#821: OpenAI's proof of Erdős's totient conjecture — x^(1−o(1)) smooth shifted primes for every exponent — 68 pages read by ten hostile seats, sources checked, the decisive identity rebuilt by computer

The problem (erdosproblems.com/821; community database status OPEN as of our 11 October refresh, last touched 2025-08-31; Formal Conjectures erdos_821, research open; OEIS A014197). Let g(n) be the number of m with φ(m) = n. Is it true that for every ε > 0 there are infinitely many n with g(n) > n^1−ε? Pillai showed g is unbounded; Erdős (1935, §3) showed g(n) > n^c infinitely often for some c > 0, by the construction that still carries the whole subject: take many primes p whose p − 1 has only small prime factors, multiply k of them together, and notice that the k-fold products have totients Π(p − 1) that are all smooth and all below x^k — far fewer values than products — so some value is hit enormously often. The exponent is the supremum C of the c that work; Pomerance (1980, Popular values of Euler's function) states the conjecture as C = 1 and attributes it to Erdős. Records for C: Baker–Harman 1998 proved x/(log x)^C₁ primes p ≤ x with P⁺(p − 1) ≤ x^0.2961, hence C ≥ 0.7039; Lichtman (2022) pushed the smoothness exponent to 15/(32√e) = 0.2843… with x/(log x)^O(1) primes, hence C ≥ 0.7156. Erdős's transfer (Pomerance's Theorem B states it) says: if for every δ > 0 there are x^1−o(1) primes p ≤ x with P⁺(p − 1) ≤ x^δ, then C = 1. So the question is really a question about smooth shifted primes, and below exponents of about 0.284 nothing of that strength was known: a sieve on {p : p − 1 smooth} — equivalently, primes among 2u + 1 with u smooth — has level of distribution ½ and no bilinear ("Type II") information, and the parity phenomenon stops a sieve with only that from ever detecting primes.

What OpenAI claims. Weighted dilation graphs, smooth shifted primes and totient fibers (24 September 2026, family 011 of the 6 October release, 68 pages, no Lean — the release's formalization catalogue has no entry for this family). Theorem 1.2: for every fixed 0 < δ < ¼, the number of primes p with 2x < p ≤ 5x and P⁺(p − 1) ≤ x^δ is at least x^1−o(1) (the o(1) may depend on δ; no uniformity as δ → 0 is claimed). Theorem 1.1: Erdős's conjecture, C = 1 — and the exponent 1 is best possible by the elementary bound g(n) ≪ n^1+η. The same family's other preprint (the Poisson–Dirichlet law for prime predecessors, 74 pages, the Ford–Konyagin–Luca conjecture) takes this paper's Theorems 3.5 and 4.1 as inputs, so anything wrong here is wrong there too.

How the proof is built (plain version). It is an asymptotic sieve for primes in the style of Friedlander–Iwaniec, run on the sequence N_u = 2u + 1 where u carries a weight A(u) that prescribes its factorization: r₀ primes from each of ≍ log log x geometric bands (the largest band stops below x^δ, the smallest at (log x)²⁰), plus "group" primes of size exp((log x)^0.1…0.2) (small groups) and exp((log x)^0.3…0.4) (big groups) whose distinct divisors the weight marks, plus one slot holding an integer with no prime factor below W = exp(√log x). Because every prime slot is charged its reciprocal mass, the candidate set has mass x/(log x)^O(1) — smooth integers are plentiful; the whole difficulty is detecting primes among the 2u + 1.

The sieve (§7). Type I information comes free from the bilinear structure in u — split off one band prime of size between x^a and x^1−a and use the multiplicative large sieve — and gives level of distribution x^½−κ/2 (Proposition 7.2). A Brun–Hooley block sieve (Lemma 2.9, after Ford–Halberstam) removes composites with a prime factor below x^b₁, as a lower bound. Composites whose least prime factor p lies between x^b₁ and x^½−κ are counted by Buchstab's identity: the cofactor m = N_u/p must be rough above x^γ ≈ p, and the paper replaces that roughness test by a proxy — a function constant on W-rough integers in each short logarithmic cell, count-matched to the test on every full cell (Lemma 7.3) — using the Type II estimate below; the proxy's size is a Buchstab density (Lemma 7.4), and integrating those densities over the least prime factor reproduces Buchstab's identity (7.18) exactly: the composites subtract D_b1(1) − 1 while the preliminary sieve gave D_b1(1), leaving the one-prime term — a positive constant times X_A/log x — on primes. Composites with two prime factors near √x are bounded by a two-variable sieve whose constant does not depend on the width κ of that window, so κ can be taken small. The singular series is the expected 2Π_p≥3(1 − 1/(p − 1)²): u is coprime to every prime below (log x)²⁰, so p | 2u + 1 happens with probability 1/(p − 1).

Type II (§6, Theorem 6.1). For m ≍ U, n ≍ V, UV ≍ x and both scales between x^b* and x^1−b*: Σ A(u)Ψ(u/x)α_mβ_n over mn = 2u + 1 is ≪ x(log x)^−D, with β arbitrary and bounded and α supported on W-rough integers and satisfying a "discrepancy hypothesis" (5.4): its harmonic sums over every interval, against every Dirichlet character of modulus ≤ (log x)^B and every Mellin twist m^iu with |u| ≤ (log x)^B, are ≤ (log x)^−B. That is the shape of Friedlander–Iwaniec's bilinear-form hypothesis — one side arbitrary, the other orthogonal to all low-complexity structure — the hypothesis that breaks parity, and the paper's claim is to prove it for this sequence. The proof splits u = eh with e just below the scale of m by a smooth slot-by-slot allocation of the band primes, applies Cauchy–Schwarz in (e, n) — the diagonal is paid by placing e a large power of log x below U — and turns an off-diagonal pair mn = 2eh + 1, m′n = 2eh′ + 1 into a shifted pair by an exact determinant identity: z = m′h and z + k = mh′ with k = (m′ − m)/2e a nonzero integer of polylogarithmic size (§6.5). The endpoints are functions F₀(z), G₀(z + k) of the group-free parts — each a convolution of a W-rough m-sequence (the original α at the first endpoint; arbitrary at the second), a W-rough integer in an interval [x^τ, x^η] with η < ¼, and a bounded residual — times the mark weights.

Shifted correlations (§5, Theorem 5.1). Σ Ψ(z/Z) z^ia₁(z + k)^ia₂ F₀(z)G₀(z + k)W_ℓ(z)W_ℓ(z + k)/z ≪ (log x)^−D₀ whenever the first endpoint's m-coefficients satisfy (5.4). This is a logarithmically averaged two-point correlation — the shape of the two-point Chowla/Elliott problem Tao solved for multiplicative functions — for sequences that are not multiplicative but are invariant under multiplication by group primes. The engine that replaces Tao's entropy decrement is new: (i) the correlation at shift k is lifted (§5.2) to a pairing in a weighted dilation graph whose vertices are integers n carrying ordered lists of M = J + ℓ distinct group-prime divisors per group and whose edges move n → n + kD, with D a product of J "shared" labels copied from the list plus "free" labels; the 1/n weight turns shared labels into independent draws with law ∝ 1/p, and the raw pattern (all labels shared) is exactly the original correlation; (ii) Theorem 3.5, transference: if the ideal operator — the same edges with labels genuinely independent, repetitions allowed — has norm ≤ (log x)^−E_id uniformly in the additive frequency, then the physical graph's closed-walk moment of length N = 2⌈(log x)^0.52⌉ is ≤ (log x)^−EN; the proof replaces the integer root by independent residues (CRT, Lemma 3.6), expands each big prime's divisibility pattern into one "lifespan" with ghost births and terminations, builds a Fock-type memory space in which a prime pays its reciprocal once, at birth (Lemma 3.7, an exact identity), shows transfers between memory and the active lists are rare because the omitted small-group labels randomize the shift (Lemma 3.9), reduces the clean edges to the ideal operator by Fourier analysis on the residues (Lemma 3.10), and restores the distinct-birth rule by an equality-rank expansion in which few equalities cost only a few edges their saving and many equalities pay a reciprocal prime each (§3.8); Corollary 3.11 converts the moment into cancellation for pairings whose first endpoint does not see its marks; (iii) Theorem 4.1 constructs, for any target E_id, signed comparison patterns — free probe slots in two blocks of big groups, kernels built from logarithmic cells and Dirichlet characters of polylogarithmic conductor (Lemma 4.3) — whose sum with the raw pattern has ideal norm ≤ (log x)^−E_id: an Efron–Stein decomposition of symmetric test vectors by probe coordinates, permutation symmetry across ≍ log log x groups to make the fully mean-zero components small, and an exact alternating cancellation of everything else; (iv) hence the original correlation equals minus the sum of correlations at the larger shifts kD_AD_B (D_A, D_B products of free big primes) up to any power of log; (v) each of those is bounded by the circle method — minor arcs by a bilinear exponential-sum bound in the two independent free products (Lemma 4.2); major arcs u/r with r polylogarithmic by the local Fourier energy of the first endpoint at width 1/H, H ≈ D_AD_B = exp((log x)^0.3…0.4) (§5.4–5.8), which is a Matomäki–Radziwiłł-style argument: extract one small-group prime from the marks (an exact identity, (5.12)), convert to a Dirichlet-polynomial mean value over |t| ≤ Y/H by Mellin inversion, let the small-prime polynomial handle ordinary times, show by high moments that the exceptional times occupy Y^o(1) unit intervals (Lemma 5.2) and use a sparse mean square there (Lemma 5.3), get cancellation for |t| ≥ (log x)^B₀ from the long W-rough factor's polynomial via a Brun–Hooley sieve polynomial and van der Corput (Lemma 5.4), and for |t| ≤ (log x)^B₀ from the discrepancy hypothesis itself. The paper states the order in which every constant is chosen (Convention 2.1, (3.37), §4.3, §5.8, Remark 6.2, §7.7) and says the order is not circular.

What we checked ourselves before the seats reported, from a full read of the 68 pages: the Buchstab identity (7.18) (pick the minimum coordinate under the symmetric simplex measure; l/l! = 1/(l − 1)!); the final accounting (7.22) from (7.11), (7.18), (7.19), with the singular series as V₁/V; the direction of every subtraction in §7.5–7.6 (composites over-counted, the sieve count a lower bound); the unit class of 2u + 1 ≡ 0 (mod d) and the (log x)⁻¹⁸ coprimality correction in Proposition 7.2; Corollary 3.11's spectral-Hölder and trace argument and its exponents; the exponent bookkeeping in (3.37) and (3.47) (the per-edge saving G must beat the per-visit ghost cost C_v; a rank-r equality collection spoils at most 2r edges and r < r₀ = o(N)); the Efron–Stein fraction (4.22)–(4.23) and the exact cancellation (4.26); the minor-arc arithmetic of §5.3 from Lemma 4.2 (the cell width only helps); the determinant identities (6.16) and the coprimality (m, k) = 1; and that the arbitrary residual coefficients at either endpoint enter the energy bound (5.11) only through ℓ² norms and trivial bounds, so Theorem 5.1's one-sided hypothesis is the one its proof uses. None of that is a check of the fifteen pages of §3 — that is what the seats were for.

The paper as written — a hostile read by ten seats. We refereed the 68-page preprint the way we refereed OpenAI's Hindman and hypercube Ramsey papers: tool-equipped Opus seats with disjoint scopes over the mathematics (§3 the transference theorem; §4 the comparison kernels; §5.1–5.3 the lift to the dilation graph and the circle method; §5.4–5.8 the local Fourier energy bound; §6 the Type II estimate; §7–8 the sieve and the deduction of Theorem 1.1), one over the bibliography and the live world, one whole-paper seat told to build a counterexample to Theorem 5.1 inside its hypotheses and to draw the global parameter graph, one seat that checked every exact identity and small-case claim by brute force with negative controls, and — because the first nine all pointed at the same page — a tenth, briefed blind to the others, that built the memory identity of Lemma 3.7 as an exact computation on a toy instance. Each was told that the base rate for a machine-written proof of a famous open problem is low, that its job was to find the sentence where it fails, to fetch every cited theorem from its current source, to track every constant's dependencies, and to say which steps it verified line by line and which it only skimmed. Every objection carries its quote in the bundle.

What the seats found, in order of weight. The §3 seat — the section every other seat named as the single point of failure — re-derived every displayed identity and inequality of the transference theorem, drew up a complete ledger of every cost in the length-N path sum (only the ghost operations and the small-group projection cost a power of log per visit, both paid by (3.37); the store/promote transfers are killed by the choice of J, the density change and the collision ban are additive and of size exp(−(log x)^c), the equality-rank phases touch o(N) edges, the memory truncation is a tail of size exp(−Ω(log²x))), and found no hidden loss exponential in N. It coded the physical path sum and the memory-space operator product independently from their definitions and found Lemma 3.7's memory identity exact to 10⁻¹⁵ on thirteen toy instances exercising shared and free labels, ghosts, stores, promotions and terminations; four deliberate mutations (wrong termination sign, no V_g at ghost births, no ρ in step (ii), no distinct-birth rule) break it, two rules the paper states are vacuous because the ban and distinct births imply them, and letting a group prime divide the shift k breaks it — so the hypothesis the paper uses there is load-bearing and used correctly. Its one exposition finding: the bijection behind Lemma 3.7 is a two-paragraph sketch that leaves three things to the reader (why the two exclusion rules are automatic, why an unterminated store or a birth at the last visit contributes zero, why free labels are not births); ℓ ≥ 1 and the law of the free draws are left implicit. Because nine seats had pointed at that one page, a tenth seat, briefed blind to the other reports, coded both sides of the memory identity (3.17) from the paper's definitions as two independent programs in exact arithmetic over ℚ(√2, i) — the physical path sum with independent uniform residues on one side, Π b_p times the ghost-and-edge operator product on the memory space with distinct births enforced on the other — and found them exactly equal in all nineteen informative settings it could build (one to three big groups, raw and comparison patterns, two coinciding free labels, N = 2 and 4, four damping values, four phases), with nothing in the paper changed; removing any one rule (distinct births for fresh slots or ghosts, the 1/b!, μ̃ for μ, the ρ in step (iv), √q at the end visits, the end terms in b_p, the free-label ban, the free-draw law) breaks it, and it discarded its own first toys as nearly vacuous once it attached coverage counts to every run. It then wrote out the backward induction behind (3.44) for three edges with a pivot dying at visit 1 and its dependent born at visit 3, confirmed C_h = C_a + C_v works and enters only through an o(N) term, that "at most 2r edges affected" is a valid upper bound even for blocks inside one edge or one ghost batch, and that "distinct performed birth values" is exactly the distinct-birth rule with shared labels, promotions and free draws correctly excluded. The §4 seat wrote the raw-versus-comparison cancellation out for an explicit instance and confirmed it by an exact toy computation — the identity holds only for the paper's asymmetric assignment (A labels from the target, B labels from the source, which the text calls "input"), so the asymmetry in (4.2) is load-bearing and the word "input" is a drafting hazard; it also found that Theorem 4.1's first paragraph is vacuous without its "More precisely" clause (a comparison equal to minus the raw pattern is admissible as written), which §5.3 avoids by using the precise form. The lift seat wrote the physical pairing out from the definitions for one small and one big group and showed by hand, and then to 10⁻¹⁸ on toy instances with mutation controls, that the raw pattern is exactly the original correlation (5.5) once the shared labels are summed, up to the overlap error the paper states; it found one inequality the paper never writes, C₄ > max(C, C₇) (the ideal frequency range must cover the Mellin cutoff), true and one line to fix. The energy seat re-derived every displayed step of the Matomäki–Radziwiłł-style argument, confirmed the small-mark extraction (5.12) is exact for ordered distinct lists (a toy with a non-constant mark function mismatched only on square multiples, exactly the error set the paper bounds), recomputed the van der Corput exponent recursion exactly, and found no cycle in the order of choices. The Type II seat verified the determinant parametrization is a genuine bijection on the W-rough support (toy-enumerated; the coprimality (m, k) = 1 is necessary and automatic there, with an explicit failure without it), the endpoint identification by direct substitution, and that no step needs a discrepancy hypothesis on β or on the residual coefficients; it noted that §6 never uses the "arbitrary second endpoint" freedom and that Theorem 5.1's uniformity in the shift k and the scale Z is used but not listed. The sieve seat re-derived Buchstab's identity (7.18) and checked it numerically to 10⁻¹⁶ from the simplex definition with perturbed-identity controls, verified every subtraction's direction, and pointed out that §7 uses Theorem 6.1 at its most unbalanced end (b* ≈ b₁/3 with b₁ of order 1/200), the regime where §3–5 should be tested. The whole-paper seat tried nine counterexample constructions to Theorem 5.1 — Archimedean and character pretenders of small and large conductor, a constant-sign Dirichlet inverse, scale-dependent pretenders of the kind that break the unaveraged Elliott conjecture, Liouville encodings, random multiplicative functions, structured residuals at the first endpoint — and found each stopped by one of three things (the discrepancy hypothesis, invariance under group primes together with the marks, or the trivial coefficient on the compulsory long rough factor); it drew the global parameter graph (no cycle, including C₄ ← K₀, which nothing chosen before K₀ uses) and observed that the proof uses nothing about the second endpoint beyond invariance and a size bound, so it proves a strictly stronger one-sided statement than it claims. The numerics seat found no identity that fails. The citations seat found all 23 references exist as cited, every arXiv version and date correct, 21 of 22 theorem-number locators quoted and matching, and two things worth knowing: Pomerance's Theorem B is cited for a transfer whose hypothesis (a positive proportion of smooth shifted primes) is stronger than what Theorem 1.2 delivers — harmless, §8 proves its own — and none of the cited divisibility-graph works (Tao, Helfgott–Radziwiłł, Pilatte, Matomäki–Radziwiłł–Tao) underwrites the new step, all being proved for multiplicative functions; the paper says so itself.

One thing we checked that no seat was asked. Because β in Theorem 6.1 is arbitrary, one might fear the theorem proves too much — a Chowla-type cancellation of λ(2u + 1) along the sequence. It does not: λ restricted to W-rough integers violates the discrepancy hypothesis already on the single dyad [W, 2W], where the harmonic sum of the primes is ≈ 1/√log x, not (log x)^−B. The hypothesis is exactly as strong as the proxies of Lemma 7.3 satisfy and no stronger.

Novelty and the world (11 October; details and URLs in the bundle's world/). No human proof of x^1−o(1) smooth shifted primes for arbitrary δ, or of C = 1, anywhere we could look (arXiv title/abstract sweeps, Semantic Scholar citations of Baker–Harman and Lichtman, MathOverflow, Hacker News, Mathstodon, the press); Lichtman's 0.2843… is still the best exponent in print. The preprint is not on arXiv; it was never revised after the 6 October initial commit; openai/math has issues and discussions turned off. OpenAI's own Lean library has one 312-line sorry-free file for this family, TotientFibers/FiniteFiber.lean, which formalizes the conditional step of §8 — given enough primes with smooth predecessors, some fiber is large — and nothing about the count; we compiled it under --trust=0 against the Mathlib build we had from the #172 replay and its main theorem depends on propext, Classical.choice, Quot.sound (bundle replay/finitefiber/). No third-party formalization of Theorem 1.1 or 1.2 exists (GitHub code search for erdos_821 finds only Pillai's lim sup and failed Epoch attempts; Yao Xu, who formalized OpenAI's #172 and #181 proofs, has nothing for this family). erdosproblems.com/821 shows no status under the site-wide policy, was last edited 1 October 2025 and does not mention OpenAI; the teorth database says OPEN; Formal Conjectures has erdos_821 research open and its 11 October pull request marking twelve OpenAI claims solved left 821 out; Bloom's 8 October post lists the claim under his blanket "I have checked nothing" and says "This proof has not been formalised" — it is one of four such among the twenty-five. Downstream users already lean on it conditionally (a repository on #824 says "if the argument of WD is correct … Sections 3–5 have not been independently verified"; a Zenodo note assumes "the graph-transference, comparison, long-prime, and positive extraction estimates"); Will Sawin's MathOverflow answer says only that the statement fits the probabilistic heuristics. No public comment from Lichtman, Granville, Ford, Pomerance, Baker, Harman, Maynard, Matomäki or Radziwiłł on the proof. The companion Poisson–Dirichlet paper in the same family imports Theorems 3.5 and 4.1, Corollary 3.11, Lemmas 3.4 and 4.3 from this one.

Verdict on the paper as written: HOLDS AS READ, ten seats out of ten; no mathematical gap found. What the read did find, in order of weight: the bijection behind the memory identity is sketched, not written (true as stated; verified by computer twice, independently); the ideal frequency range C₄ > max(C, C₇) is used but never written; Theorem 4.1's first paragraph is vacuous without its "More precisely" clause and "input" means source throughout §4; Theorem 5.1's uniformity in the shift and the scale is used in §6 but not listed; the proof of Theorem 5.1 uses nothing about the second endpoint beyond invariance and a size bound, so it proves a strictly stronger one-sided statement than it claims — a single-scale, power-of-log Elliott-type cancellation of the first endpoint along dilated progressions, which is the real content beyond the state of the art and has no precedent in the cited literature; Pomerance's Theorem B is cited for a transfer whose hypothesis is stronger than what Theorem 1.2 supplies (§8 proves its own); ℓ ≥ 1, the law of the free draws, and "C_bal is independent of κ" (the text says "of the gap") are left implicit or misworded; §8 requires 4δ < ε where 3δ < ε is used; and several dozen mislabelled cross-references ("Theorem" for lemmas and corollaries) and overloaded symbols (C₄, C₆, B₀, S, D, h, θ each mean two to four things) that make the order of choices error-prone to verify. Every one of these has its quote in the bundle, and every one, where the seat could decide, resolved in the paper's favour. The place every seat was told to hunt — a cost exponential in the path length N inside Theorem 3.5, a cycle in the order of the forty-odd constants, a counterexample to Theorem 5.1 inside its own hypotheses — came back empty. What this is not: a human expert's reading. The seats' own skimmed lists — the column bound of Lemma 3.8, the backward induction with retained pivots behind (3.44), the ghost-batch phase symmetrization, the constants in (4.20), the multi-group and permutation aspects of Lemma 3.7 (checked by hand, not by computer), Buchstab 1937 and Montgomery–Vaughan 1974 unfetched — are the edge of this read. A referee that finds nothing is weaker evidence than a kernel that accepts a proof, and for this paper no kernel has seen anything but the last page.

Labels. The theorem: (c) plausible/unverified by a human — ten tool-equipped machine seats with disjoint scopes, each briefed to refute, found no mathematical gap; two of them, independently and blind to each other, rebuilt the decisive identity as an exact computation and could make it fail; no Lean exists for anything but the §8 transfer, which is (d) computer-assisted and kernel-checked on our machine. The citation checks, the Buchstab and determinant identities, and our own spot reads are (a)/(b), ours. Nothing mathematical here is ours; the theorem is OpenAI's. This entry does not change the scoreboard: #821 is not closed in the literature by our standard (no formal proof, no human referee), and the community database still says OPEN.