Openai 2026 weighted dilation graphs smooth shifted primes totient fibers

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theorem_1_1: The claimed resolution of Erdős's conjecture on the largest totient fibers: for every ε > 0 infinitely many n have more than n^(1-ε) preimages under Euler's function, derived from Theorem 1.2 by products of smooth-predecessor primes and a pigeonhole count; Problem 821's question, unverified here.

theorem_1_2: The claimed count of smooth shifted primes: for every fixed δ in (0,1/4) at least x^(1-o(1)) primes p in (2x,5x] have p-1 free of prime factors above x^δ, which the manuscript derives from a weighted-dilation-graph transference theorem, a Type II estimate and a sieve on primes 2u+1; the input to Theorem 1.1 and the claimed resolution of the smooth-predecessor conjecture, unverified here.

OpenAI, Weighted dilation graphs, smooth shifted primes and totient fibers, OpenAI Math Release preprint, September 24, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision adc7f1241), folder preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026; the held PDF, paper.pdf in the release, is retained as openai_2026_weighted_dilation_graphs_smooth_shifted_primes_totient_fibers.pdf, and the release's TeX bundle sits beside paper.pdf in that folder.

@misc{OAI:Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026,
  author = {{OpenAI}},
  title = {{Weighted dilation graphs, smooth shifted primes and totient fibers}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026/paper.pdf}{OAI:Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026}},
  year = {2026}
}

Attestation, recorded as the source's own statements and not as this corpus's review: the release's root README says its manuscripts were "produced by an internal OpenAI model", that the collection "includes results at different stages of verification", that not all have Lean formalizations, and that "some of the unformalized results could have issues". The manuscript's own README carries only the title, author line "OpenAI", the date and the citation block, and adds no statement about human assistance or verification. The manuscript names no author beyond "OpenAI", carries no arXiv identifier and no journal. No refereed publication, arXiv version or independent review of the manuscript is recorded here and nothing on this card is independently reviewed.

The release's Lean catalog (lean/formalization.yaml) lists no formalization for this manuscript, and no lean/docs page exists for its family; the release folder holds no verification/ directory.

Companions in the release's family: The Poisson--Dirichlet law for prime predecessors (its card), which the introduction (p. 2) says takes the graph and ideal-kernel estimates of Sections 3--4 here as inputs to its own determinant-graph estimate and its own extraction of primes; and Prime predecessors with an even number of prime factors (September 17, 2026), a third member of the same release family that is not held in this library and that the manuscript does not cite.

Read status: claims checked for Theorem 1.1, Theorem 1.2 and Lemma 8.1, read clause by clause in the TeX source (sections/00-introduction.tex, lines 10--53, and sections/07-totient-conclusion.tex, lines 10--16; PDF pp. 1 and 65) on 2026-10-07; the statements of the intermediate theorems (3.5, 4.1, 5.1, 6.1) and propositions (7.1, 7.2) were read as statements, and the proofs of Sections 3--8 were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

The manuscript is 68 pages: eight sections and a reference list of 23 items. Theorems are numbered within sections, and every lemma, proposition, corollary and definition shares the theorem counter. In the PDF every cross-reference to a lemma, proposition, corollary or definition prints as "Theorem" (for example "Theorem 7.1" for Proposition 7.1 on pp. 56 and 65, "Theorem 2.9" for Lemma 2.9 on pp. 57--63, "Theorem 8.1" for Lemma 8.1 on p. 66); the statements themselves print with their own labels, and this card uses the printed statement labels.

• Section 1, Introduction (pp. 1--4; sections/00-introduction.tex). Defines $g(n)=\#\{m\ge1:\varphi(m)=n\}$ and states Theorem 1.1: for every real $\varepsilon>0$ there are infinitely many $n$ with $g(n)>n^{1-\varepsilon}$. The manuscript says this "resolves positively" Erdős's conjecture on the largest totient fibers, citing Pomerance 1980 (p. 84) for the formulation $C=1$, with $C$ the least upper bound of the exponents $c$ for which infinitely many $n$ have $g(n)>n^c$, attributed there to Erdős 1956; the elementary bound $g(n)\ll_\eta n^{1+\eta}$ makes the exponent optimal. States Theorem 1.2: for every fixed $0<\delta<1/4$, the primes $p\in(2x,5x]$ with $P^+(p-1)\le x^\delta$ number at least $x^{1-o(1)}$, the $o(1)$ depending on $\delta$; hence infinitely many primes with $P^+(p-1)\le p^\varepsilon$ for every $\varepsilon>0$, which the manuscript calls the smooth-predecessor conjecture (Erdős 1956; Lichtman 2022, introduction). The count is said to hold for every fixed $\delta>0$ by monotonicity, with no uniformity as $\delta\to0$ asserted or needed. The literature paragraph records Baker--Harman 1998 (exponent $0.2961$ for shifted primes, multiplicity exponent $0.7039$), Lichtman 2022 (count $x/(\log x)^C$ for $\beta>15/(32\sqrt e)=0.2843\ldots$, multiplicity exponent $0.7156$), the Dickman-law prediction of a positive proportion (Granville 2008), and Bharadwaj--Rodgers 2026, whose Poisson--Dirichlet law for shifted primes the manuscript says is conditional on Elliott--Halberstam; it notes that its own $x^{1-o(1)}$ gives the full power exponent but that a positive proportion would be stronger. The passage from smooth shifted primes to large fibers is attributed to the Erdős--Pomerance method (Pomerance 1980, Theorem B) and reproved in Section 8. One paragraph names the Carmichael construction of Alford, Granville and Pomerance 1994 as a further application of smooth predecessors, claiming nothing about Carmichael numbers. Subsection 1.1 describes the mechanism (primes $2u+1$ with $u$ of prescribed factorization; a transference theorem comparing a physical divisibility graph with an independent-label operator; comparison kernels; shifted correlations; a Type II estimate; a sieve), comparing it with the divisibility-graph work of Matomäki--Radziwiłł--Tao 2016, Tao 2016, Helfgott--Radziwiłł 2021 and Pilatte 2026. Subsection 1.2 gives the dependency figure (Theorems 3.5 and 4.1 feed 5.1, which feeds 6.1, which with the Section 7 sieve bounds gives 1.2, which gives 1.1) and the order in which constants are chosen.
• Section 2, Notation and preliminary estimates (pp. 4--10; sections/01-preliminaries.tex). Fixes $L=\log x$, $T=\log L$, $W=\exp(\sqrt L)$, the smoothness and roughness vocabulary and Convention 2.1 on parameter order and uniformity. Records as external inputs Theorem 2.2 (Siegel--Walfisz for moduli up to $(\log y)^C$, constants "need not be effective", cited to Tao's 254A notes) and Theorem 2.3 (multiplicative large sieve, Montgomery--Vaughan II, Theorem 19.16), and derives Corollary 2.4 (harmonic character sums over primes in intervals above $\exp(L^c)$ save any power of $L$). Proves Lemma 2.5 (fixed divisor moments), Lemma 2.6 (an elementary Dirichlet-polynomial mean square), Lemma 2.8 (Fourier separation with log-power frequency truncation) and Lemma 2.9 (a Brun--Hooley block sieve with level $z^{4h+2}$ and coefficients bounded by one, after Ford--Halberstam 2000) with Corollary 2.10 (an upper polynomial for rough integers); Lemma 2.7 (van der Corput derivative tests) is cited to Montgomery--Vaughan II.
• Section 3, Transference for dilation graphs (pp. 10--26; sections/02-transference.tex). Definition 3.1 fixes small and big prime groups ($\exp(L^a)\le p\le\exp(L^b)$ and $\exp(L^c)\le p\le\exp(L^d)$ with $0<a<b<c<d<0.47$, $O(T)$ groups, harmonic masses in $[v_-,v_+]$), Definitions 3.2 and 3.3 the ideal operator $\mathcal T(\Theta)$ on independent labels and the physical operator $\mathcal A$ whose labels divide the integer at their vertex and whose edges shift by $kD$. Theorem 3.5 (local transference): each fixed $E>0$ admits some $E_{\mathrm{id}}>0$, depending on $E$ and the group data, such that if $\sup_\Theta\|\mathcal T(\Theta)\|\le L^{-E_{\mathrm{id}}}$, then for every sufficiently large fixed $J$ the averaged moment $\langle u_n,(\mathcal A\mathcal A^*)^Ru_n\rangle$ over $n\in[X,2X)$ is at most $L^{-EN}$ with $R=\lceil L^{0.52}\rceil$, $N=2R$. The proof replaces the integer root by independent residues (Lemma 3.6), builds a memory Hilbert space recording prime lifespans (Lemma 3.7, exact identity; Lemma 3.8, absolute bounds), shows transfers between memory and active lists are rare (Lemma 3.9), reduces clean edges to the ideal norm by Fourier analysis (Lemma 3.10), and restores distinct births by an inclusion-- exclusion over equality rank. Corollary 3.11 converts the moment into a pairing bound against a mark-independent endpoint.
• Section 4, Small ideal kernels at every frequency (pp. 27--34; sections/03-ideal-kernels.tex). Theorem 4.1 constructs, for any $E_{\mathrm{id}}$ and frequency exponent, a raw pattern and signed comparison patterns with coefficients of total size $L^{C_1}$ such that the symmetrized ideal operator has norm at most $L^{-E_{\mathrm{id}}}$, uniformly in the additive frequency and for $|\zeta|\le L^{C_4}$, with complexity fixed before $J$. Inputs: Lemma 4.2 (an elementary integer bilinear bound near rationals), Lemma 4.3 (comparison kernels from log cells and Dirichlet characters, via Theorem 2.2), and the Efron--Stein orthogonal decomposition (1981) with a permutation-symmetry bound.
• Section 5, Lifted shifted correlations (pp. 35--45; sections/04-shifted-correlations.tex). Defines the marked weight $W_{\mathbf t}$ (display (5.1)), the endpoint form (5.3) with a long rough-integer factor in $[x^\tau,x^\eta]$, $\eta<1/4$, and the discrepancy hypothesis (5.4) on the coefficient $\alpha$: harmonic sums over intervals in $[1,x^2]$ twisted by characters of modulus at most $L^B$ and by $m^{iu}$, $|u|\le L^B$, are at most $L^{-B}$. Theorem 5.1 (lifted shift cancellation): for every $D_0$ there is $B$ such that the harmonic shifted correlation of two endpoints at shift $k$, $0<|k|\le L^C$, is $O(L^{-D_0})$. The proof lifts to the physical graph, applies Corollary 3.11 with Theorem 4.1, treats comparison shifts by minor arcs (Lemma 4.2) and major arcs (a Mellin reduction, Lemma 5.2 on exceptional times with a prime-polynomial moment after Soundararajan 2009, Lemma 5.3 a sparse mean square, Lemma 5.4 cancellation of the long rough factor by derivative tests), and uses (5.4) only at low frequencies; the split into ordinary and exceptional times is compared with Matomäki--Radziwiłł 2016.
• Section 6, A Type II estimate (pp. 45--53; sections/05-type-ii.tex). Defines the band weight $a_Q$ on products of primes from geometric bands $\mathcal Q_j$ with $r_0$ slots each and one rough-integer slot, and $A(u)=a_Q(u_*)W_\ell(u)$. Theorem 6.1: for dyadic $U,V$ with $UV\asymp x$ and $x^{b_*}\le U,V\le x^{1-b_*}$, the rough-slot exponent $\eta_*$ below $b_*/4$, coefficients bounded by $L^C$, $\alpha$ supported on $W$-rough $m$ and satisfying (5.4) with a sufficiently large $B$, the sum of $A(u)\Psi(u/x)\alpha_m\beta_n$ over $mn=2u+1$ is $O(xL^{-D_*})$, with no hypothesis on $\beta$. The proof splits $u=eh$ smoothly, applies Cauchy's inequality, parametrizes the off-diagonal by an exact determinant identity ($m'-m=2ek$, $h'-h=kn$, $mh'-m'h=k$) and reduces to Theorem 5.1; Remark 6.2 records the order of precisions.
• Section 7, Extracting primes with smooth predecessors (pp. 53--65; sections/06-prime-extraction.tex). Proves Theorem 1.2 with $0<\delta<1/4$ fixed. Chooses $q=1/2$, $\ell=1$, band constants $c_1=1$, $c_2=6/5$, $c_3=3/2$, $c_4=9/5$, $s'=1/(r_0(c_1+c_2))$ with $c_2s'<\delta$, so that every prime factor of a supported $u$ lies in $[L^{20},x^\delta]$. Proposition 7.1 (candidate mass): $X_A=\sum_uA(u)\Psi(u/x)\gg xL^{-C_A}$ and $A(u)\le\exp(C\sqrt L)$. Proposition 7.2 (Type I distribution): the remainders of $N_u=2u+1$ in progressions to odd squarefree moduli up to $x^\vartheta$, $\vartheta<1/2$, sum to $O(xL^{-D}+X_AL^{-18})$, by the large sieve and Theorem 2.2. Lemma 7.3 (proxy discrepancy): a primality or roughness test minus its cell-constant proxy on $W$-rough integers satisfies (5.4). Lemma 7.4 (proxy density bounds): Buchstab densities $D_\gamma(w)$ in logarithmic coordinates, the identity $\int_{b_1}^{1/2}D_\alpha(1-\alpha)\,d\alpha/\alpha=D_{b_1}(1)-1$ and the sieve upper bound for $D_{b_1}(1)$. The extraction sieves $N_u$ below $x^{b_1}$ (Lemma 2.9), subtracts composites with least prime factor up to $x^{1/2-\kappa}$ through Theorem 6.1 and the proxies, bounds the balanced composites (two prime factors near $x^{1/2}$) by a two-variable sieve with a constant independent of $\kappa$, and lists the order of all parameter choices; the surviving mass $\gg X_A/L$ on primes and the pointwise bound give $x^{1-o(1)}$ distinct primes $p=2u+1\in(2x,4x+1]$ with $P^+(p-1)=\max(2,P^+(u))\le x^\delta$.
• Section 8, Large fibers of the totient function (pp. 65--66; sections/07-totient-conclusion.tex). Lemma 8.1: each fiber is finite and $g(n)\ll_\eta n^{1+\eta}$. Proof of Theorem 1.1 from Theorem 1.2 by products of $k=\lfloor X^{2\delta}\rfloor$ smooth-predecessor primes and a count of possible totient values (displays (8.1), (8.2)); see the Theorem 1.1 page.
• References [1]--[23] (pp. 66--68). The external inputs the proofs rest on are Siegel--Walfisz and Mertens (cited to Tao's 254A lecture notes, 2014), the multiplicative large sieve and the derivative tests (cited to an undated author-hosted draft of Montgomery--Vaughan II), Montgomery--Vaughan 1974 (compared, not used), the Brun--Hooley sieve (Ford--Halberstam 2000, reproved here), the Efron--Stein decomposition lemma (1981), Buchstab's identity (1937) and the prime number theorem with log-power error. The manuscript flags nothing as numerical, computer-assisted or conditional; its constants are not asserted to be effective, since the Siegel--Walfisz input of Theorem 2.2 is stated with constants that "need not be effective".

Bears on

• Problem 821: claimed resolution. Theorem 1.1 is the problem's question in the affirmative, for every $\varepsilon>0$ and with the same $g(n)$; the claim rests on Theorem 1.2 and is unverified here, and the page's open status rests on acceptance evidence, not on this card.
• Problem 1057: background only. The manuscript names the Alford--Granville--Pomerance construction as an application of smooth shifted primes (p. 2) and claims nothing about $C(x)$; its Theorem 1.2 gives $x^{1-o(1)}$ primes with $P^+(p-1)\le x^\delta$, weaker than the positive-proportion hypothesis $\pi(x,x^{1-E})\ge\gamma_1(E)\pi(x)$ that the AGP theorem on its card requires, so no Carmichael bound follows from this manuscript alone; the page's status is untouched and nothing here is verified.
• Baker and Harman 1998: claimed stronger exponents. Theorem 1.2 claims every fixed smoothness exponent $\delta>0$ against the card's $0.2961$ (Theorem 1), though with the weaker count $x^{1-o(1)}$ in place of $x/(\log x)^{C_1}$, and Theorem 1.1 claims the multiplicity exponent $1-\varepsilon$ against Corollary 1's $0.7039$; unverified here.
• Lichtman 2022: claimed stronger exponents. Theorem 1.2 against Theorem 1.1 there ($\beta>15/(32\sqrt e)$, count $x/(\log x)^C$) and Theorem 1.1 against Corollary 1.3 there (exponent $0.7156$), with the same count caveat; unverified here.