Unresolved estimates

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A refuted finite Dirichlet inequality

For a finite distinct-modulus singleton family, let $B$ be its delayed union and $\beta$ the density of its full periodic union. The assertion

$$\sum_{n\in B}n^{-s}\le\beta\zeta(s)\qquad(s>1) \tag{D}$$

would imply the general problem positively. Indeed, monotone convergence over finite families bounds the normalized infinite Dirichlet series above by $\beta_\infty=\lim\beta_k$, while each fixed finite prefix supplies the matching lower limit as $s\downarrow1$. The Hardy–Littlewood/Karamata Tauberian theorem for nonnegative coefficients then gives logarithmic density $\beta_\infty$ for the excluded union. Equivalently one applies the Laplace–Stieltjes form to $\sum_{\log n\le t,n\in B}1/n$. The analogous Tauberian passage is the one used in the Davenport–Erdős 1936 proof, whose Theorem 1(a) proof was checked in the supplied full source. However, the prime-constellation construction now refutes (D) for finite distinct prime families and $s>1$ arbitrarily close to $1$. The implication above remains correct, but (D) is not an available route to the problem.

For $s\ge2$, (D) is elementary and does not approach the missing limit. If $Q$ is the least modulus, then $B\subseteq[Q,\infty)$ and $\beta\ge1/Q$. Also $Q\sum_{n\ge Q}n^{-s}$ decreases with integer $Q$: its consecutive difference is $\sum_{n\ge Q+1}n^{-s}-Q^{1-s}\le0$, by the integral estimate for $s\ge2$. Its value at $Q=1$ is $\zeta(s)$, proving (D) in this range.

The stronger normalized compression comparison with the zero classes is false. Take moduli

$$D=\{2,3,5,7,11,13,25\}$$

with zero residues except $4\pmod{25}$, and take $X=29$. For the zero union $M=M(D)$,

$$d(M)=\frac{809}{1001},\qquad H_M(29)=\frac{29970107}{10810800}.$$

The shifted class adds just $29$ below $X$, and adds periodic density $48/5005$. Exact subtraction gives

$$\frac{H_M(29)+1/29}{809/1001+48/5005} -\frac{H_M(29)}{809/1001} =\frac{41902022}{21605821425}>0.$$

This refutes that normalized comparison, not (D).

Nor can (D) be proved separately for every first-kill increment. The old classes $0\pmod2$, $0\pmod3$, $1\pmod4$, $5\pmod6$ leave only $7\pmod{12}$. Adding $0\pmod7$ has incremental periodic density $1/84$, but its first-kill set contains $7$, and $1/49>\zeta(2)/84$.

Coherent quotient sieves can also have large harmonic spikes

The following tempting bound is false:

$$H_S(Y)\ll d(S)\log(zY+2) \tag{F}$$

for all coherent finite sieves $S={t\ge1:t\not\equiv\alpha\pmod d \ (d\in D)}$ with $D\subseteq[2,z]$. Coherence alone does not supply the missing first-kill charge.

Proof of failure. Set $z=Y=N$ and take $D$ to be the primes at most $N$. For $p=2,3$, choose the forbidden residue uniformly modulo $p$. For every prime $p\ge5$, independently choose it uniformly from

$$R_p=\{\lceil\sqrt p\rceil,\ldots,p-1\}.$$

CRT realizes every resulting selection as one common integer $\alpha$. Regardless of the selection, $d(S)=\prod_{p\le N}(1-1/p)$.

For a fixed integer $t$, a prime $p>t^2$ cannot remove it, since its allowed forbidden residues all exceed $t$ and $p>t$. For the other primes the hitting probability is at most $1/(p-\lceil\sqrt p\rceil)$. The infinite product

$$c_0=\prod_{p\ge5} \frac{1-1/(p-\lceil\sqrt p\rceil)}{1-1/p}$$

is strictly positive: every factor is positive, and each is $1-O(p^{-3/2})$. Independence and the classical Euler-product bound $\prod_{p\le x}(1-1/p)^{-1}\ll\log(2x)$ give

$$\Pr(t\in S)\ge c_0\prod_{p\le\max(3,t^2)}(1-1/p) \gg\frac1{\log(t+2)}.$$

Consequently $\mathbb EH_S(N)\gg\log\log N$, so some coherent selection has that harmonic mass. On the other hand, $\prod_{p\le N}(1-1/p)^{-1}\ge H_N$, since every integer at most $N$ is $N$-smooth. Therefore $d(S)\log(N^2+2)=O(1)$, contradicting (F). The Euler-product upper bound was checked in the Davenport–Erdős 1951 source before equation (12). $\square$

This mechanism is related to the original reciprocal-admissible-set construction in this mathematical discussion; the argument above was independently checked in full.

It is realizable as an actual coherent first-kill quotient. After the prime moduli $p\le N$, choose any prime $Q>N$ and prescribe

$$\alpha\equiv0\pmod Q,\qquad \alpha\equiv Qb_p\pmod p,$$

where $b_p$ are a successful selection from the construction. At the top modulus $Q$, every earlier modulus is active and the first-kill set is exactly $\{Qt:t\in S\}$. Its density is $d(S)/Q$, but its harmonic mass up to $QN$ is $H_S(N)/Q\gg(\log\log N)/Q$. Choosing $Q<2N$ by Bertrand's postulate makes the putative bound $e_Q\log(QN)$ only $O(1/Q)$. This is not an infinite counterexample: repeating an ever larger prime sieve forces the full survivor density to zero.

Bounded-quotient first kills: still unproved in general

For fixed $K$, it would be useful to establish logarithmic density zero for

$$\bigcup_i(E_i\cap[n_i,Kn_i]).$$

For first endpoints, this is a shifted primitive-set question: selected $r_i\in[n_i,2n_i)$ must satisfy $r_j\not\equiv r_i\pmod{n_i}$ whenever $i<j$. Primitive moduli and the structured residue families in structured cases are controlled, but no general endpoint theorem was proved.

A new recurrence obstruction shows that finite-horizon privacy is much weaker: a set of natural density arbitrarily close to one can assign distinct $q(r)\le r<2q(r)$ so that all its first $c\sqrt{\log r}$ positive progression steps miss the set. Every assigned progression nevertheless returns eventually. Thus infinite privacy cannot be replaced by a bounded-return estimate or by that particular growing return horizon.

A cutoff-preserving compression to primitive subsets is already false. The pairs

$$(4,6),(5,5),(6,8),(7,7),(8,13),(9,9),(10,11),(12,12),(16,16)$$

satisfy these first-kill conditions. The endpoint set has reciprocal sum larger by $1/48$ than the largest reciprocal sum of a primitive subset of $[4,16]$, attained by $\{4,5,6,7,9,11,13\}$. Indeed the difference replaces $1/4$ by $1/8+1/12+1/16$. For the primitive optimum, $5$ outweighs any compatible selection from $\{10,15\}$, then $4$ outweighs any compatible selection from $\{8,12,16\}$; next $6$ outweighs $12$ and $7$ outweighs $14$. The remaining isolated available integers are $9,11,13$. This only rules out this particular compression, not the endpoint statement.

Further checks after renewed work

Even coherent, pairwise-coprime moduli do not permit harmonic compression to zero residues. With $D=\{2,3,5\}$, common shift $\alpha=12$, and $X=17$, the delayed system $0\pmod2$, $0\pmod3$, $2\pmod5$ loses $5$ and gains $7,17$ relative to $M(D)$. Consequently

$$H_B(17)-H_{M(D)}(17)=\frac17+\frac1{17}-\frac15=\frac1{595}>0.$$

Both periodic union densities are $11/15$. This refutes the additive comparison $H_B-\beta H_X\le H_{M(D)}-d(M(D))H_X$ as well as the normalized comparison, even under coherence. It does not refute the direct possible bound $H_B\le\beta H_X$.

Exact protected-root accounting

For any increasing-modulus family, put

$$C_i=[a_i]_{n_i},\qquad F_i=C_i\setminus\bigcup_{j<i}C_j, \quad 0\le a_i<n_i.$$

On positive integers, $E_i=F_i\cap[n_i,\infty)$. The only possible positive point of $F_i$ below $n_i$ is $a_i$. Therefore, if

$$R=\{a_i>0:a_i\notin\bigcup_{j<i}C_j\},$$

then the delayed excluded union is exactly

$$B=\left(\bigcup_i C_i\right)\cap\mathbb N\ \setminus R.$$

Indeed every integer in the full union has a first full covering class; if that class is inactive, all later classes are also inactive. This identity is valid for infinite families as well as finite ones.

For a finite family, the direct harmonic bound is equivalently $H_U(X)-\beta H_X\le H_R(X)$. The finite prime-constellation construction refutes this exact compensation inequality. The identity itself remains valid, including with repeated moduli; a sublogarithmic-error variant has not been proved or refuted here.

Dirichlet compression to zero residues fails as well

The preceding finite harmonic counterexample does not by itself settle a comparison of Dirichlet series. However, the quotient-sieve construction above gives a separate rigorous obstruction: for some $s>1$ arbitrarily close to $1$, even a coherent pairwise-coprime finite family can have

$$D_B(s)-\beta\zeta(s) > D_{M(D)}(s)-d(M(D))\zeta(s), \qquad D_E(s)=\sum_{n\in E}n^{-s}. \tag{3}$$

This is still not a counterexample to the direct global bound (D).

Keep the old zero classes at primes $p\le N$. In a quotient sieve, set $b_2=b_3=0$ and independently choose $b_p\in\{\lceil\sqrt p\rceil,\ldots,p-1\}$ for $p\ge5$. Let $S_N=\{t\ge1:t\not\equiv b_p\pmod p\ (p\le N)\}$. The previously proved product estimate now gives, for $(t,6)=1$,

$$\Pr(t\in S_N)\gg1/\log(t+2)$$

uniformly in $N$. By Tonelli,

$$\mathbb E D_{S_N}(s) \gg\sum_{(t,6)=1}\frac{t^{-s}}{\log(t+2)}.$$

The right side tends to infinity as $s\downarrow1$: at $s=1$ it diverges, as is seen by restricting to the two reduced residue classes modulo $6$, and monotone convergence applies. Choose $s$ close enough that the lower bound exceeds $3$. For this fixed $s$, choose $N$ sufficiently large that

$$D_{S_N^0}(s)=\zeta(s)\prod_{p\le N}(1-p^{-s})<2, \quad S_N^0=\{t:(t,\prod_{p\le N}p)=1\}.$$

The Euler product tends to $1/\zeta(s)$, so this choice is possible. A realization of the $b_p$ has $D_{S_N}(s)>3$.

Writing $P=\prod_{p\le N}p$, CRT supplies arbitrarily large positive $q$ with

$$q\equiv1\pmod6, \qquad q\equiv-6b_p^{-1}\pmod p\quad(5\le p\le N).$$

These congruences force $(q,P)=1$. Choose $q>\max(N,6)$ and append the single class $6\pmod q$. Its delayed first-kill set is $\{qt+6:t\in S_N\}$, because all earlier prime moduli are active there. Its Dirichlet contribution is

$$q^{-s}\sum_{t\in S_N}(t+6/q)^{-s}.$$

Along the permitted arbitrarily large $q$, dominated convergence makes the inner sum tend to $D_{S_N}(s)>3$. Appending $0\pmod q$ instead contributes $q^{-s}D_{S_N^0}(s)<2q^{-s}$. Thus the former contribution is larger for large enough $q$. The completed periodic densities of these two systems are equal, since $q$ is coprime to $P$, proving (3). All moduli are distinct and pairwise coprime, so the systems are coherent by CRT.

The old zero-sieve deficit has not been shown to be overcome by this small positive increment. This argument rules out additive and normalized Dirichlet comparison to the zero system, not the global inequality (D).

Two additional endpoint and charging checks

For a first-endpoint family $r_i\in[q_i,2q_i)$ satisfying the avoidance condition above, write $\alpha=\overline{\operatorname{ld}}(\{r_i\})$ and let $\beta$ be the limiting periodic density of its full congruence unions. The proved endpoint compression bound gives $\alpha\le\beta$. Avoidance of every fixed prefix gives $\alpha\le1-\beta$, hence $\alpha\le1/2$. If every $r_i\ge\lambda q_i$, then the first bound sharpens to $\lambda\alpha\le\beta$, and $\alpha\le1/(1+\lambda)$. These bounds do not imply zero logarithmic density.

Cylinder containment antichains alone cannot prove that zero-density conclusion. Let $D=\{q\ge2:\sigma(q)<2q\}$. For every $q\in D$, recursively choose a residue avoiding all earlier classes whose moduli properly divide $q$. This is possible because

$$\sum_{\substack{d\mid q\\1<d<q}}\frac1d =\frac{\sigma(q)}q-1-\frac1q<1.$$

Classes with comparable selected moduli are therefore disjoint. Yet $D$ has lower natural density at least $2-\zeta(2)>0$: each nondeficient integer contributes at least one to $\sigma(n)/n-1$, whereas

$$\frac1N\sum_{n\le N}\left(\frac{\sigma(n)}n-1\right) =\sum_{2\le d\le N}\frac{\lfloor N/d\rfloor}{Nd} \le\zeta(2)-1.$$

Deleting the single integer $1$ has no effect on the lower density. The construction imposes no avoidance of earlier nondividing moduli and is not an endpoint counterexample.

Finally, a positive global survivor density cannot repair a false individual first-kill estimate. Replacing each $(q,a)$ in a spike example by $(Mq,Ma)$ confines all exclusions to $M\mathbb N$ and leaves survivor density at least $1-1/M$, while multiplying both first-kill densities and harmonic masses (with rescaled cutoff) by $1/M$. The local discrepancy persists; a successful charge must account for the relevant fibers rather than only the total survivor density.

A containment antichain can use every sufficiently large modulus

Section 6.1 of the third independent record strengthens the deficient-modulus example above. Set $N=10^{16}+1$. For every successive integer $q\ge N$, choose $a_q\pmod q$ outside all earlier classes whose moduli properly divide $q$. Such a residue exists: otherwise those finitely many distinct divisor-modulus classes would cover every residue modulo $q$, and hence all integers. This contradicts Hough's Theorem 1, which bounds the least modulus of a finite distinct covering system by $10^{16}$. The digest and the actual theorem statement were checked.

Classes with comparable selected moduli are therefore disjoint, although every modulus $q\ge N$ occurs and $\sum_{q\ge N}1/(q\log q)$ diverges. Dilating $(q,a_q)$ to $(Mq,Ma_q)$ confines the full union to $M\widehat{\mathbb Z}$ and gives modulus natural density $1/M$. In fact both its full Haar union and its delayed union have density $1/M$: before dilation, the selected primes alone have divergent reciprocal sum and hence delayed deletion density one.

Thus even this particularly dense containment antichain is not a counterexample. Containment avoidance says nothing about earlier nondividing moduli. The two-colored construction supplies a different, stronger profinite-private phenomenon, but neither construction supplies comparable positive integer private witnesses.

Later progress and obstructions

The finite bounded-rank estimate proves natural density for arbitrary residues when $\Omega(n_i)$ is uniformly bounded. It also proves natural density for coherent residues with bounded $\omega(n_i)$, allowing unbounded exponents. The general coherent case and the arbitrary unbounded-rank singleton case remain unresolved.

The accompanying rank and moment obstructions rule out two extensions of that proof. Divisor-face capacities, even with their exact harmonic cutoffs and primitive distinct moduli, do not imply rank-independent control. The usual first-kill Haar potential does not have any uniform $L^p$ bound for $p>1$, even for zero residues. The transversal-prime argument separately rules out one coherent high-rank construction suggested by those abstract capacity examples.

Private-witness reduction for endpoints

For $q_i\le r_i\le Kq_i$ satisfying $r_j\not\equiv r_i\pmod{q_i}$ whenever $i<j$, the endpoint set partitions into at most $\lceil K\rceil$ sets $R_c$ such that

$$[r_i]_{q_i}\cap R_c=\{r_i\}\quad\text{whenever }r_i\in R_c.$$

Indeed, later endpoints never lie in the $i$th class. Every earlier endpoint is at most $Kq_i$, so that class contains at most $\lceil K\rceil$ endpoints in total, including $r_i$. Greedily color the indices, avoiding the colors of earlier endpoints in the current class. There are at most $\lceil K\rceil-1$ forbidden colors. Earlier classes automatically avoid the new endpoint. Thus the endpoint-nullity question reduces to a distinct-modulus family whose assigned classes each have a private endpoint. Nullity for that private-witness problem is still unproved.

The remaining private-witness problem reduces further to $q_i\le r_i<2q_i$. Partition by the finitely many values $t=\lfloor r_i/q_i\rfloor$, and in each part replace $q_i$ by $tq_i$. The new moduli are distinct, the new cylinders are contained in the old ones, and $tq_i\le r_i<2tq_i$. Thus the private-witness property is preserved.

Private positive witnesses can still have very small individual increments

For $L\ge3$, take

$$(q_j,r_j)=(2^j,3\cdot2^{j-1})\quad(1\le j\le L), \qquad (Q,R)=(2^L+1,2^{L+1}).$$

All witnesses lie in $[q,2q)$, and all moduli are distinct and increasing. The dyadic cylinders are the disjoint classes of exact valuations $v_2(n)=j-1$; the last witness $R$ avoids them. The last cylinder contains no earlier witness: if $Q\mid R-r_j$, then $0<R-r_j<2Q$ forces $R-r_j=Q$, or $r_j=2^L-1$, impossible for $L\ge3$. Thus every witness is private in the entire finite family.

Nevertheless the final first-kill density is

$$e_Q=\frac1{Q2^L},$$

because the old survivors are $0\pmod{2^L}$ and $\gcd(Q,2^L)=1$. In particular no uniform individual lower bound $e_i\gg1/(q_i\log q_i)$ holds even for globally private positive endpoints. This does not disprove endpoint nullity or any appropriately amortized charge; it only rules out this individual charging argument.

Purely profinite private witnesses are insufficient

For every prime $p$, $e\ge1$, and $1\le j<p$, consider

$$C_{p,e,j}=[jp^{e-1}]_{jp^e}.$$

The moduli are globally distinct. Give this cylinder the profinite witness whose $p$-adic coordinate is $jp^{e-1}$ and whose other prime coordinates are zero. Each cylinder contains exactly its own witness: another prime's coordinate is zero, and for the same prime the valuation and next digit distinguish $(e,j)$. These witnesses have exactly one accumulation point, $0$. Indeed outside any neighborhood $[0]_m$ there are only finitely many of them.

Their modulus set nevertheless has positive lower logarithmic density:

$$\sum_{\substack{q\le X\\q=jp^e,\ j<p}}\frac1q \ge\sum_{p\le\sqrt X}\frac{H_{p-1}}p \ge\frac{\log2}{2}\log X-O(1).$$

For the last estimate, $H_{p-1}\ge\log p$ and $\vartheta(y)\ge y\log2-O(\sqrt y\log y)$ suffice by partial summation. The latter follows elementarily because the central binomial coefficient divides $\operatorname{lcm}(1,\ldots,2n)$, while prime-power terms beyond the first contribute $O(\sqrt y\log y)$ to the logarithm of that lcm.

Every finite subsystem also has positive integer private witnesses by CRT, but with no required size bound. There are no positive integer private witnesses for this entire infinite system: every positive $n$ belongs to $C_{p,1,n}$ for every prime $p>n$. Thus compactness and abstract private witnesses cannot replace the quantitative requirement that the witnesses be positive integers comparable to their moduli. This is not an endpoint counterexample.

Positive survivor measure does not make essential moduli sparse

The preceding profinite-private example also refutes the stronger proposed assertion that positive survivor Haar measure forces the set of moduli with nonempty full first-kill cells to have logarithmic density zero. Replace each pair $(q,a)$ in that example by $(2q,2a)$. Distinctness and all private profinite witnesses are preserved by dilation; hence every finite first-kill cell is nonempty. All cylinders now lie in the even fiber. The original family has full Haar union measure one already from the classes $1\pmod p$ (take $e=j=1$), so the dilated family's survivor Haar measure is exactly $1/2$. Its modulus set still has positive lower logarithmic density. This is not a family of comparable integer private endpoints, and supplies no density counterexample.

Even conditional contraction for a new prime fails

Take old zero classes at every prime at most $53$, append $2\pmod{59}$, and put $X=179$. The new first kills below $X$ are $61,179$, so

$$59H_E(179)=\frac{14160}{10919}.$$

The old survivors are $1$ and the $25$ primes from $59$ through $179$. Grouping these primes successively into groups of $5,10,10$ gives

$$H_{A_{\rm old}}(179) \le1+\frac5{59}+\frac{10}{79}+\frac{10}{131} =\frac{786236}{610591}<\frac{14160}{10919}.$$

Thus the tempting conditional contraction $H_E(X)\le H_{A_{\rm old}}(X)/59$ is false. Its Haar analog is exact because the new prime is coprime to the old moduli. All moduli here are distinct primes, and the residues are coherent by CRT.

A limitation of unconditioned weak-type bounds

Failure of a uniform estimate

$$\sup_{\substack{\text{finite singleton families }F\,,\ \mu(U_F)\le b\\X\ge2}} \frac{H_{B_F}(X)}{H_{\lfloor X\rfloor}}\longrightarrow0 \qquad(b\downarrow0)$$

would supply small-footprint epochs for a counterexample: a footprint less than $1/Y$ automatically has every modulus greater than $Y$, so successive anomalous blocks can be installed beyond any recovery cutoff.

The converse has not been established. Conditioning on a fixed old period $P$ expands a tail class into classes modulo $\operatorname{lcm}(P,q)$. This introduces a bounded multiplicity depending on $P$, and changes activation by a factor at most $P$. An unconditioned bound with modulus of continuity $f(b)$ would therefore give a bound with a factor depending on $P$. As the old prefix grows, $P$ may grow arbitrarily faster than its omitted Haar measure decreases. Hence this localization does not by itself turn $f(b)\to0$ into the required conditioned-tail theorem.

Comparison with the first independent record

The independent investigation was treated as unverified input, not as a replacement for the established arguments. Its finite-truncation and summable cases, first-kill decomposition, compression, and first-$K$-hit arguments agree with the proofs in basic reductions and compression and blocks. Its dyadic second-moment obstruction is also already covered there.

The main disagreement is substantive: its proposed sharp inequalities (7.1) and (7.2) are false, not merely unproved. The established finite prime-constellation counterexamples refute both. Small finite checks failing to find counterexamples do not support retaining either estimate as a proof target. The weaker uniform additive $o(\log X)$ harmonic target is still open.

The following checked additions were retained:

• Section 5's arbitrary-common-shift interval theorem is correct. Common-shift blocks gives a finite bound and its stronger, uniform harmonic corollary for a single coherent dyadic block, including unbounded quotients. Accumulation across arbitrarily many blocks remains uncontrolled.
• Section 7's finite zero-residue Dirichlet sandwich is correct; a self-contained valuation proof is in basic reductions.
• Section 8's affine-endpoint nullity is correct. Comparing its smooth-part argument with structured residues also removes the coprimality hypothesis from the fixed rational-translate theorem. General private-endpoint nullity remains unproved.

Section 8's deficient-modulus construction needs an arithmetic repair: exclude modulus $1$, and count

$$\sum_{\substack{d\mid q\\1<d<q}}q/d=\sigma(q)-q-1<q.$$

The displayed divisor-sum equality in the independent input is incorrect as written. Its improved density bound $2-\zeta(2)$ is valid and has been incorporated above. This only supplies a containment antichain, not the avoidance of all earlier cylinders needed for endpoint nullity.

Finally, its failed attempts to refine Wang's reused moduli are superseded by the proved arbitrary-divisor-refinement bound. Its generalized-tower and biased-label ideas supply no verified new construction. The former's synchronization difficulty is partially quantified by colored towers; neither mechanism controls the general conditioned incremental footprint.

Latest checked partial results

For coherent residues, the power-cutoff argument now gives a uniform bound for selected first-kill cells of total Haar mass $\tau$, provided their moduli are at most $H^{1-\eta}$ on an interval of length $H$: their relative count is $O(1/(\eta\log(2/\tau)))$. This is a conditioned estimate with no constant depending on an old period. It proves natural density for every fixed power delay and for the original activation under the stated reciprocal-shell sparsity criterion. The general near-diagonal layer is not controlled.

For residues $\lfloor\theta q\rfloor$ with $\theta$ badly approximable, bounded-quotient first kills have natural density zero. A Liouville-slope construction shows that replacing badly approximable by irrational in this natural-nullity theorem is false: it gives globally private endpoints with positive upper natural density, but logarithmic density zero. This does not refute the general logarithmic endpoint target.

Finally, the fixed-rank product obstruction shows that divisor-face capacities alone remain insufficient even for distinct primitive moduli with at most three distinct prime factors. The bounded-$\omega$ question for actual arbitrary-residue systems is not settled by these abstract weights. The later one-unbounded-prime-power argument does prove natural density for arbitrary singleton residues when $\omega(q)\le2$. The subsequent squarefull-kernel truncation proves natural density for all bounded-$\omega$ singleton systems, and for the larger class with bounded exponent-one prime rank. The abstract weights do not meet the additional arithmetic constraints used in that proof.

Comparison with the second independent record

The independent investigation was checked as unverified input. Its unresolved status agrees with the current research, but several of its proposed targets have already been refuted or its partial results strengthened. Its text is preserved unchanged; references there to companion notes/ and research/ files are not available in this research folder and are not used as evidence.

Agreements and results already covered

Sections 1--4 agree with the established finite-truncation arguments, finite compression, exact harmonic comparison for multiples, and the sharpened bounded-quotient bound in basic reductions and compression and blocks. They concern the total periodic footprint, not the incremental footprint inside an old survivor set.

Section 7's counterexample to a uniform local large-quotient counting bound is correct. Besicovitch's averaged dyadic-window theorem suffices for its subsequence choice, but the established quotient-only natural obstruction is stronger: the footprint tends to zero for all sufficiently large base scales, primitive reduction is allowed, and the large quotients can remain uniformly subpower. Thus this section adds no new theorem and no logarithmic-density counterexample.

The Wang objections in Sections 4 and 8.1 are also correct. The existing arbitrary-divisor-refinement bound already rules out a larger class of repairs, with no upper bound on the multipliers or quotients. Biased labels and translated windows in Sections 8.1--8.2 supply no checked new infinite construction.

Sharp comparisons: false, not open

The finite harmonic target (5.1) and Dirichlet target (8.2) are both false, by finite sharp counterexamples. The energy construction additionally gives an elementary harmonic refutation with unbounded excess. Failed small searches in Section 9 do not support retaining these exact targets.

Their hypothetical sufficiency is correct: at cutoff $X$ only moduli $q\le X$ are active, so an exact harmonic bound for that finite family would squeeze the infinite union to its Haar candidate. For the Dirichlet version, the nonnegative Laplace--Stieltjes Tauberian passage already recorded at the start of this note is valid; it is not a further unresolved step. The obstacle is the false finite premise. A uniform additive $o(\log X)$ harmonic comparison remains open.

The example $(2,3,5)$ in Section 5 is already recorded above. Section 6's $(2,3,35)$ example is also exact and has been retained in the harmonic-constant comparison. It proves a strictly positive change in the asymptotic harmonic constant and a Dirichlet recentering failure near $1$. The stopped-coordinate model of Section 8.4 is valid, but this same example refutes its proposed domination of the actual Dirichlet trace. The example's constant is still below $\beta\gamma$; the separate changing-family Dirichlet violations do not alone settle that fixed-family constant question. No floating-point evidence is needed for these checks.

Entropy, conditioning, and positive regimes

Section 8.3's entropy summation is valid. With $S=\sum_{q_i\le X}e_i\le1$ and $R=\sum_{q_i\le X}1/q_i\le H_{\lfloor X\rfloor}$, log-sum gives

$$\sum_{q_i\le X}e_i\log\frac1{q_i e_i} \le S\log(R/S) \le\log H_{\lfloor X\rfloor}+1/e.$$

The zero-mass case is interpreted as zero. This checks the entropy calculation in the Chojecki source, Lemma 5.2 and Corollary 5.3, read after its digest. It does not supply the source's missing sublogarithmic charge, nor the different empirical first-kill entropy bound. The tower discussion supplies no new amortized estimate.

The warning about period-dependent conditioning is still necessary for general residues. For coherent systems, however, the later period-independent reduction already overcomes that particular localization loss, conditional on the still-unproved finite harmonic small-footprint estimate.

Sections 8.5--8.6 give correct positive cases, both subsumed by finite rational patterns. The abstract $S_M$-stability lemma has been checked and retained with a proof; it also allows repeated moduli and arbitrary finite activation thresholds in its bounded-order application. This is an alternative structural formulation, not a new solution of the unrestricted singleton problem. No general conditioned logarithmic-tail estimate has resulted from this review.

Comparison with the third independent record

The independent investigation was read as unverified input and compared with the established topical arguments. Its mathematical text is preserved unchanged. Its linked companion files are absent from this research folder; all additions below have proofs here rather than relying on those links.

Agreements and stronger existing results

Sections 2--4 agree with basic reductions and compression and blocks, including the coefficient-one multiples inequality and bounded-witness nullity for summable full footprints. The negative-integer-center correction in Section 2.4 is correct and can be infinite: with every prime modulus and center $-1$, every point $p-1$ is omitted by activation. Its primitive exception argument proves logarithmic nullity. The density conclusion is already subsumed by finite rational patterns and signed branch-separated centers.

The Wang distinctness objection is correct. The established arbitrary-divisor-refinement bound is stronger than the input's bounded-scale refinement obstruction and handles unbounded multipliers and quotients. Section 5's original common-shift estimate is also correct but has a narrower interval range than common-shift blocks.

The tower and quotient-sieve discussion in Section 6.2 correctly keeps the proposed global charge conjectural. None of the biased-label or rare-progression-word heuristics in Section 6.5 supplies a verified singleton epoch. The newer conditioned endpoint blocks and overlapping preparations go farther on finite preparations, but still lack a long active trace.

Checked additions

The two briefly reported window calculations were proved from their displayed mechanisms in weighted prime-residue windows. For one dyadic block, with $r_p$ permitted residues at each selected prime, the finite interval bound is

$$|U\cap I|\ll T/h+(T/Q+Q)m^2/h^2, \quad h=\sum_p\frac1{pr_p},\quad m=\#\{p\text{ used}\}.$$

Keeping the modulus-window error makes this uniform in arbitrarily large $T$. Truncation then proves the stronger harmonic bound $\sup_X H_B(X)/H_{\lfloor X\rfloor}\le\min(1,C/\mathcal H)$ for the full prime-support parameter. Completing missing prime supports with singletons gives $O(R/\log\log Q)$ when all local support sizes are at most $R$. The input's power-law exponent $\delta_*=0.086071\ldots$ is also verified, using two Bonferroni truncations and a factorial moment. These results require only the stated prime-level coherence, not coherence at higher prime powers. They do not give an inverse theorem or a general conditioned-tail bound.

Other useful verified additions are the nearby-multiple injection failure, the rough-window reciprocal support bound, and the every-large-modulus antichain. The all-prime $(2,3,23)$ example is an exact, smaller variant of the $(2,3,35)$ recentering obstruction. It refutes weighted comparison with zero residues, not comparison with the full periodic density itself.

Dirichlet targets and unsupported charges

The substantive disagreement is Section 7's status of the exact finite Dirichlet bound: it is false, not open, by finite sharp counterexamples. The input's unsuccessful small searches do not alter that conclusion. Its weaker uniform assertion

$$F_{\mathcal F}(s)/\zeta(s)\le d_{\mathcal F}+\eta(s), \qquad \eta(s)\to0\quad(s\downarrow1), \tag{U}$$

remains unproved and unrefuted. Its sufficiency is correct: monotone convergence over finite families and fixed-prefix lower bounds give the normalized infinite Dirichlet limit, and the nonnegative Laplace--Stieltjes Tauberian passage gives logarithmic density. The threshold-sensitive inclusion--exclusion and digamma formulas in Section 7.1 are correct for each fixed family; their remainder is not uniform in the family and supplies no proof of (U).

The speculative reciprocal-cost bound $H_B(X)\le\beta\log X+C\sum_q1/q$ has neither a proof nor a refutation in these notes. The energy example has reciprocal sum and harmonic excess of the same order; its failed linear reciprocal charge concerns a different, entropy quantity. Prime-only unbounded excess likewise does not refute this cost bound, because its old prime reciprocal sum grows. Distinctness alone gives only $\sum_{q\le X}1/q\le H_{\lfloor X\rfloor}$, not an $o(\log X)$ error. No new reduction is asserted here.

Section 7.4's decreasing-step observation is valid when its periodic least-modulus function is defined on all integers. It gives no charging estimate; the exact protected-root identity above and affine first-kill flow already expose the missing transfer. The period-dependent conditioning warning remains valid for general residues, whereas the established coherent conditioning reduction does remove that particular loss in the coherent case.

A finite epoch already refutes the multi-residue uniform analog

Section 7.3 correctly notes that one Wang block only gives a normalized Dirichlet contribution of order $1/\log Q$. However, one need not invoke an infinite multi-epoch construction to refute the multi-residue analog of (U). A single finite epoch suffices.

Take the source's blocks with indices $j=a,\ldots,2a$. By Wang's Lemma 3.4, checked after its digest, their disjoint endpoint sets lie in $[(11/10)2^j,(19/10)2^j]$, have size at least $(3/8)2^j$, and are active. Each therefore has harmonic mass at least $15/76$. Their combined periodic footprint is at most $\sum_{j\ge a}\eta_j\to0$. Put $t=1/(a\log2)$. Every selected endpoint satisfies $n^{-t}\ge e^{-3}$ for large $a$, so the delayed finite union has

$$\frac{F_a(1+t)}{\zeta(1+t)} \ge\frac{15e^{-3}(a+1)}{76\zeta(1+t)},\qquad \liminf_{a\to\infty}\frac{F_a(1+t)}{\zeta(1+t)} \ge\frac{15e^{-3}}{76\log2}>0.$$

This directly contradicts a uniform vanishing error for multiple residues. The repeated-modulus obstruction remains, so it gives no counterexample to the singleton version (U) or to Problem 25.

Comparison with the fourth independent record

The independent investigation was checked as unverified input against the linked topical notes. Its mathematical text is preserved unchanged. The review adds a two-class positive result and a small moment certificate, but no general conditioned-tail estimate or singleton counterexample. The auxiliary files named by the input are absent from this research folder and were not used as evidence.

Sharp comparisons and the checked two-class boundary

The main disagreement is the status of Section 5's inequalities (H) and (D): both are false in general, not open, by the established finite sharp counterexamples. The energy example independently refutes (H) with unbounded additive excess. Failed finite searches do not change either conclusion. A uniform sublogarithmic harmonic error, or the vanishing normalized Dirichlet error (U) above, remains open. The nonnegative Tauberian passage from an appropriate Dirichlet limit to logarithmic density is already justified at the start of this note; it is not the unresolved part of that route.

The input's two-class Dirichlet proof is correct. Moreover, its intersection shift loss is a sub-sum of the larger-modulus shift loss, so the argument works for every nonnegative decreasing summable weight when the classes are compatible. This proves both sharp (H) and (D) for families of at most two classes. The full proof and precise weight restriction are retained in the two-class proposition. Comparison with the zero union already fails for three compatible classes; the result supplies no many-class extension.

Both Section 6 calculations were checked by exact finite-period counting and rational arithmetic. For moduli $4,7,9,10,18$, changing the last residue from $0$ to $4$ at $X=100$ gives

$$u_0=\frac7{15},\quad u_1=\frac{17}{35},\qquad u_0h_1-u_1h_0 =\frac{113383068679483}{482750393681556000}>0.$$

For moduli $4,9,11,210$, changing the last residue from $0$ to $48$ at $X=1000$ preserves density $274/693$, while the additional points outside the first three classes change from $\{210\}$ to $\{258,678\}$. Thus

$$h_{48}-h_0=\frac1{258}+\frac1{678}-\frac1{210} =\frac{601}{1020390}>0.$$

These are correct recentering obstructions, not violations of (H) at those cutoffs. Their qualitative conclusions are already covered by the smaller three-modulus example and the harmonic-constant obstruction.

Capacity, windows, and Wang refinements

Sections 1--3 agree with basic reductions and compression and blocks, including the coefficient-one harmonic capacity and the factor $H_K$ for the first $K$ hits. The input correctly confines these estimates to the whole periodic footprint. They do not imply that every counterexample must use unbounded quotients: arbitrary conditioned bounded-quotient first kills remain uncontrolled, as the endpoint reduction makes precise.

Section 4's Wang distinctness objection is correct. In particular, the endpoint interval is shorter than every assigned modulus, so a legal singleton block with those endpoints would have $\mu(U)H_{\lfloor19Q/10\rfloor}\ge15/76$. The label-incidence count is also valid, with the intended bound

$$O\!\left(Q\,2^k4^{-(k+1)(k/2-\sqrt k)}+2^k\right)=o(Q).$$

This writes the intended product explicitly in place of the input's comma after $Q$. The arithmetic skeleton and endpoint bounds were checked in Wang's full source, Lemmas 3.1 and 3.4, after its digest. The existing arbitrary-divisor-refinement theorem is stronger than the input's refinement failures: it controls all cutoffs and unbounded multipliers and quotients.

Section 7's second-moment proof is valid uniformly in its integer translate, but common-shift blocks already allow the wider interval range $H/N\to\infty$. The weighted prime-residue theorem gives, for fixed $K,C>1$, the quantitative strengthening

$$\sup_R\sum_{\substack{N\le n\le N^C\\ q\mid n-R\text{ for some }N\le q\le KN}}\frac1n \ll_{K,C}\frac{\log N}{\log\log N}.$$

Indeed split the modulus window into finitely many dyadic blocks and apply its one-residue harmonic bound; any inactive points lie below $KN$ and cost at most $\log K+O(1)$. This still does not allow summation over arbitrarily many windows or conditioning on old survivors.

Exact conditioning formulas and their scope

Section 8's formulas are correct. On a surviving prefix fiber $n=L(u-1)+r$, with $1\le r\le L$, write $g=(q,L)$ and $m=q/g$. A tail cylinder meets the fiber exactly when $a_q\equiv r\pmod g$; then it pulls back to one class modulo $m$, since $(m,L/g)=1$. Its actual activation is

$$gm\le L(u-1)+r,$$

not $m\le u$. For a fixed $m$, possible original moduli have $g\mid L$ and $(m,L/g)=1$, giving the exact upper multiplicity

$$\prod_{\substack{p^\ell\parallel L\\p\nmid m}}(\ell+1)\le\tau(L).$$

If $R$ is the set of surviving prefix fibers and $\nu_r$ the full tail measure on the pulled-back fiber, the exact average is

$$\frac1L\sum_{r\in R}\nu_r =\mu(U_{\rm tail}\setminus U_{\rm prefix}),$$

as in (8.3). The harmonic change of variables is harmless for fixed $L$: all terms $1/(L(u-1)+r)-1/(Lu)$ are nonnegative, and in fact their total over all fibers and $u\ge1$ is

$$\lim_{M\to\infty}(H_{LM}-H_M)=\log L.$$

This sharpens the input's valid $H_L$ error bound, but does not fix the separate period-dependent multiplicity and activation losses.

The input's quantifier warning remains necessary for general residues: shrinking incremental measure need not compensate for growing prefix periods. For a concrete example, take zero classes at distinct primes $2^j\le p_j<2^{j+1}$, $j\ge1$. Their survivor measure $\delta=\prod_j(1-1/p_j)$ is positive, and the incremental tail $\epsilon_k=\mu(U\setminus U_k)$ tends to zero, but

$$L_k\epsilon_k \ge \frac{\delta L_k}{p_{k+1}} \ge \delta\,2^{k(k+1)/2-k-2}\longrightarrow\infty.$$

These summable systems already have natural density; the example only defeats the naive period-dependent bound.

For coherent systems the already established period-independent conditioning reduction does overcome this particular loss, conditional on the still-open unconditioned small-footprint estimate. The input does not supersede that reduction or the more specialized prepared-block bounds.

First-kill charges and a small second-moment failure

The first-kill representation and tower example in Section 9 check out; the tower is exactly Chojecki's Proposition 6.3. The inequality $1/r\le2\log(q/m)$, for $r=m/(m,q)\ge2$ and $m<q$, is also valid, but has no accompanying conservation or multiplicity bound. It does not prevent repeated charges to the same predecessor.

For each fixed infinite family, dominated convergence gives

$$\mathcal E_X:=\sum_{q_i\le X}e_i\log q_i=o(\log X), \qquad \sum_i e_i\le1.$$

The proposed (9.1), with an unrestricted system-dependent $o(\log X)$ remainder, is therefore equivalent to the desired candidate-density conclusion: also $\sum_{q_i\le X}e_i\to1-\delta$, and the finite-prefix lower bound supplies the opposite inequality. No estimate for that remainder was provided. Dropping it and requiring an absolute implied constant would give a refuted uniform linear-energy bound, whereas (9.1) as written is not refuted by the known sublogarithmic excess.

The final four-prime calculation is exact and is retained as a small second-moment certificate. Its potential $V/H_X$ is precisely the overlapping potential already known to have unbounded higher moments, not the disjoint entropy potential or the capped prime-stage fractions. None of those distinct unresolved targets is settled by this calculation.

Private canonical witnesses can occur in a full finite cover

The pairs

$$(q,r)=(2,2),(3,3),(4,5),(6,7),(12,23)$$

have distinct moduli and satisfy $q\le r<2q$, with every selected witness private in the entire finite family. Their full cylinders are

$$[0]_2,\quad[0]_3,\quad[1]_4,\quad[1]_6,\quad[11]_{12},$$

and cover every residue modulo $12$. The first covers the even residues; the others cover, among the odd residues, respectively $3,9$, $1,5,9$, $1,7$, and $11$. Direct substitution of $2,3,5,7,23$ shows that each is in only its own cylinder.

Thus privacy of canonical positive endpoints does not imply positive uncovered Haar measure, even with the exact size restriction. This finite family cannot be extended by another private endpoint, so it does not refute infinite logarithmic endpoint nullity.

Nor does full-cylinder disjointness follow from privacy. The disjoint-selector bound therefore cannot be applied to these first-kill families. In Croot's prime-factor iteration, full nonintersection forces another modulus to share an additional prime with a chosen modulus after common coordinates have been fixed. Endpoint avoidance does not imply that prime-sharing assertion. A separate profinite-fiber obstruction also rules out deducing the required return solely from bounded endpoint and modulus marginals. No private-endpoint nullity theorem or positive-logarithmic-density private family was obtained.

Affine rough shadows can overlap with arbitrarily large multiplicity

One attempted analog of the primitive-set proof attached to a private endpoint $q+a$ the points $a+qt$ whose quotient $t$ has no small prime factor. These shadows are neither disjoint nor of bounded multiplicity.

Fix $k\ge1$, set $P=\prod_{p\le k}p$, and choose $Q>20kP$ and $k$ distinct primes $Q\le q_1<\cdots<q_k\le1.1Q$. Such intervals exist arbitrarily far out: if every interval $[1.1^j,1.1^{j+1})$ beyond some point contained fewer than $k$ primes, the prime reciprocal sum would converge. Put $a_i=iP$ and $r_i=q_i+a_i$. Then $q_i\le r_i<2q_i$, and the distinct witnesses have diameter less than $Q$. Hence every witness is private in the whole family.

Nevertheless, CRT produces arbitrarily large $x$ for which all $(x-a_i)/q_i$ are integers having no prime factor at most $2Q$. At each $q_i$, impose $x\equiv a_i+q_i\pmod{q_i^2}$, so the $i$th quotient is $1\pmod{q_i}$. For $j\ne i$, it is also nonzero in the $j$th quotient, since $0<|a_i-a_j|<Q\le q_i$. At a prime $p\le k$, impose $x\equiv1\pmod p$; all $a_i$ are zero modulo $p$. At each remaining prime $p\le2Q$ other than the $q_i$, choose a residue outside $\{a_1,\ldots,a_k\}\pmod p$, possible because $p>k$. These prime-power conditions are compatible by CRT. They describe a positive-density progression of $x$ on which all $k$ shadows overlap.

Thus bounded-multiplicity disjointification cannot repair this rough shadow argument. The later rough-shadow energy construction also rules out a uniform quadratic charge: globally private canonical endpoints can have bounded total shadow measure and unbounded second moment. Neither obstruction rules out a different fractional allocation. In particular, the possible private-endpoint bound $\sum_i1/(q_i\log(2q_i))<\infty$ remains unproved and unrefuted by this construction. Such a bound would imply logarithmic endpoint nullity, but it has not been established here.

Globally private canonical endpoints can have quadratic-small cells

Global privacy does not restore a pointwise estimate $qe_q\gg(\log q)^{-C}$, for any fixed $C$. The following finite families have a final cell of density exactly $1/(q(q+1))$.

The four pairs

$$(d,s)=(2,2),(3,3),(4,5),(6,7)$$

have globally private witnesses and leave only $-1\pmod{12}$ uncovered. For $j=0,\ldots,k-1$, install the lifted pairs

$$q_{j,d}=12^j d,\qquad r_{j,d}=12^j(s+1)-1.$$

Each witness satisfies $q_{j,d}\le r_{j,d}<2q_{j,d}$. The $j$-th block lies in the old survivor fiber $-1\pmod{12^j}$, and leaves $-1\pmod{12^{j+1}}$. Witnesses in a block are private within it; earlier witnesses lie outside its parent fiber, while later ones survive its classes. Thus all witnesses are globally private, and the moduli are distinct and strictly ordered block by block.

Put $M=12^k$ and append the zero class at $q=M-1$, with witness $r=q$. Its endpoint is an old survivor. The old moduli are at most $M/2$, and their witnesses are at most $2M/3-1<q$, so none of the old positive witnesses is in the new zero class. The entire family remains globally private, with the exact original activation. Since $(M,M-1)=1$, its final first-kill density is

$$e_q=\mu([-1]_M\cap[0]_{M-1}) =\frac1{M(M-1)}=\frac1{q(q+1)}.$$

This is another local obstruction, not a refutation of any summed entropy estimate or of private-endpoint logarithmic nullity.

Final status of the private-endpoint attempt

No positive-upper-logarithmic-density family of globally private integer witnesses $q\le r_q<2q$ with distinct moduli was found. Nor was logarithmic nullity proved. A stronger finite assertion considered during the continuation was

$$\sup_{\mathcal F}\frac1{\log X} \sum_{\substack{q\le X\\q\in\mathcal F}}\frac1q \longrightarrow0,$$

where the supremum is over finite globally private canonical families. This remains an unproved target, not an estimate available for use. It would imply endpoint nullity, since $1/r_q\le1/q$.

The attempted arguments based only on containment antichains or private profinite witnesses are insufficient for the already documented binary-colored example. Rough-shadow multiplicity and quadratic-energy bounds are ruled out by the explicit examples above and in rough-shadow energy. These obstructions do not disprove the displayed finite target or the original endpoint assertion. Mixed-idempotent coding likewise failed to control incomparable earlier moduli, as recorded in the preparation notes. No theorem converting the height restriction on the witnesses into the needed logarithmic bound was obtained.