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Purpose and standing. This page draws the connections from the author-recorded reconstructions in the Problem 354 folder to the rest of the corpus. Each section names its exact target with the target's current assessment, states what the reconstructed mechanism gives when transported, with a proof where one is given here, states what it does not give and why, and names the next investigation with what each outcome would establish. Every new statement below is author-recorded, and no status or tier changes. The earlier section subjects were independently reviewed on 2026-09-28T13:50:48Z in seven focused reports, graded in the connections grade, which assigns no tier. The current page also includes the separate dyadic-root deduction L33; those reports do not cover that deduction or its consequences here. L33 retains its own author-recorded standing. The reconstruction reviews in the folder cover their named reconstruction subjects, not every transfer on this page. The notation is the folder's: for $\alpha,\beta>0$,

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

a set is complete if every sufficiently large integer is a sum of distinct elements and strongly complete if it stays complete after every finite deletion; a normalized pair has $N=\lfloor\beta\rfloor<M=\lfloor\alpha\rfloor<2N$ and $N\ge2$, with weights $a_i=\lfloor2^i\alpha\rfloor$, $b_i=\lfloor2^i\beta\rfloor$, conversions $u_i=a_{i+1}-2a_i$, $v_i=b_{i+1}-2b_i$ (the binary digits of $\alpha$ and $\beta$ at position $i+1$), events at the positions after a nonzero conversion, prefix sums $P_n$, moduli $D_n=\gcd(a_n,b_n)$ and the modular run invariant $h_n$, all as on the normalization page; $K_*$, $C_M$ and the hypotheses of Section 3 are those of the Theorem 5.1 page. A dyadic rational is a nonzero integer times a power of $2$. Hegyvári's conjecture, as the problem page and Fan's preprint record it, says that $A_{\alpha,\beta}$ is complete whenever $\alpha/\beta$ is not a power of $2$ and at least one of $\alpha,\beta$ is not a dyadic rational; Fan's condition (1.5) for a set $A\subseteq\mathbb N$ is $\sum_{a\in A}\|a\theta\|=\infty$ for every $\theta\in\mathbb R\setminus\mathbb Z$, and Fan's constants $M_\rho$ and $M_\rho^*$ are as on the Remark 4.2 page.

Problem 354: Hegyvári's conjecture at a rational ratio

Target. Problem 354, whose Current assessment records the first question as answered yes on a Conjectures.io-accepted Lean proof for base $2$ and irrational ratio, lists Hegyvári's conjecture as open, and names its rational-ratio cases as untouched by every 2026 source (remaining gap (5)). The case in which exactly one of $\alpha,\beta$ is a dyadic rational is Hegyvári's own (erdosproblems.com-attested, the 1989 paper not held), so the open cases are those with $\alpha/\beta$ rational, not a power of $2$, and neither coefficient a dyadic rational. After multiplying one coefficient by a power of $2$, which replaces $A_{\alpha,\beta}$ by a subset whose strong completeness implies that of the original set, the ratio lies in $(1,2)$; write it as $p/q$ in lowest terms, so $q<p<2q$ and $q\ge2$, and put $\gamma=\beta/q$, so that $\alpha=p\gamma$ and $\beta=q\gamma$. Both coefficients are dyadic rationals exactly when $\gamma$ is one: if $p\gamma,q\gamma$ are dyadic then so is $\gamma=x(p\gamma)+y(q\gamma)$ for integers $x,y$ with $xp+yq=1$.

Where irrationality enters the reconstructed argument (author-recorded reading of the folder's pages). Item 5 of the normalization page (the event set is infinite) uses only that the conversions are not eventually zero, which fails exactly when both coefficients are dyadic rationals; items 1--4 and 6 and the reduction hold for every pair whose ratio is not a power of $2$. The three finite lemmas and Theorem 5.1 with its certificate use nothing about the ratio: the constant $K_*=2q(p-1)+4(p+q)+64$ of the Section 3 hypotheses is computed from the reduced pair $(p,q)$ of the layer at which the theorem is applied, and only the length condition (5.2), which bounds that pair by $p\le a_n<2^n(M+1)$, brings in the layer index. Step 2 of the theorem page uses Theorem 5.1 and the infinitude of events only. The finite-event decay holds for every normalized pair. Irrationality is consumed in exactly two places: the windows page needs good rationals of unbounded denominator, and Step 2 of the bounded-spacing page needs $\theta\notin\mathcal C_k$.

What transports. The descent survives with a fixed constant.

Lemma 354.1 (bounded event spacing from incompleteness; author-recorded). Let $(\alpha,\beta)$ be a normalized pair that is not a pair of dyadic rationals. If the normalized sequence is incomplete, then only finitely many pairs of consecutive events $n<m$ satisfy $m-n\ge2n+C_M$, and there is $n_0$ such that every $n\ge n_0$ has an event in $(n,4n]$.

Proof. If the conversions vanished from some index $n_0$ on, then $\alpha=a_{n_0}2^{-n_0}$ and $\beta=b_{n_0}2^{-n_0}$ as in item 5 of the normalization page, against the hypothesis; so the event set is infinite. Step 2 of the theorem page now applies word for word, since it consumes only Theorem 5.1 and the infinitude of events. $\square$

Proposition 354.2 (the descent at a rational ratio; author-recorded). Let $(\alpha,\beta)$ be a normalized pair with rational ratio $p/q$ in lowest terms, and let $\gamma=\beta/q$. Put

$$K_*(p,q)=2q(p-1)+4(p+q)+64,\qquad L(p,q)=\lceil\log_2K_*(p,q)\rceil-1+\lceil\log_2p\rceil .$$

If $\gamma$ is not a dyadic rational and the binary expansion of $\gamma$ contains infinitely many runs of at least $L(p,q)$ consecutive zeros, then every sufficiently large integer lies in $\bigcup_nP_n$.

Proof. (i) By the Bézout remark above, $\gamma$ not dyadic means that $\alpha,\beta$ are not both dyadic, so the conversions are not eventually zero and every common zero run, an index interval on which $u_i=v_i=0$, is finite.

(ii) Exactness. For every $s$ the integer $\delta_s=qa_s-pb_s$ equals $-q\{2^s\alpha\}+p\{2^s\beta\}$, because $q\cdot2^s\alpha=p\cdot2^s\beta$. The digits of $\alpha$ at positions $s+1,\ldots,s+L'$ vanish exactly when $\{2^s\alpha\}<2^{-L'}$, and likewise for $\beta$. So on a common zero run of length $L'\ge\log_2p$ starting at $s$, $|\delta_s|<p\,2^{-L'}\le1$, hence $\delta_s=0$: $a_s/b_s=p/q$, and as $\gcd(p,q)=1$ the reduced pair of layer $s$ in the Section 3 hypotheses is $(p,q)$ itself, with $d=D_s$.

(iii) From runs of $\gamma$ to common runs. Let $\{2^s\gamma\}<2^{-L}$ with $L=L(p,q)$. Since $p\{2^s\gamma\}<p\,2^{-L}\le1$, the fractional part of $2^s\alpha=p\cdot2^s\gamma$ is $p\{2^s\gamma\}<2^{-(L-\lceil\log_2p\rceil)}$, and the same holds for $\beta$ because $q<p$. So the conversions vanish on $[s,s+L_0)$ with $L_0=L-\lceil\log_2p\rceil=\lceil\log_2K_*\rceil-1$; note $L_0\ge\log_2K_*-1>\log_2(4p)-1>\log_2p$, as $K_*>4p$.

(iv) One descent. Let $[s',s'+L'-1]$ be the maximal common zero run containing $[s,s+L_0)$, so $L'\ge L_0$; it is finite by (i), so the conversion at index $s'+L'$ is nonzero, and by (ii) layer $s'$ has reduced pair $(p,q)$. The Section 3 hypotheses hold at $n=s'$ with $\ell=L'+1$ and $K=2^\ell\ge2^{L_0+1}=2^{\lceil\log_2K_*\rceil}\ge K_*(p,q)$. Theorem 5.1 gives $h_t\le\max(0,h_{s'}-1)$ for all $t\ge s'+L'+4$, and completeness if $h_{s'}\le1$.

(v) Iteration. The runs of $\gamma$ start at positions tending to infinity and each maximal common run is a finite interval, so there are infinitely many maximal common runs of length at least $L_0$; choose a sequence of them with $s'_{j+1}\ge s'_j+L'_j+4$. If the sequence were incomplete, then $h_{s'_j}\ge2$ for every $j$ and $h_{s'_{j+1}}\le h_{s'_j}-1$, a strictly decreasing sequence of integers bounded below by $2$. Hence the sequence is complete. $\square$

Unlike the irrational case, no relation between the run length and its position is needed: the constant $K_*(p,q)$ is fixed by the ratio, and the length condition (5.2) is not used.

Corollary 354.3 (almost every coefficient on a rational-ratio line; author-recorded). Let $\alpha,\beta>0$ have a rational ratio that is not a power of $2$; let $p/q$ be the reduced form of the unique $2^k\alpha/\beta$ in $(1,2)$ and $\gamma=\beta/q$. If $\gamma$ is not a dyadic rational and its binary expansion contains infinitely many runs of $L(p,q)$ zeros, then $A_{\alpha,\beta}$ is strongly complete. In particular, for each fixed rational ratio in $(1,2)$, the set $A_{p\gamma,q\gamma}$ is strongly complete for Lebesgue-almost every $\gamma>0$, since almost every real number is normal to base $2$ (Borel; imported) and so contains every finite block of digits infinitely often.

Proof. Fix a finite $F\subseteq\mathbb Z$. Items 1--3 of the normalization page give $u,v\ge0$ with $(2^u\alpha,2^v\beta)$ normalized and all values above $\max(F\cup\{0\})$; only the exclusion of the value $1$ for $2^k\alpha/\beta$ used irrationality there, and it holds because the ratio is not a power of $2$. The normalized ratio is the $2^k\alpha/\beta$ in $(1,2)$, that is $p/q$, and the normalized pair's $\gamma$ is $2^v\beta/q=2^v\gamma$, whose expansion is that of $\gamma$ shifted, so it is not dyadic and keeps infinitely many runs of $L(p,q)$ zeros. Proposition 354.2 makes the normalized sequence complete, and the reduction on the normalization page turns this into completeness of $A_{\alpha,\beta}\setminus F$. $\square$

For the smallest ratios the constants are $L(3,2)=8$ ($K_*=92$), $L(5,4)=10$ ($K_*=132$) and $L(5,3)=9$ ($K_*=120$): for instance $A_{3\gamma,2\gamma}$ is strongly complete for every non-dyadic $\gamma$ whose binary expansion has infinitely many runs of eight zeros. These pairs have two irrational coefficients whenever $\gamma$ is irrational, so they lie outside Hegyvári's proved case and outside the irrational-ratio theorem of the Conjectures.io-accepted proof and of the Yu--Chen manuscript.

Lemma 354.4 (finite certificate; author-recorded). Let $(\alpha,\beta)$ be a normalized pair. If for some $n\ge1$ the set $P_n$ contains the $b_n$ consecutive integers $A,A+1,\ldots,A+b_n-1$, then every integer at least $A$ lies in some $P_t$; in particular $A_{\alpha,\beta}$ is complete and the interleaved sequence of the problem is complete for this pair.

Proof. List the weights of indices at least $n$ in increasing order, $c_0=b_n<c_1=a_n<c_2=b_{n+1}<\cdots$; by item 4 of the normalization page $c_{j+1}\le2c_j$. If a set of subset sums contains the $\Lambda$ consecutive integers from $A$ and $c\le\Lambda$ is a further weight, adjoining it gives the $\Lambda+c$ consecutive integers from $A$. Starting from $\Lambda_0=b_n=c_0$ and adjoining $c_0,c_1,\ldots$ in turn, the lengths satisfy $\Lambda_{j+1}=\Lambda_j+c_j\ge2c_j\ge c_{j+1}$, so every weight is admissible and the lengths tend to infinity. The last sentence is the reduction on the normalization page. $\square$

This is the abutting case of the mesh lemma of the Lemma 2.2 page with gap $1$: the $b_n$ consecutive integers have span $b_n-1$, one short of the lemma's hypothesis, which the count argument above replaces, while the digit-budget page applies the lemma with span at least the next weight; it turns completeness of one pair into a finite computation. The evidence of the lead computes the subset sums exactly and records the following certificates, each for a pair outside every held source: $(48/5,32/5)$, that is $(3\gamma,2\gamma)$ with $\gamma=16/5$, at depth $n=9$ from $1423$ with $b_9=3276$; $(48/7,32/7)$ at depth $9$ from $577$ with $b_9=2340$; $(80/9,64/9)$, ratio $5/4$, at depth $7$ from $445$ with $b_7=910$; $(6\sqrt2,4\sqrt2)$ at depth $7$ from $183$ with $b_7=724$; $(5\sqrt2,3\sqrt2)$, ratio $5/3$, at depth $3$ from $18$ with $b_3=33$; and every pair $(3\gamma,2\gamma)$ with $\gamma=2^T/d$ for the thirty-two odd denominators $5\le d\le99$ not divisible by $3$, normalized by the least $T$, by depth $12$: twenty-nine of them by depth $9$, and $d=23$, $89$ and $91$, the pairs $(192/23,128/23)$, $(768/89,512/89)$ and $(768/91,512/91)$, at depths $10$, $12$ and $11$, from $2139$, $11036$ and $4323$ with $b_{10}=5698$, $b_{12}=23563$ and $b_{11}=11522$. The rational coefficients have periodic expansions without zero runs of the required length (eight, nine or ten), and $\sqrt2$ is not known to have infinitely many such runs, so none of these pairs is known to be covered by Proposition 354.2; each certificate is a proof of completeness for its own pair and nothing more. The control pair $(\sqrt{50},\sqrt{20})$, of irrational ratio, certifies at depth $6$, and Hegyvári's incomplete multiset $(5/2,5)$ receives no certificate to depth $16$, as it must not.

What it does not give.

• The short-block coefficients. Proposition 354.2 acts through a maximal common zero run of the conversions of length at least $\lceil\log_2K_*\rceil-1$ at an exact layer, which a run of $L(p,q)$ zeros in $\gamma$ guarantees; a coefficient with only finitely many such runs is outside the descent. Exact layers occur exactly at the $s$ with $\{2^s\gamma\}<1/p$, possibly nowhere ($\gamma=24/7$ at ratio $3/2$ has none: $\delta_s$ runs through $2,1,1$), and at an exact layer a common zero run of that length forces $\{2^s\gamma\}<2^{-\lceil\log_2K_*\rceil}$. For a rational $\gamma$ whose denominator in lowest terms has odd part $v$, $\{2^s\gamma\}\ge1/v$ for all large $s$, so every rational $\gamma$ with $v\le2^{\lceil\log_2K_*\rceil}$ ($v\le128$ at ratio $3/2$), in particular every rational pair certified below, and uncountably many irrationals are outside the descent, while $\gamma=1/511$ at ratio $3/2$ has infinitely many runs of exactly eight zeros and is covered by Corollary 354.3. For the coefficients outside, Hegyvári's conjecture stays open except pair by pair through Lemma 354.4, since Theorem 5.1 needs $K\ge K_*$ for its coefficient interval (Step 1) and node connection (Step 4).
• The rest of the irrational-case argument does not transfer, for two precise reasons. First, a rational $\theta=p/q$ has only finitely many good rationals: a reduced $p'/q'\ne\theta$ has $|\theta-p'/q'|\ge1/(qq')$, so $|\theta-p'/q'|<1/q'^2$ forces $q'<q$; hence the digit budget (DB), which needs a good denominator at least $2^n\beta$, is empty for large $n$, and the crossing denominator $D(b)$ of the windows page does not exist for $b>q$, so (10.2) is unavailable. Second, $\theta=p/q$ lies in $\mathcal C_k$ as soon as $k$ is at least the number of ones in the binary expansions of $p$ and of $q$: with $2^J$ the largest power of $2$ in $p$, both $p/2^J$ and $q/2^J$ lie in $\mathcal S_k$ and $q/2^J>1/2$. So the neighborhood argument of Step 2 of the bounded-spacing page fails, and it fails for a reason: at $\theta=3/2$, whenever $\{2^s\gamma\}\in[1/4,1/3)$ the layer $s$ is exact, the conversions at $s$ and $s+1$ are $(1,1)$ and $(1,0)$, and $\delta_{s+2}=4\delta_s+qU-pV$ with $U=3$, $V=2$ vanishes again: a return between exact layers costing two events, for every such $s$. Under a rational ratio the bounded event spacing of Lemma 354.1 is therefore not contradictory by itself; the finite-event decay still holds but only bounds the missing count by a polynomially small fraction under bounded spacing.
• No uniformity over pairs. Corollary 354.3 is a full-measure statement on each rational-ratio line; the exceptional set contains every pair with rational coefficients whose $\gamma$ has odd denominator part at most $2^{L(p,q)}$, and Hegyvári's 1991 dichotomy (for fixed $\alpha$, the incomplete $\beta$ form a set of measure $0$ or $\infty$) is neither used nor sharpened.

Next investigation. Two directions, recorded as the lead rational_ratio_dyadic_floor_descent. First, a descent for coefficients with bounded runs: for rational $\gamma=m/v$ with $v$ odd the conversions are eventually periodic with period the order of $2$ modulo $v$, so a block of consecutive 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; whether a coefficient interval and a connection certificate exist for such blocks, in place of the exact block $d\{px+qy\}$, is the question. A positive outcome would prove Hegyvári's conjecture for every pair with rational coefficients and rational ratio, and the general bounded-run case would remain; a negative outcome, an incomplete pair found by the search below, would refute the conjecture. Second, the certificate search of Lemma 354.4 is cheap and exact; extending it over families of rational and quadratic coefficients turns each positive case into a proved instance, and a pair with no certificate to a large depth is the first candidate for an incompleteness argument through Folkman-type gaps.

Outcome. Transfers: Lemma 354.1 for every non-dyadic normalized pair, Proposition 354.2 and Corollary 354.3 for almost every coefficient on every rational-ratio line, and Lemma 354.4 with certified pairs; open for the short-block coefficients, with the two obstructions above.

Problem 354: the second question at bases up to the plastic number

Target. The second question of Problem 354, "What if $2$ is replaced by some $\gamma\in(1,2)$?", whose reading the Current assessment leaves unfixed: under "for every $\gamma$" it is claimed no by Geneson's Corollary 12, graded sound by the internal review recorded on its problem page, its sign-off pending. Under "for some $\gamma$", Kitamura's proof at $\sqrt\varphi$ remains pending. The separate author-recorded L33 deduces strong completeness of every single ray at $\gamma=2^{1/m}$, $m\ge2$, from the accepted dyadic theorem. In particular $\sqrt2$ supplies the existential conclusion; its claim page, separate from this author argument, awaits an outside sign-off. For $\gamma\in(1,2)$ and $\alpha,\beta>0$ write

$$A^\gamma_{\alpha,\beta}=\{\lfloor\alpha\gamma^n\rfloor, \lfloor\beta\gamma^n\rfloor:n\ge0\}\setminus\{0\};$$

completeness of this set implies completeness of the problem's interleaved sequence, by the choice of one index per value as on the theorem page. The folder index names the door: Fan's Theorem 1.1 has $M_\rho=2$ for $1<\rho\le\rho_0=1.3247\ldots$, the real root of $x^3=x+1$, so at such a base every set with two elements in each large $\rho$-adic interval and (1.5) is strongly complete, on the preprint's claim.

What transports. The per-interval count is a theorem; the divergence condition is the premise.

Proposition 354.5 (two elements per $\gamma$-adic interval; author-recorded). Let $\gamma\in(1,2)$ and $\alpha,\beta>0$. For all sufficiently large $k$, each of the two rays has exactly one term in $I_k=(\gamma^k,\gamma^{k+1}]$, and $|A^\gamma_{\alpha,\beta}\cap I_k|$ equals $2$ if $\alpha/\beta\notin\gamma^{\mathbb Z}$ and $1$ otherwise.

Proof. Let $j$ be the integer with $\gamma^j<\alpha\le\gamma^{j+1}$ and $\alpha^*=\alpha\gamma^{-j}\in(1,\gamma]$. For $n\ge0$ and $m=n+j$, $\alpha\gamma^n=\alpha^*\gamma^m\in(\gamma^m,\gamma^{m+1}]$. Since $(\alpha^*-1)\gamma^m\to\infty$, there is $m_0$ with $\alpha^*\gamma^m\ge\lfloor\gamma^m\rfloor+1$ for $m\ge m_0$, and then $\gamma^m<\lfloor\alpha^*\gamma^m\rfloor\le\gamma^{m+1}$: the term of index $m-j$ lies in $I_m$. Every other term of index $n'\ge0$ lies in $I_{n'+j}$ if $n'+j\ge m_0$ and is at most $\gamma^{m_0}$ otherwise, so for $k\ge\max(m_0,j)$ the $\alpha$-ray has exactly one term in $I_k$, namely $\lfloor\alpha^*\gamma^k\rfloor$. The same holds for $\beta$ with $\beta^*\in(1,\gamma]$. If $\alpha^*\ne\beta^*$ then $|\alpha^*-\beta^*|\gamma^k\ge1$ for large $k$ and the two floors differ; if $\alpha^*=\beta^*$ they coincide for every $k$. Finally $\alpha^*=\beta^*$ exactly when $\alpha/\beta$ is an integer power of $\gamma$, by the uniqueness of the representative in $(1,\gamma]$. $\square$

Conditional deduction 354.6 (author-recorded). Let $\gamma\in(1,\rho_0]$ and $\alpha,\beta>0$ with $\alpha/\beta\notin\gamma^{\mathbb Z}$. Assume (P1) Fan's Theorem 1.1 as stated on the Remark 4.2 page (a preprint claim whose proof, Sections 2--4 of the source, is not reconstructed in the corpus) and (P2) that $A^\gamma_{\alpha,\beta}$ satisfies (1.5). Then $A^\gamma_{\alpha,\beta}$ is strongly complete, and the interleaving of the two floor sequences is complete. Argument: $u_\gamma=\lceil\gamma(\gamma-1)\rceil=1$ because $0<\gamma(\gamma-1)<\varphi(\varphi-1)=1$, and $v_\gamma=\lceil\gamma^3/(\gamma+1)\rceil=1$ because $\gamma^3\le\gamma+1$ on $(1,\rho_0]$; so $M_\gamma=\min\{3,2\}=2$, Proposition 354.5 supplies $|A^\gamma_{\alpha,\beta}\cap(\gamma^k,\gamma^{k+1}]|=2=M_\gamma$ for all large $k$, and (P1) applies with $\rho=\gamma$, $M=2$ under (P2). Premise (P2) follows from (1.5) for either single ray, since a superset of a set with divergent sums has divergent sums, and it is necessary for strong completeness by the observation reconstructed on the Remark 4.2 page; so (P2) is exactly what a proof of strong completeness at base $\gamma$ must supply.

Proposition 354.7 (Corollary 12 at every reciprocal Pisot or Salem base with $P(1)\le-5$; author-recorded). Let $\gamma\in(1,2)$ be a Pisot or Salem number whose minimal polynomial $P$ is reciprocal, $x^{\deg P}P(1/x)=P(x)$, with $P(1)=-q$ for an integer $q\ge5$. Then there are $\alpha,\beta>0$ with $\beta/\alpha\ne r\gamma^k$ for all $r\in\mathbb Q$ and $k\in\mathbb Z$ such that every term of both floor sequences $\lfloor\alpha\gamma^n\rfloor$, $\lfloor\beta\gamma^n\rfloor$ is even. Hence $\alpha/\beta$ is irrational, neither sequence is a tail of the other, the interleaving and the set $A^\gamma_{\alpha,\beta}$ are incomplete, and $A^\gamma_{\alpha,\beta}$ fails (1.5) at $\theta=1/2$.

Proof. The proof on the Corollary 12 page with $q$ in place of $5$. Proposition 11 of the Theorem 9 page, stated there for every Pisot or Salem number with $P(1)=-q$, $q\ge3$, gives $\eta>0$ with $3/(4q)<\{\eta\gamma^n\}<5/(4q)$ for $n\ge1$; put $\alpha=2\eta\gamma$ and $\beta=\alpha(1+\gamma)$. The $\alpha$-terms are even since $5/(4q)<1/2$; the $\beta$-terms are even since the sum of two fractional parts lies in $(3/(2q),5/(2q))\subset(0,1/2)$, which is where $q\ge5$ enters. The ratio condition uses only that $P$ is reciprocal and minimal, through the automorphism $\gamma\mapsto\gamma^{-1}$ of $\mathbb Q(\gamma)$, and the tail and parity conclusions follow as on that page. $\square$

Geneson's base is the case $q=5$. Every Salem number has a reciprocal minimal polynomial (a standard fact, not needed here since reciprocity is assumed). At every such base in $(1,\rho_0]$, premise (P2) fails for the pair constructed and the pair is incomplete, so no route can make every irrational-ratio pair complete there; at every such base in $(1,2)$ the every-base reading of the second question is answered no, as Corollary 12 answers it at one base.

What it does not give.

• The two-per-interval estimate alone does not establish premise (P2). At the dyadic-root bases, the separate L33 gives strongly complete single rays, so this route is unnecessary there. Premise (P2) fails for the pairs constructed in Proposition 354.7. At a rational base $\gamma=p'/q'\in(1,\rho_0]$ (such as $5/4$, $9/7$ or $13/10$), its failure at $\theta=1/2$ for a single ray means that $\lfloor t\gamma^n\rfloor$ is even for all large $n$, that is $\{\eta\gamma^n\}<1/2$ for all large $n$ with $t=2\eta$, a question of the same type as Mahler's open question on $\{\xi(3/2)^n\}$; no held source addresses it, and (1.5) at irrational $\theta$ is a further condition. At a transcendental base nothing is known here either way.
• The exceptional pairs $\alpha/\beta\in\gamma^{\mathbb Z}$. There $A^\gamma_{\alpha,\beta}$ is a single ray with one element per interval, Theorem 1.1 does not apply, and the problem's multiset has every sufficiently large value twice; at a rational base the exceptional set is empty, because $\gamma^{\mathbb Z}\subseteq\mathbb Q$ while $\alpha/\beta$ is irrational.
• Standing. The deduction inherits (P1): the criterion is a preprint claim read at statement level, and no reconstruction or review of Theorem 1.1 exists in the corpus.
• Redundancy at a base where every ray is complete. If every single sequence $\lfloor t\gamma^n\rfloor$ is complete at base $\gamma$, then every such sequence is strongly complete (its tails are the sequences of the coefficients $t\gamma^K$), so (P2) holds for every pair, but the two-ray question is then trivial. The route has content only at a base where (1.5) can be established for every ray without proving completeness; none is known here.

Next investigation. Fix one rational base in $(1,\rho_0]$ and attempt (1.5) for every single ray, beginning with rational $\theta$: for every $q\ge2$ and every $t>0$, infinitely many $n$ with $q\nmid\lfloor t\gamma^n\rfloor$. A positive outcome, together with (P1), gives strong completeness of $A^\gamma_{\alpha,\beta}$ for every pair with irrational ratio at that base, an answer to the reading "for some $\gamma$" in which the pair, not a single ray, carries the completeness; a negative outcome, a coefficient with all late terms divisible by $q$, gives an incomplete single ray at a rational base below the golden ratio (the ray of the coefficient $t\gamma^{n_0}$, every term of which is divisible by $q$), which is new for Problem 349 and closes the single-ray sufficient condition at that base; premise (P2) for the irrational-ratio pairs then needs a direct argument, or a second bad coefficient with irrational ratio to the first, which would give an incomplete pair as Proposition 354.7 does. This is recorded as the lead small_base_two_ray_distance_divergence.

Outcome. Transfers conditionally: Proposition 354.5 is proved, deduction 354.6 makes the divergence premise (P2) and the criterion (P1) explicit, and Proposition 354.7 marks the bases where (P2) fails and extends the negative answer to the every-base reading to each of them.

Problem 349: single rays and the digit recurrence

Target. Problem 349 asks for which $t,\alpha>0$ the sequence $\lfloor t\alpha^n\rfloor$ is complete. Its page records the status open on six partial claim pages, none of which settles the question: Graham's 1964 determination is accepted, and van Doorn's preprint, Geneson's Salem-base counterexample and three others are pending. The held sources record Graham's 1964 characterization for $0<t<1$, $1<\alpha<2$, van Doorn's 2026 extension settling every base $\alpha\ge\varphi$ together with the elementary integer-base cases (van Doorn's Propositions 1 and 2: no base outside $[1,2]$ and, at base $2$, only $t=2^{-k}$ gives a complete sequence), and Geneson's Theorem 9, which refutes the conjectured completeness for all $t>0$ and $1<\alpha<\varphi$ at one Salem base. The folder index suggests the Yu--Chen mechanism might bear on integer-base instances.

What transports. At $\gamma=2^{1/m}$, $m\ge2$, L33 applies the accepted dyadic theorem to indices $j+mr$ and $j+1+ms$. The coefficient ratio is $\gamma^{-1}$, which is irrational. A late tail has strictly increasing positive floors, so the two indexed rays produce distinct values and survive every finite deletion. Thus every $t>0$ gives a strongly complete single ray at these bases.

This grouping is special to a base with an integral power equal to $2$. The ungrouped Yu--Chen recurrence does not give the arbitrary-base classification. The integer-base instances are settled and trivial in the direction that mechanism proves: at base $2$ a single ray has $a_{j}-\sum_{i<j}a_i=a_0+\sum_{i<j}u_i>0$ (the identity of item 6 on the normalization page), so its subset sums below $a_n$ number at most $2^n$ while $a_n\ge a_02^n$, and the mesh lemma's hypothesis, a span at least the next weight, never holds; the interlacing of two rays, $b_i<a_i<2b_i\le b_{i+1}$, is what makes every weight at most twice its predecessor, and the whole argument rests on it. At a non-integer base the digit recurrence $a_{i+1}=2a_i+u_i$ with $u_i\in\{0,1\}$ is absent, and with it the exact doubling blocks, the coefficient interval $d\{px+qy\}$, the certificate and the modular invariant. The single-ray analog of Lemma 354.4 is van Doorn's Lemma 5 (held), which the corpus already records.

A separate reading of scope concerns Proposition 11 on the Theorem 9 page is stated and proved for every Pisot or Salem number whose minimal polynomial has $P(1)\le-3$, so at every such base in $(1,\varphi)$, if any besides Geneson's exists, there are arbitrarily large $t$ with every $\lfloor t\gamma^n\rfloor$ even and the sequence incomplete; the source states Theorem 9 at one base and Proposition 11 in this generality, so this adds no mathematics.

What it does not give. L33 does not classify arbitrary pairs $(t,\alpha)$ and gives no uniform effective completeness threshold. The loss of an exact doubling recurrence remains an obstruction at general noninteger bases; it is not an obstruction at the dyadic roots after regrouping.

Next investigation. An extension to other bases needs an actual replacement for that recurrence or a different completeness criterion. The negative outcome of the previous section's investigation would add an incomplete ray at a rational base.

Outcome. The dyadic theorem transfers to all coefficients at the bases $2^{1/m}$ with the standing of L33. The Geneson reconstruction bears on incomplete rays exactly as its source card records. Neither gives the full parameter classification.

Problem 348: deletion thresholds

Target. Problem 348 asks for which $0\le m<n$ a complete sequence stays complete after removing any $m$ elements and fails after removing any $n$. Its page records the status claimed (answered) on Geneson's pending full claim, the preprint's classification $m\in\{0,1\}$; the Geneson card records Theorem 1, that classification, as a preprint claim read for structure only. The folder index names the problem only as a further target of the same preprint.

What transports. Nothing. The reconstructed pages concern Theorem 9 and Corollary 12, not Theorem 1, and the Yu--Chen mechanism produces completeness after every finite deletion: the sets $A_{\alpha,\beta}$ of its theorem satisfy the first property for every $m$ and the second for no $n$, so they are examples for no pair $(m,n)$. The problem's content, the exclusion of $m\ge2$, is an incompleteness mechanism (Geneson's central-interval Theorem 2 and Lemma 4), for which the folder holds no reconstruction and the descent argument has no counterpart.

What it does not give. Any of the classification, in either direction.

Next investigation. None from this folder. A reconstruction of Geneson's Sections 2--4 would be a separate compilation task; the card records that this proof was read for structure only.

Outcome. Fails; the only link is the shared source.

Problem 254 and the threshold $M_2^*$

Target. Problem 254 asks whether $A\subseteq\mathbb N$ with $|A\cap(x,2x]|\to\infty$ and $\sum_{n\in A}\|\theta n\|=\infty$ for every $\theta\in(0,1)$ is complete. Its page records the problem as solved, with the question proved, on Fan's full claim, accepted on Boris Alexeev's built and audited Lean formalization of the preprint's six-per-interval version; the preprint's Corollary 1.2, at five elements in every large dyadic interval, is not built, and Snyder's Lean proof of the statement as posed is a pending full claim. The Fan card records Corollary 1.2, strong completeness under (1.5) and at least five elements in every large dyadic interval, as the preprint's confirmation of the conjecture in strong form, with the proof of Theorem 1.1 unread. The folder reconstructs Fan's Remarks 4.1 and 4.2, which use Corollary 1.2 as a statement, so the bound $M_2^*\le5$ below rests on that unbuilt corollary; the built six-per-interval criterion certifies only $M_2^*\le6$.

What transports. Two things, neither toward the proof.

• Sharpness of the growth hypothesis. The reconstructed Remark 4.1 at base two proves that $A=\{2^k+1:k\ge1\}$ satisfies (1.5), has exactly one element in every $(2^k,2^{k+1}]$ with $k\ge1$, and is incomplete. So the hypothesis $|A\cap(x,2x]|\to\infty$ of Problem 254 cannot be weakened to one element in every large dyadic interval, while Corollary 1.2 weakens it to five; the exact threshold $M_2^*$ lies in $\{2,3,4,5\}$ and is open in the source.
• A two-per-interval family. By the reconstructed Remark 4.2, every $A_{\alpha,\beta}$ with $\alpha/\beta$ not a power of $2$ and one coefficient not a dyadic rational satisfies (1.5) and has two elements in every large dyadic interval. The Yu--Chen theorem, if it stands, makes these sets strongly complete when the ratio is irrational, and Corollary 354.3 does so for almost every coefficient on each rational-ratio line: a family outside the reach of Corollary 1.2 and consistent with $M_2^*=2$, the value that Remark 4.2 shows would imply Hegyvári's conjecture. Fan's Proposition 4.3 (its statement read in the canonical conversion of the source's Section 4) says that the preprint's three-component criterion cannot give $M_2^*\le4$ and that a different mechanism may be needed; the descent is a different mechanism, but it consumes the digit recurrence of two rays, not the per-interval count, so it says nothing about a general set with two elements per dyadic interval.

Also, item 6 of the normalization page, the gap bound $\operatorname{gap}(P_n)\le N$ from $c_{j+1}-\sum_{i\le j}c_i\le c_0$, is the two-ray instance of the Burr--Erdős lemma that Fan records as Lemma 2.1 (bounded $\Delta(S)$ makes the subset sums syndetic); the corpus holds neither Burr--Erdős nor a general proof.

What it does not give. Nothing toward Problem 254 itself: the folder holds no reconstruction of Theorem 1.1 or of the three-component criterion (Theorem 2.2) behind it, and the descent does not extend to sets without the digit structure.

Next investigation. None from this folder. The compilation task for Problem 254 is a reconstruction of Fan's Theorem 2.2 with Sections 3--4, which would also fix the standing of premise (P1) above; for $M_2^*$, the question whether a two-per-interval set with (1.5) can be incomplete has no candidate in the corpus, every incomplete two-ray set known with one coefficient not a dyadic rational (Hegyvári 1989, Jiang--Ma, Fang--He, all erdosproblems.com-attested) having a ratio that is a power of $2$ and hence a single ray as a set, while a pair of dyadic rationals fails (1.5).

Outcome. Transfers the sharpness example and the two-per-interval family as author-recorded context for $M_2^*$; fails as a route to Problem 254.

Problem 124: mixed power sets

Target. Problem 124 asks whether, for integers $3\le d_1<\cdots<d_r$ with $\sum_i1/(d_i-1)\ge1$, every large integer is a sum of at most one element from each $P(d_i,0)$, the sums of distinct powers of $d_i$, and the same with $P(d_i,k)$ when the $d_i$ are coprime. Its page records the status open on four partial claim pages: Burr, Erdős, Graham and Li 1996 and Bergelson and Simmons 2015 are accepted, and Alexeev's Lean proof of the first question and Fan's Theorem 1.5 are pending. The Fan card records Theorem 1.5 (v5, p. 7) at statement level only; the theorem says that for a nonempty finite $D\subseteq\mathbb N_{\ge2}$ with no two elements powers of the same integer, $\bigcup_{d\in D}\{d^n\}$ is strongly complete whenever $D=D_1\cup D_2\cup C$ is a partition into nonempty parts with $\gcd(C)=1$ and $\sum_{d\in D_i}1/(d-1)\ge1$ for $i=1,2$, one component fewer than the Bergelson--Simmons theorem as the source states it. The folder index names the problem only as a further target of the same preprint.

What transports. Nothing. The objects are single-coefficient rays $\{d^n\}$ at distinct integer bases at least $3$, without floors; the descent needs two rays at base $2$ with the digit recurrence and the interlacing $b_i<a_i<2b_i\le b_{i+1}$, none of which a power set has. Fan's Theorem 1.5 is not reconstructed, and its proof (the source's Section 7) runs through the three-component criterion, not through anything in the folder. The only shared step is elementary: the bounded-$\Delta$ argument that makes subset sums syndetic, which is Fan's Lemma 2.1 and, for two rays, item 6 of the normalization page; that is a resemblance of tools, not a transfer of a result.

What it does not give. Anything about either question of the problem.

Next investigation. None from this folder; the compilation task is a reconstruction of Theorem 1.5 with Theorem 2.2.

Outcome. Fails.

No further consumer

The mechanism at its actual scope. The reusable content of the folder is a descent: for two interlaced base-$2$ rays, a long exact doubling block followed by a nonzero digit lowers the modular run invariant of every later layer, with a finite certificate making the descent uniform; its inputs are the interlacing, the digit recurrence and a reduced ratio $(p,q)$ at the block's first layer with $K\ge K_*(p,q)$. A consumer would be recognized by a set that is a union of two rays $\lfloor2^n\alpha\rfloor$, $\lfloor2^n\beta\rfloor$ up to finitely many elements, or by a question about subset sums of such a set.

The search. The problems, research and theory trees were searched for complete and strongly complete sequences, subset sums of distinct terms, floors of geometric sequences, dyadic intervals and the named authors. Beyond the targets above, the corpus holds: Problem 343 and Problem 344 (subcompleteness under a density condition; proved), Problem 345 (thresholds of completeness of the powers $n^k$; open, on polynomial sequences), Problem 346 (strongly complete sequences that lose completeness under every infinite deletion; solved), Problem 347 (a sequence with ratio tending to $2$ whose cofinite subsequences have subset sums of density $1$; proved), Problem 351 (strong completeness of $p(n)+1/n$; proved), Problem 253 (disproved by Cassels), Problem 246 and Problem 123 (sums of distinct $a^kb^l$ and $a^kb^lc^m$; proved) and Problem 125 (disproved). None consumes the descent: the resolved ones need nothing, Problem 345 concerns polynomial sequences, and the two-ray sets are not examples for Problem 347, since their sorted terms have ratios alternating near $\alpha/\beta$ and $2\beta/\alpha$ and so do not tend to $2$; for Problem 346, refuted by a counterexample, the ratios decide nothing, since its hypotheses ask that completeness be lost under every infinite deletion, which the descent does not address and which fails for the pair $(3\sqrt2,3)$ after the deletion of $\{3\cdot2^m:m\ge4,\ 4\mid m\}$; a single ray, whose ratios tend to $2$, has at most $2^n$ subset sums below $a_n\ge\lfloor\alpha\rfloor2^n$ and both problems are resolved. The separate L33 and its research account concern dyadic-root single rays, using the accepted two-ray theorem through exponent regrouping.

Outcome. No further consumer of the direct descent is identified among these targets. The separate dyadic-root application to Problem 349 is described above; it does not supply an arbitrary-base descent.