Rational-ratio dyadic floors by permanent descent

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evidence/: Exact finite completeness certificates for specific two-ray dyadic floor sets with rational ratio, recomputed by one entry point: the pinned pairs, a sweep over thirty-two rational coefficients, and an incomplete control pair.

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This lead's notes that cite no source are author-recorded research, not claims.

Target and status evidence

For $\alpha,\beta>0$ let

$$A_{\alpha,\beta}=\{\lfloor2^n\alpha\rfloor,\lfloor2^n\beta\rfloor:n\ge0\} \setminus\{0\}.$$

Hegyvári's conjecture, as recorded on Problem 354 and in Fan's preprint, says that $A_{\alpha,\beta}$ is complete (every sufficiently large integer a sum of distinct elements) whenever $\alpha/\beta$ is not a power of $2$ and at least one of $\alpha,\beta$ is not a dyadic rational, a nonzero integer times a power of $2$. The problem's Current assessment records the irrational-ratio case as answered yes on a Conjectures.io-accepted Lean proof, the case of exactly one dyadic rational as Hegyvári's (erdosproblems.com-attested, paper not held; Fan's preprint, held, records it as Hegyvári's on p. 4), and the remaining rational-ratio cases as untouched by every 2026 source. The target here is exactly those cases: $\alpha/\beta$ rational, not a power of $2$, neither coefficient a dyadic rational. After multiplying one coefficient by a power of $2$ the ratio is $p/q\in(1,2)$ in lowest terms, and $\alpha=p\gamma$, $\beta=q\gamma$ with $\gamma=\beta/q$ not a dyadic rational.

Route and established inputs

The route is the permanent descent of the Yu--Chen manuscript, reconstructed on the Theorem 5.1 page of the Problem 354 folder with the normalization and the three finite lemmas (2.1, 2.2, 2.3), whose inputs are the interlacing of the two rays and the base-$2$ digit recurrence, not the ratio. The connections page proves, as author-recorded statements:

• Lemma 354.1: for every normalized pair not consisting of two dyadic rationals, incompleteness forces bounded event spacing (an event in $(n,4n]$ for every large $n$).
• Proposition 354.2: at a rational ratio $p/q$ the descent constant $K_*(p,q)=2q(p-1)+4(p+q)+64$ is fixed at every exact layer, so a zero run of the fixed length $L(p,q)=\lceil\log_2K_*\rceil-1+\lceil\log_2p\rceil$ in the binary expansion of $\gamma$ produces one descent of the modular run invariant; infinitely many such runs give completeness.
• Corollary 354.3: $A_{\alpha,\beta}$ is strongly complete whenever $\gamma$ is not a dyadic rational and has infinitely many runs of $L(p,q)$ zeros, hence for Lebesgue-almost every $\gamma$ on each rational-ratio line ($L(3,2)=8$, $L(5,4)=10$, $L(5,3)=9$).
• Lemma 354.4: if the subset sums of the first $n$ layers of a normalized pair contain $b_n=\lfloor2^n\beta\rfloor$ consecutive integers, the pair is complete; the evidence certifies $(48/5,32/5)$, $(48/7,32/7)$, $(80/9,64/9)$, $(6\sqrt2,4\sqrt2)$, $(5\sqrt2,3\sqrt2)$ and thirty-two further pairs $(3\gamma,2\gamma)$ with $\gamma=2^T/d$, all outside every held source, and none of them known to be covered by Corollary 354.3: the rational coefficients have no zero run of the required length, and $\sqrt2$ is not known to have infinitely many.

Provisional premises

The descent is consumed through the author-recorded reconstruction of Theorem 5.1 and its finite certificate (rechecked by the folder's evidence); the manuscript is unrefereed; the reconstruction of that page has a filed focused review in the folder's evidence, graded faithful and sound with one correction and assigning no tier. The transport imports items 1--6 and the reduction of the normalization page, Lemmas 2.1--2.3, Theorem 5.1 with the Appendix A certificate (finite data rechecked by the folder's evidence) and Step 2 of the theorem page, each at the standing its own page records; Lemma 354.4 and the certificates rest on the normalization page alone. The a.e. statement imports Borel's normal number theorem. Everything here is author-recorded; its review standing is stated under Current review.

The consequential gap

Coefficients $\gamma$ with only finitely many maximal common zero runs of length at least $\lceil\log_2K_*\rceil-1$ at exact layers: every rational $\gamma$ whose denominator has odd part at most $2^{\lceil\log_2K_*\rceil}$ ($128$ at ratio $3/2$), in particular every rational pair certified by the evidence, and uncountably many irrationals, while $\gamma=1/511$ at ratio $3/2$, with infinitely many runs of eight zeros, is covered by Corollary 354.3. For them exact layers occur only at the $s$ with $\{2^s\gamma\}<1/p$, possibly nowhere, and the common zero runs after them are too short for Theorem 5.1, and the irrational-case continuation of the Yu--Chen argument does not apply for two reasons recorded on the connections page: a rational ratio has finitely many good rational approximants, so the digit budget and the sparse windows are empty, and a rational ratio lies in the compact set $\mathcal C_k$ of ratios of sparse binary sums, so returns between exact layers can cost two events (the witness at ratio $3/2$: $\{2^s\gamma\}\in[1/4,1/3)$). Hegyvári's conjecture therefore stays open for these coefficients, pair by pair certifiable but not uniformly proved.

Next step

Two bounded investigations. First, a descent for periodic conversions: for rational $\gamma=m/v$ with $v$ odd, a block of layers satisfies $a_{n+i}=2^ia_n+r_i$ with the correction $r_i=\lfloor2^i\{2^n\alpha\}\rfloor$, the integer whose binary digits are the eventually periodic conversions; decide whether the coefficient interval of Step 1 and the node connection of Step 4 of the Theorem 5.1 page have analogs for such blocks. A positive outcome proves the conjecture for every pair of rational coefficients with rational ratio; a negative outcome locates the obstruction and leaves the search below as the only tool. Second, extend the certificate search over families of rational and quadratic coefficients; each positive case is a proved instance, and a pair without a certificate to a large depth is the first candidate for an incompleteness argument through Folkman-type gaps, which would refute the conjecture.

Current review

Reviewed. The route was first independently reviewed on 2026-09-28 in the review records [[research/erdos_354/evidence/verify/connections_problem_354_hegyv_ri_s_conjecture_at_a_rational__review|the rational-ratio review]] (the section of the connections page that proves the route's statements, with this lead) and [[research/erdos_354/evidence/verify/connections_no_further_consumer_review|the no-further-consumer review]] (the scope of the mechanism, with both leads), and graded in the grade, whose outcome for this lead was needs_update; its corrections C2, C3 and C4 were applied to this page, the connections page and the evidence entry point before the re-review below.

The route was then re-reviewed blind as it stood on 2026-10-02. The subject was an extraction of this page and the connections page on that date, with the review-status fields and this section removed, kept outside the repository; a first attempt at the re-review, whose extraction carried the earlier verdicts, was stopped before completion and its output discarded. The re-review is the record [[research/erdos_354/evidence/verify/lead_rational_ratio_dyadic_floor_descent_rereview|the blind re-review]], graded in [[research/erdos_354/evidence/verify/lead_rational_ratio_dyadic_floor_descent_rereview_grade|the re-review grade]], which records pass. The graded verdicts of the statements the route rests on: Lemma 354.1, Proposition 354.2, Corollary 354.3 with Borel's theorem as its imported premise, and Lemma 354.4 are each refutation-failed, as are the constants $L(3,2)=8$, $L(5,4)=10$ and $L(5,3)=9$, the two obstructions with the two-event witness, and the certificate figures of all thirty-nine pairs, under the reviewer's and the grader's independent exact recomputations; the clause of "Route and established inputs" placing the certified pairs outside Corollary 354.3 was proof defective; statement unresolved for the two square-root pairs, since whether $\sqrt2$ has infinitely many zero runs of the required length is open; and the first sentence of "The consequential gap" was refuted-as-stated as wording only, since without "at least" it placed $\gamma=1/511$ at ratio $3/2$, which the paragraph itself calls covered, in the gap set. The grade's outcome for this lead is reviewed once its corrections land, and they were applied, so the current text differs from the reviewed text at these places: the fourth bullet of "Route and established inputs" (R1: none of the certified pairs is known to be covered by Corollary 354.3), the first sentence of "The consequential gap" (R2: runs of length at least $\lceil\log_2K_*\rceil-1$), the sentence added to "Provisional premises" naming the imported reconstruction inputs (S1), the correction $r_i$ of "Next step" (S3), the Fan locator in "Target and status evidence" (S4), and, on the connections page, the coverage clause after Lemma 354.4 (R1), the mesh-lemma sentence after Lemma 354.4 (S2) and the correction $r_i$ of "Next investigation" (S3). No tier is assigned and no status changes.