Ho's p=2 large-partial-sum theorem: independent examination

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Subject and independence

Mathematical verdict: refutation-failed. The frozen proof of Theorem 1.6 at $p=2$, including the stated consequence for the example question in Problem 995, survives the attacks below. This is a whole-proof examination of that specialization, not an assertion about every $p$ or the rest of the paper. A nonconsumed transcription discrepancy is recorded under Source fidelity. No essential missing lemma or repair was needed for the assigned conclusion.

The reviewer is an independent mathematical examiner; the disclosed model is gpt-6-astra. The examination date is 2026-10-15 UTC. The mathematical subject was frozen on 2026-10-15 at 05:52:55.296610 UTC. Its canonical locations are:

• wiki/problems/discrepancy/E0995/_index.md, Statement;
• wiki/problems/discrepancy/E0995/claims/2026_04_20_ho.md, the supplied mathematical statement, consequence, Covers and Depends on passages;
• library/analysis/ho_2026_counterexamples_lacunary_dilates_via_dyadic_spike/ho_2026_counterexamples_lacunary_dilates_via_dyadic_spike.md, Notation, Theorem 1.6, Sections 2–4, and the proof of Theorem 1.6 in Section 7.

The source is Boon Suan Ho, Counterexamples for lacunary dilates via dyadic spike blocks, arXiv:2604.18535v2. The source mathematics and construction are Ho's. The independent examination and derivations in this report are attributed to Plasma AI.

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The mathematical evidence is the derivation and attempted refutations in this report. No numerical experiment, executable theorem check, formal proof, or cached success output is used. There is consequently no computational reproduction leg to run. The retained text suffices to repeat the mathematical assessment from an ordinary clone.

Restatement and conventions

Equip $\mathbb T=\mathbb R/\mathbb Z$ with normalized Lebesgue measure. There are a real measurable representative $f$ on $\mathbb T$, with $\int f=0$ and $\int |f|^2<\infty$, a fixed sequence of positive integers $n_1<n_2<\cdots$ with $n_{j+1}/n_j\geq2$, and a measurable set $X\subseteq\mathbb T$ of measure one such that

$$\forall x\in X\ \forall\varepsilon>0,\qquad \limsup_{N\to\infty} \frac{\sum_{j=1}^{N}f(n_jx)} {N(\log N)^{1/2-\varepsilon}}=+\infty.$$

The same $f$, sequence, and full-measure set work for every $\varepsilon$. The numerator is signed, not replaced by its absolute value. $N$ counts selected summands; it is neither the largest selected exponent nor the largest dilation. Logarithms are natural, and limits concern sufficiently large integer $N$, so the initial undefined logarithmic expressions do not enter.

Taking $\varepsilon=1/4$ contradicts the proposed $o(N\sqrt{\log\log N})$ estimate almost everywhere for this pair. This answers the example question in the supplied catalog statement. It does not give an optimal general growth estimate for its open-ended first sentence. Although the catalog statement does not require mean zero, a mean-zero counterexample lies within its class.

The intermediate depth, layer, digit, exponent, spacing and count parameters used below are integers. The source's finite sums and digit windows require these discrete types. The actual stage construction realizes them: $\ell,L,T,U,D,M$ are chosen as integers and $d$ can be chosen by the ceiling of a base-two logarithm. We do not rely on a claim that the auxiliary digit-window lemmas hold for arbitrary real depths or offsets.

Weakest steps and independent derivations

1. A rare central hit must survive every negative term

Write $\rho=2^{-d}$, using $\rho$ to distinguish the spike probability from the theorem's exponent $p=2$. On the interval of length $\rho$ the spike is $h=\sqrt{2^d-1}$; elsewhere it is $-g=-1/\sqrt{2^d-1}$. Directly,

$$\rho h=(1-\rho)g,\qquad \rho h^2+(1-\rho)g^2=1.$$

Thus the spike is centered with variance one. Apart from dyadic rationals, $\phi_d(2^v x)$ is determined by the bits in $[v+1,v+d]$. The probability of any prescribed values of finitely many distinct binary digits is $2^{-r}$, where $r$ is their number: refine to the partition by the largest digit and count the intervals. This proves the finite independence used here for all window positions, rather than inferring it from numerical samples.

For the source's Fourier argument, integration of the interval indicator gives, at nonzero integer frequency $r$,

$$\widehat{\mathbf1_{[0,2^{-d})}}(r) =\frac{1-e^{-2\pi i r2^{-d}}}{2\pi i r}.$$

It vanishes when $2^d$ divides $r$. Dilation by $2^v$ places the nonzero coefficients only at frequencies with valuation in $[v,v+d-1]$. The zero coefficient is zero. Disjoint such bands therefore give orthogonality by Parseval, as used in Lemma 3.1 and Proposition 4.3.

Put $\alpha=\sqrt{\lambda/L}$ and $F=\alpha\sum_{q=1}^L\phi_d(2^{U+qD}\,\cdot)$. The within-block windows and bands are disjoint for $D\geq d+2$. Consequently $\int F=0$ and $\|F\|_2^2=\lambda$. Moreover

$$\sqrt{\lambda L}\,g \leq \sqrt{\frac{2\lambda L}{2^d}} \leq\frac{\lambda}{4\sqrt2 B}.$$

Here $2^d\geq64B^2L/\lambda$ and $2^d-1\geq2^{d-1}$ were used. In particular the source's absolute lower-floor constant can be taken to be $C_3=1$, uniformly in all stage parameters.

Expanding a trial sum gives weights

$$w_h=\#\{(q,r):1\leq q\leq L,\ 1\leq r\leq\ell,\ q+r=h\}.$$

Counting the pairs gives $\sum_h w_h=L\ell$. For $\ell+1\leq h\leq L+1$, each $r\in[1,\ell]$ gives the allowed $q=h-r$, so $w_h=\ell$. This verifies the central range, including its two endpoints.

The deliberately hostile configuration is exactly one central hit and every other spike in the trial at its negative value. Its weighted value is

$$\alpha\{\ell h-(L\ell-\ell)g\}.$$

Any additional hit increases this value. Even discarding the positive $\alpha\ell g$ gives the source's lower estimate

$$\sum_{r=1}^{\ell}F(2^{M+rD}x) \geq \ell\{\alpha h-\sqrt{\lambda L}\,g\} \geq (4\sqrt2-1)B\ell >2B\ell.$$

The last bound uses $\alpha h\geq4\sqrt2 B$ and $\lambda/(4\sqrt2 B)\leq B$. Thus the attempted cancellation by all noncentral terms fails with an absolute margin.

There are $n_c=L-\ell+1$ independent central windows. For $1\leq\ell\leq L/8$ we have $7L/8\leq n_c\leq L$, and

$$\frac{\lambda}{128B^2L}<\rho\leq\frac{\lambda}{64B^2L}.$$

Let $u=n_c\rho\leq1/64$. Then $1-(1-\rho)^{n_c}\geq1-e^{-u}\geq u/2$. It follows that the success probability is at least $c_0\lambda/B^2$ with the single absolute choice $c_0=7/2048$. No constant deteriorates as $L$, the depth, the offset, or the stage number grows.

The local statement has no admissible integer trial when $L<8$; that case is vacuous. Actual stages have $\ell\geq20$ and $L\geq160$. An allowed negative $M$ causes no noninteger dilation: $M+D\geq0$ implies $M+rD\geq0$, and $U+M+hD\geq U+D\geq0$ for $h\geq2$.

2. Independence must yield the same function at all selected dilations

There are two distinct independence questions.

The undilated block $F_k$ uses the bit windows

$$J_{k,q}=[U_k+qD_k+1,\ U_k+qD_k+d_k], \qquad 1\leq q\leq L_k.$$

Their largest coordinate is $V_k^{\max}+1$. Since $U_k=V_{k-1}^{\max}+1$, the first coordinate of the new block is $V_{k-1}^{\max}+D_k+2$, beyond every coordinate of the old blocks. Within a block consecutive windows are disjoint because $D_k=d_k+2$. Thus the $F_k$ are mutually independent centered variables. This conclusion uses the actual bit sets; disjoint Fourier supports alone would not imply independence.

For a trial, all the relevant translated spike windows, including more than the central ones, lie between

$$U_k+M_{k,t}+2D_k+1 \quad\hbox{and}\quad H_{k,t}=U_k+M_{k,t}+(L_k+\ell_{k,t})D_k+d_k.$$

By (4.11), the first such coordinate of the next trial is $H_{k,t}+2D_k+3$, strictly beyond this range. The second inequality in (4.10) places the first range of a new stage beyond $\Omega_{k-1}$, and (4.14) bounds all previous ranges. Hence the central-hit events for distinct trials are mutually independent. No independence between a block and its success event, or between an event and all the other blocks, is asserted or needed.

Now use the source's explicit choice $a_k=2^{-k-2}$ at $p=2$, so $\lambda_k=a_k$ and $\sum_k\lambda_k=1/4$. The disjoint Fourier bands give, for every finite tail,

$$\left\|\sum_{k=m}^{n}F_k\right\|_2^2 =\sum_{k=m}^{n}a_k.$$

Completeness gives an $L^2$ limit, with integral zero by continuity of the integral. The permitted convergence theorem for independent centered variables with summable variances gives almost-sure convergence of the same series. The almost-sure and $L^2$ limits agree almost everywhere, since $L^2$ convergence implies convergence in probability and that limit is unique.

Here is an explicit representative and exceptional-set reconstruction. Let $E$ be the measurable full-measure set on which the series converges to a finite value, with the dyadic rationals removed. Define $f$ to be that sum on $E$ and zero on its complement, as a function on $\mathbb T$. It is real, measurable, periodic when pulled back to $\mathbb R$, belongs to $L^2$, and has integral zero. With the source's finite constant

$$\mu=C_3\sum_i\frac{a_i}{B_i},$$

every finite block sum is at least $-\mu$. Removing any one block leaves a convergent series on $E$, and each of its finite sums has the same lower bound. Therefore, simultaneously for all $k$ and all $y\in E$,

$$f(y)\geq-\mu,\qquad f(y)-F_k(y)\geq-\mu.$$

For a positive integer $n$, the map $y\mapsto ny$ on $\mathbb T$ preserves Lebesgue measure, by splitting $[0,1)$ into its $n$ inverse branches and changing variables. In particular it pulls back null sets to null sets. Thus

$$X_0=(\mathbb T\setminus D_{\rm dyad}) \cap\bigcap_{j\geq1}\{x:n_jx\in E\}$$

has measure one. On $X_0$ the preceding bounds apply at every selected dilation and for every omitted block. This is a countable intersection; there is no uncountable exceptional-set deletion hidden here. Changing the representative of the $L^2$ class would likewise change none of these sums outside a countable union of inverse images of a null set.

At a successful trial endpoint $N=P+\ell$, the preceding $P$ summands, the current block on the $\ell$ new summands, and the remaining blocks on those new summands give exactly

$$\sum_{j\leq N}f(n_jx) \geq -\mu P+2B_k\ell-\mu\ell =2B_k\ell-\mu N.$$

The ordered exponent construction ensures that these are precisely the first $N$ summands, without an intervening selection from another trial. Since $P\leq\ell/20$, division by $N$ yields at least $2B_k/(1+1/20)-\mu\geq B_k-\mu$. This lower bound controls the signed sum regardless of dependence among its different contributions.

3. Large trial counts must not destroy the logarithmic gain

The most dangerous interaction is between the number of trials needed for almost-sure success and the final number of selected terms. Both increase when $B_k$ increases. A bound involving the much larger selected exponents would not suffice.

At the beginning of a fixed stage let $P=P_{k,1}$, already determined. Because $P_{k,t}$ is an integer and $\eta^{-1}=20$, the ceiling in (4.3) is redundant:

$$\ell_{k,t}=20(P_{k,t}+1),\qquad P_{k,t+1}+1=21(P_{k,t}+1).$$

Consequently the actual count satisfies the exact identity

$$N_k^*+1=(P+1)21^{T_k}.$$

Large gaps in the exponent sequence add no selected terms. With $A_k=\Gamma\log(k+1)/a_k$ and $T_k=\lceil A_kB_k^2\rceil$,

$$\log N_k^* \leq\log(P+1)+(A_kB_k^2+1)\log21 \leq K_k B_k^2\quad(B_k\geq100),$$

where one possible choice is

$$K_k=A_k\log21+ \frac{\log(P+1)+\log21}{100^2}.$$

$K_k$ is finite and independent of the proposed value of $B_k$. It may depend on $k$ and previous stages; the proof needs no uniform bound on that dependence.

There is also a future-independent bound on the floor. For any allowed future choices $B_i\geq100$, both $\mu$ and the source's $\overline\mu_k(B)$ are at most

$$M_*=\frac{C_3}{100}\sum_i a_i=\frac{C_3}{400}.$$

In particular this bound does not assume that future stages already exist. It follows directly from the three sums in (7.4), with $\lambda_i=a_i$ and $B\geq100$.

Set $\delta_k=1/(4(k+2))$ and $\beta_k=1/2-\delta_k>0$. For $B\geq\max(100,2M_*)$ the preceding estimates imply

$$\frac{B-\overline\mu_k(B)}{(\log N_k^*)^{\beta_k}} \geq \frac{B^{2\delta_k}}{2K_k^{\beta_k}} \longrightarrow+\infty \quad\hbox{as }B\to\infty.$$

This bound holds despite all jumps of the ceiling in $T_k$. Thus a finite $B_k$ satisfying (7.5) exists at every fixed stage. For example, choosing an integer beyond the displayed bound is legitimate; no observation of $x$ or of trial successes enters the choice.

After such a choice, $T_k$ is a finite positive integer, every recursive count and length is finite, and so is $L_k$. Writing $R=64B_k^2L_k/a_k$, the integer $d_k=\lceil\log_2 R\rceil$ satisfies $R\leq2^{d_k}<2R$. Then $D_k=d_k+2$ and $U_k$ are integers. The two strict lower bounds on $M_{k,1}$ concern only finite integers, so an integer satisfying both exists. The remaining offsets and bookkeeping quantities are finite. This verifies the recursion without assuming its termination or its eventual conclusion.

The selected exponents are positive and strictly increasing. The first one exceeds $m_0^{\rm last}=0$; within a trial the increment is $D_k$; between trials (4.11) places the next first exponent strictly after the previous last; and (4.10) does the same between stages. Their powers of two are therefore positive integers with successive ratio at least two.

The independent-trial probability estimate now gives

$$\Pr(S_k^c) \leq\exp\!\left(-c_0T_k\frac{a_k}{B_k^2}\right) \leq(k+1)^{-8},$$

since $\Gamma\geq8/c_0$. First Borel–Cantelli gives a measure-one event $X_1$ on which every sufficiently large stage has a successful trial. This implication needs no independence between different stages. Let $X=X_0\cap X_1$.

For $x\in X$, choose a successful trial in each sufficiently large stage and denote its endpoint by $N_k(x)$. These endpoints are strictly increasing between stages and at least $20$, so $\log N_k(x)>1$. The endpoint lower bound and (7.5) give

$$\frac{\sum_{j\leq N_k(x)}f(n_jx)} {N_k(x)(\log N_k(x))^{\beta_k}}\geq k.$$

In making the denominator comparison with $N_k^*$, the numerator is positive: (7.5) already implies $B_k-\overline\mu_k(B_k)>0$, and $\mu\leq\overline\mu_k(B_k)$. Thus the direction of that comparison is justified.

Now fix any $\varepsilon>0$, after fixing $x\in X$. Eventually $\delta_k<\varepsilon$, so $(\log N_k(x))^{1/2-\varepsilon} \leq(\log N_k(x))^{\beta_k}$. The same positive sums and the same endpoints give a ratio at least $k$. This argument also covers $\varepsilon\geq1/2$. It proves the claimed positive infinite limsup on one $X$ for every $\varepsilon$, without intersecting an uncountable collection of separately obtained full-measure sets.

For the catalog implication, at $\varepsilon=1/4$ put $t=\log\log N$. Then

$$\frac{(\log N)^{1/4}}{\sqrt{\log\log N}} =\frac{e^{t/4}}{\sqrt t}\longrightarrow+\infty.$$

If the catalog-normalized sums tended to zero at any $x\in X$, multiplying them by $\sqrt{\log\log N}/(\log N)^{1/4}$ would make the theorem-normalized sums tend to zero there, contradicting their infinite limsup. Defining the restriction of the periodic representative on $[0,1]$ gives exactly $f(\{\alpha n_j\})=f(n_j\alpha)$. Integer translation of $\alpha$ leaves these values unchanged, and countably many translates of a null subset of $[0,1)$ are null in $\mathbb R$. The consequence therefore has the catalog's almost-everywhere meaning.

Strongest attack and attempted refutations

The strongest attack was to make the stage selection circular or quantitatively impossible. Increasing $B_k$ increases $T_k$, which makes the trial lengths and the end count grow exponentially. At the same time, the denominator in (7.5) and an apparently unknown future floor both depend on the construction. Ignoring either dependence could falsely manufacture the claimed gain.

This attack fails for the precise reasons in the third derivation: the count is exactly $(P+1)21^{T_k}-1$, its logarithm is bounded by $K_kB_k^2$ with $K_k$ independent of the proposed $B_k$, and the future floor is bounded before future choices by $C_3/400$. The resulting lower bound grows as $B_k^{2\delta_k}$, with strictly positive exponent at each fixed stage. Although that exponent tends to zero with $k$, no uniform choice in $k$ is being claimed: the recursive choices occur after the previous stages are fixed.

Other concrete attacks were:

• Put every spike except one central hit at its most negative value. The exact weighted value in the first derivation still exceeds the required positive signal; the lower floors cannot cancel it.
• Look for a shared bit at adjacent block and trial boundaries, or confuse orthogonality with independence. The explicit coordinate comparisons in the second derivation rule out those overlaps for the families actually required to be independent.
• Let convergence fail at a selected dilation, or change the representative on a null set. The explicit countable pullback defining $X_0$ handles both possibilities.
• Replace signed sums by absolute values, or reverse a denominator comparison with a negative numerator. Positivity is obtained before the comparison, and no absolute-value conclusion is substituted.
• Let the successful endpoints depend on $\varepsilon$, creating incompatible exceptional sets. $X$ and the stage endpoints are fixed first; decreasing the logarithmic exponent proves every later $\varepsilon$ assertion on the same set.
• Try to infer the catalog conclusion from a large value in the wrong normalization. The exact logarithmic ratio above tends to infinity in the catalog's count variable $N$.

These are mathematical attacks on arbitrary legal stages and on the infinite construction, not tests of finitely many favorable parameter values.

The explicit constants, exact count recurrence, parameter types, and common exceptional set above are routine reconstructions of inferences in the frozen proof. They do not replace a failed step with a different proof. The source supplies the particular sequence $a_k=2^{-k-2}$ used here. Summability alone would not imply $a_k\leq1$ for an arbitrary positive sequence; using the already supplied sequence satisfies that required bound. No extra hypothesis has been added to the theorem.

Premises and reading depth

The dependency census was checked against the frozen statement, proof, and consequence passages on 2026-10-15 UTC. It contains no native L-claim premise or quoted L-claim definition. This follows from the actual argument, not merely the claim page's Depends on sentence. The commissioned dependency channel therefore has no ledger rows to read. No live ledger was accessed, no pending native grade is assumed, and no native dependency acceptance order is needed.

All source entries below refer to Ho's v2 and to the retained source extract. “Proof verified” means verified in this examination at the specified scope; it does not assert a separate accepted grade for those auxiliary results.

The exact external interfaces are:

• Lebesgue measure and probability foundations. Measurability of finite digit events and countable limits, probabilities of binary cylinders, countable unions of null sets, and measure preservation by integer circle dilations. The specific cylinder and inverse-branch applications are derived above.
• Hilbert-space and Fourier foundations. Parseval/orthogonality, completeness of $L^2$, continuity of the integral on this probability space, and uniqueness of limits in probability. They turn the disjoint bands and summable squared norms into the stated $L^2$ limit.
• Convergence of independent square-summable series. Independent centered real variables with finite variances whose sum is finite have an almost-sure finite series sum. Here the variables are exactly $F_k$; their independence, centering and variance $a_k$ have been checked.
• First Borel–Cantelli. If $\sum_k\Pr(A_k)<\infty$, almost surely only finitely many $A_k$ occur. Here $A_k=S_k^c$ with $\Pr(A_k)\leq(k+1)^{-8}$. Second Borel–Cantelli appears in the source's unused Proposition 4.3(i), but is not a premise of the assigned route.

These are standard foundations expressly permitted by the commission. No additional literature was read to establish them, and no independent reconstruction of their general proofs is claimed. There is no Rosenthal interface at $p=2$. Lemma 2.2, the Fourier-tail conclusions of Lemma 3.1, and the auxiliary thresholds $Q_k$ do not enter the assigned conclusion. The source's reference to Section 1.2 for interpretation of parameters requires no missing theorem: their mathematical definitions are present in the supplied notation and construction.

Checklist

Erdos-specific checklist

Canonical failure modes

Canonical named patterns

The two neighboring disciplines are also covered: the probability calculation does not count correlated samples as independent evidence, and each proof interface is restricted to the coverage recorded in the premise table. This is neither a computational batch nor a second independent cycle; neither a structurally different computational reproduction leg nor a comparison with prior-cycle attacks is applicable.

Source fidelity and retained witness

The print comparison used the PDF at the canonical library path with the same stem as the transcription, identified by the commission as v2. Theorem 1.6 on printed p.4; the supplied Sections 2–4 on pp.7–16; and Section 7 through the proof's final sentence on pp.21–23 were inspected as page images rendered at 300 dpi. Boundary crops excluded preceding material on pp.4, 7 and 21 and excluded Remark 7.1 on p.23. A crop before the next section on p.17 contained only the running header; it contributed no additional proof text. No other source sections were read. The full supplied text was available; no print access gap affects an essential $p=2$ inference.

There is one explicit, nonconsumed transcription discrepancy: display (3.8) in the supplied source extract has the right-hand coefficient $C_2^2\lambda Q/N$, whereas the PDF on printed p.9 has $C_2\lambda Q/N$. The retained equation image shows the printed coefficient. This is a discrepancy in the extraction's transcribed text, not in Ho's printed formula. Neither version of that Fourier-tail estimate is used in the assigned proof, so it supplies no refutation or missing premise for Theorem 1.6 at $p=2$. The frozen extract has been preserved without correcting it.

The source text and equation image retain Boon Suan Ho's attribution and the source's CC-BY-4.0 license. Reading context about $p>2$ and Fourier tails does not extend this report's mathematical verdict to those results. The v1 result numbering and publication-history parentheticals are likewise outside the commissioned mathematical conclusion.

Verdict and grading

refutation-failed for the entire assigned specialization of Ho v2 Theorem 1.6 at $p=2$ and for its stated negative answer to the $o(N\sqrt{\log\log N})$ example in E995.

The report covers the periodic representative, all selected-dilation null sets, quantifier order, positive signed limsup, actual selected-term count, stage recursion, both required independence arguments, uniform constants, external foundations, and the catalog implication. No essential mathematical coverage is left unfinished within that scope. There is no computational or kernel warrant, no all-$p$ verdict, and no finding about an optimum, the Fourier-tail theorem, E996, or Remark 7.1.

This mathematical verdict does not grade the report, promote a native claim, change catalog standing, or assert journal or community acceptance. A grader distinct from the source author, reviewer, and any collaborator must assess the complete frozen report and its independence under the commissioned contract.