Independent investigation of Erdős Problem 25

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Status and scope

This investigation did not resolve the problem. No proof of existence of logarithmic density in full generality, and no counterexample satisfying the distinct-modulus hypothesis, was obtained.

Only the supplied local problem and sources were used. No supplied problem page or source was edited. The claims below are separated into proved facts, explicit calculations, unsuccessful constructions, and unproved targets. Computational searches are not used as proofs of a general assertion.

The main useful results of this investigation are a self-contained periodic compression argument, a sharp bound for one or finitely many early hits per distinct modulus, a two-class Dirichlet inequality, and explicit examples showing that several tempting harmonic compression assertions are false.

1. Exact problem, conventions, and elementary cases

Let

$$1\le q_1<q_2<\cdots,\qquad 0\le a_i<q_i,$$

and put

$$B_i=\{n\ge q_i:n\equiv a_i\pmod{q_i}\} =\{q_i t+a_i:t\ge1\},\qquad B=\bigcup_i B_i,\qquad A=\mathbb N\setminus B.$$

In particular, the first eligible point is (q_i+a_i), including (q_i) when (a_i=0). The smaller representative (a_i>0) is not removed by this congruence. This convention is maintained throughout these notes.

For a set (S\subseteq\mathbb N), natural density, when it exists, is

$$d(S)=\lim_{X\to\infty}\frac{\#(S\cap[1,X])}{X}.$$

Write

$$L_S(X)=\frac1{\log X}\sum_{\substack{n\le X\\n\in S}}\frac1n, \quad \underline{\operatorname{ld}}(S)=\liminf L_S(X), \quad \overline{\operatorname{ld}}(S)=\limsup L_S(X).$$

Logarithmic density exists precisely when the latter two quantities agree. Natural density implies logarithmic density with the same value; the converse is not asserted. Set (H_N=\sum_{n\le N}1/n).

For a finite initial segment, define (A^{(k)}=\mathbb N\setminus \bigcup_{i\le k}B_i). Beyond (q_k), this set is periodic modulo

$$L_k=\operatorname{lcm}(q_1,\ldots,q_k).$$

Thus its natural and logarithmic densities agree; denote the common value by $\delta_k$. These decrease to $\delta\in[0,1]$, and

$$\overline d(A)\le\delta, \qquad \overline{\operatorname{ld}}(A)\le\delta.$$

Consequently $\delta=0$ already implies natural density zero. The difficult case necessarily has $\delta>0$.

The following standard positive cases were checked against Chojecki's full source, Sections 2–3, and have elementary proofs:

• If $\sum_i1/q_i<\infty$, then (A) has natural density $\delta$. Indeed, the activation convention gives $\#(B_i\cap[1,X])\le\lfloor X/q_i\rfloor$, so the infinite tail has upper natural density at most $\sum_{i>k}1/q_i$. This justifies the infinite union bound, rather than merely citing finite additivity.
• If the moduli are pairwise coprime, then $\delta_k=\prod_{i\le k}(1-1/q_i)$. A convergent reciprocal sum is the previous case; a divergent sum gives $\delta=0$.
• For zero residues, the Davenport–Erdős theorem gives logarithmic density for the union of multiples and its complement. Natural density can fail. The relevant full 1936 and 1951 source passages were examined. Their divisibility closure and smooth-factor identities cannot simply be transferred to arbitrary residues.

2. Periodic union compression: proved

For a finite family of full residue classes let

$$U=\bigcup_i[a_i]_{q_i},\qquad U_0=\bigcup_i[0]_{q_i}.$$

Their densities are proportions in the cyclic group modulo the least common multiple. Equivalently, use Haar measure $\mu$ on the profinite integers. Then

$$\boxed{\mu(U)\ge\mu(U_0).} \tag{2.1}$$

Here is a complete finite proof. Write the cyclic group, by CRT, as a product of prime-power groups. Each congruence class is a product of cosets of subgroups in these coordinates. In one coordinate replace every coset by the subgroup of the same cardinality containing zero. Fix the other coordinates. The participating old fibers have union cardinality at least their largest cardinality. The replacement fibers are nested subgroups, so their union has exactly that largest cardinality. The union therefore cannot increase under this replacement. Repeat for every prime coordinate. The resulting union is exactly (U_0), proving (2.1).

This argument also allows repeated moduli. Distinctness enters the later capacity bounds, not (2.1).

The supplied Filaseta–Ford–Konyagin–Pomerance–Yu full source, Introduction, paragraph beginning “Similar to the definition of $\delta^-(S)$,” explicitly attributes the equivalent maximum-uncovered-density statement to Rogers. Thus (2.1) is not claimed as a new theorem.

Limitation: (2.1) compares completed periodic unions. It does not compare their delayed finite harmonic sums. Section 6 gives exact counterexamples to two stronger comparisons that were considered.

3. Sharp harmonic capacity for moduli and early hits: proved

3.1 Harmonic inequality for a finite set of multiples

Let (D\subseteq[1,X]\cap\mathbb N) be finite, let

$$M(D)=\bigcup_{q\in D}q\mathbb N, \qquad d_0=d(M(D)),\qquad N=\lfloor X\rfloor.$$

Then

$$\sum_{\substack{n\le X\\n\in M(D)}}\frac1n\le d_0 H_N. \tag{3.1}$$

Proof: for each prime (p\le N), independently choose a nonnegative integer

$$\mathbb P(V_p=j)=(1-1/p)p^{-j},$$

and set (Z=\prod_{p\le N}p^{V_p}). Every positive integer (n\le N) has probability (c/n), where (c=\prod_{p\le N}(1-1/p)). Thus conditioning

on (Z\le N) gives exactly the probability weights (1/(nH_N)).

The event (Z\in M(D)) is increasing in all the valuations, whereas

(Z\le N) is decreasing. Independent-coordinate association therefore gives

$$\mathbb P(Z\in M(D)\mid Z\le N) \le\mathbb P(Z\in M(D))=d_0.$$

For completeness, the association used here follows in one coordinate from

$$\operatorname{Cov}(f(V),g(V)) =\tfrac12\mathbb E[(f(V)-f(V'))(g(V)-g(V'))]\ge0$$

for increasing (f,g), with (V') an independent copy; induction gives the product version. Apply this to the increasing event and the complement of the decreasing event. The equality with (d_0) is the usual independent Haar valuation distribution. This proves (3.1).

3.2 One modulus supplies only one reciprocal term

If (U) consists of one selected class for each distinct (q\in D\subseteq [1,X]), and (u=\mu(U)), then (2.1) and (3.1) give

$$\boxed{\sum_{q\in D}\frac1q\le u H_{\lfloor X\rfloor}.} \tag{3.2}$$

Indeed, (D\subseteq M(D)), and distinctness permits counting each (q) only once. This is a uniform finite-family inequality.

In particular, if at most one endpoint (x_q\in[q,X]) is selected per modulus, then

$$\sum_q\frac1{x_q}\le u H_{\lfloor X\rfloor}. \tag{3.3}$$

The endpoint need not even satisfy its selected congruence for (3.3).

For the first (K) eligible representatives of every class, the union bound and (q t+a\ge qt) give

$$\boxed{ \sum_{\substack{n\le X\\ n=q t+a_q\text{ for some }q,\ 1\le t\le K}}\frac1n \le H_K\,u H_{\lfloor X\rfloor}. } \tag{3.4}$$

Only moduli (q\le X) need be included in (U) here. With at most (s) classes per modulus the corresponding bound has an extra factor (s).

Precise limitation: (3.4) controls boundedly many early hits of a family whose whole periodic union has small measure. It does not control the unshadowed tail relative to a previous family. It also has the factor (H_K). It would be an overstatement to conclude from it alone that every possible counterexample must use unbounded quotients. What it rules out is a small-whole-footprint construction obtaining a fixed positive fraction of $\log X$ from a fixed number of hits per modulus.

3.3 Dirichlet version

For $s>1$, the distribution (n^{-s}/\zeta(s)) has independent prime valuations with tails (p^{-as}), coordinatewise dominated by Haar valuation tails (p^{-a}). The same increasing-event argument gives

$$\sum_{n\in M(D)}n^{-s}\le d_0\zeta(s).$$

Consequently, for distinct moduli and endpoints (x_q\ge q),

$$\sum_q x_q^{-s} \le\sum_q q^{-s} \le\sum_{n\in M(D)}n^{-s} \le\mu(U)\zeta(s). \tag{3.5}$$

This also agrees with the finite Dirichlet inequality in the checked Davenport–Erdős 1936 source.

4. Why the supplied multi-residue counterexample does not transfer

Wang's full source, particularly Lemmas 4–7 and Section 4, was inspected. Its finite block at (Q=2^j) has at least (3Q/8) endpoints in

$$[11Q/10,19Q/10],$$

with assigned moduli in $[19Q/20,21Q/20]$, all strictly smaller than their endpoints, and completed union measure at most

$$\eta_j=\exp(-k_j/100),\qquad k_j=2\lfloor\sqrt j/8\rfloor.$$

The same modulus is used for many different residues. Thus this is not a distinct-singleton block for Problem 25.

There is a rigorous obstruction stronger than merely observing these repetitions. If such a block were singleton, each modulus could witness at most one of these endpoints: the endpoint interval has diameter (4Q/5), smaller than every assigned modulus. Thus (3.3) would give

$$\frac{15}{76} \le\sum_{m\in E_j}\frac1m \le\mu(U_j)H_{\lfloor19Q/10\rfloor}.$$

Thus its footprint would have to be $\gg1/\log Q$, whereas Wang's $\eta_j$ is (o(1/\log Q)). A direct singleton analog with those parameters is impossible. This does not rule out a construction whose different-scale footprints overlap substantially.

Two attempted repairs failed more concretely:

1. Wang's label sets (S) have $|S|=k/2+O(\sqrt k)$, and a modulus with label (S) must be divisible by $d_S=\prod_{i\in S}p_i$, with (p_i>4^{k+1}). The number of available distinct moduli in a fixed-ratio interval of size (Q) with these incidences is at most

$$O\!\left(Q\sum_S\frac1{d_S}+2^k\right) \le O\!\left(Q,2^k4^{-(k+1)(k/2-\sqrt k)}+2^k\right)=o(Q).$$

Retaining the incidence skeleton cannot supply a positive fraction of (Q) different moduli. 2. Refining each old class to a class modulo a distinct multiple of its modulus keeps the new full class inside the old one, but loses capacity. The available modulus density is at most $\sum_S1/q_S=O(2^k/Q)$. Accumulating enough reciprocal mass over an enormous multiplier range also pushes the endpoint range beyond the old finite periodic structure, destroying the desired local anomaly. No successful refinement was found.

A further attempted repair used sparse labels of probability (1/p), so that endpoint label patterns occur in the same proportions as prime divisibility patterns of moduli. With one marked residue modulo each prime, these labels are just divisibility of a common translate (n-R). This fixes the counting mismatch but returns to divisor-window behavior rather than Wang's rapidly decaying footprint. No adequate multi-scale construction was obtained from this idea.

Wang's recovery-and-epoch mechanism itself is sound for its stated multi-residue blocks: a fixed finite past is eventually periodic, later moduli can be chosen larger than a recovery cutoff, and long epochs collect harmonic deletion. For Problem 25, such an argument would still need legal singleton blocks and separation of their modulus ranges. No infinite singleton construction was specified here, so none is claimed to satisfy those obligations.

5. Candidate finite domination inequalities and a two-class result

For a finite distinct-singleton family let (U_{\rm tr}) denote its delayed union, and (U) its full periodic union. Two strong possible targets are

$$\sum_{\substack{n\le X\\n\in U_{\rm tr}}}\frac1n \le\mu(U)H_{\lfloor X\rfloor}, \tag{H}$$

and

$$\sum_{n\in U_{\rm tr}}n^{-s}\le\mu(U)\zeta(s) \qquad(s>1). \tag{D}$$

Neither (H) nor (D) was proved or refuted in general. The direct one-endpoint inequality is not a proof of either statement for all hits. Splitting by the quotient (t) loses an additional harmonic or zeta factor.

(H) would settle the problem: apply it at cutoff (X) to all moduli at most (X), and combine the resulting upper bound for (B) with the lower bound from every fixed finite initial union. A general proof of (D) would give the analogous Dirichlet limit; a nonnegative Tauberian step would then be needed to state the logarithmic-density conclusion.

Both assertions are false if arbitrary multiplicity is permitted. Wang's finite blocks have eligible harmonic mass bounded below by (15/76), but $\mu(U_j)H_{2Q}\to0$. For (D), take (s=1+1/\log Q). Their Dirichlet mass stays bounded below absolutely, while $\mu(U_j)\zeta(s)\to0$. Thus an argument ignoring denominator multiplicity cannot prove the required result.

A proof of (D) for two classes

For $s>1$, define the shift loss

$$\ell_s(q,a)=\sum_{t\ge1}\bigl((qt)^{-s}-(qt+a)^{-s}\bigr)\ge0.$$

A single delayed class has Dirichlet mass $q^{-s}\zeta(s)-\ell_s(q,a)$, so (D) holds for one class.

Consider distinct moduli (m<n). If the classes are incompatible, their union has density (1/m+1/n), and the single-class bounds suffice. Otherwise let (L=\operatorname{lcm}(m,n)), and let (b\in[0,L)) be the common CRT representative. The union density is the same as for the two zero classes.

If (b=0), the intersection mass is (L^{-s}\zeta(s)). If (b\ge n), the first intersection point is eligible, and

$$\sum_{t\ge0}(b+tL)^{-s}\ge L^{-s}\zeta(s).$$

In both cases inclusion–exclusion bounds the union mass by that of the two zero classes. If (0<b<n), then (b=a_n), and the intersection mass is

$$L^{-s}\zeta(s)-\ell_s(L,b).$$

Since (L\ge n),

$$\ell_s(L,b)\le\ell_s(n,b)$$

term by term: (x^{-s}-(x+b)^{-s}) decreases with (x). Hence the intersection's shift loss is absorbed by the (n)-class shift loss. Again the union mass is no larger than the zero-class union mass. The increasing valuation argument in Section 3.3 bounds the latter by its density times $\zeta(s)$. This proves (D) for two classes.

For three or more classes, alternating higher intersections obstruct this simple absorption. No conserving extension was found.

6. Exact counterexamples to stronger compression ideas

These calculations were independently recomputed with rational arithmetic before this record was written. They do not disprove (H).

6.1 Normalized harmonic mass is not maximized by zero residues

At (X=100), compare moduli $4,7,9,10,18$, first with all residues zero, then with only the residue modulo 18 changed to 4. Let the eligible harmonic sums be (h_0,h_1), and periodic union densities (u_0,u_1). Exact enumeration in one common period and through (X=100) gives

$$u_0=\frac7{15},\quad u_1=\frac{17}{35}, \quad h_0=\frac{12037863923159}{6746328388800}, \quad h_1=\frac{17081934714855317}{9195245593934400}.$$

In particular,

$$u_0h_1-u_1h_0 =\frac{113383068679483}{482750393681556000}>0.$$

Thus (h_1/u_1>h_0/u_0). Periodic compression is not monotone for this normalized harmonic ratio.

6.2 Even at equal periodic density, shifting can increase harmonic mass

At (X=1000), compare

$$0\pmod4,\quad0\pmod9,\quad0\pmod{11},\quad a\pmod{210}$$

for (a=0) and (a=48). Both full unions have density (274/693). Outside the first three classes, the last class adds only (210) in the first system, and adds (258,678) in the second. All three displayed points respect activation (n\ge210). Therefore

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

This also refutes a proposed comparison of the additive harmonic deficit against the all-zero system.

Targeted exhaustive or stochastic finite searches found no counterexample to (H), including families of divisors of several moderately large common periods. These searches cover only finite cases, have no established extremal completeness, and are not evidence on which any theorem here depends. The scripts and detailed search notes are not retained.

7. A uniform obstruction to common-translate narrow-window blocks

The following partial result was proved during the investigation. It rules out a particular proposed construction, not all compatible-residue systems.

Proposition. Fix (K,C>1). Uniformly over integers (R=R_N),

$$\sum_{\substack{N\le n\le N^C\\ \exists q\in[N,KN]\cap\mathbb N:\ q\mid n-R}}\frac1n =o(\log N). \tag{7.1}$$

Discarding ineligible points (n<q), or taking only a subset of the moduli, can only decrease the left side.

Here are the details. For fixed $\alpha>0$, first consider an integer interval of length (H\ge N^{1+\alpha}). Choose $z=N^\epsilon$, where (2\epsilon<\min(1,\alpha)), and put

$$s_z(v)=\sum_{p\le z}1_{p\mid v},\qquad \lambda=\sum_{p\le z}\frac1p=\log\log N+O_\epsilon(1).$$

For any interval (J) of (T) integers, expansion of the square and CRT give

$$\sum_{v\in J}(s_z(v)-\lambda)^2\ll T\lambda+z^2. \tag{7.2}$$

Each one-prime or two-prime count has error (O(1)); the same estimate holds with any subset of the primes omitted and its own reciprocal mean. The prime reciprocal asymptotic used here is the classical one also used in the supplied sieve literature.

Only (O_K(N/\lambda)) moduli (q\in[N,KN]) have (s_z(q)<3\lambda/4). Their progressions cover (O_K(H/\lambda)) points. Only (O(H/\lambda)) points of the translated interval have (s_z(n-R)\ge3\lambda/2).

For a remaining modulus and point write (n-R=qt), and set

$$X_q(t)=\sum_{\substack{p\le z\\p\nmid q}}1_{p\mid t}, \qquad \lambda_q=\sum_{\substack{p\le z\\p\nmid q}}\frac1p.$$

Uniformly for (q\le KN), $\lambda_q=(1-o(1))\lambda$. Indeed, splitting prime divisors of (q) at $\log N$ gives

$$\sum_{p\mid q}\frac1p \le\sum_{p\le\log N}\frac1p +\frac{\log(KN)}{(\log N)(\log\log N)} =O(\log\log\log N)=o(\lambda).$$

Since (s_z(n-R)=s_z(q)+X_q(t)), the remaining points have (X_q(t)<3\lambda/4). Their (t)'s occupy an interval of length (H/q+O(1)), so (7.2) bounds their number by

$$O\!\left(\frac{H}{q\lambda}+\frac{z^2}{\lambda^2}\right).$$

Summing over (q\in[N,KN]) gives (O_K(H/\lambda)), since (Nz^2=o(H)). All these estimates are uniform in the translate (R).

The initial range $[N,N^{1+\alpha}]$ has harmonic mass at most $\alpha\log N+O(1)$. Split the rest through (N^C) into dyadic intervals. Each contributes (O_{K,\alpha}(1/\lambda)), and there are (O_C(\log N)) intervals. First let (N\to\infty), then $\alpha\downarrow0$, proving (7.1).

This shows why simply translating the zero classes from a fixed-ratio modulus window does not create a sufficiently long logarithmic deletion epoch. It does not handle arbitrary independently chosen residues, growing window ratios, or the general conditioned-tail problem.

8. Conditional tails: exact reduction and a quantifier gap

An initially attractive proposal was to prove only a uniform continuity bound for families with a small full periodic union, then apply it to the unshadowed tail after conditioning on a finite prefix. The conditioning step has a nontrivial loss.

Fix a prefix of period (L), with allowed residue representatives (r\in R\subseteq{1,\ldots,L}). Write

$$n=L(u-1)+r,\qquad u\ge1.$$

For a later modulus (q), put (g=(q,L)) and (m=q/g). The class meets this fiber iff (a\equiv r\pmod g); if compatible it pulls back to one class modulo (m), since $(m,L/g)=1$.

The activation condition remains exactly

$$gm\le L(u-1)+r. \tag{8.1}$$

It implies only (m\le Lu), not the original-form condition (m\le u).

Moreover, distinct original moduli may produce the same quotient modulus. For a fixed (m), the possible (g\mid L) satisfy $(m,L/g)=1$, giving the exact upper multiplicity

$$\prod_{\substack{p^\ell\parallel L\\p\nmid m}}(\ell+1) \le\tau(L). \tag{8.2}$$

If (\nu_r) is the full tail-union measure inside the pulled-back fiber, then

$$\frac1L\sum_{r\in R}\nu_r =\mu(U_{\rm tail}\setminus U_{\rm prefix}). \tag{8.3}$$

The harmonic change of variables has bounded error:

$$0\le\sum_{u\in S}\left(\frac1{L(u-1)+r}-\frac1{Lu}\right) \le\frac1r$$

for every subset (S) and cutoff. Separate (u=1), and for (u\ge2) bound the difference by $(L-r)/(L^2u(u-1))$. Summing over fibers costs at most (H_L). The finitely many discrepancies of the initial delayed prefix also have bounded harmonic mass.

These formulas rigorously account for distinctness, multiplicity, and activation. They do not give the hoped-for tail estimate. For example, a hypothetical singleton estimate with coefficient (f(\mu(U))\to0) typically becomes one with factors depending on (L), such as $\tau(L)f(L\epsilon)+O(L\tau(L)\epsilon)$, where $\epsilon$ is the unshadowed tail measure. Although $\epsilon_k\to0$ as the prefix grows, its periods (L_k) may grow much faster. One cannot infer that

$$L_k\epsilon_k\to0\quad\text{or}\quad \tau(L_k)f(L_k\epsilon_k)\to0.$$

Thus weak unconditioned continuity was not shown sufficient. This is an identified gap, not an established local-to-global reduction. Further formulas were recorded in a companion note that is not retained.

9. First-kill decomposition and unproved charging ideas

Let

$$E_i=B_i\setminus\bigcup_{j<i}B_j.$$

They are disjoint, eventually periodic, and their densities satisfy

$$e_i=d(E_i)=\delta_{i-1}-\delta_i,\qquad \sum_i e_i=1-\delta.$$

For (j<i), compatibility is (a_i\equiv a_j\pmod{(q_i,q_j)}). A compatible earlier class induces one forbidden residue on (t) modulo

$$r_{ij}=\frac{q_j}{(q_i,q_j)}$$

in the representation (n=a_i+q_i t), (t\ge1). Thus (E_i) is a translate-dilate of a finite periodic quotient sieve of density (q_i e_i). There is no extra activation condition from earlier moduli here, since (n\ge q_i>q_j).

Chojecki's full source, Sections 4–6, was checked for this representation and its tower example. In that example, with $(u,6)=1$,

$$m_j=u2^{r-j}3^j\quad(0\le j\le r),$$

and residues

$$a_r=0,\qquad a_j\equiv u3^r(1+2^{r-j-1})\pmod{m_j}\quad(j<r),$$

the top first-kill set is

$$E_{m_r}=\{u3^r(1+2^r t):t\ge0\}.$$

Its density is (1/(u6^r)), but its first point contributes (1/(u3^r)). This disproves a naive individual bound using only the first-kill density and an entropy-sized error. It does not disprove a global bound: the preceding scaffold has reciprocal scale (O(1/(u2^r))), larger than that first spike by a factor of order $(3/2)^r$. Disjoint copies of this gadget do not provide the missing amplification.

One weaker sufficient global target considered was

$$\sum_{\substack{n\le X\\n\in B}}\frac1n \le \left(\sum_{q_i\le X}e_i\right)\log X +O\!\left(\sum_{q_i\le X}e_i\log q_i\right)+o(\log X). \tag{9.1}$$

This is unproved. Its attraction is that for every fixed infinite system,

$$\frac1{\log X}\sum_{q_i\le X}e_i\log q_i\longrightarrow0$$

by dominated convergence, since $\sum e_i\le1$. If (9.1) were proved, the finite-prefix lower bound would give logarithmic density (1-\delta) for (B), hence $\delta$ for (A).

A possible backward charge uses compatible (m<q). Write

$$r=\frac m{(m,q)},\qquad s=\frac q{(m,q)}>r.$$

If (r=1), the earlier class contains the later class, so the later class has no first kills. Otherwise (r\ge2), and

$$\frac1r\le2\log(s/r)=2\log(q/m).$$

Indeed (s\ge r+1) and $\log(1+1/r)\ge1/(r+1)\ge1/(2r)$. This compares the relative size of a quotient restriction to a decrease in logarithmic modulus. No conserving charge scheme was found. Naively summing intersections with later classes charges the same earlier class repeatedly, potentially on logarithmically many scales.

Other unfinished ideas included a second-moment capacity argument, entropy of the least killing modulus, and prime-coordinate martingale approximation. None supplied the required global estimate. In particular, the elementary second-moment choice (h_q=\sum_{n\le X,,n\in E_q}1/n) and

$$V(\omega)=\sum_q qh_q1_{[a_q]_q}(\omega)$$

has $\mathbb EV=\sum_qh_q$, but its second moment involves

$$\sum_{q,r}(q,r)\,1_{a_q\equiv a_r\pmod{(q,r)}}h_qh_r.$$

The hoped-for upper bound by $H_X\sum_qh_q$ is false already for the zero classes modulo (2,3,5,7) at (X=10). Here

$$(h_2,h_3,h_5,h_7) =\left(\frac{137}{120},\frac49,\frac15,\frac17\right),$$

and exact calculation gives

$$\mathbb EV^2-H_{10}\sum_qh_q =\frac{329873}{6350400}>0.$$

Consequently that direct Cauchy–Schwarz route is not a proof of (H).

10. Remaining mathematical uncertainty

There is no established control here of all large-quotient hits from a distinct-modulus family, nor of their unshadowed part after a finite prefix. There is also no construction making those hits produce a fixed logarithmic oscillation while the periodic survivor densities remain bounded away from zero.

In particular, none of the following has been established:

• the sharp finite inequality (H) or its general Dirichlet counterpart (D);
• Chojecki's global charge condition or the weaker target (9.1);
• a valid singleton replacement for Wang's multi-residue blocks;
• an infinite legal congruence system with unequal upper and lower logarithmic densities.

The auxiliary notes with more detailed calculations and failed attempts (compression, early-hit capacity, and the common-translate narrow-window proposition; Dirichlet approaches and their multiplicity obstruction; exact finite examples and searches; quotient and activation formulas; label and matching attempts) are not retained.

Where such a note used informal language suggesting that bounded-hit capacity alone forces every possible counterexample to use unbounded quotients, the qualification in Section 3.2 of this consolidated record is the intended precise statement. The obstruction is proved for small-whole-footprint bounded-hit blocks; the general conditioned setting remains unresolved.

Final status: incomplete investigation, not a solution of Problem 25.