Gpt 5 5 pro 2026 totient fibre extremes

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theorem_2_1: The asymptotic for the largest ratio of a maximal to a minimal totient preimage over totient values up to x, with the upper bound from the extremal order of m/phi(m) and the lower bound from a Linnik-prime construction; the site's accepted resolution of Problem 694.

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GPT-5.5 PRO, Totient fibre extremes. A five-page note whose title page prints the author line "GPT-5.5 PRO" (p. 1; the running head of pp. 2 and 4 repeats it) and names no person; the PDF's information dictionary carries no author field and gives a creation timestamp of 2 May 2026, 15:00 UTC; the text prints no date, affiliation, arXiv identifier or journal. It is hosted in the GitHub repository Shashi456/erdos-formalizations as Erdos/P694/proof.pdf beside its TeX source (proof.tex, whose line \author{GPT-5.5 Pro} produces the author line), an informal note, a Lean development and a checker report; the repository's main was at commit 286f856aa3fc08957b80950fd18a45aab8d045ea (2026-05-14) when the copy was taken.

The retained folder-name PDF is that file, five pages with a complete text layer, read on the rendered page images. Provenance: retrieved from https://github.com/Shashi456/erdos-formalizations/blob/main/Erdos/P694/proof.pdf as part of the survey download set of September 2026; 223,768 bytes. No notice is printed in the file; the hosting repository's LICENSE file and its README section "Apache 2.0 — matches Mathlib and formal-conjectures" name the Apache License 2.0 for the repository (https://github.com/Shashi456/erdos-formalizations, read 2026-10-02), the only term stated for the copy.

Attribution as the sources state it. The note credits "GPT-5.5 PRO" and no one else. The repository's README says "Proof by Liam Price + GPT-5.5 Pro, May 2026" and "Formalization assembled incrementally with Claude Code subagents, May 2026"; the plby/lean-proofs file below lists "Informal authors: GPT-5.5 Pro, Liam Price"; the Formal Conjectures docstring says "GPT-5.5 Pro (prompted by Price) has proved". The card records these sentences as the sources' own and claims no independent check of the argument. The site's label for Problem 694 is SOLVED (LEAN); no refereed publication, arXiv version or written independent review is held. The note states no license, and the repository copy held records none for it.

Read status: claims checked for Lemma 1.1, Theorem 2.1 and Proposition 3.1, read clause by clause on the page images of pp. 1--5; the proofs were read in full and their steps followed; nothing here is independently reviewed.

Contents

• Abstract (p. 1): with $f_{\max}(n)$ and $f_{\min}(n)$ the largest and smallest $m$ with $\phi(m)=n$, "if the maximum is taken over totient values $n\le x$, then $\max_{n\le x,\,n\in\phi(\mathbb N)}f_{\max}(n)/f_{\min}(n)=(e^\gamma+o(1))\log\log x$"; the upper bound from the extremal order of $m/\phi(m)$, the lower bound from a construction using Linnik's theorem, and a permanence observation.
• Section 1, Preliminaries (pp. 1--2): the interpretation $\mathcal R(x):=\max_{n\le x,\,n\in\phi(\mathbb N)}f_{\max}(n)/f_{\min}(n)$, nonempty for $x\ge1$ since $1=\phi(1)=\phi(2)$; the fiber-finiteness bound $\phi(m)^2\ge m/2$, so $\phi(m)=n$ implies $m\le2n^2$; the "standard unconditional estimates" used, $\vartheta(y)=\sum_{p\le y}\log p\sim y$, Mertens' product theorem $\prod_{p\le y}(1-1/p)^{-1}\sim e^\gamma\log y$, and Linnik's theorem, absolute $C,L>0$ with the least prime $\ell\equiv1\pmod a$ satisfying $\ell\le Ca^L$ ($L\ge1$); Lemma 1.1, $\max_{1\le m\le T}m/\phi(m)=(e^\gamma+o(1))\log\log T$, proved through primorials $N_k=\prod_{i\le k}p_i$ and $\log N_k=\vartheta(p_k)\sim p_k$.
• Section 2, The asymptotic formula (pp. 2--4): Theorem 2.1, $\mathcal R(x)=(e^\gamma+o(1))\log\log x$ as $x\to\infty$; the upper bound through $M/m\le M/\phi(M)$ and $M\le2x^2$; the lower bound through $P_y$, $A_y$, the Linnik prime $\ell$, $U_y=(\ell-1)/A_y$, $Q_y$, $a_y=\ell Q_y$ and $b_y=P_yU_yQ_y$ with $\phi(a_y)=\phi(b_y)=n_y$, $b_y/a_y=(P_y/A_y)(\ell-1)/\ell=(e^\gamma+o(1))\log y$, $\log n_y\le(2L-1+o(1))y$ and the choice $y=\log x/(4L)$.
• Section 3, A permanence observation (pp. 4--5): Proposition 3.1, if $a>b$ and $\phi(a)=\phi(b)=n$ then infinitely many distinct totient values $N$ have $f_{\max}(N)/f_{\min}(N)\ge a/b$ (take $N_r=(r-1)n$ for primes $r\nmid ab$); "the existence of one nontrivial totient fibre implies the existence of infinitely many nontrivial totient fibres".
• No reference list is printed.

Compiled scope

The whole note was read (five pages). Theorem 2.1 is compiled as a statement with the proof pointer on its page; Lemma 1.1 and Proposition 3.1 are described above and have no pages of their own. The three analytic inputs (the prime number theorem in the form $\vartheta(y)\sim y$, Mertens' product theorem, Linnik's theorem) are taken at statement level on p. 1 and were not checked here. The note's $\mathcal R(x)$ restricts the maximum to totient values, as the site's parenthetical does.

Formal artifacts (read statically, not built)

• Shashi456/erdos-formalizations, Erdos/P694/Proof.lean (143,271 bytes, 2,817 lines; repository main at 286f856a, 2026-05-14): "STANDALONE VERSION ... Trust boundary: Mathlib core (propext, Classical.choice, Quot.sound) + mertens_product + linnik_dvd", both declared as axiom (lines 417 and 430); R (x : ℕ) : ℝ (line 1116), the supremum over totient values n ∈ Set.Icc 1 x of sSup {m | φ m = n} / sInf {m | φ m = n}; theorem totient_fibre_extremes : Tendsto (fun x : ℕ => R x / (Real.exp Real.eulerMascheroniConstant * Real.log (Real.log x))) atTop (𝓝 1) (line 2648); permanence_step and infinitely_many_collisions (Section 3, whose header comment, line 2702, reads "This section is fully proved — no sorries, no axioms beyond Mathlib"); the alias erdos_694_asymptotic; eleven closing #print axioms lines with no recorded output. The repository README and safeverify/report.json (both in the survey download set) are the repository's own trust-boundary table and checker record: the two named axioms beyond the core three for the asymptotic theorems, core only for the permanence theorems; the README lists deviations from the note (the Landau lemma and the height bound avoid the prime number theorem through primorial_le_4_pow; Linnik invoked modulo $A_YP_Y$; Proposition 3.1 strengthened to an equality of cross products) and names the note compact_cayley_proof.pdf in its file table while the repository tree holds proof.pdf.
• plby/lean-proofs, src/latest/ErdosProblems/Erdos694.lean (15,922 bytes, 353 lines; history ending at commit 8a203599bb60145f390e9a7cca3c436f1514edf8, 2026-08-25, "Make Erdos694 unconditional without Linnik"): header leanprover/lean4:v4.33.0 mathlib v4.33.0, "Formalization status: Unconditional: standard Lean axioms only", "Informal authors: GPT-5.5 Pro, Liam Price; Formal authors: Claude Code 4.7, GPT-5.5 Pro, Pawan Sasanka Ammanamanchi", URLs naming the thread's post 6202, an online-editor read link and the Shashi456 file; imports the modules Erdos694/Unconditional and Erdos694/LinnikConstruction (the chain Core, PrimeProducts, SmallModuli, Height, Unconditional held; SmallModuli imports ErdosProblems.Erdos387.UniformAnalyticInputs, not held, for Erdos387.shiftedSiegelWalfiszLower, which the root's comment calls "the proved uniform prime-counting theorem"); the same R (in Core, line 1093); proves totient_collision_construction (line 40, from unconditional_totient_collision_construction in Unconditional), R_lower_bound (181), totient_fibre_extremes (198) and erdos_694 (326), the Tendsto statement above; no sorry or axiom in the root or the held modules; eleven #print axioms lines without recorded output. The repository's v4.29.1 copy (144,465 bytes, 2,898 lines) carries the header "Conditional on: mertens_product; Conditional on: linnik_dvd" and imports ErdosProblems.Axioms; the repository's summary page says the archived copies "retain the Linnik axiom" and describes the unconditional route (distinct primes in a dyadic interval congruent to one modulo each prime divisor of $A$, coefficient $e^\gamma D/(D+1)$ for each $D$).
• google-deepmind/formal-conjectures, FormalConjectures/ErdosProblems/694.lean (2,489 bytes): erdos_694, a sorry statement quantifying fmax, fmin with IsGreatest and IsLeast hypotheses and asserting sSup {...} = (exp eulerMascheroniConstant + o x) * log (log x); not textually the Tendsto theorem (the problem page records the reading).

None was built, kernel-checked or audited here; the #print axioms outputs and the checker report are the repositories' own records.

Bears on. #694: Theorem 2.1 answers the problem's "Investigate" with the asymptotic $(e^\gamma+o(1))\log\log x$ for the maximum over totient values $n\le x$; the site's label SOLVED (LEAN) rests on this note and the developments above. Proposition 3.1 is a permanence statement about collisions, of the same kind as the unique-preimage statement the collection's file records as erdos_694.variants.inf_unique ("Erdős has proved that if there exists an integer $n$ for which $\phi(m)=n$ has exactly one solution, then there must be infinitely many such $n$"); no implication between the two is asserted here.

Results.

• Theorem 2.1 (p. 2): as $x\to\infty$, $\mathcal R(x)=(e^\gamma+o(1))\log\log x$.