Separate grade of the analytic aliquot-collision review

Grade: pass for Theorems A and B and every mathematical consequence specified below, with mathematical verdict refutation-failed. The original assertion about what is known concerning equivalence of (D) has void affirmative verification: the report correctly withholds it. An unqualified certification of every sentence of the frozen subject is therefore void. The excluded motivational exposure is immaterial to this review under the content test, for the reasons given below. This ruling concerns this report alone.

The mathematical acceptance and tier assertion are subject to the durable adoption conditions at the end of this grade. Neither condition (D) itself nor a solution of E0955 is established. The separately supplied later wording is not silently substituted for the original subject.

Subject, grader and governing contract

Grader: separate independent mathematical grader, commissioned model GPT-6-astra, distinct from both the subject author and the reviewer, with no participation in constructing the subject or producing its review. Reviewer: separate independent mathematical reviewer, commissioned model GPT-6-astra. Both assignments began in their own fresh contexts. A shared model is not the asserted basis of independence; separation of contexts, restricted reading and the evidence assessed below are. Examination date: 2026-10-16 UTC. Original subject and report freeze date: 2026-10-16 UTC. Project mathematics is attributed to Plasma AI.

The native subject is wiki/research/leads/aliquot_collision_density/_index.md, Target and conventions, The density-zero form is false, and An equivalent form of the problem, through the final semiprime separation inequality before Exact census. The supporting countermodel is from wiki/problems/arithmetic_functions/E0955/_index.md, Exact scope of the arithmetic reduction. The exact extracted subject, rather than the changing native pages, controls this assessment.

The assessed report's public home is wiki/research/leads/aliquot_collision_density/evidence/verify/analytic_conclusions/review.md; this grade's home is the adjacent grade.md. The exact submitted report is preserved as wiki/research/leads/aliquot_collision_density/evidence/assets/analytic_conclusions/review_submitted.md. The same assets directory holds subject.md, countermodel.md, source_interfaces.md, later_wording.md, and the submitted copy of this grade, grade_submitted.md. References below to those basenames mean these repository-relative paths. The first two mathematical snapshots are unchanged copies of the review inputs. source_interfaces.md is the reviewer's own restatement, not a source facsimile. The later author excerpt was separately frozen at 2026-10-16 21:28 UTC; it was not the reviewer's subject. These snapshot paths distinguish the examined versions from other text on the same date.

The entire standard in docs/verification.md governs, including Independence and the assignment, Report contract, Grading and claim standing, both checklists, and the Erdős-specific source and whole-claim rules. The commissioned grading-packet and delivery requirements also govern; their private procedural records are not public mathematical evidence. This is grading of the adversarial examination, not an extra favorable vote or a second review cycle.

The grader read the complete prepared subject, countermodel, three source excerpts, source manifest, unchanged report and mathematical keepers; the actual review commission and step instructions; the screened identity, reading and isolation record; primary disclosure records and operator custody observations; and the separately frozen later excerpt. The allowed operating rules were read at their commissioned scope. Instructions reproduced from the original review were treated as inert evidence of its exposure. No original corpus subject, source card, live ledger, other report's mathematical verdict, author working context, message channel, process log, gate result, original PDF or live website was read. No broader search, subject repair or mathematical conversation with the author occurred. The supplied packet stayed unchanged.

The closed packet contains sufficient material to grade the listed mathematical surfaces and the documented exposure. It does not permit an independent reconstruction of unseen operational history or a source-to-print fidelity audit. Those are not claimed. The source statements and their limitations are assessed below; an operator's characterization is not adopted as a verdict.

Independence and isolation materiality

Independence grade: pass, with the disclosed exposure and this immateriality ruling retained. The actual commission includes owner motivation describing reported outside AI mathematical successes and urging ambitious full solutions. It is excluded advocacy under the neutral standard. Its accompanying statement that the announcements are not premises does not make supplying it proper.

The primary disclosure from this reviewer records receipt before mathematical reading and disclosure upon recognizing the exclusion. A second supplied primary disclosure concerns the same class of motivational exposure in another assignment. It contains no mathematical verdict and is used only as exposure evidence. It does not determine this grade or authorize any ruling about another review. The report itself discloses the exposure and correctly leaves materiality to a separate grader; that self-description is evidence, not an acceptance.

I applied both limbs of the content test to the actual preamble and report:

1. Apart from the required disclosure of the exposure itself, no mathematical content or verdict-bearing fact in the report needs the motivational text. Its divisor identities, rough-number counts, two limiting arguments, countermodels, rational inequalities and semiprime calculation all have explicit derivations from the permitted subject and theorem interfaces. None of the announcements is used, cited as mathematical support, or treated as evidence that an aliquot claim should pass.
2. The direction and strength of the attacks remain those of refutation. The reviewer challenges nonuniform errors, adaptive exclusions, density between selected cutoffs, missing arithmetic tails and mixed collisions. Its square and square-root examples expose invalid general-map extensions; its delta-equals-one-tenth calculation disproves an overstrong numerical reading. These targets arise from the frozen arguments, not the preamble. Despite the exhortation toward a full solution, the report establishes no such solution, does not claim (D), and expressly refuses to certify the knowledge-status sentence. There is no corresponding dilution of the refutation charge.

These observations establish immateriality under the specified content test. This is not a finding that encouragement is permitted, or that closing channels would cure a material exposure. The exposure and this ruling must both remain visible at adoption.

The operator's inspected settings, the recorded empty subscription tables and the reviewer's screened reading record support channel isolation. They do not by themselves prove what was read. I weighed them together with the primary communications, the actual assignment and the report's content. The grader's own subscription table was also empty, and no incoming channel was opened.

The supplied post-input changed-path record lists only the reviewer's private continuation, report and three keeper files. It lists no subject, source or corpus edit. The subject and countermodel keepers agree byte-for-byte with the supplied inputs. The continuation action confined to the reviewer's own records therefore does not enlarge authored scope or compromise independence under the adapter. A separately disclosed path-naming deviation concerned writes to the correct authorized report and interface files; the supplied record establishes no outside write or extra reading from it. That operational deviation is immaterial here. The operator supplied these custody facts, not the conclusion of this paragraph; no unseen history is asserted to have been independently inspected.

Exact mathematical scope and grades

Throughout, inputs are positive integers, $x\ge1$ is real, $s(n)=\sigma(n)-n$, and $s(1)=0$. The only input with value zero is 1. The collision count is $C_S(x)=\sum_{m\ge0}r_{S,x}(m)^2$, with ordered pairs and the diagonal included. An admitted or excluded set may depend on x. All decimal constants in the arguments denote exact rationals. The products over p and divisor sums over q below are over primes; $\pi$ counts primes, and $\gamma$ is the Euler–Mascheroni constant.

Let $m_w=\prod_{p\le w}(1-1/p)$. For $0<\delta\le1/10$ put $z=1/\delta$, $Z=100z$ and

The constructed class $V_\delta$ consists of $n=pk$, where p lies in that prime window, k has no prime factor at most Z, and $\sum_{q\mid k}1/(q-1)\le\delta/10$. The four assertions in B are, with their quantifiers retained:

1. Every fixed set of natural density zero has a preimage of natural density zero under s.
2. For $F_x(T)=x^{-1}\#\{n\le x:r_x(s(n))>T\}$, $\lim_{T\to\infty}\limsup_{x\to\infty}F_x(T)=0$.
3. For every $\varepsilon>0$, some finite K and threshold work for every sufficiently large x: at most $\varepsilon x$ inputs can be removed to leave collision count at most Kx.
4. For every $\varepsilon>0$, some $\eta>0$ and threshold work simultaneously for every sufficiently large x and every set B of at most $\eta x$ nonnegative integer values: its preimage among inputs at most x has size at most $\varepsilon x$.

No assertion that any one of B's four conditions is true is part of the four-way equivalence. A target in (4) may depend on x and have arbitrarily large values. Finally $D(\delta)=\{n\text{ composite}:s(n)\ge\delta n\}$, and (D) is the hypothesis that for each fixed $\delta>0$ there is a finite $K_\delta$ with $C_{D(\delta)}(x)\le K_\delta x$ eventually.

The table covers every substantive analytic consequence in the scoped original subject. Its final row is not a load-bearing premise of any preceding row. The qualified examination of the entire mathematical argument passes; a blanket claim that all frozen prose, including that row, has been verified is void. This distinction preserves both the useful examination and the actual gap.

Attacks and derivations assessed by the grader

A and its exclusion-cost consequences

The report's main attack targets the lack of an error bound uniform in the roughness threshold. Replacing a fixed delta by a moving $\delta(x)$ would invalidate the displayed error argument. The retained proof does not do that. For each fixed Z, periodic counting gives $R_Z(t)=m_Zt+O_Z(1)$ for every real $t\ge0$, and the exact weighted identity is

Higher powers of q are counted once in both sides. The accumulated error is $O_Z(\log\log y)$, since $\sum_{q\le y}1/(q-1)\le\log(1/m_y)+1$. The main term is at most $m_Zy/\lfloor Z\rfloor$; for $Z\ge1000$ this is at most $1.01m_Zy/Z$. Markov at $\delta/10$ removes at most $0.101m_Zy+o_\delta(y)$, leaving more than the claimed $0.89m_Zy-o_\delta(y)$.

The unique small prime p identifies each constructed input, including the case $k=1$. The divisor-product bound is unaffected by prime powers. The exponential estimate follows from $e^t\le(1-t)^{-1}$ for $0\le t\le1/100$, so the output-ratio bound is at most $1.91\delta+0.198\delta^2\le1.9298\delta<2\delta$ and at least delta. Thus the constructed values are positive, and their number at scale x is at most $\lfloor2\delta x\rfloor$.

For fixed $a<b$, partial summation and PNT give $\sum_{az<p\le bz}1/p\sim\log(b/a)/\log z$; the PNT error is uniformly small on this particular interval because its left endpoint tends to infinity. This is not uniformity in arbitrary moving a or b. Mertens at $100z$ then gives $d_\delta\log^2(1/\delta)\to0.89e^{-\gamma}\log(18/11)$.

I checked the report's exact arithmetic certificate rather than relying on a printed decimal evaluation. The harmonic-sum integral estimate gives $\gamma<H_5-\log5-1/12=11/5-\log5$. The four positive logarithmic-series terms total $122492/76545$, exceeding $8/5$ by $20/76545$. Hence $\gamma<3/5$. The alternating exponential sum through degree five is $34297/62500=0.548752$, a strict lower bound for $e^{-3/5}$. The two logarithmic-series terms for $\log(18/11)$ give $36008/73167$;

Consequently the coefficient exceeds

The series have the remainder signs used in the report. No floating-point or unretained executable result is needed. The adverse endpoint example is also correct: at $\delta=1/10$ only p=7 occurs, $d_{1/10}\le89/2625$, whereas $\log10<12/5$ gives $0.2/\log^2 10>5/144>89/2625$. Thus the tempting all-delta estimate really fails. The frozen assertion expressly requires sufficiently small delta, so this witness tests the qualification without refuting the theorem.

For each fixed small delta, any $o(x)$ exclusion leaves $(d_\delta-o(1))x$ constructed inputs. Cauchy–Schwarz gives the lower limit $d_\delta^2/(2\delta)\ge0.02/(\delta\log^4(1/\delta))$. Given a desired collision ratio, choose delta first and then x. This defeats adaptive exclusions and proves eventual divergence, not merely a subsequence.

For an $\varepsilon x$ exclusion, subtract epsilon before squaring, and only use $d_\delta>\varepsilon$. The lower limit dominates each term and hence their supremum without an attained maximum or a uniform x-threshold. Choosing $\delta=\exp(-\sqrt{0.1/\varepsilon})$ eventually lies in the valid range and gives $d_\delta\ge2\varepsilon$. The logarithm of the asserted lower bound multiplied by $\varepsilon^a$ is $\sqrt{0.1/\varepsilon}+(a+2)\log\varepsilon-\log2\to\infty$ for every fixed $a\ge0$. The constant-growth consequence is therefore in the claimed $C/x$ units and conditional on an admissible upper constant. The report also supplies the upper proportion bound by dropping only the bound on f(k), while retaining roughness, and supplies the semiprime density-zero argument. No consequence is being supplied solely by the grader to fill a missing review.

B and the two arithmetic interfaces

The most serious union attack is that each sparse piece can contain values far above its lower cutoff, creating unexpectedly dense intermediate intervals. The report answers exactly that risk. Failure of tightness supplies arbitrarily large witnesses at every integer threshold k. For a fixed L,

because any remaining n has $n>x/\sqrt L$ and no prime divisor at most $\sqrt L$. Choose L to make the coefficient at most $\eta/4$, and then choose witnesses $x_k\ge k^2Lx_{k-1}$ absorbing the fixed error. The sets $A_k=\{m\ge x_k/L:r_{x_k}(m)>k\}$ retain capture at least $\eta x_k/2$, while $|A_k|<x_k/k$.

For every sufficiently large y, take the last k with $x_k/L\le y$. Later sets contribute nothing; counting every element of every earlier set is an upper bound even when its values are enormous. Therefore

This establishes natural density zero at all large y. It does not assume bounded values, disjoint pieces, or a rate of tightness. The witness construction is valid at each stage because arbitrarily large witnesses remain available; integer parts handle real witness points. Its image target has positive values, so the convention at zero is harmless.

The sufficiency direction, (3) implies (1), needs an upper-output interface. Finite divisor summation gives

Inputs with $s(n)>Lx$ thus number at most $2x/L$. With epsilon and L fixed, the surviving capture from a density-zero target below Lx is $\sqrt{Kx\,|A\cap[0,Lx]|}=o(x)$. Taking the upper density and only then letting epsilon decrease and L increase proves the needed implication. Arbitrary x-dependent exclusions are allowed.

The two model attacks have real force. For $g(n)=n^2$ the fibers have size one but the sparse squares capture all inputs; the upper-output interface fails. For $f(n)=\lfloor\sqrt n\rfloor$, a density-zero target captures at most $(2y+1)|A\cap[1,y]|=o(x)$ with $y=\lfloor\sqrt x\rfloor$, whereas all but $O_T(1)+O(\sqrt x)$ inputs lie in fibers exceeding any fixed T; the lower-output interface fails. These are valid counterexamples to the respective unsupported general-map transfers, not to s. The actual arithmetic argument establishes both interfaces independently.

The finite-fiber arrows are also complete. Strict slack in the limiting statement gives eventual heavy-input mass below epsilon. Removing those fibers bounds every retained multiplicity by T. Conversely a heavy-value set has cardinality less than $x/T$, exactly the target-size uniformity needed in (4). The review's additional direct check of (3) implies (2) is sound: $F_x(T)\le\varepsilon+\sqrt{K/T}$, using surviving counts and the heavy-value cardinality bound. It requires no hidden control of output sizes. All four conditions, their quantifiers and the corrected-route consequence are covered.

C and D

PPT controls the capture maximum at a scale $x^{-1/2+h(x)}$ depending on x; B(4) requires its upper limit to vanish as a fixed target fraction tends to zero. One cannot exchange those limiting operations or multiply an unspecified little-o error by an uncontrolled number of target pieces.

The supplied witness verifies that failure at the asserted abstract scope. For $a_j=\lfloor j^{3/2}\rfloor$, the sequence increases strictly from 1, and the fiber of $T(n)=\max\{a_j:a_j\le n\}$ is $[a_j,a_{j+1}-1]$. Its length is at most $\tfrac32\sqrt{j+1}+1$. If $a_j\le x$, then $j<(x+1)^{2/3}$ and, for $x\ge1$, $j+1<3x^{2/3}$. Since $\sqrt3<2$, the length is less than $4x^{1/3}$, including the final truncated fiber. Thus all targets of at most $x^{1/2+h(x)}$ values capture at most $4x^{5/6+h(x)}=o_h(x)$. The image has at most $(x+1)^{2/3}$ elements up to x, $0\le n-T(n)\le4n^{1/3}$, and every input maps into that image. All hypotheses of the defeated abstract inference are verified. That abstract inference is refuted-as-stated by this witness; the frozen subject's claim that the witness defeats it has verdict refutation-failed. The report neither imports this map into the arithmetic function s nor prohibits arguments with additional arithmetic information.

For D, a prime divisor $p\le1/\delta$ of a composite n implies $s(n)\ge n/p\ge\delta n$. The complement is rough apart from finitely many primes, so the claimed upper density follows by residue counting. Choosing its bound strictly below epsilon turns upper-density control into the required bound for every sufficiently large x. If (D) holds, its threshold for this fixed delta can then be combined with that one. No existence of (D) is hidden in the composition.

The prime inputs in $V_\delta$ form a finite set for each fixed delta; all other members lie in $D(\delta)$. Deleting those primes preserves A's lower proportion and its pigeonhole bound, giving the stated necessary growth of $K_\delta$.

For distinct primes $p<q$, membership of pq in D forces $\delta\le1/p+1/q+1/(pq)\le3/p$. Given p and the value m, q is uniquely $m-p-1$. A square has value p+1 and adds at most one representative. Writing u for semiprime fiber counts and v for the others gives

The word “contribute” in the frozen paragraph is justified as the semiprimes' own sum of squares: the very next inequality handles mixed pairs. The report makes this reading explicit and checks the factor two. It does not certify a separate bound of $M_\delta x$ for all mixed pairs. This resolves the possible composition failure without changing a statement or discarding collisions.

Premises, source identity and reading depth

I derived the consumed premises from the frozen statements and proofs, including the consequence sentences. There is no native L-claim consumed as an established premise. The source countermodel is rederived rather than accepted because of a prior standing label. All definitions are meaningful without a pending result. Accordingly the native dependency list is empty on the grading date, 2026-10-16; no live or prospective ledger standing is substituted for a required reading. No pending grade or author computation supports acceptance.

The source manifest identifies the canonical transcription paths, editions and locators; the retained source_interfaces.md preserves those identities and the exact consumed mathematical statements. I checked their agreement with the supplied excerpts, not with original artifacts outside the permitted reading. Both reviewer and grader have claims checked depth restricted to those excerpts and applications. Original proofs, publication-to-manuscript identity beyond the supplied provenance, transcription-to-print fidelity and the supplied retrieval's execution are not independently verified here. The review's stated limits are accurate. The commission's wording about primary statements does not enlarge them.

Ordinary reliance on these identified external theorem interfaces is permitted by the neutral standard and does not require native L wrappers or recursive proof reconstruction. Original source-proof audits and PDF fidelity were expressly excluded, so their absence is not a missing required proof leg for this grade. They acquire no accepted coverage from it. The unsupported literature-wide knowledge assertion is different: none of these interfaces addresses it, and it remains withheld.

The acceptance order is acyclic: first conventions, elementary residue/divisor facts and the external interfaces; then A and its exclusion consequences; then B using the elementary residue fact and Mertens, without A's divergence theorem; then C's comparison using B and PPT, with the countermodel proved independently; then D's sufficiency using B and its necessary-growth consequence using A's construction. D's semiprime estimate is elementary. (D) is an explicit hypothesis of a conditional result, not a tier-bearing premise claimed to exist.

Complete report-contract grade

The report's five mathematical groups are A, A-epsilon, B, C and D. Each has its own restatement, derivations, attacks, premises and full verdict; the common checklist explicitly applies to each group with distinctions stated. Grouping consequences in those sections does not omit their individual examination.

Canonical failure modes

Each row grades the report's examination of all five groups. “Inapplicable” is an explicit finding about the named mechanism, not an unexamined surface.

Named canonical patterns

Erdős-specific checklist

The neighboring statistical discipline is inapplicable: there are no samples or claims of probabilistic independence. Evidence coverage under docs/anatomy.md and docs/evidence.md is satisfied by preserved proofs and witnesses at the stated scope. A structurally different computational reproduction leg is not required for this noncomputational batch. This grade is not a second independent review cycle and makes no multiple-cycle reliability claim.

Original knowledge assertion, later wording and exact correction

The original sentence is “It is not known to be equivalent.” Its natural subject is equivalence of (D) with the target. That is a literature-wide knowledge assertion. A one-way proof does not establish it: failure to supply a converse here neither shows that no converse is known nor disproves equivalence. The packet contains no source survey bearing on this assertion. The report's withholding is therefore correct, and its absence of coverage is material to any request to certify that sentence or all frozen prose without qualification. It is immaterial to the validity of A, B and the other rows because none consumes it. This is a content-based ruling, not acceptance of the operator's label.

The separately frozen later sentence, “The argument does not establish the converse.”, makes the narrower argument-level observation. Inspection of the retained proof confirms that it proves (D) implies B(3) and that it does not prove B(1) or B(3) implies (D). Necessary lower bounds on constants do not supply that missing implication. The reviewer already explains this limitation, but the later author's bytes were not its frozen subject.

I authorize precisely the following correction outside a statement field, in the D paragraph of An equivalent form of the problem, immediately after “and hence the problem.” and before “The class”. The old bytes are:

It is not known to be
equivalent.

Replace that single occurrence with these bytes:

The argument does not establish the converse.

The line break in the old span is part of the supplied extraction. Reflow of surrounding whitespace is mechanical; no further wording change is authorized by this correction. The supplied later D excerpt, apart from its extraction wrapper and terminal blank lines, agrees exactly with the original D excerpt after this replacement. Thus pass applies to its eligibility for this specific body-prose correction rule. It is void as a claim of a fresh whole-statement review of later text: no such review occurred, and no inherited verdict covers other later changes.

Under docs/verification.md, Grading and claim standing, the acceptance of the unchanged mathematical statements remains in force after exactly this correction if the owning page discloses where current prose differs from the reviewed subject and cites this grade's condition. The original subject and report must remain preserved; the report must never be rewritten to say that it reviewed the later sentence. This corrects unsupported surrounding characterization without changing Theorem A, Theorem B, the displayed definition or hypothesis (D), or any proved consequence. It gives no warrant for necessity or for an equivalence involving (D).

Any change to a statement: field, or substantive change to the whole theorem statement in a page without such a field, is a new frozen subject requiring a fresh blind review and separate grade. The correction exception must not be used to replace a statement or add a stronger consequence. No other later page content, including routes outside this commission, acquires standing here.

Accepted standing and durable adoption conditions

As the eligible separate grader, I accept the review for all mathematical surfaces marked pass; refutation-failed above and assert tier 1 for those exact propositions at that scope, effective only on durable adoption of this warrant. This is acceptance of a whole statement in each listed row, not a promotion of an unreviewed fragment to a larger claim. It is not tier 2, a source proof audit, a tier for (D)'s unproved antecedent, or an acceptance of an E0955 solution. No tier is warranted for the original knowledge assertion or a new whole statement written after the review.

The integrator may record this grade's accepted judgment and supply no new tier assertion of its own. The following conditions remain obligations of adoption:

1. File the report and this grade at the named verification homes and retain the exact submitted subject, countermodel, review, grade and necessary mathematical keeper interfaces at the named assets paths. Retain the separately frozen later excerpt as the evidence for the correction's limited applicability. These files, the retained reasoning and the public references needed to read them must resolve from an ordinary clone. Private custody alone is not a public verification warrant.
2. Preserve each external interface's actual reading depth and provenance. The source-interface keeper must continue to be labeled a restatement. No publication, original-proof, PDF or broader source-fidelity coverage may be added by filing. Mechanical public-path maintenance is allowed under the standard, with its disclosed mapping and unchanged subject, scope and verdict.
3. Retain the motivational exposure and this immateriality ruling in the filed record's index and the owning standing account. Preserve the distinct author, reviewer and grader roles and the independence facts; keep raw communications and operational records private.
4. The owning standing account must limit acceptance to the passed mathematical propositions and state that (D) and the full problem remain unproved by this package. Any description of the entire scoped prose as accepted must first implement exactly the authorized knowledge-sentence correction and disclose it as differing from the reviewed subject. All other accepted statements, definitions, proofs and inputs must remain the reviewed ones, apart from permitted mechanical maintenance; the supplied later excerpt establishes that comparison only for its D span. The original assertion remains uncertified in its preserved snapshot.
5. Preserve the submitted records while enumerating any publication maintenance allowed by the standard. A changed mathematical check, theorem statement or unapproved consequence is not covered by this grade. Custody and adoption checks belong to the integrator; they were not replaced by a repository gate or by the private delivery.

All required mathematical and report-contract checks for the passed rows are complete within the commissioned source interfaces. No mathematical executable was used, so no code, runtime output or numerical rerun remains owed. The remaining obligations are durable adoption and the specified scope disclosures; the original knowledge assertion and any stronger or later whole statement remain outside favorable verification. The retained abstract counterexamples refute only their stated general inferences, while the scoped analytic results have verdict refutation-failed.