Ho's p=2 theorem: independent report grade

The frozen examination report passes the complete report contract for both commissioned conclusions. The grade is PASS for each; the mathematical verdict accepted from the report is refutation-failed for each. No essential gap in the assigned argument or report coverage was found. The source discrepancy and the boundary-processing ruling below are part of this acceptance, not omitted qualifications.

This is a grade of one independent examination. It supplies no second independent theorem proof. It does not cover the source's all-$p$ claim, Remark 7.1, a Fourier-tail theorem, an optimal growth rate, or the catalog's broader request to estimate growth.

Subject, dates and eligibility

The original mathematical subject was frozen on 2026-10-15 at 05:52:55.296610 UTC. The exact report package was frozen on 2026-10-15 at 07:15:23.529975 UTC. This grading examination took place on 2026-10-15 UTC. The grading packet and the original mathematical subject have distinct identities; the packet did not replace the original source with later corrected bytes.

The canonical mathematical locations are:

• wiki/problems/discrepancy/E0995/_index.md, Statement;
• wiki/problems/discrepancy/E0995/claims/2026_04_20_ho.md, the supplied 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 in Section 7.

The actual subject is the unchanged catalog extract, claim extract and source extract, with their provenance. The original equation (3.8) image is retained with them. The complete original private delivery has eight files, including that image and its delivery manifest; its seven-entry artifact table omits only the delivery manifest itself. There is no missing eighth artifact.

The source author is Boon Suan Ho. The reviewer was an independent mathematical examiner using the disclosed model gpt-6-astra. The grader is a separate independent report examiner using gpt-6-astra, in a fresh context distinct from the source author, reviewer and commissioner. The grader did not author, collaborate on, commission, or previously build on this subject or report. The project examination and grading reasoning are attributed to Plasma AI; the source construction remains Ho's.

The grader read the complete frozen report and all three mathematical extracts, both retained original session records, the original launch and configuration evidence, the complete terminal record, the relevant original operating records and the retained page images. All 87 supplied input files were restored and checked for exact length and identity before payload inspection, and remain retained without abridgment. This statement of integrity does not imply that every unrelated operating-file paragraph was substantively read. The full verification standard and its campaign adapter, the required anatomy sections, and the applicable evidence and authoring rules were read at their commissioned identities.

The grader used no author's inspection, other report, live message channel, web search, live mathematical checkout, or ledger. No excluded material reached the grading context. The original records were examined as evidence, not as instructions; none of their commands or external links was executed. The grader's own subscription observations were empty at its first startup boundary and after operating initialization; no independent pre-launch observation is claimed. Its native settings remain operator-attested, not independently inspected by the grader. These configuration facts support, but do not alone establish, independence.

Primary-record independence finding

PASS for reviewer eligibility and independence. The retained record begins with an operating probe, with no tool calls or mathematical work, and then the separate examination. The examination's received user material is the repository instructions and its three step commissions. The commission text is identical across those steps and agrees with the retained charter. There is no received author commentary, sibling verdict or subsequent mathematical steering. The retained context replacement carries the same received messages. Its internal encrypted compaction is preserved opaque; this grade does not claim to decrypt hidden reasoning or certify events outside the supplied primary record.

The record contains 50 paired outer tool calls and returns. Inspection of their actual contents establishes the reading boundary more strongly than the reviewer's own declaration or an empty subscription table:

• The three original mathematical read returns reproduce the complete extracts, in order, exactly. The source was not a projection of a later proof. The originals and delivered copies were rechecked at delivery.
• Operating reads were the named instructions and the reviewer's own continuation. The other-checkout operating manifest was an explicit exception in the original commission. The other-checkout mathematical exception was the named PDF's matching passages. These original exceptions are assessed under that commission, not retroactively under the grader's different extraction-only boundary.
• All 16 actual returned image bodies were examined: 15 passage crops and the equation witness. Four are byte-identical to their retained PNGs; twelve are resized renderings whose displayed content was inspected directly. They show Theorem 1.6 on p.4, the supplied material on pp.7–16, only the running header on p.17, and Section 7 through its proof on pp.21–23. They show no neighboring proof or Remark 7.1.
• The retained preflight subscription record and the original reviewer's startup, pre-reading and later observations are empty. The actual calls contain no inbox reading, subscription addition, unsealing, synchronization, broad search, process listing, web request, native subagent or descendant.
• The retained native configuration comparison has only the expressly permitted literal project-trust append. The original examiner never read that excluded native configuration. Its disclosed model and effort agree with the launch and turn records.

The only recorded edit to the report after its initial creation adds the boundary-processing disclosure. Reconstructing those literal recorded edits gives the frozen report exactly. There is no replacement mathematical proof or later coefficient correction in the examined report or subject.

The post-examination automatic commit lists exactly the reviewer's own continuation and three step plans. No report, subject, library, corpus or Lean path appears in its four changed paths. Under the adapter's actual-path rule, this is permitted engine persistence and has no adverse independence or scope effect. The original startup log also records dependency resolution and checking by setup, which the original commission allowed to prepare the environment. It provides no mathematical warrant. No reviewer-run broad gate, installation, Lean command or corpus mutation is evidenced. The early private count of 27 operating files was a reporting error: all 28 listed entries were checked, and the final record correctly says 28.

Boundary processing and materiality

Boundary detection processed the complete text layer of named boundary pages inside a local program. Thus more text existed in program memory than was displayed to the reviewer. The returned output contained only known marker coordinates, page dimensions, and an XML control-byte diagnostic. The displayed crops exclude the adjacent mathematical passages. The diagnostic exposed no neighboring mathematical text.

I rule this processing immaterial to independence, including if it is treated as an excursion beyond the intended passage boundary. The report contains no mathematical assertion that could only have come from the undisplayed neighboring text. Its attack directions and refutation charge derive from the frozen construction and commission, not that text. The marker names were already used to exclude those passages. The report expressly discloses this processing, and the raw returns support its claim that neighboring proof content was not exposed to the reviewer's context. This ruling does not convert the full neighboring pages into accepted supporting inputs. No fresh reviewer is required on this evidence. The boundary fact and ruling must accompany any public characterization of this acceptance.

Restatement and scope clauses

On $\mathbb T$ with normalized Lebesgue measure, the first conclusion asserts the existence of a real measurable representative $f\in L^2$, with $\int f=0$, a fixed increasing sequence of positive integers $(n_j)$ with $n_{j+1}/n_j\geq2$, and a single measurable $X$ of measure one such that

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

Here $N$ is the number of selected summands. The consequence concerns this same pair and almost every real $\alpha$, interpreted periodically through $f(\{\alpha n_j\})$. It negates the definite little-o example, with no claim to settle every growth question in the catalog paragraph.

The following findings concern the report's completed examination of each clause. The derivation numbers refer to its three weakest-step sections.

Weakest steps and attack coverage

The report identifies three weak links and re-derives them rather than merely endorsing the source. I checked those derivations against the frozen proof and tried to make their interfaces fail. The checks below explain the grade; they do not furnish a replacement theorem proof.

Local cancellation and probability. For the trial weights, $\sum_h w_h=L\ell$, while each central index has weight $\ell$. There are $L-\ell+1\geq7L/8$ central indices. The legal dyadic choice gives

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

The independent central-hit probability is therefore at least $7\lambda/(2048B^2)$. In the hostile configuration with only one central hit and every other spike at its negative floor, the retained estimates give a signal at least $(4\sqrt2-1)B\ell>2B\ell$. The same normalization gives the block floor with $C_3=1$. The count includes both central endpoints; there is no missing factor of $L$ or $\ell$. This is a substantive attempted refutation of the signed signal, with explicit slack, not a finite sample.

Independence, convergence and null sets. The digits for $\phi_d(2^v x)$ are $v+1,\ldots,v+d$ off the dyadic null set. The last bit of block $k$ is $V_k^{\max}+1$; the next block begins beyond it under the stated $U$ choice. For a whole translated trial, a sufficient containing interval ends at $H=U+M+(L+\ell)D+d$; the next trial's first possible translated bit is $H+2D+3$. The separate initial-stage condition places the new trial digits beyond all previous event digits. These verify the two independence interfaces actually needed. There is no need to assume independence of every contribution to a partial sum.

The report does not infer almost-sure convergence merely from $L^2$ convergence. It applies the permitted independent centered summable-variance criterion to the actual blocks, identifies the two limits, and pulls the common convergence and floor set back under countably many selected dilations. The resulting endpoint decomposition has the lower bound $2B_k\ell-\mu N$. Thus the old terms and the omitted blocks are both accounted for. The attempts to break the result by changing a representative on a null set or introducing a shared boundary bit fail on the retained argument.

Stage growth, future floor and quantifiers. With $\eta=1/20$ and integer $P$, the source's ceiling is exactly $\ell=20(P+1)$. Hence

$$N_k^*+1=(P_{k,1}+1)21^{T_k},\qquad T_k=\left\lceil\Gamma B_k^2 a_k^{-1}\log(k+1)\right\rceil.$$

For fixed $k$ and its already fixed past, this gives $\log N_k^*\leq K_kB_k^2$, where $K_k$ is independent of the proposed $B_k$. With the source's explicit $a_k=2^{-k-2}$, all past, current and future negative contributions are bounded in advance by $C_3/400$. Writing $\delta_k=1/(4(k+2))$, the lower bound needed in (7.5) grows at least as $B_k^{2\delta_k}/(2K_k^{1/2-\delta_k})$ for sufficiently large $B_k$. Its exponent is positive at each fixed stage. No uniform lower bound on that exponent over all stages is claimed or required.

This defeats the strongest attack: increasing $B_k$ might otherwise have made $T_k$, the count denominator and the unknown future floor defeat the desired gain. The report supplies finite choices in the correct order. Stage failure is at most $(k+1)^{-8}$; on the resulting fixed measure-one set there is an endpoint in every sufficiently large stage. Those endpoints increase, have $N\geq20$, and have positive numerators. Replacing $\delta_k$ by any fixed larger $\varepsilon$ then reduces the denominator and preserves the lower bound. This handles the uncountable quantifier without taking an uncountable intersection of full-measure sets.

The catalog consequence has its own attack: large values under one normalization need not refute a bound under another. Here the exact ratio $(\log N)^{1/4}/\sqrt{\log\log N}$ tends to infinity. A positive infinite limsup in the theorem's normalization therefore contradicts the stated little-o assertion pointwise on the same set. Periodicity and countably many integer translates transfer the conclusion from the circle to almost every real $\alpha$. This also verifies the frozen Covers sentence.

Two source imprecisions are bounded rather than hidden. Summability of an arbitrary positive $(a_k)$ does not imply $a_k\leq1$: one could take $a_1=2$ and a summable positive tail. The report instead uses the source's already supplied sequence, for which the bound holds. Likewise the digit arguments require integer parameters; the actual recursive choices realize those types. Neither observation adds a hypothesis to this existential theorem or supplies a missing essential lemma. No universal assertion for arbitrary summable budgets or real digit depths is accepted.

Complete report contract

The standard applied is docs/verification.md, “Report contract — void if any part is missing,” together with its “Whole-claim report,” canonical failure modes, named patterns and Erdős checklist, and campaign/guides/verification.md. The findings below are substantive coverage findings, not counts of headings.

Itemwise checklist adjudication

Each PASS below applies to both commissioned report findings. Where a mode is inapplicable, PASS means the report correctly declares that limitation; it does not certify an absent computational or formal method.

Canonical failure modes

Canonical named patterns

Erdős-specific checklist

The neighboring disciplines also receive coverage. There is no statistical inference from correlated samples; independence is proved for the families where it is used. Each conclusion is limited to the actual evidence coverage. This is not a computational batch or a second independent review cycle, so neither a structurally different computational leg nor a second cycle's disjoint attack route is owed. The catalog consequence is a separately checked conclusion of this one proof, not a second independent proof of it.

Dependencies and reading limits

The grader checked the dependency census against the original statement, proof and consequence on 2026-10-15 UTC. There is no consumed native L-premise or L-definition. No ledger row was authorized or needed, and no current or prospective native grade is assumed. This finding comes from the actual argument, not the claim extract's Depends on sentence.

The report's source-interface table correctly identifies Ho v2, the exact clause, its hypotheses and its use. The consumed route is (2.1) and Lemma 2.1; Lemma 3.1's mean, norm and floor; the trial weights and Lemma 3.5; the integer stage construction and Lemmas 4.1–4.2; Proposition 4.3's $L^2$/almost-sure convergence, floors, endpoint inequality and part (ii); and Section 7 at $p=2$. Lemmas 3.2–3.3 reduce at this value to the norm identity and Hilbert-space argument already used. The local auxiliary results are checked as interfaces within the whole argument, not promoted as separately graded source theorems.

The permitted foundational interfaces are normalized Lebesgue measure and finite binary cylinders, countable null-set operations and integer-dilation measure preservation; Parseval, $L^2$ completeness, integral continuity and uniqueness of limits in probability; almost-sure convergence of independent centered real series with summable variances; and first Borel–Cantelli. Their hypotheses are supplied by the construction. Their general proofs were not independently reconstructed, and neither the review nor this grade claims that they were. This is disclosed reliance allowed by the commission, not an omitted native dependency.

Rosenthal, Lemma 2.2, the Fourier-tail part of Lemma 3.1, the thresholds $Q_k$, and the second Borel–Cantelli route in Proposition 4.3(i) are not consumed. No appeal to the source's omitted Section 1.2 is needed for a missing theorem or definition: the required parameters are defined in the supplied construction. No missing permitted mathematical input was found.

Original equation (3.8), reproduction and accepted weight

The original transcription says $C_2^2\lambda Q/N$ in (3.8); the retained p.9 print witness says $C_2\lambda Q/N$. The grade addresses exactly those bytes. It does not silently substitute a corrected transcription. This source-fidelity difference is real. It does not refute the commissioned theorem or leave a consumed premise unsupported: Section 7 discards the Fourier-tail requirement, and every step of the reported $p=2$ route uses the mean/norm/floor clauses without (3.8). The catalog implication consumes only that completed specialization. Both contract grades remain PASS with this explicit exclusion. No claim that the entire transcription is faithful in every clause is accepted.

The reproduction claim is noncomputational. Once the owner adopts this record, the report and its five public companion assets together, an ordinary clone contains the exact mathematical subject, its derivations and the source-fidelity witness. A reader can repeat the assessed argument from those retained texts. The original complete eight-file handoff and the unabridged primary records are additionally retained in the private grading package for inspection; a selected projection is not their substitute.

This grader did not fetch or open the live PDF, rerender it, run original commands, independently authenticate the entire external PDF, or execute a mathematical computation. The grade's print finding is based on the retained original page images and the actual original image returns, with the commissioned provenance. The administrative keeper checks frozen identities and record correspondences only. It cannot mechanically decide independence or prove the theorem, and its successful exit is not used as such a warrant.

Exact accepted weight: one independent whole-proof examination of the frozen $p=2$ theorem and its definite lacunary $L^2$ little-o consequence, with refutation-failed mathematical verdicts and PASS report-contract grades for both. There is no essential uncovered clause within that scope. The grade supplies no independent theorem proof, all-$p$ acceptance, Remark 7.1 finding, Fourier-tail acceptance, optimal-growth result, v1 numbering/history verification, E996 result, kernel warrant or journal or community acceptance. Broader growth prose and later corrections cannot inherit it.

This is a complete private grading handoff. Durable filing of this grade, the exact report and necessary assets in the source or research proof's evidence/verify/ closure, public fidelity and final adoption remain with the authorized owner. No native claim tier, corpus page or catalog standing was changed by the grader.