing condition at this reading. It does not certify undisplayed acceptance narratives or future standing.

The interfaces agree on repeated terms, tie order, lifts, $r$ forward gaps, integer $n\ge r$, natural logarithms and absolute parameter dependence. L163 has the exact extended-ratio convention. L164 requires no division by a minimum. Their half-open counting-arc conventions are recorded but no lemma from their omitted proofs is consumed. No full premise proof re-audit or batch promotion is part of this grade. No external classical-paper theorem, historical quotation or source-credit assertion is used as a premise; original publication reading depth is unread.

Complete report-contract rulings

The governing requirements are docs/verification.md, Report contract, Whole-claim report, Independence and the assignment, and Grading and claim standing, together with its verification adapter. The original report uses identifiable sections for all seven governing parts.

Governing Erdos checklist

Shared canonical failure modes

Named patterns

Evidence coverage remains analytic composition from accepted native interfaces. The statistical discipline is inapplicable. There is no computational batch or second examination cycle requiring a structurally different numerical route. This is grading of the existing examination, not a replacement cycle.

Preservation, reproduction and limits

The exact original eight-file delivery, its original input and delivery manifests, the historical reviewer's commission and continuation, and the supplied complete primary provenance have been preserved without alteration. Split originals were concatenated only in their declared order with no added separator. Original input copies agree with the returned mathematical extracts; the original delivery inventory agrees with the preserved originals. All existing [cut] markers remain. No omitted source was reconstructed and no later mathematical bytes are included in the accepted subject.

For mathematical reproduction from an ordinary clone, the eight linked files are the complete required package. Read the statement, proof and premise interfaces; verify the finite cyclic count and two dyadic prefix calculations in keeper Sections 1–2; check the uniform quantifiers in Section 3; derive the tail identities and order inequalities in Section 4; check each divergence and family inequality in Section 5; and compare the exact modified Statement and Notes with Section 6. Section 7 supplies that route in the original keeper. The grader followed these derivations. No private program, numerical output or original publication is needed for these mathematical checks. The primary provenance supports the independence facts summarized here and is preserved separately from the public mathematical evidence.

The original private mechanical success counts were not adopted as theorem certificates. The grader independently matched calls and returns, inspected displayed coverage, checked original custody and compared the permitted current interfaces. Parser outputs distinguish displayed bytes from internal command records and do not infer unread bytes from successful exit codes. The original material is retained with those findings and their limits.

Accepted scope. PASS covers the whole frozen L165 examination, including the conjunction, both families' extremal and divergence consequences, and the implication at the supplied mean-normalized modified formulation. Its accepted mathematical verdict remains refutation-failed. There is no unresolved contract or mathematical coverage gap within that scope.

This grade does not re-audit L163/L164 proofs, certify historical wording or source credit, answer the literal printed first two expressions, prove a coefficient $1/100$, establish optimality, attainment or equality between families, or grant Lean credit or catalog acceptance. It supplies no warrant for text written after the frozen delivery. Preservation and filing must retain the complete assessed package; a favorable summary alone is insufficient.