Collision Counts for the Sum of Proper Divisors

../

evidence/: Exact census of collisions s(n) = s(n') through 10^10 in classes cut out by s(n)/n, by the number of prime factors and by the largest prime factor, with an independent small-range reference and external count checks.

large_prime_class_average: Exact local factors and a Brun–Titchmarsh compatibility argument bound the averaged sieve input G1 of the large-prime class, after removing few inputs, except for common divisors of s(a) and s(a') that divide a'-a and exceed every prime factor of both cofactors by a fixed power; that core is open.

rough_prime_local_density: For at least three free primes on each side, the collision polynomial has uniformly summable local factors away from its coefficients. After a negligible removal of small cofactors, their full local majorants have the required reciprocal-weighted average. The count of prime solutions in integer boxes remains unproved.

uniform_two_form_sieve: An elementary Selberg-weight construction proves the uniform sieve input G0 for the large-prime class, including nonprimitive forms, the interval enlargement and a remainder absorbed uniformly in the main term.

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This lead's results that are not drawn from a cited source are Plasma AI's, credited to Plasma AI.

Current review

The independent review of 2026-10-16, accepted by a separate grade, found Theorems A and B and every specified analytic mathematical consequence through the semiprime separation inequality refutation-failed. This records the grader's tier-1 judgment for those exact propositions at the source interfaces retained in the verification record. It establishes neither (D) nor a solution of Problem 955. The numerical census, the G0/G1/G2 route and the rough-prime classes are outside that examination and retain author standing.

The frozen subject contains an unsupported assertion about what is known concerning equivalence of (D). Affirmative certification of that assertion is void; no disproof is given. The current D paragraph makes only the grade-authorized body correction to "The argument does not establish the converse." Other examined mathematical statements and proofs are unchanged, apart from standing descriptions. The later excerpt is separately identified and was not the reviewer's subject.

The reviewer received excluded motivational advocacy, which the separate grader ruled immaterial to this examination under the content test. The exposure and ruling remain disclosed. The external PNT, Mertens and PPT premises were checked at their supplied statement interfaces; original source proofs and PDF fidelity are not certified.

Target and conventions

The target is Problem 955, whose page records the general assertion as open: for $s(n)=\sigma(n)-n$, does every set $A\subseteq\mathbb N$ of natural density zero have a preimage $s^{-1}(A)$ of natural density zero? Nothing on this page changes that status. Here inputs are positive integers, and $s(1)=0$.

For $x\ge1$ and $m\ge0$ write $r_x(m)=\#\{n\le x:s(n)=m\}$. For a set $S$ of admitted inputs, which may depend on $x$, write $r_{S,x}(m)$ for the number of $n\in S\cap[1,x]$ with $s(n)=m$, $R_S(x)=\#(S\cap[1,x])$, and

$$C_S(x)=\sum_m r_{S,x}(m)^2 =\#\{(n,n')\in(S\cap[1,x])^2:\ s(n)=s(n')\}.$$

Pairs are ordered and the diagonal $n=n'$ is included. So $C_S(x)-R_S(x)$ counts the ordered pairs of distinct inputs with equal values, and $(C_S(x)-R_S(x))/2$ the unordered ones.

The route recorded here is the second-moment route: if few pairs collide, Cauchy–Schwarz bounds the number of inputs that any sparse target can capture. The sections below show which form of that route is false, which form is exactly as strong as the problem, and what an exact computation shows about each.

The density-zero form is false

The proposed form removes a density-zero set of degenerate inputs, such as the semiprimes with $s(pq)=p+q+1$ for distinct primes $p,q$, and asks for $C_S(x)=O(x)$ for the rest. No density-zero exclusion can give this.

Theorem A. Let $E_x\subseteq[1,x]$ satisfy $|E_x|=o(x)$ and put $S_x=[1,x]\setminus E_x$. Then $C_{S_x}(x)/x\to\infty$.

Proof. Fix $\delta\in(0,1/10]$ and put $z=1/\delta$ and $Z=100z$. Let $\mathcal R_Z$ be the set of positive integers with no prime factor at most $Z$ (so $1\in\mathcal R_Z$), let $m_Z=\prod_{q\le Z}(1-1/q)$, and let $f(k)=\sum_{q\mid k}1/(q-1)$, summed over the primes $q$ dividing $k$.

A class with small $s(n)/n$. Let $V_\delta$ be the set of $n=pk$ with $p$ prime, $z/1.8<p\le z/1.1$, $k\in\mathcal R_Z$ and $f(k)\le\delta/10$. Since $p<Z$, $p\nmid k$, so $\sigma(n)/n=(1+1/p)\,\sigma(k)/k$ and

$$1\le\frac{\sigma(k)}{k}\le\prod_{q\mid k}\Bigl(1-\frac1q\Bigr)^{-1} \le e^{f(k)}\le1+0.11\,\delta,$$

so $\delta\le1/p\le s(n)/n\le(1+1.8\delta)(1+0.11\delta)-1\le2\delta$. The pair $(p,k)$ is determined by $n$, since $p$ is the only prime factor of $n$ at most $Z$.

Its size. $\mathcal R_Z$ is a union of residue classes modulo $\prod_{q\le Z}q$, so $\#(\mathcal R_Z\cap[1,y])=m_Zy+O_Z(1)$. For a prime $q>Z$, the members of $\mathcal R_Z$ divisible by $q$ are exactly the $qk'$ with $k'\in\mathcal R_Z$. Hence

$$\sum_{\substack{k\le y\\ k\in\mathcal R_Z}}f(k) =\sum_{Z<q\le y}\frac{\#(\mathcal R_Z\cap[1,y/q])}{q-1} \le m_Zy\sum_{m>Z}\frac1{m(m-1)}+O_Z(\log\log y) \le\frac{1.01\,m_Zy}{Z}+O_Z(\log\log y),$$

using $\sum_{m>Z}1/(m(m-1))=1/\lfloor Z\rfloor$ and $Z\ge1000$. By Markov's inequality at most $(10/\delta)\,1.01\,m_Zy/Z+o(y)\le0.11\,m_Zy+o(y)$ members of $\mathcal R_Z\cap[1,y]$ have $f(k)>\delta/10$. Summing over the finitely many $p$,

$$\#(V_\delta\cap[1,x])\ge(d_\delta-o(1))\,x,\qquad d_\delta=0.89\,m_Z\sum_{z/1.8<p\le z/1.1}\frac1p.$$

As $\delta\to0$, Mertens' product theorem gives $m_Z\sim e^{-\gamma}/\log(1/\delta)$ and the prime number theorem gives $\sum_{z/1.8<p\le z/1.1}1/p\sim\log(18/11)/\log(1/\delta)$. Since $0.89\,e^{-\gamma}\log(18/11)>0.24$, every sufficiently small $\delta$ has $d_\delta\ge0.2/\log^2(1/\delta)$. The convergence is slow, because $\log Z=\log(1/\delta)+\log100$; the bound is asymptotic and is not claimed for every $\delta\le1/10$.

Pigeonhole. The set $N_x=(V_\delta\cap[1,x])\setminus E_x$ has at least $(d_\delta-o(1))x$ members, and their values $s(n)$ lie in $[1,2\delta x]$. By Cauchy–Schwarz over those values,

$$C_{S_x}(x)\ge\sum_{m\le2\delta x}r_{N_x,x}(m)^2\ge\frac{|N_x|^2}{2\delta x} \ge\frac{(d_\delta-o(1))^2}{2\delta}\,x .$$

So $\liminf_x C_{S_x}(x)/x\ge d_\delta^2/(2\delta)\ge 0.02/(\delta\log^4(1/\delta))$ for every small $\delta$, which is unbounded as $\delta\to0$. $\square$

The mechanism is general: the inputs of $V_\delta$, whose least prime factor is near $1/\delta$ and whose other prime factors are large, have their values in $[1,2\delta x]$ although they range over $[1,x]$, and they form a proportion of order $1/\log^2(1/\delta)$ of all inputs. The degenerate families that an exclusion is meant to remove, such as the semiprimes, are only the extreme case. The same argument with $|E_x|\le\varepsilon x$ gives

$$\liminf_{x\to\infty}\frac{C_{S_x}(x)}{x} \ge\sup_{\delta:\ d_\delta>\varepsilon}\frac{(d_\delta-\varepsilon)^2}{2\delta},$$

and taking $\delta=\exp(-\sqrt{0.1/\varepsilon})$, where $d_\delta\ge2\varepsilon$ for small $\varepsilon$, shows that the right side is at least $(\varepsilon^2/2)\exp(\sqrt{0.1/\varepsilon})$. Any constant $K$ in the corrected form below must therefore grow faster than every power of $1/\varepsilon$.

This proof has the independent review and grade described above. Its external inputs are the prime number theorem and Mertens' product theorem, at the retained statement-interface reading depth.

An equivalent form of the problem

The second-moment route loses nothing in principle once the exceptional set may have small positive density and the constant may depend on it.

Theorem B. The following are equivalent.

1. Every $A\subseteq\mathbb N$ of natural density zero has $s^{-1}(A)$ of natural density zero.
2. The fiber sizes are tight: $\lim_{T\to\infty}\limsup_{x\to\infty}x^{-1}\#\{n\le x:r_x(s(n))>T\}=0$.
3. For every $\varepsilon>0$ there are $K$ and $x_0$ such that every $x\ge x_0$ has a set $E_x\subseteq[1,x]$ with $|E_x|\le\varepsilon x$ and $C_{[1,x]\setminus E_x}(x)\le Kx$.
4. For every $\varepsilon>0$ there are $\eta>0$ and $x_0$ such that $\#\{n\le x:s(n)\in B\}\le\varepsilon x$ for every $x\ge x_0$ and every set $B\subseteq\mathbb N\cup\{0\}$ of at most $\eta x$ integers.

Proof. (2 ⇒ 3) Given $\varepsilon$, choose $T$ and $x_0$ with $\#\{n\le x:r_x(s(n))>T\}\le\varepsilon x$ for $x\ge x_0$, and let $E_x$ be that set. Every value taken on $[1,x]\setminus E_x$ is taken at most $T$ times there, so $C_{[1,x]\setminus E_x}(x)\le T\,R_{[1,x]\setminus E_x}(x)\le Tx$.

(3 ⇒ 1) Let $A$ have density zero, and fix $\varepsilon>0$ and $L\ge1$. For $x\ge x_0$ the inputs $n\le x$ with $s(n)\in A$ are of three kinds. At most $\varepsilon x$ lie in $E_x$. Those with $s(n)>Lx$ have $\sigma(n)/n>L+1$, and

$$\sum_{n\le x}\frac{\sigma(n)}n=\sum_{d\le x}\frac1d\Bigl\lfloor\frac xd \Bigr\rfloor\le\zeta(2)\,x$$

bounds their number by $2x/L$. The rest have values in $A\cap[0,Lx]$, and Cauchy–Schwarz bounds them by $(|A\cap[0,Lx]|\cdot Kx)^{1/2}=o(x)$. So the upper density of $s^{-1}(A)$ is at most $\varepsilon+2/L$ for all $\varepsilon$ and $L$.

(1 ⇒ 2) Suppose (2) fails. The quantity in (2) is nonincreasing in $T$, so there is $\eta>0$ such that for every $T$ arbitrarily large $x$ have $\#\{n\le x:r_x(s(n))>T\}>\eta x$. Write $m_w=\prod_{p\le w}(1-1/p)$ and fix $L\ge4$ with $L^{-1/2}+m_{\sqrt L}\le\eta/4$. An input $n\in(x/\sqrt L,x]$ with $s(n)<x/L$ has $\sigma(n)/n<1+L^{-1/2}$, hence no prime factor at most $\sqrt L$; so $\#\{n\le x:s(n)<x/L\}\le(\eta/4)x+O_L(1)$. Put $x_0=1$ and choose inductively $x_k$ with $\#\{n\le x_k:r_{x_k}(s(n))>k\}>\eta x_k$, $x_k\ge k^2L\,x_{k-1}$ and $O_L(1)\le\eta x_k/4$, and let

$$A_k=\{m\ge x_k/L:\ r_{x_k}(m)>k\},\qquad A=\bigcup_kA_k .$$

Then $\#\{n\le x_k:s(n)\in A\}\ge\eta x_k-(\eta/2)x_k$, so $s^{-1}(A)$ has upper density at least $\eta/2$. On the other hand $|A_k|<x_k/k$, because $\sum_mr_{x_k}(m)\le x_k$, so $\sum_{j<k}|A_j|<\sum_{j<k}x_j\le k\,x_{k-1}$. If $k$ is the largest index with $x_k/L\le y$, then $A\cap[1,y]\subseteq A_1\cup\dots\cup A_k$ and

$$\frac{|A\cap[1,y]|}{y}\le\frac{L}{x_k}\Bigl(\frac{x_k}{k}+k\,x_{k-1}\Bigr) \le\frac{L}{k}+\frac1k ,$$

so $A$ has density zero and (1) fails.

(2 ⇔ 4) If (2) holds, choose $T$ with $#{n\le x:r_x(s(n))>T}\le\varepsilon x/2$ for large $x$ and put $\eta=\varepsilon/(2T)$: a set $B$ of at most $\eta x$ values captures at most $\varepsilon x/2$ inputs in fibers larger than $T$ and at most $T\eta x$ others. If (4) holds, take $T=1/\eta$; the set $B=\{m:r_x(m)>T\}$ has fewer than $x/T=\eta x$ members, so it captures at most $\varepsilon x$ inputs. The quantity in (2) is therefore at most $\varepsilon$ for every $T\ge1/\eta$, and $\varepsilon$ is arbitrary. $\square$

This proof has the independent review and grade described above. Statement 4 places the problem beside [[../library/arithmetic_functions/pollack_2018_divisor_sum_fibers/theorem_1_2|Pollack, Pomerance and Thompson's Theorem 1.2, which proves the same uniform control for targets of total size at most $x^{1/2+\epsilon(x)}$]]: the problem asks for it at size $\eta x$, with $\varepsilon\to0$ as $\eta\to0$. The problem page's countermodel shows that control at the smaller size cannot be extended by an abstract argument; Theorem B says that control at linear size is exactly what is needed.

Statement 3 is the corrected form of the route. Its natural explicit version uses the sets

$$D(\delta)=\{n\ \text{composite}:\ s(n)\ge\delta n\}.$$

An input outside $D(\delta)$ is $1$, a prime, or has no prime factor at most $1/\delta$, so the complement of $D(\delta)$ has upper density at most $\prod_{p\le1/\delta}(1-1/p)\to0$. The bound

$$\text{(D)}\qquad C_{D(\delta)}(x)\le K_\delta\,x\quad\text{for every fixed }\delta>0\text{ and all large }x$$

therefore implies statement 3, by choosing $\delta$ with $\prod_{p\le1/\delta}(1-1/p)<\varepsilon$ and $E_x=[1,x]\setminus D(\delta)$, and hence the problem. The argument does not establish the converse. The class $V_\delta$ of Theorem A, apart from its finitely many primes, lies in $D(\delta)$, so the pigeonhole step gives $K_\delta\ge0.02/(\delta\log^4(1/\delta))$ for small $\delta$, and $K_\delta\to\infty$. The semiprimes in $D(\delta)$ are harmless. For distinct primes $p<q$, $s(pq)=p+q+1\ge\delta pq$ forces $p\le3/\delta$, and $s(p^2)=p+1$ comes from one prime square at most; so each value is taken by at most $\pi(3/\delta)+1$ semiprimes in $D(\delta)$, and they contribute at most $(\pi(3/\delta)+1)x$ to the sum of squares, which $(r_1+r_2)^2\le2r_1^2+2r_2^2$ separates from the rest.

Exact census through 10^10

The census computes $R$ and $C$ exactly for composite inputs $n\le x$ in the classes

$$D_k=\{n\ \text{composite}:\ 2^ks(n)\ge n\},\qquad DT_k=\{n\in D_k:\ \Omega(n)\ge3\},$$

for $k=0,\dots,14$, and for all composites, written $Q$, and the composites with $\Omega(n)\ge3$, written $T$; $\Omega$ counts prime factors with multiplicity. So $D_k=D(2^{-k})$, $D_0$ is the set of abundant and perfect composites, and $T$ is the proposed density-zero exclusion: it removes $1$, the primes and every input with exactly two prime factors, including the prime squares with $s(p^2)=p+1$. By Landau's estimate at most $O(x\log\log x/\log x)$ inputs $n\le x$ are removed, so the excluded set has density zero, as the proposed route requires. The checkpoints are $x=\lfloor10^{3+j/4}\rfloor$, $j=0,\dots,28$, so the range reaches $10^{10}$. All arithmetic is exact. The program and its checks are described in the evidence record.

The ratio $C/x$ at each decade, rounded to four decimals:

At $x=10^9$ the exact restricted values are $C_T=2323154413$ and $R_T=788363929$, with $211636071$ excluded inputs. At $x=10^{10}$ the exact values are $C_T=25949106637$ with $R_T=8051171045$, $C_{D_0}=4143350299$ with $R_{D_0}=2476238927$, and $C_{D_4}=21185514129$ with $R_{D_4}=8188172635$. The proportions of $n\le10^{10}$ outside each class are $0.1949$ for $T$, $0.7524$ for $D_0$, $0.2997$ for $D_2$, $0.1812$ for $D_4$ and $0.1280$ for $D_6$.

Reading. No numerical tolerance for leveling off was fixed in advance. This page reads "levels off" as decade increments of $C/x$ that shrink from one decade to the next, and "does not level off" as increments that do not shrink. The reading was chosen when the computation was made, so the census is exploratory evidence, not a test of a prediction stated in advance.

• The ratio for $Q$ rises by a factor of about $1.8$ from $10^9$ to $10^{10}$. Balanced semiprimes alone force superlinear collisions: there are $\gg x/\log^2x$ distinct products $pq$ with $\sqrt x/2<p<q\le\sqrt x$, and their values $p+q+1$ occupy $O(\sqrt x)$ integers. Cauchy–Schwarz therefore gives $C_Q(x)\gg x^{3/2}/\log^4x$. This aggregate argument uses the prime number theorem; it asserts no estimate for any individual Goldbach sum.
• $T$ does not level off. Its decade increments from $10^6$ to $10^{10}$ are $0.228$, $0.229$, $0.243$ and $0.272$. Theorem A predicts unbounded growth for every density-zero exclusion, but its pigeonhole bound is far below this observed rate at these scales; the observed rate is not derived here. The growth sits in inputs with small $s(n)/n$. At $10^6$ the classes $T$ and $DT_6$ differ by seven inputs, the prime cubes $p^3$ with $64<p<100$, and their ratios agree to four decimals; at $10^{10}$ the ratios differ by $0.435$, and $DT_6$ alone decelerates.
• Every $D_k$ with $k\le7$ and every $DT_k$ with $k\le6$ levels off over the last three decades. From $10^6$ to $10^{10}$ the increments of $D_0$ are $0.0098$, $0.0055$, $0.0030$, $0.0015$; of $D_4$ $0.0434$, $0.0251$, $0.0152$, $0.0085$; of $DT_2$ $0.0708$, $0.0457$, $0.0315$, $0.0219$; of $DT_6$ $0.2230$, $0.1440$, $0.0956$, $0.0757$. Classes with larger $k$ are still in transition: the restriction $s(n)\ge2^{-k}n$ removes nothing when $x\le4^k$, since $s(n)>\sqrt n$ for composite $n$.
• Large fibers are rare in the restricted classes. At $10^{10}$ the largest value multiplicity is $47$ in $D_0$ and $68$ in $D_4$, against $124$ in $T$; the number of admitted inputs lying in fibers of more than eight inputs, divided by $x$ rather than by $R_S(x)$, is $0.0007$ in $D_0$, $0.0128$ in $D_4$ and $0.0472$ in $T$.

Split by the largest prime factor. A second run of the census flags the inputs whose largest prime factor exceeds $\sqrt n$, a set of density $\log2$, which is the model of the class S1 of the sketch below; the other composites model S2. The table gives the ordered off-diagonal pairs per input range, $(C-R)/x$, inside S1, inside S2, and across the two (both orders), within $D_k$:

For $k\le4$ the S1 and cross components level off, the S2 components for $k\le2$ level off or decrease, and the S2 component for $k=4$ stays between $0.093$ and $0.099$. The collisions inside S2 are far fewer than inside S1. Without the restriction the S2 part does not level off: over all composites with no prime factor above $\sqrt n$, $(C-R)/x$ is $0.1267$, $0.1667$, $0.2158$ and $0.2779$ at $10^6,\dots,10^9$, with growing increments, again from inputs with small $s(n)/n$.

The census cannot prove any asymptotic statement, and over these few decades it cannot separate convergence of $C/x$ from growth like $\log\log x$. It does show that the proposed density-zero route fails its own test already at these scales, and that nothing in the corrected classes, or in their split by the largest prime factor, contradicts (D).

Proof sketch of the corrected route

Target. Statement 3 of Theorem B, through the sufficient bound (D) or another family of exceptional sets whose upper densities tend to zero. Theorems A and B have the independent review described above. The argument in this section remains conditional on G1, whose trimmed form is reduced to Gap G1-core below, and on G2. Its coefficient-uniform sieve input G0 has a complete author-recorded proof.

Splitting. For a disjoint union of two admitted classes, $(r_1+r_2)^2\le2r_1^2+2r_2^2$, so bounds for collisions inside the classes control collisions between them too. A finite number of classes is allowed, with that number depending on the discarded proportion.

Class S1: a prime factor above $x^{1/2+\eta}$. Fix $\delta>0$ and $0<\eta<1/2$, put $y=x^{1/2-\eta}$, and let $S_1(x)$ consist of the composite $n=aP\le x$ with $P$ prime, $P>x^{1/2+\eta}$ and $s(n)\ge\delta n$. Then $P\nmid a$, $2\le a<y$, and

$$s(aP)=s(a)P+\sigma(a).$$

Since $\sigma(a)/a\le1+\log a$, the identity $s(aP)/(aP)=s(a)/a+\sigma(a)/(aP)$ implies $s(a)\ge\delta a/2$ for sufficiently large $x$, uniformly over these cofactors. The same cofactor $a$ gives no off-diagonal collision because $s(a)>0$.

For $a\ne a'$ put $g=\gcd(s(a),s(a'))$ and $h=\sigma(a')-\sigma(a)$. A collision requires

$$s(a)P-s(a')P'=h,\qquad g\mid h.$$

If $h=0$, there is no collision in these ranges: both large primes exceed $s(a)$ and $s(a')$ for large $x$, so $s(a)P=s(a')P'$ forces $P=P'$ and $s(a)=s(a')$; together with $h=0$ this forces $a=a'$. For $h\ne0$ and $g\mid h$, write the integer solutions as

$$P=u+\beta t,\quad P'=v+\lambda t,\qquad \beta=s(a')/g,\quad\lambda=s(a)/g.$$

The admissible parameters lie in an interval contained in one of length $L=xg/(a\,s(a'))$. This is an upper length supplied by the first prime's range, not a lower bound on the actual intersection of the two prime ranges. It satisfies $L\ge x^{2\eta}/(1+\log x)$, so it is possible to enlarge the interval to this length when seeking an upper bound.

For these pairs let $\omega_{a,a'}(p)$ be the number of residues $t$ modulo $p$ where at least one of the two forms vanishes, and define

$$\mathfrak S_z(a,a')=\prod_{p\le z} \Bigl(1-\frac{\omega_{a,a'}(p)}p\Bigr)\Bigl(1-\frac1p\Bigr)^{-2}.$$

A different integer solution $(u,v)$ translates $t$ and leaves this quantity unchanged. Pairs with no solution, and pairs with $h=0$, are omitted from the sum below; no singular series is needed for them.

Uniform sieve input G0. For every eligible pair,

$$N(a,a')\ll_\eta \frac{xg}{a\,s(a')\log^2x}\,\mathfrak S_x(a,a').$$

The coefficient-uniform two-form argument proves this by an explicit Selberg-weight construction. Its interval error is bounded by $z^2(1+\log z)^{10}$, uniformly in all coefficients. With $z=L^{1/4}$ and $L\ge x^{2\eta}/(1+\log x)$, this error is absorbed in the main term, whose singular series has an absolute positive lower bound whenever the forms are admissible. Nonprimitive forms and the obstruction $\omega(2)=2$ give no prime pair in the stated range. The proof also justifies the interval enlargement and the comparison of singular-series cutoffs. G0 therefore needs no separate error sum and no unexamined sieve theorem. It is author-recorded and outside the analytic review.

Set $\mathcal A_\delta(y)=\{2\le a\le y:s(a)\ge\delta a/2\}$ and

$$\Lambda_\delta(y;z)= \sum_{\substack{a,a'\in\mathcal A_\delta(y),\ a\ne a'\\ h\ne0,\ g\mid h}} \frac{g\,\mathfrak S_z(a,a')}{aa'}.$$

Summing G0 gives

$$C_{S_1}(x)\ll_\eta x+ \frac{x}{\delta\log^2x}\Lambda_\delta(x^{1/2-\eta};x).$$

Gap G1 (averaged singular series). The missing estimate is $\Lambda_\delta(x^{1/2-\eta};x)\ll_{\delta,\eta}\log^2x$ for every fixed $\delta,\eta$. The prime cutoff and the cofactor cutoff are distinct in this statement. For statement 3 of Theorem B a trimmed form suffices: the same bound after removing a set of cofactors $a<y$ with $\sum1/a\le\varepsilon'\log y$, whose inputs number at most $2\varepsilon'x$. The trimmed form is not G1 as stated for every cofactor, and it does not prove (D) for the whole class.

The averaged estimate for the large-prime class reduces the trimmed form to one estimate, at author standing. Its local lemmas are complete. For an eligible pair and a prime $p$, with $e_1=v_p(s(a))$, $e_2=v_p(s(a'))$, $e=v_p(g)$ and $f=v_p(h)$, the factor of $g\,\mathfrak S_x(a,a')$ is $p^ep/(p-1)$ when $e_1\ne e_2$ or $f>e$, and $p^e(1-(p-1)^{-2})$ otherwise; a pair with $e_1\ne e_2$ and $f>e$ has no prime solutions. If $v_p(\sigma(a))>v_p(a)$ and the same holds for $a'$, the factor is exactly $p^{v_p(a)}p/(p-1)$, and the pair is excluded unless $v_p(a)=v_p(a')$. Under independent geometric valuation frequencies $\Pr(v_p(a)=e)=(1-1/p)p^{-e}$ this factor has mean

$$\sum_{e\ge0}(1-1/p)^2p^{-2e}\,p^e\frac p{p-1}=1,$$

which is why the small primes balance. The page then restricts to good cofactors: their two largest primes exceed $y^\nu$, $(a,\sigma(a))$ has no prime factor above $W=\log\log y/(4\log\log\log y)$, and $\sigma$ satisfies the small-prime condition above at the cofactor $a/P(a)$; the others carry reciprocal weight $O(\nu\log(1/\nu)\log y)$. For good cofactors, every divisor $d$ of the part of $g$ above $W$ fixes the largest prime of $a$ modulo $d$, and the congruence $a\equiv a'\pmod d$ restricts the second largest prime through a quadratic congruence. Removing a further set of reciprocal weight $o(\log y)$ ensures that no prime above $W$ divides both $s(a)$ and $P(a)-P_2(a)$. The quadratic roots representing retained cofactors are then simple, giving at most $2^{\omega(d)}$ classes modulo $d$, including all its prime powers. With Brun–Titchmarsh this bounds by $O(\log^2y)$ every part of the trimmed sum except the divisors $d$ with $d\ge\max(P(a),P(a'))^{1-\kappa}$, for a fixed small $\kappa>0$. The density of these good cofactors and the averages of the small-prime and singular-series weights are proved there, with constants not computed explicitly.

Gap G1-core. Bound by $O(\log^2y)$ the part of the trimmed sum coming from common divisors $d$ of $s(a)$ and $s(a')$ that divide $a'-a$ and satisfy $d^{1/(1-\kappa)}\ge\max(P(a),P(a'))$. When such a $d$ exceeds $P(a)$, the prime $P(a)$ is the least positive residue of $-\sigma(c)s(c)^{-1}$ modulo $d$, where $c=a/P(a)$. The estimate asks that these least residues fall in short initial intervals no more often than their length predicts, on average over the pairs. Counting arguments on the linked page remove further core divisors: those for which a cofactor has its largest prime above $d^{1-\kappa''}$, for a small fixed $\kappa''$; those for which the two cofactors' largest primes have product at least $d^{1+\epsilon}$; and those for which one cofactor's two largest primes have product at least $d^{1+\epsilon}$. The simple-root condition makes these removals valid without a restriction on the radical of $d$. What remains are large divisors exceeding, up to the factor $d^\epsilon$, the product of the two largest of these primes. It is not a local-factor question; it is the same kind of input as the slice estimate G2-bal below.

The reduced coefficients $u=s(a)/g$, $v=s(a')/g$ of an eligible pair satisfy $\max(u,v)>\prod_{p\le W}p/(1+2/\delta)$, by Lemma 7 of the linked page. In particular $s(a)=s(a')$ gives no retained eligible pair for large $y$. A descent must therefore control scaled collisions with unbounded reduced coefficients and the extra congruence, rather than just ordinary cofactor fibers. Removing the largest primes produces affine equations with a generally nonzero right side; no contracting recurrence is derived. Luca and Pomerance avoid this regime only through cofactors $m=q\ell$ with $\ell\le x^{1/10}$, as recorded on the linked page.

What S1 would give. G0 and G1 imply $C_{S_1}(x)\ll_{\delta,\eta}x$. G0 with the trimmed form of G1 implies the same bound after removing at most $\varepsilon x$ inputs, for each $\varepsilon>0$. The upper-output truncation in Theorem B then gives $\#\{n\in S_1(x):s(n)\in A\}=o(x)$ for every density-zero $A$. The number of integers with a prime factor above $x^{1/2+\eta}$ is $(\log(2/(1+2\eta))+o(1))x$, by summing $\lfloor x/P\rfloor$ over those primes; the factor is unique. The discarded condition $s(n)<\delta n$, together with the primes, has upper density tending to zero as $\delta\to0$. Therefore G0 and either form of G1 for every $\delta,\eta$ would bound the upper density of $s^{-1}(A)$ by $1-\log2$. This is conditional partial progress, not a conclusion proved by the census or by this sketch.

Class S2: the remaining inputs. Define $S_2(x)=(D(\delta)\cap[1,x])\setminus S_1(x)$. The unrestricted complement of the large-prime class has density $1-\log(2/(1+2\eta))$, tending to $1-\log2$ as $\eta\to0$; it cannot be discarded as a density-zero family.

The S1 argument supplies no uniform long parameter interval for S2. On input intervals comparable to $x$, cofactors can exceed $\sqrt x$; for pairs with small $g$ the upper interval length is then only $O(1)$ or smaller. This defeats that particular uniform application of the sieve, not every possible use of a large prime factor or another parametrization.

Gap G2. Prove $C_{S_2}(x)\ll_{\delta,\eta}x$, or replace this estimate by another decomposition that proves Theorem B's statement 3. The section on rough-prime classes below gives such a decomposition. It reduces the smooth inputs to explicit collision equations in boundedly many primes, proves that the uncovered proportion tends to zero, and identifies a failure of summing a unit error over every pair of short-slice fixed parts.

Standing and composition. Theorems A and B and the implication (D) $\Rightarrow$ E0955 have the independent review and separate grade described above; (D) itself remains unproved. G0 has the author-recorded uniform sieve proof linked above; G1 is reduced, in its trimmed form, to Gap G1-core; G2 is the unresolved smooth-input estimate. For every $\delta$, G0 and G1 together with G2 at one fixed $\eta\in(0,1/2)$ would give (D), because $C_{D(\delta)}(x)\le2C_{S_1}(x)+2C_{S_2}(x)$; with the trimmed forms of G1 and G2 they would give statement 3 of Theorem B instead. Theorem B then supplies the image truncation and the density conclusion. Theorem C below replaces this split by rough-prime classes whose collision bounds would give statement 3 without (D). The finite census proves none of G0, G1, G2 or the class bounds. There is no full proof of E0955 here.

The density-zero exclusion route is closed by the independently reviewed Theorem A at its stated source interfaces. Changing which density-zero family is removed cannot reopen it. Allowing an arbitrarily small positive discarded proportion gives the distinct, still-open route of Theorem B; establishing that reformulation does not establish its equivalent asymptotic assertion.

Rough-prime classes

This section gives a decomposition that meets statement 3 of Theorem B directly. Outside a set of at most $\varepsilon x$ inputs, every input is a small cofactor times a bounded number of distinct large primes, so G2 becomes a finite family of explicit collision equations. The section describes the estimates needed for these equations and the limitation of summing unit errors over all short slices. Theorem C, the identities and Proposition D below are complete author-recorded arguments, outside the review described above. The full-equation local calculation gives an alternative that can omit every fixed finite list of small prime counts; its remaining gap is the global count G2-full below. The gaps G2-quad, G2-bal and G2-full are open.

Classes. Fix $0<\theta<\tau<1$ and $\delta>0$, and put $y=x^\theta$. For $n\le x$ let $b(n)$, the product of the prime powers $p^a\,\|\,n$ with $p\le y$, be its $y$-smooth part, and let $e(n)=n/b(n)$ be its $y$-rough part. For $k\ge1$ let $\mathcal S_k(x)$ be the set of $n\in D(\delta)\cap[1,x]$ with $b(n)\le x^\tau$ whose rough part $e(n)$ is squarefree with exactly $k$ prime factors. Writing $P(b)$ for the largest prime factor of $b$, with $P(1)=1$, its members are

$$n=bP_1\cdots P_k,\qquad P(b)\le x^\theta<P_k<\dots<P_1,\qquad b\le x^\tau,$$

and $k<1/\theta$, since $x^{k\theta}<e(n)\le x$. Since $\sigma(e)/e=\prod_i(1+1/P_i)\le\exp(x^{-\theta}/\theta)$, membership in $D(\delta)$ forces $s(b)\ge\delta b/2$ for large $x$; in particular $b\ge2$.

Theorem C. (i) For every $x$ with $x^\theta\ge2$, with an absolute implied constant,

$$\#\Bigl((D(\delta)\cap[1,x])\setminus\bigcup_{1\le k<1/\theta} \mathcal S_k(x)\Bigr) \le\frac\theta\tau\,x+O\Bigl(\frac{x}{\tau\log x}\Bigr) +\frac{2x}{x^\theta}+x^\tau .$$

(ii) Suppose that there is $\tau\in(0,1)$ such that for every $\delta>0$, every $\theta\in(0,\tau)$ and every $k<1/\theta$ there are $K$ and $x_0$ with $C_{\mathcal S_k(x)}(x)\le Kx$ for all $x\ge x_0$. Then every set of natural density zero has a preimage under $s$ of natural density zero.

Proof. (i) An input $n\in D(\delta)\cap[1,x]$ in no class has $b(n)>x^\tau$, or a rough part that is not squarefree, or $e(n)=1$. Since $\log b(n)$ is the sum of $\log p$ over the prime powers $p^j\mid n$ with $p\le y$,

$$\sum_{n\le x}\log b(n)=\sum_{p\le y}\log p\sum_{j\ge1} \Bigl\lfloor\frac{x}{p^j}\Bigr\rfloor \le x\sum_{p\le y}\frac{\log p}{p-1}=x\,(\theta\log x+O(1))$$

by Mertens' estimate $\sum_{p\le y}(\log p)/p=\log y+O(1)$, and Markov's inequality bounds the inputs of the first kind by $(\theta/\tau)x+O(x/(\tau\log x))$. A rough part that is not squarefree is divisible by $p^2$ for some prime $p>y$, which happens for at most $\sum_{p>y}x/p^2\le x/(y-1)\le2x/y$ inputs. If $e(n)=1$ and $b(n)\le x^\tau$, then $n\le x^\tau$. Every other $n$ lies in $\mathcal S_k(x)$, with $k$ the number of prime factors of $e(n)$.

(ii) Let $0<\varepsilon<1$. Choose $\delta$ with $\prod_{p\le1/\delta}(1-1/p)<\varepsilon/3$ and put $\theta=\varepsilon\tau/3$. An input outside $D(\delta)$ is $1$, a prime, or a composite with no prime factor at most $1/\delta$, because a composite $n$ with a prime factor $p\le1/\delta$ has $s(n)\ge n/p\ge\delta n$. The last kind is a union of residue classes modulo $\prod_{p\le1/\delta}p$, so for large $x$ at most $(\varepsilon/3)x+o(x)$ inputs lie outside $D(\delta)$. Let $E_x$ be the complement in $[1,x]$ of the union of the classes. By (i), $|E_x|\le\varepsilon x$ for large $x$. There are at most $1/\theta$ classes, and $(r_1+\dots+r_L)^2\le L(r_1^2+\dots+r_L^2)$ gives

$$C_{[1,x]\setminus E_x}(x)\le\frac1\theta\sum_{1\le k<1/\theta} C_{\mathcal S_k(x)}(x)\le\frac{K}{\theta^2}\,x$$

for large $x$, with $K$ the largest of the finitely many class constants. This is statement 3 of Theorem B. $\square$

Since $\mathcal S_k(x)\subseteq D(\delta)$ and $C_S$ increases with $S$, (D) for every $\delta$ implies the hypothesis of (ii). The converse is not shown, and the hypothesis of (ii) is not shown to follow from the problem. The logarithmic prime sum in (i) has an elementary proof. The Chebyshev bound established in the sieve argument gives $\sum_{p^j\le N}\log p\ll N+\sqrt N\log N\ll N$. For an integer $N\ge2$, counting prime factors in $N!$ and replacing each floor by its argument gives

$$\log N!=\sum_{p^j\le N}\Bigl\lfloor\frac{N}{p^j}\Bigr\rfloor\log p =N\sum_{p^j\le N}\frac{\log p}{p^j}+O(N).$$

Integral bounds give $\log N!=N\log N-N+O(\log N)$, and $\sum_{p,j\ge2}(\log p)/p^j=\sum_p(\log p)/(p(p-1))<\infty$. This proves the stated logarithmic prime sum, also for real $y$ by taking $N=\lfloor y\rfloor$. Thus the only external asymptotic input of Theorem C is Mertens' product theorem. The reduction fixes the shape of the inputs that need collision bounds and proves none of those bounds.

The class equations. For $n=bP_1\cdots P_k\in\mathcal S_k(x)$, multiplicativity of $\sigma$ gives

$$s(n)=\sigma(b)\prod_{i\le k}(P_i+1)-b\prod_{i\le k}P_i ,$$

an equation of degree $k$ in the primes whose coefficients $\sigma(b)$ and $b$ are at most $x^{\tau+o(1)}$. For pairwise coprime $d,e_1,e_2$, expanding $s(de)=\sigma(d)\sigma(e)-de$ gives the exact identity

$$s(de_1)\,s(de_2)=s(d)\,s(de_1e_2)+d\,\sigma(d)\,s(e_1)\,s(e_2).$$

Put $X_i=s(bP_i)=s(b)P_i+\sigma(b)$. Since $s(P)=1$ for a prime $P$, the identity with $d=b$, $e_1=P_1$ and $e_2=P_2$ gives the two-prime identity

$$s(b)\,s(bP_1P_2)=X_1X_2-b\,\sigma(b).$$

With $e_1=P_1$ and $e_2=P_2P_3$ instead, the two-prime identity for $P_2P_3$ and $s(b)(P_2+P_3+1)=X_2+X_3-2\sigma(b)+s(b)$ give

$$s(b)^2\,s(bP_1P_2P_3)=X_1X_2X_3-b\,\sigma(b)(X_1+X_2+X_3) +b\,\sigma(b)\bigl(2\sigma(b)-s(b)\bigr).$$

So inside $\mathcal S_1$ a collision is a linear equation in two primes. Inside $\mathcal S_2$ it is the split quaternary quadric

$$s(b')X_1X_2-s(b)X_1'X_2'=s(b')\,b\,\sigma(b)-s(b)\,b'\sigma(b')$$

in four variables of the form $s(b)P+\sigma(b)$. Inside $\mathcal S_3$ it is a cubic in six such variables, and inside $\mathcal S_k$ an equation of degree $k$ in $2k$ primes.

Full equations with at least three primes per input. The local-density argument gives a possible replacement for the linear and quadratic slices. Fix $0<\beta<1/2$. The number of prime divisors of a uniform $n\le x$ in $(x^\theta,x^\beta]$ has mean $M=\log(\beta/\theta)+o(1)$ and variance at most $M+o(1)$, by expanding its first two moments. Thus, for every fixed $K<M$, the proportion of inputs with at most $K$ prime factors above $x^\theta$ is at most $(M+o(1))/(M-K)^2$. This proportion tends to zero as $\theta\to0$. Theorem C's proof therefore still reaches Theorem B(3) if its class estimates are required only for $k\ge3$: the small-$k$ classes can be included in the discarded proportion.

For $b,b'\le y=x^\tau$, the linked page removes a set of reciprocal weight $o(\log y)$ so that $v_p(\sigma(b))>v_p(b)$ for $p\le W$ and $p\nmid(b,\sigma(b))$ for $p>W$, where $W=\log\log y/(4\log\log\log y)$, and likewise for $b'$. This removes $o(x)$ class inputs. Let $\rho_{p^r}(b,b')$ be $p^r$ times the probability that the two class polynomials agree modulo $p^r$ when their $2k$ variables are independent units. For $k\ge3$ the page proves

$$\rho_{p^r}(b,b')=1+O_k(p^{-2}) \quad\bigl(p>k,\ p\nmid b\sigma(b)b'\sigma(b')\bigr)$$

uniformly in $r$, and $\rho_{p^r}\le1+O_k(p^{-1})$ at every prime where each coefficient pair is primitive. At a small prime, unequal valuations of $b,b'$ prevent an integer collision; equal valuations $e$ give $\sup_r\rho_{p^r}=p^e p/(p-1)$. The corresponding local majorant $\mathfrak U_k(b,b')$, defined there using these suprema and set to zero for incompatible small-prime valuations, satisfies

$$\sum_{b,b'\ {\rm retained}}\frac{\mathfrak U_k(b,b')}{bb'} \ll_k\log^2y.$$

The continuous output density obtained by weighting each prime variable with $dP/\log P$ is $O_{\delta,\theta,k}(1/(b\log x))$. On any fixed output interval $[0,Cx]$, the local majorants times the overlaps of these densities therefore have total $O(x)$.

Gap G2-full. Bound the off-diagonal prime solutions of the full class equations by these local and continuous majorants with aggregate error $O(x)$, uniformly over the growing cofactors and the prime boxes. The diagonal contributes at most $x$. The proved local average does not control this error. The full leading homogeneous form has a singular locus of codimension four regardless of $k$, even though the affine critical points at the coefficient-free primes are isolated. Thus increasing the number of free prime variables supplies no automatic point-counting theorem. With G2-full for all the retained classes, Theorem C, the small-$k$ removal and Theorem B would finish the whole target without G1-core or G2-quad. No such global estimate is proved here.

Slices. Assume $\tau<1/2$ from here on; Theorem C(ii) allows any $\tau\in(0,1)$. For a pair $(n,n')$ in one class and $1\le j\le k$, the $j$-slice fixes the parts $d=n/(P_1\cdots P_j)$ and $d'=n'/(P_1'\cdots P_j')$ and leaves the $j$ largest primes of each input free. The colliding pairs with given fixed parts are the prime solutions of the class equation with $b$ replaced by $d$ and $k$ by $j$, the free primes exceeding $P(d)$ and $P(d')$ respectively. There are about $x/d$ free $j$-tuples on one side and $x/d'$ on the other, up to logarithmic factors, and the common value has size about $x$. A heuristic count, not a proved estimate, therefore gives about $x/(dd')$ solutions for each pair of fixed parts, before logarithmic and local factors. Any bound for a class obtained by bounding each pair of fixed parts separately, with an error of at least one per pair, is at least the number of such pairs.

• The linear slice, $j=1$, is the S1 computation with cofactors $m=d$ and $m'=d'$, with two changes. When $h=0$ a collision with $P\ne P'$ forces $P\mid s(m')$, and a form vanishing identically modulo $p$ allows only $P=p$ or $P'=p$; these cases give $O(\log x)$ solutions per pair of fixed parts and need a count of their own, which is not made here. Otherwise the G0 argument, with a sieve exponent smaller than $\theta/2$, bounds a pair of fixed parts when $mm'\le x^{1-\epsilon}$, because the parameter interval then has length at least $x^{\epsilon}g/(1+\log x)$, and the analogue of G1 over the range $mm'\le x^{1-\epsilon}$ sums those bounds. Every pair in $\mathcal S_1$ is of this kind, with $mm'\le x^{2\tau}$.
• The quadratic slice, $j=2$, is the quaternary quadric with $b$ replaced by $d$. When $P_2P_2'\le x^{1-\epsilon}/(dd')$, fixing $P_2$ and $P_2'$ too reduces the pair to the linear slice. Otherwise each free prime lies within a factor $x^{\epsilon}\sqrt{dd'}$ of $\sqrt{x/d}$ or of $\sqrt{x/d'}$, on its own side. When the right side vanishes, for example when $d=d'$, the quadric becomes $s(d')X_1X_2=s(d)X_1'X_2'$, whose solutions include the diagonal and the coincidences between products of two such variables.
• The full slice, $j=k$, leaves only $b$ and $b'$ fixed, so $dd'\le x^{2\tau}$ and the heuristic count per pair is large, but the equation has degree $k$ in $2k$ primes.

Gap G2-quad. An upper bound of the heuristic order for the prime solutions of the quaternary quadric, uniform in fixed parts with $dd'\le x^{\gamma}$ for some fixed $\gamma>0$ and with congruence conditions sufficient for a four-dimensional upper-bound sieve, together with the matching average of its local factors and a count of the vanishing-right-side components. With $\tau<\gamma/2$, G0, the G1 analogue for pairs that reduce to the linear slice, and a bound for that slice's exceptional prime cases, it would settle $\mathcal S_2$, whose pairs all have $dd'\le x^{2\tau}$. None of these is proved here. A count of points on a quaternary quadric with error a power below its main term, such as a delta-method estimate, is a natural tool, but no uniform version has been checked against these coefficients and boxes. The right side is $\Delta(d,d')=d\sigma(d)s(d')-d'\sigma(d')s(d)$, the integer that governs large common divisors in the large-prime average. Its Lemma 5 shows that $\Delta(d,d')=0$ forces $d=d'$ when both fixed parts satisfy $v_p(\sigma(d))>v_p(d)$ for $p\le W$, have $(d,\sigma(d))$ free of primes above $W$, and have the same part supported on primes up to $W$. Fixed parts failing these conditions are not covered.

Proposition D (the cost of summing errors over all short slices). Let $J\ge1$ and $k\ge2J$, put $\alpha=1/(k+1/2)$, and fix $\iota>0$ with $k\alpha(1+\iota)<1$; then $c'=1-2J\alpha(1+\iota)>0$. Let $B_k(x)$ be the set of $n=mp_1\cdots p_k$ with $x/2<n\le x$, $m$ even and $p_1,\dots,p_k$ distinct primes in $(x^\alpha,x^{\alpha(1+\iota)}]$. Then $B_k(x)\subseteq D(1/2)$ and $\#B_k(x)\ge(c_k-o(1))x$ with $c_k=(\log(1+\iota))^k/(4\cdot k!)$. For every $j\le J$, every $j$-slice fixed part of a member of $B_k(x)$ exceeds $x^{(1+c')/2}/2$, so every pair of them has $dd'>x^{1+c'}/4$, and for large $x$ these fixed parts form at least $(c_k/2)^2x^{1+c'}$ pairs.

Proof. Since $\alpha>1/(k+1)$, every such $n$ has $m\le x^{1-k\alpha}<x^\alpha$, so $p_1,\dots,p_k$ are exactly the prime factors of $n$ above $x^\alpha$ and the representation is unique. For $p_1>\dots>p_k$ the $j$-slice fixed part is $d=n/(p_1\cdots p_j)>x^{1-j\alpha(1+\iota)}/2\ge x^{(1+c')/2}/2$. An even composite $n$ has $s(n)\ge n/2$. Counting the even $m$ in $(x/(2p_1\cdots p_k),\,x/(p_1\cdots p_k)]$,

$$\#B_k(x)\ge\frac x4\sum_{p_1<\dots<p_k}\frac1{p_1\cdots p_k} -\pi\bigl(x^{\alpha(1+\iota)}\bigr)^k ,$$

where Mertens' theorem gives $\sum_{x^\alpha<p\le x^{\alpha(1+\iota)}}1/p =\log(1+\iota)+o(1)$, the distinctness condition costs $o(1)$ because $\sum_{p>x^\alpha}p^{-2}\to0$, and the subtracted term is at most $x^{k\alpha(1+\iota)}=o(x)$. A fixed part determines $n$ up to the choice of $p_1,\dots,p_j$, so for large $x$ there are at least $(c_k/2)\,x^{1-j\alpha(1+\iota)}\ge(c_k/2)\,x^{(1+c')/2}$ distinct fixed parts, and at least $(c_k/2)^2x^{1+c'}$ pairs. $\square$

So for $j\le J$ the members of $B_k$ have heuristic count below $4x^{-c'+o(1)}$ for every pair of $j$-slice fixed parts, and those pairs exceed $x$ in number by a power. A summation that charges an error of at least one to every pair of represented fixed parts therefore cannot bound their collisions by $O(x)$. This does not count how many pairs remain after additional arithmetic eliminations: such eliminations require their own argument. Their positive density means that they cannot be deleted once $\varepsilon<c_k$. For $\delta\le1/2$ and $\theta<\alpha$, a member of $B_k$ outside the exceptions of Theorem C(i) lies in $\mathcal S_{k'}(x)$ for some $k'\ge k$, and its $j$ largest primes are still $p_1,\dots,p_j$. Those exceptions number at most $(\theta/\tau)x+o(x)$ and there are fewer than $1/\theta$ classes, so once $\theta/\tau<c_k/2$ some class contains at least $\theta c_kx/3$ members of $B_k$, and the counts above apply to them with a factor of order $\theta^2$. The all-pairs error sum already fails for the linear slices of $B_2$, and for the quadratic slices of $B_k$ with $k\ge4$. A proof confined to separate slice bounds with such an error must allow arbitrarily large slice degree as $\varepsilon\to0$. Proposition D concerns only that summation rule. It does not show that the collisions of $B_k$ are numerous, and it does not exclude other arguments.

Gap G2-bal. Bound the collisions of inputs whose large primes are comparable, such as $B_k$ for $k\ge4$, by an argument that does not bound each pair of short-slice fixed parts separately. Two forms of the missing input are visible.

• Detect slice solutions on average. With $Q=P+1$ and $Q'=P'+1$, the identity $s(mP)=m+(P+1)s(m)$ turns a collision of $n=mP$ and $n'=m'P'$ into $s(m)Q-s(m')Q'=m'-m$. With $g=\gcd(s(m),s(m'))$, a solution requires $g\mid m'-m$ and $Q\equiv((m'-m)/g)\,(s(m)/g)^{-1}$ modulo $s(m')/g$, with $Q-1$ a prime at most $x/m$ and $Q'-1$ prime. When $s(m')/g>x/m+1$, $Q$ is the least positive residue of that class. For cofactors of size about $x^{1/2+\kappa}$ with small $g$, the residue must fall in an interval of relative length about $x^{-2\kappa}$. An approach using a Fourier majorant of this interval and uniform termwise bounds would seek exponential sums of these residues over pairs of cofactors, at frequencies up to about $x^{2\kappa}$, with a saving of about $x^{2\kappa}$ and with both prime conditions. This is a sufficient route to the expected count, not a necessary bound on each frequency. A bound saving a fixed small power reaches only cofactors slightly above $x^{1/2}$. The inputs $B_k$ have cofactors near $x^{1-\alpha}$, so $\kappa$ is near $1/2-\alpha$. With $M\approx x^{1-\alpha}$ cofactors on each side they need a saving of about $x^{1-2\alpha}=M^{(1-2\alpha)/(1-\alpha)}$ in sums of about $M^2$ terms, close to square-root cancellation when $k$ is large. These are Kloosterman fractions whose numerator, inverse and modulus are all arithmetic functions of the cofactors. No estimate for them is proved here.
• A level of distribution beyond $x^{1/2}$. Counting one input's largest prime with a sieve and its partners through the residues of $s(n')$ needs the values $s(n')$ in progressions to moduli $s(m)e$ of size about $x^{1-\alpha}$ or more when that input lies in $B_k$. The same holds when the two-prime identity turns one side into a divisor count, with moduli of size about $\sqrt{bx}$ or larger. A collision bound gives such distribution through the large sieve only on average over moduli up to about $x^{1/2}$, and for general sequences the large sieve cannot do better. In outline, an Elliott–Halberstam-type estimate for the values of $s$, with explicit local main terms, would bound every pair whose first cofactor is at most $x^{1-2\epsilon}$. Its main term would need only the local densities of $s(n')$ modulo $q$, in place of the pairwise average G1. This outline is not a proved implication: the hypothesis would need an exact formulation, and the repetition of one modulus $s(m)e$ for many cofactors $m$ would need its own bound.

The missing estimates concern the distribution at scale one of $s(n)$ on inputs with several large prime factors of comparable size and none above about $x^{1/2}$. They remain unproved here; Proposition D does not refute them. A route that proves tightness without bounding collisions class by class is not affected by Proposition D. Gap G1-core in the large-prime class is the same kind of requirement at the cofactor scale: there the modulus is a common divisor of $s(a)$ and $s(a')$, and the residue whose least representative must rarely lie in a short initial interval is $-\sigma(c)s(c)^{-1}$. In both places Brun–Titchmarsh-type counting works exactly when some free prime has a range longer than the modulus.

The averaged sieve error, with its weights retained. There is a precise form of the first proposed estimate. Fix a rough-prime class $\mathcal S_k(x)$ and write each member uniquely as $mP$, where $P$ is its largest prime. For a fixed eligible $m$, the allowed $P$ form the primes in an interval $I_m$: the conditions $P>\max\{x^\theta,P(m)\}$ and $mP\le x$ are interval conditions, and $s(mP)\ge\delta mP$ supplies at most another upper endpoint. The remaining class conditions depend only on $m$. For large $x$, every such cofactor has $s(m)\ge\delta m/2$.

Consider the pairs $m\ne m'$ with $h=\sigma(m')-\sigma(m)\ne0$, $g\mid h$, and two primitive, locally admissible forms $P=u+\beta t$, $P'=v+\lambda t$, where $g=(s(m),s(m'))$, $\beta=s(m')/g$ and $\lambda=s(m)/g$. Call these the regular pairs. Let $J_{m,m'}$ be their actual parameter interval, of length $\ell_{m,m'}\le xg/(m s(m'))$, with the same endpoint conventions as $I_m,I_{m'}$. Let $f_{m,m'}(t)$ be the product of the forms. The prime-pair count for all other off-diagonal cofactor pairs is denoted $C_{\rm deg}(x)$. This count includes the $h=0$ case and the possible prime values of nonprimitive forms; it is not silently discarded.

Fix $0<\kappa<\theta$ and set $z=x^\kappa$. Use, for each regular pair, the explicit weights $\lambda_d^{(m,m')}$ of the uniform sieve argument, and put

$$\begin{aligned} R_q(m,m')&=\#\{t\in J_{m,m'}\cap\mathbb Z:q\mid f_{m,m'}(t)\} -\frac{\ell_{m,m'}\omega_{m,m'}(q)}q,\\ \mathcal E_z(x)&= \sum_{\substack{(m,m')\\\text{regular}}}\ \sum_{d,e<z} \lambda_d^{(m,m')}\lambda_e^{(m,m')}R_{[d,e]}(m,m'). \end{aligned}$$

The sums on $d,e$ use the squarefree support specified with those weights. They are finite sums of explicitly given real numbers; the remainders and the total $\mathcal E_z$ are signed. All relevant primes exceed $z$, so summing the finite sieve inequality gives, for an absolute constant $C$,

$$C_{\mathcal S_k(x)}(x) \le x+C_{\rm deg}(x)+ \frac{Cx}{\delta\kappa^2\log^2x} \sum_{\substack{(m,m')\\\text{regular}}} \frac{g\,\mathfrak S_x(m,m')}{mm'}+\mathcal E_z(x).$$

The $x$ term covers equal cofactors, which force $P=P'$. The main term uses only the proved denominator bound and the comparison of singular series cutoffs. It is valid even when $\ell_{m,m'}<1$; unlike G0, it does not absorb the remainders individually. Consequently, a $O(\log^2x)$ bound for the displayed local-factor sum, $C_{\rm deg}(x)=O(x)$ and the upper bound $\mathcal E_z(x)\le Kx$ for some fixed $K$ would prove the class estimate in Theorem C. The constants may depend on $\delta,\theta,\tau,k,\kappa$. None of these three bounds for general rough-prime classes is established here. Bounding each $|R_q|$ by $\omega(q)$ and summing absolute weights produces an error majorant of at least one per regular pair, from $d=e=1$. Proposition D explains why charging that bound to every represented cofactor pair cannot work; a reduction to regular pairs must quantify which pairs it removes. Merely displaying the signed sum supplies no cancellation. Its explicit weights specify an analytic estimate to attack without assuming that the residues are independent.

Other whole-target attempts and their failure points. The proof of Pollack, Pomerance and Thompson's finite-target theorem (2017 manuscript, Section 2, pp. 3–4) writes $n=de$ with $d$ the largest divisor below $\sqrt x$. For fixed output $a$, the identity $a=\sigma(d)s(e)+s(d)e$ determines $s(e)$ modulo $s(d)/(d,\sigma(d))$. Its count is a bound for each output; multiplying that bound by the number of targets retains the square-root restriction. Averaging it without a new estimate for which residues actually occur does not give Theorem B(4).

Luca and Pomerance's positive-density range theorem (15-page preprint, Section 3, pp. 6–13) does supply a linear collision bound on selected classes. Its inputs have $n=mp\in(x/2,x]$ with $m\le x^{7/15}$, hence $p>x^{8/15}/2$. Its proof therefore does not bound the positive-density complement in G2. Its further prime ranges are used in the local-factor argument and cannot be removed just by summing cofactor boxes. These two source proofs were read in their transcriptions for the stated uses; no new verification of their external premises or PDF fidelity is asserted.

For a counterexample, the large localized fibers of the same Pollack–Pomerance–Thompson paper (Section 3, pp. 4–6) do not themselves supply positive input density. Common multiplication does not amplify an equal-value fiber into another one: if $s(n_i)=a$ and $r>1$ is coprime to every $n_i$, then

$$s(rn_i)=\sigma(r)a+s(r)n_i.$$

Since $s(r)>0$, these values are distinct for distinct $n_i$. This exact identity rules out that amplification step only. Different multipliers or another construction are not excluded. No positive-upper-density preimage of a density-zero target is constructed here.

Next discriminating tests

• Full equations. Gap G2-full can bypass the large-divisor core and the quadratic slice. The prime-power local densities, their reciprocal-weighted cofactor average and the continuous main term are supplied by the full-equation calculation. The missing test is an integer prime-solution estimate for $F_{\sigma(b),b,k}(\mathbf P)=F_{\sigma(b'),b',k}(\mathbf P')$ with $k\ge3$, $b,b'\le x^\tau$ and all $P_i,P_i'>x^\theta$, whose total error is $O(x)$ after summing the boxes and cofactors. It must include off-diagonal solutions sharing some prime variables. A count for one fixed $b,b'$ or one fixed $k$ does not cover every class needed as the discarded proportion tends to zero. Complete residue counts alone do not bound this error.
• Large common divisors. Gap G1-core is the first target in the large-prime class. In the core, a divisor $d$ of $s(a)$ with $d\ge P(a)^{1-\kappa}$ has complementary divisor $s(a)/d\ll c\,P(a)^\kappa\log y$, where $c=a/P(a)$, so the core partners of $a$ lie among $a+js(a)/\lambda$ for divisors $\lambda$ of $s(a)$ of that size. Enumerating them without using $d\mid s(a')$ overshoots $\log^2y$ by about the average number of such $\lambda$, a factor of order $\log y$ in a random model. A proof must recover that factor from the condition $d\mid s(a')$, for example by bounding, on average over $\lambda$ and $j$, the number of $a$ for which $s(a+js(a)/\lambda)$ is divisible by $s(a)/\lambda$. Finite sums say little about this regime: $W=\log\log y/(4\log\log\log y)$ is below $1$ unless $y$ exceeds $10^{2000}$.
• Quadratic slice. For cofactors $b,b'\le x^\tau$, count the prime solutions of the quaternary quadric in the balanced box, with the congruence conditions of a four-dimensional upper-bound sieve, and separately its components with vanishing right side. Report the error term against the main term, of order $x/(bb'\log^2x)$ up to local and doubly logarithmic factors, as $b$ and $b'$ grow. A uniform bound of that order, with G0, the G1 analogue for pairs that reduce to the linear slice, and a bound for that slice's exceptional prime cases, settles $\mathcal S_2$ through G2-quad. It leaves the balanced inputs $B_k$ with $k\ge4$, so it cannot complete the route.
• Balanced classes. The first family forcing Proposition D's all-pairs error obstruction for quadratic slices is $B_4$, whose cofactors $n/P_1$ lie near $x^{7/9}$. For these inputs, seek an upper bound of order $x$ for the explicit signed remainder $\mathcal E_z(x)$ above, together with the local-factor sum and the degenerate count appearing in the same finite inequality. This states exactly what an averaged use of the linear sieve would need. Fourier estimates for the short residue intervals are one possible tool; uniform bounds on each individual frequency are not necessary. A result for $B_4$ alone would still need extension to every rough-prime class required by Theorem C. No such estimate is supplied here.