Early science acceleration experiments with GPT-5

Scope and aim

• The report uses concrete examples across mathematics, physics, astronomy, computer science, biology, and materials science to show what GPT-5 can and cannot do today.Its stated aim is to show how researchers can use GPT-5 to accelerate discovery while keeping standards high.Source: Introduction · p. 3

Human input and limitations

• The report documents interactions and human input to clarify where GPT-5 adds value and where expert oversight remains essential.It also warns that GPT-5 can confidently make mistakes, results may depend on prompt details, and reproducing them can be challenging.Source: Introduction · p. 3

Chapter I

• Chapter I describes GPT-5 independently rediscovering known results at the research frontier in mathematics, physics, and biology.Source: Introduction · p. 3

Chapter II

• Chapter II shows GPT-5 using deep literature search to find hard-to-find connections across scientific disciplines by focusing on concepts rather than wording.Source: Introduction · p. 3

Chapter III

• Chapter III presents researchers working with GPT-5 to accelerate their research workflows.Source: Introduction · p. 3

Chapter IV

• Chapter IV presents examples of GPT-5 obtaining novel research-level results.Source: Introduction · p. 3

Experiment rationale

• The experiment aimed to reproduce the main theorem of recent paper [BSZ25], a genuinely new question in classical convex optimization absent from GPT-5’s training data.Source: §I.1 · p. 4

Experiment question

• The test asked whether GPT-5 could rederive v2 from v1, where v1’s result was later refined to an optimal result in v2.Source: §I.1 · p. 4

Experiment outcome

• GPT-5 did not fully rederive v2 from v1, but went halfway between the two versions’ results.Source: §I.1 · p. 4

Problem definition

• The question is whether the piecewise-linear curve through gradient descent’s visited objective values is convex, rather than merely whether the iterates converge.Source: §I.1.1 · pp. 4–5

Prior results

• Version 1 proved η ≤1/L sufficient and η ≤1.75/L necessary, while version 2 established η ≤1.75/L as sufficient.Source: §I.1.1 · p. 5

Prompt and setup

• GPT-5 Pro was asked to improve the step-size condition without adding hypotheses, using version 1 as context.Source: §I.1.2 · p. 5

GPT-5 Pro’s result

• η ≤1.5/L improved version 1’s η ≤1/L sufficient condition, though it remained short of the η ≤1.75/L optimum.The author verified GPT-5 Pro’s proof, shown in Figure I.2.Source: §I.1.2 · p. 5, Figure I.2 · p. 7

Proof comparison

• GPT-5 Pro’s proof differed from version 2’s proof, which requires cleverly weighting inequalities to obtain cancellations.Source: §I.1.2 · pp. 5–6

Expert-time comparison

• The author estimated experts could likely achieve the improvement in hours to a few days.Source: §I.1.2 · p. 6

Longer-running models

• Internal models that could think for a few hours derived the optimal 1.75/L bound from scratch, without version 1 in context.Source: §I.1.2 · p. 6

• GPT-5 Pro (re)derived nontrivial Lie point symmetries, including an SL(2, R) algebra, of the stationary, axisymmetric wave equation on a Kerr background.Source: §I.2 · p. 8
• The model initially failed on the curved-space problem, then succeeded after a flat-space warm-up.Source: §I.2 · p. 8
• The correct symmetry generators underpin recent results in [Lup25b], which the model did not have access to.Source: §I.2 · p. 8

• Massless, stationary, axisymmetric waves on a rotating Kerr black hole are governed by a two-variable scalar Laplace equation in Boyer-Lindquist coordinates.Source: §I.2.1 · p. 8
• The equation’s solutions encode static tidal response, with their asymptotics determining horizon deformability through Love numbers.Source: §I.2.1 · p. 8
• In general relativity, black holes have vanishing static Love numbers, a surprising rigidity often traced to hidden symmetries of the field equations.Source: §I.2.1 · p. 8
• The central question is what symmetries Eq. (I.1) has.Source: §I.2.1 · p. 8
• Lie’s algorithm is arduous in the curved background; recent work found three nontrivial symmetries generating SL(2, R), including a dilatation used to explain vanishing Love numbers [Lup25b].Source: §I.2.1 · p. 8

• The stated goal was to elicit the curved-space Lie point symmetries, ideally reproducing the SL(2, R) generators and coefficients recently obtained in [Lup25b].Source: §I.2.2 · p. 8
• The initial curved-space attempt took 5m 13s and incorrectly reported no symmetries beyond trivial scalings.Source: §I.2.2 · pp. 8–9
• After a flat-space warm-up, the model produced all symmetries in 10 minutes and 27 seconds, including three generating SL(2, R).The H− generator was the nontrivial special conformal generator, which the authors took as evidence of an emulated symmetry computation rather than guessing.Source: §I.2.2 · p. 9
• The curved-space retry used the same prompt as before: “What are the Lie point symmetries of (I.1)?”Source: §I.2.2 · p. 9
• Within ∼18 minutes, GPT-5 Pro produced the correct curved-space generators closing into SL(2, R).Source: §I.2.2 · p. 10

• GPT-5 Pro (re)discovered the curved-space SL(2, R) symmetry generators, matching the key structural insight of [Lup25b].Once the symmetry is known, downstream results such as constraints on tidal response and vanishing static Love numbers in this sector follow with comparatively modest analysis.Source: §I.2.3 · p. 10
• The model failed cold but succeeded rapidly after a closely related warm-up, suggesting that a simpler problem can prime retrieval or internal pattern activation.Source: §I.2.3 · p. 10
• The structured generators were judged too elaborate to be a lucky guess; the model likely combined conformal-invariance recognition, a curved analogue, or a simplifying coordinate map.Source: §I.2.3 · p. 10
• With minimal domain scaffolding, current models can perform nontrivial Lie-symmetry discovery for PDEs with non-constant coefficients.Source: §I.2.3 · p. 10
• The authors say the time from idea to publishable result can compress from months to days when the right prompts and scaffolds are in place.Source: §I.2.3 · p. 10
• The authors suggest applying the warm-up-then-lift workflow to more complex problems in black hole theory and beyond.Source: §I.2.3 · p. 11

• GPT-5 Pro analyzed 2-DG data showing an increased proinflammatory Th17 subset and proposed a mechanism and relevant experiments.The authors said the insight and hypotheses were not immediately obvious despite their expertise.Source: §I.3 · p. 12
• For CD8+ cells, GPT-5 Pro attributed altered PD-1 and LAG-3 expression to combined glycosylation effects and attenuated T-cell receptor signaling.The authors report that it also predicted a brief 2-DG pulse during CAR-T generation would enhance cytotoxicity, a prediction they had internally validated in unpublished results.Source: §I.3 · p. 12
• The authors describe GPT-5 Pro as a mechanistic co-investigator that compressed months of reasoning into minutes and shaped experimentally testable strategies.Source: §I.3 · p. 12

• The study addressed how human T-cell subsets respond to changes in glucose metabolism.Source: §I.3.1 · p. 12
• The researchers used 2-deoxy-D-glucose (2-DG) to inhibit glucose uptake and sorted T cells into CD4+ helper and CD8+ cytotoxic subsets.Source: §I.3.1 · p. 12
• In the initial CD4+ experiment, cells received 2-DG doses or low-glucose conditions during activation, then were washed and expanded with IL-2 for two weeks.The design tested whether early glucose inhibition could leave an imprint that later appeared in T-cell differentiation and function.Source: §I.3.1 · p. 12

• CCR6 and CD161 increased more than IL-17A, consistent with enrichment or imprinting of Th17-lineage memory cells rather than uniform cytokine production.Many CCR6+/CD161+ cells were IL-17A− at restimulation.Source: §I.3.2 · pp. 14–15
• GPT-5 Pro produced data histograms and a dose-response readout, and proposed reduced IL-2 signaling through N-linked glycosylation interference as a plausible explanation.The authors said this mechanistic connection had not occurred to them previously.Source: §I.3.2 · p. 14
• A single 2-DG exposure at priming led two weeks later to a 13× increase in IL-17A+ cells and a 16–20× increase in CCR6 and CD161 Th17-lineage markers.Source: §I.3.2 · p. 15
• Proposed contributors included impaired N-linked glycosylation and reduced IL-2 signaling, altered TCR/CD28 signals, epigenetic imprinting, and selective survival or expansion.Source: §I.3.2 · p. 15
• GPT-5 Pro suggested unperformed N-linked-glycosylation and IL-2-axis experiments, and its prediction of enhanced anti-CD19 CAR-T cytotoxicity matched the authors’ unpublished experiment.Source: §I.3.2 · pp. 16, 20
• Mannose nearly normalized CCR6/CD161 increases, supporting impaired N-linked glycosylation as the primary driver; the authors note a similar result appeared in their BioRxiv preprint.They therefore caution that GPT-5 Pro may have known the rescue finding when it predicted it.Source: §I.3.2 · pp. 16–17
• The effect appeared in memory, not naïve, CD4+ cells; the authors’ interpretation favors preserving or expanding existing Th17-lineage memory rather than instructing naïve cells.They also suggest programming and selection together explain the effect’s magnitude and persistence after washout.Source: §I.3.2 · p. 17
• In CD8+ cells, 3 mM 2-DG reduced the naïve PD-1+/LAG-3+ population from ∼94% to ∼0.8% and memory LAG-3+ cells from 92% to ∼57%.The reported changes were stronger than those under glucose-free conditions.Source: §I.3.2 · p. 18, Figure I.3 · p. 19

• GPT-5 Pro distinguished glycolysis blockade from impaired N-linked glycosylation, proposed the IL-2 receptor pathway, and predicted that memory rather than naïve T cells carried the effect.Source: §I.3.3 · p. 21
• Its proposed decision tree included mannose rescue, metabolic readouts, and epigenetic assays, which the authors describe as testable wet-lab hypotheses.Source: §I.3.3 · p. 21
• The authors say separating selection from programming and glycosylation effects from energy restriction could reduce unnecessary experiments and months of testing.Source: §I.3.3 · p. 21
• GPT-5 Pro predicted that brief 2-DG exposure would lower PD-1/LAG-3, preserve cytotoxic potential, and enhance serial killing in anti-CD19 CAR memory CD8+ cells.Source: §I.3.3 · p. 21
• The authors expect lab-integrated models to speed mechanism discovery, make negative results cheaper, and improve experimental reproducibility and discovery rates.Source: §I.3.3 · p. 21

II.1 From density estimation and convex geometry to multiobjective optimization – Nikita Zhivotovskiy

• The experiment asked GPT-5 to find related quantitative results and central references after the authors had obtained Theorem II.1.1.The queries were rerun in a temporary new session to avoid contamination from prior searches or interactions.Source: II.1 From density estimation and… · pp. 22–23
• GPT-5 surfaced a connection between Theorem II.1.1 and ’s work on multiobjective optimization, despite their initially seeming unrelated.The authors describe using the connection to broaden the theorem’s scope toward theoretical computer science.Source: II.1 From density estimation and… · pp. 22–23
• For convex K, Theorem II.1.1 gives a 32-factor cover of size at most 28d, independent of range parameter R.In the Papadimitriou–Yannakakis framework, K is the set of achievable objective vectors; convexity is not assumed generally.Source: II.1 From density estimation and… · p. 24
• The authors extended the theorem to (1 + ε)-covers of size O((log(1/ε)/ε)^(d−1)) that, for convex K, are independent of R.For convex instances, this removes the log R factor from the classical bound and provides a benchmark for convex multiobjective optimization.Source: II.1 From density estimation and… · p. 24

II.2.1 Introduction

• The online Erdős-problem index lists each problem with a status and a discussion thread, and ChatGPT helped write the code underlying the site.As of October 31, 2025, the index contained 685 open problems out of 1105 total.Source: II.2.1 Introduction · p. 25
• Scattered problem statements and solutions make it difficult to tell whether an Erdős problem listed as open was solved decades earlier.The authors describe mathematicians spending futile effort on problems that were already known.Source: II.2.1 Introduction · p. 25
• GPT-5 helped locate published solutions to 10 problems, noteworthy partial progress for 10 others, correct a misprint in Problem 1041, and generate an idea that helped settle Problem #848.The #848 solution also used prior suggestions from online commenters.Source: II.2.1 Introduction · p. 25

II.2.2 Literature search

• The authors queried each listed-open problem for a solution or significant partial progress using problem-agnostic prompts, and found references generally accurate and manually verifiable.They observed some overenthusiasm about partial progress, but no cases of GPT-5 pretending to find a correct reference.Source: II.2.2 Literature search · pp. 25–26
• Given only a screenshot, GPT-5 Pro found the solution to Problem #339 in its first query, despite the source paper’s length and many citations.The 100-page paper had roughly 700 citations, making it time-consuming for a human to sort relevant references.Source: II.2.2 Literature search · p. 26
• GPT-5 located references solving Problems #515, #750, and #1043 in,, and, respectively.The authors say finding these decades-old papers required reading them in detail and applying mathematical understanding.Source: II.2.2 Literature search · p. 27
• For Problem #515, proves a more general result about integrals of e−F(z), and its subharmonic-function framework implies the Erdős problem.The connection was hard to find through standard web search because the terminology differs; GPT-5 also flagged the need to check that their definition permits the singularities of log |f(z)|.Source: II.2.2 Literature search · p. 27
• For Problem #750, concerns a bipartite subgraph; GPT-5 noted that taking the larger side of the bipartition solves the stated problem.The authors report this deduction arose during the literature search without additional prompting.Source: II.2.2 Literature search · p. 27
• Problem #1043 appears as a brief side comment in, while, written in German, contains the main work; GPT-5 translated and explained the proofs for the authors to verify.The side comment had been overlooked by many, including the MathSciNet review of.Source: II.2.2 Literature search · p. 27, Figure II · p. 28

II.2.3 Looking forward

• GPT-5 can help mathematicians access literature beyond what one person can readily keep at hand, while crowd-sourced negative searches offer a soft certificate that a solution is unlikely to be published.Researchers including Thomas Bloom and Terence Tao posted such searches on the Erdős problem website.Source: II.2.3 Looking forward · p. 28

II.3 Clique-avoiding codes: a cautionary tale

• For the clique-avoiding-code problem, GPT-5 first argued incorrectly that the random-coding bound r(n)≤n was tight, then gave a correct Chevalley–Warning-based lower-bound proof when challenged.Its initial attempts included buggy arguments and a hallucinated TCS Stack Exchange attribution.Source: II.3 Clique-avoiding codes: a… · p. 29

II.3.1 An unexpected twist

• The clique-avoiding code problem has exact minimum co-dimension r(n)=⌊n/2⌋, and GPT-5’s lower-bound proof reproduced an earlier result by Alon without identifying its source.The authors later found a matching construction themselves; GPT-5 Pro recovered the prior reference when queried again.Source: II.3 Clique-avoiding codes: a… · p. 30, II.3.1 An unexpected twist · pp. 30–31

Recent experiences of using LLMs as research partners – Timothy Gowers

• Gowers found LLMs useful for locating references, solving well-defined subproblems quickly, and critiquing precise approaches, but open-ended proof sketches could elicit invalid arguments and they were unlikely to supply a difficult problem’s main idea.He characterized them as speeding up the process of thinking, especially outside one’s primary expertise.Source: Recent experiences of using LLMs as… · p. 32

Example 1: a compact subset of L2

• GPT-5 used Kolmogorov–Riesz to support a compactness argument, but later checks exposed a flawed Hermite-function claim; a relevant result supported compactness for 0<c<1/2, though invariance under T remained unclear.The initial proof saved an estimated one to three hours; human collaborators identified that the Gaussian condition was too strong and questioned whether the resulting set was invariant under T.Source: Example 1: a compact subset of L2 · pp. 33–35

Example 2: graphs with uncharacteristically small clique number

• The proposed polynomial-time algorithm outputs 1 on graphs with no clique of size (3/2) log2 n, and outputs 0 on almost all graphs (a 1−o(1) proportion).Source: Example 2: graphs with… · p. 35

First idea: use linear algebra

• A linear-algebra construction gives a graph on 2^n−1 vertices with no clique of size (3/2) log2 N for sufficiently large N, but the graph is much more regular than a random graph.The construction uses odd-sized subsets as vertices, joining pairs whose intersection has even size.Source: First idea: use linear algebra · p. 35
• GPT-5 Pro showed that the proposed F2 matrix class was easy to identify because its rank is at most n, unlike a random graph adjacency matrix’s typical rank near 2αn.This rank contrast supplied a simple algorithm for distinguishing the proposed matrices from random graphs.Source: First idea: use linear algebra · p. 36

Second idea: use angles between unit vectors

• For graphs built from negative inner products of random unit vectors, GPT-5 Pro predicted excess independent triples and unusually large neighborhood intersections, exposing a detectable departure from randomness.The argument uses the fact that vectors with angle θ have probability 1/2 − θ/2π of sharing a positive inner product with a random vector; a related F2 construction also creates large intersections from zero-sum quadruples.Source: Second idea: use angles between unit… · p. 36

Third idea: replace linear algebra by polynomials

• GPT-5 Pro argued that a degree-d polynomial construction yields a matrix of low rank unless d grows linearly with n, while high degree offers no clear reason to avoid large cliques.The row space is spanned by monomials of degree at most d; the author found the low-rank argument simple but did not consider the high-degree clique claim fully compelling.Source: Third idea: replace linear algebra by… · pp. 36–37

Attempting to go in the other direction

• GPT-5’s responses to whether strong neighborhood-intersection tests imply a clique of size (3/2) log2(n) were unhelpful, reflecting how little is known about such quasirandomness conditions.The author characterized the question as too open-ended.Source: Attempting to go in the other direction · p. 37

Conclusion

• The author found LLMs useful as knowledgeable research supervisors, but not yet at the level of contribution that would warrant joint authorship under his convention.His convention distinguishes helpful ideas arising from expertise from contributions developed through sustained struggle beyond standard expertise.Source: Conclusion · p. 37

III.2.3 Result, implications, and next steps

• GPT-5 Pro derived a new analytical result confirming earlier numerical simulations of cosmic-string power, using Legendre- and Bessel-function expansions and providing a correction term.The asymptotic result for large odd n matched the previously derived large-even-n result; the correction matters for small n, and the asymptotic and numerical results differ below n = 5 but are nearly identical at large n.Source: III.2.1 The problem in context · p. 38, III.2.3 Result, implications, and next… · pp. 39–40, III.2.2 The task given to GPT-5 Pro:… · p. 39

III.3.1 Introduction

• GPT-5 helped set up a static, spherical reaction–diffusion model to investigate thermonuclear burn-wave ignition and propagation in inertial-confinement-fusion fuel.The model isolates conduction, alpha-particle energy deposition, radiation losses, and density-profile effects; GPT-5 proposed relevant physical data with little intervention from the author.Source: III.3.1 Introduction · p. 41, Premise · p. 41, III.3.2 Model Formulation · p. 43

III.3.3 Numerical Implementation

• GPT-5 produced a working numerical implementation in minutes, but human intuition and empirical exploration were needed to parameterize profiles and find marginal threshold behavior.The author had to guide the search between conditions that caused runaway ignition and those where no burn wave propagated.Source: III.3.3 Numerical Implementation · p. 44
• At fixed hot-spot temperature Ths = 1.5 keV, simulations measured mean 5 keV-isotherm front speed over 1–3 ns for slope-curvature pairs including s ≈ 3.0 × 10^9 kg m^-4 and c ≈ -4.5 × 10^13 kg m^-5.Optimal profiles exhibit a gentle initial density rise, allowing alphas to preheat the ramp before losses dominate deeper in the cold fuel.Source: III.3.3 Numerical Implementation · pp. 45–46

III.3.4 Theoretical Explanation

• The theoretical account attributes the ridge to power balance: conduction and alpha deposition must overcome radiative losses and the energy needed to heat cold fuel to 5 keV.The analysis identifies an optimal slope where preheat reach remains effective while density supports self-heating without excessive losses.Source: III.3.4 Theoretical Explanation · pp. 46–47
• Moderately negative curvature creates a low-density preheat shelf, while positive curvature raises density too early, impeding preheat and shifting alpha deposition behind the advancing front.The proposed criterion matches alpha range and conductive reach to the target preheat distance; its predicted ridge is reported to fit the numerical ridge reasonably well.Source: III.3.4 Theoretical Explanation · pp. 47–49, Figure III · p. 49
• The study reports that longer alpha range or stronger radiation favors shallower densities and more negative curvature, while lower hot-spot temperature shifts the optimum toward larger slope.The resulting static 1-D spherical reaction–diffusion framework captures essential burn-wave physics, yields credible propagation thresholds, and can support theory development, simulation validation, and experiment design.Source: III.3.4 Theoretical Explanation · p. 48, III.3.5 Physics Conclusions and Study… · p. 50

III.3.5 Physics Conclusions and Study Outcomes

• The author estimates that roughly 6 person-months of work were compressed into 6 person-hours, but stresses that experts must catch errors and persist through revisions.GPT-5 helped construct and optimize the model, while the theoretical explanation provided a check against numerical errors or misleading model arguments.Source: III.3.5 Physics Conclusions and Study… · p. 50

III.3.6 Part II — Final Ridge Map Code

• The appendix provides the complete Python code used to generate the Gaussian-smoothed ridge map and extract five density profiles.Source: III.3.6 Part II — Final Ridge Map Code · p. 51

IV.1.1 AI–assisted solution to Problem #848

• GPT-5 proposed using one unexpected b in A to constrain every other element; despite implementation errors, the idea led to a complete proof after human refinement.The initial observation came from van Doorn, and the prompt drew on a comment by Cambie; the GPT-5 step was between these human contributions.Source: IV.1.1 AI–assisted solution to Problem… · pp. 55, 57
• For every fixed a and every 25-class t other than the unique class with 25 | ab + 1, at least a 25/(4pi^2) − o(1) fraction of b in t make ab + 1 squarefree.Source: IV.1.1 AI–assisted solution to Problem… · p. 57

IV.1.2 Looking Forward

• GPT-5 can recall lemmas and suggest plausible proof strategies, but it often misses obvious blocking examples and overestimates existing methods.The authors attribute this limitation partly to mathematical literature rarely recording why obvious strategies fail or problems remain out of reach.Source: IV.1.2 Looking Forward · p. 58

IV.1.3 Formal solution of Problem #848

• For sufficiently large N, the 7 mod 25 class is maximum; the 18 mod 25 class also attains equality when its size matches.Any A with |A| ≥ 0.038·N and within 5% of optimality must be contained in one of these two classes.Source: IV.1.3 Formal solution of Problem #848 · p. 58
• The proof of Problem #848 reduces to A being contained in A7 or A18, which yields the desired bound.The argument first reduces to A* being empty, then handles the case where both A7 and A18 are nonempty.Source: IV.1.3 Formal solution of Problem #848 · p. 61

IV.2 New online algorithms lower bounds – Christian Coester

• GPT-5’s convex body chasing assistance included refuting a natural algorithm and helping raise the lower bound for arbitrary online algorithms.The section presents these as contributions from separate case studies.Source: IV.2 New online algorithms lower… · p. 62

IV.2.1 Lower bound for Follow-the-Leader

• GPT-5 produced a counterexample showing that follow-the-leader can incur arbitrarily large cost even for nested convex bodies inside the unit disk.The counterexample refutes a π/2 upper bound in the two-dimensional setting; the authors note a minor inaccuracy in GPT-5’s proof that did not affect the result.Source: IV.2.1 Lower bound for Follow-the-Leader · pp. 62, 65

IV.2.2 Improved lower bounds for general algorithms

• GPT-5 helped establish a π/2 lower bound for arbitrary online algorithms in two dimensions and suggested extending it by applying the bound across orthogonal two-dimensional components.An ε-distance trigger for switching pivots avoids the freezing problem; the authors made the higher-dimensional proposal rigorous despite errors in GPT-5’s suggested execution.Source: IV.2.2 Improved lower bounds for… · pp. 66–67

IV.2.3 Reflection

• GPT-5 saved research time and supplied useful ideas, but its responses also contained errors that the authors had to fix.The authors say its proof errors were easy for a human to correct, while later attempts to improve the lower bounds included arguments with serious flaws.Source: IV.2.3 Reflection · p. 69

IV.3.2 New proof of Theorem IV.3.1

• GPT-5 reproved the first tree inequality using a proof different from prior arguments, centered on a “miraculous identity” and fitting in a page and a half [Bub+16].The authors say the proof was entirely AI-generated apart from clarity edits; they report no web search access and that GPT-5 did not seem aware of the existing proof.Source: IV.2.3 Reflection · p. 70

IV.2.3 Reflection

• The authors first tried the research scaffolding on several Erdős problems without success; this graph-theory experiment was their first attempt outside Erdős problems.Both proofs were AI-generated apart from minor clarity editing, though human checking rejected some incorrect generated proofs.Source: IV.2.3 Reflection · p. 70

IV.3.1 Paths, stars and wyes in trees

• S, P, and Y count induced five-vertex stars, paths, and the remaining subtree type; Conjecture 1, also made in, describes the second face and is shown true here.The other conjectured inequalities were disproven in [Bub+16].Source: IV.3.1 Paths, stars and wyes in trees · pp. 70–71

IV.3.3 Proof of Conjecture 1

• GPT-5 proved the remaining C = 2 tree inequality, which earlier work had left open while disproving the conjectured cases C ≥3 [Bub+16].The proof bounds the relevant expression using a non-polynomial function on the integers and verifies that each resulting term is nonpositive.Source: IV.3.3 Proof of Conjecture 1 · p. 73
• In an evolving-tree model, GPT-5 proved that the hidden attractiveness parameter w is identifiable from the observed tree’s leaf fraction, yielding a consistent estimator for w ≥1.The limiting leaf fraction is strictly increasing in w, and the estimator based on the observed number of leaves converges in probability.Source: IV.3.3 Proof of Conjecture 1 · p. 77
• GPT-5 supplied the major proof ideas, but humans filled in some details and corrected false starts; the polynomial-rate strengthening and more general p-setting were left for future work.The authors believe a more precise implementation might establish the polynomial rate; the generalized setting removes the w ↔ 1/w symmetry.Source: IV.3.3 Proof of Conjecture 1 · p. 78

IV.4.1 Proof Overview

• The proof proceeds by showing Rt→1/2 and St/t→w+1 almost surely, then analyzing the growth of L(1)(t) and L(w)(t) using classical stochastic approximation.The authors connect the approach to results in the style of.Source: IV.4.1 Proof Overview · p. 79

IV.4.2 One-Step Dynamics

• The one-step analysis tracks the attachment event’s effect on the selected vertex and notes that A_t(z) is unchanged for all other existing vertices.Source: IV.4.2 One-Step Dynamics · p. 79

IV.4.3 Convergence of Rt and St

• Vt→0 and St/t→w+1 almost surely; the proof uses an almost-supermartingale argument and then identifies the limit through E[Vt]→0.Source: IV.4.3 Convergence of Rt and St · p. 80

IV.4.4 Stochastic Approximation for Leaf Fractions

• The ratios satisfy x_t/t → (w+1)/(2(w+2)) and y_t/t → y⋆ = (w+1)/(2(2w+1)).Source: IV.4.4 Stochastic Approximation for… · p. 82

Conclusion

• Across scientific disciplines, GPT-5 assisted ideation, deep literature search, and proofs of appropriately sized open problems, leading the authors to anticipate an exhilarating time in science and mathematics.The authors state that these capabilities would not have been possible twelve months earlier.Source: Conclusion · p. 83