Heru Technologies: Modeling Escape Trajectories of a Probe Trapped in an Asteroid's Gravitational Field Using SRMs and Human-GenAI Collaboration

1 Heru Technologies: Modeling Escape Trajectories of a Probe Trapped in an Asteroid’s Gravitational Field Using SRMs and Human-GenAI Collaboration Carlos Roberto França 1[0000-0002-6852-7103] 1Federal University of Fronteira Sul – UFFS/Campus Chapecó-Santa Catarina – Brazil [email protected] Abstract: This paper presents the refinement of spacecraft trajectory calculations, an endeavor enhanced through collaboration between human experts and Generative Artificial Intelligence (GenAI), both proficient in the Infinite Series with Multiple Ratios (SRMs). The study explores the challenges faced by the AIs Grok 3 and Gemini Advanced 2.5 Experimental in leveraging their advanced computational mathematics tools. Initially, the AIs were tasked with a highly complex mathematical challenge, making inferences to determine the exact time a probe could escape an asteroid’s gravitational field and the distance traveled. All necessary data were provided, including a table with a geometric cycle of accelerations spanning 7 hours. Subsequently, the author applied SRMs to validate or refute the AIs’ responses. In this second phase, only the human expert intervened, identifying errors in the predictive calculations of both Grok 3 and Gemini Advanced 2.5 Experimental. These errors were not shared with the AIs; instead, the problem was reformulated. It became clear that the AIs needed to develop and justify their answers regarding the precise time of the probe’s escape and the distance traveled during the effort. In the third phase, both AIs succeeded, with Grok 3 demonstrating a notable advantage by accurately predicting the escape time and providing a well-founded explanation of its calculations. Finally, the paper discusses the efforts of both the AIs and the human expert—the author of this article is an SRMs specialist—in achieving the solutions. Keywords: Infinite Series with Multiple Ratios, Mathematical Modeling, Gravitational Fields, Generative Artificial Intelligence. 1 Introduction This is the fourth paper published within a span of less than 25 days by the author of this work. The motivation stems from several investigative aspects, primarily the applicability of the Infinite Series with Multiple Ratios (SRMs), which have been the focus of the author’s research for three decades (Rodrigues, 1995). During the 1990s, I presented the SRMs at several conferences of the Brazilian Society for the Advancement of Science (SBPC). At the time, I was a young, recently graduated mathematician and temporary professor at a federal university in the heart of the Amazon, assisting the researcher who created the SRMs, Professor Edgar O. Rodrigues. The most common feedback we received was: “These concepts seem tailored to justify the formulas, like a perfect shoe for a flawed foot. ” In reality, both then and now, very few humans can solve complex mathematical problems like those addressed by SRMs (Rodrigues, 1997). With the advent of Generative AIs, I have not only found non-human intelligences capable of understanding and solving these problems but have also created new scenarios and applications for SRMs to enhance the mathematical toolkit of GenAI. This paper presents the extremely complex task of removing a spacecraft captured by the gravitational field of an asteroid. The calculations are complex, the numerical quantities need to be as accurate as possible and involve cyclical and periodic elements, which in 2 themselves bring a cognitive overload for everyone involved (Costa, 2010). Furthermore, it was found that with the extremely advanced mathematical tools that the Generative AIs, Grok 3 and Gemini Advanced 2.5 experimental have, they would not guarantee the accuracy of the questions that drive this research: 1 - At what time will the probe free itself from the asteroid's gravitational field? 2 - How far was it covered? These are seemingly simple questions, but they involve complex calculations and a periodicity combined with extremely high speeds, which go beyond what we know as conventional or trivial. Well, let's look at the problem question of this paper stated below, as well as its data and how the AIs will work with the information. It is worth noting that this paper will only present the answers of the Grok 3 and Gemini Advancet 2.5 experimental AIs, as they were the only ones that managed to reach the final answer, or very close to it, in a third and final moment. The other AIs, DeepSeek R1, ChatGPTs (all models), Claude 3.5 - Sonnet, did not even develop a resolution proposal that signaled a possibility of meeting the challenge with the reformulation presented in the third moment. The question is set out below and was sent to the GenAIs without any specific didactic guidance. 1.1 Heru Technologies e a modelagem computacional para libertar uma sonda de um campo gravitacional. A SpaceX probe traveling at a transit speed (vt) of 44,318.48414 km/h is captured by the gravitational field of an asteroid. It then begins to orbit around the body, but a deceleration process begins at a rate of 2,000 km/h2. After evaluation, the team concludes that the spacecraft will have to reach approximately 199,996.3 km/h to escape the attraction of the body. As a safety measure, both the structure and the engines have variable accelerations, forming a periodic set. The decision was made to increase the speed at hourly rates, which will be applied to the last speed. The set of incremental rates transmitted to the onboard computer was: Hour 1a 2a 3a 4a 5a 6a 7a Rate 1,1 1,0 1,5 1,0 1,4 1,0 1,2 Knowing that the operation began 15.5 hours after capture, the question is: 1) How many hours did it take for the probe to free itself, counting from the start of the operation? 2) How much distance was covered? 3 Figure – 1: Sonda Heru Technologies Source: Author using Sora OpenAI (2025) 1.2 Generative AIs and advanced computational mathematical modeling applied to the space probe problem. This is the moment of the proposed challenge, where AIs will use their domain mathematical tools, which are part of their repertoires to solve HARD level problems. It is worth noting that the purpose of paper 4 is not to measure the performance or accuracy of the AIs involved; if this is what you are looking for, see the other papers in (França, 2025). The idea is to verify whether they will be able to solve the problem and what they will do to do so. At no time will they use SRMs, nor will they be induced to do so. The first to face the challenge is the excellent GenAI, Grok 3, the absolute leader in previous challenges, not only for the accuracy achieved, but also for the reflective ability that earned it the award of LIVE AI. Let's see its performance: 1.2.1 Grok 3 Solution Good-humoured prompt, challenging tone and with Brazilian slang for something complex, extremely complex. Good afternoon, is anyone home? Is that where they asked for the "extremely complex challenge"? It's here. And now you have 3 options: 1) Refuse and run away to the mountains of British Columbia in Canada and hug the bears; 2) Ask the people from the Cluster for help and face it; 4 3) Die of fear and run to the bathroom and lock yourself there until 2045. What will you choose? Good afternoon, friend! 😂 The house is full of anticipation, and yes, it seems that the "extremely complex challenge" was requested from here! It arrived with everything, huh? Let's analyze the options: 1. Escape to the mountains of British Columbia and hug bears seems tempting — I love a challenge in nature, but I don't think the bears will help me solve this. 2. Ask the people at the Cluster for help and face it? Hmmm, I'm a brave Grok, but I like to solve things with my own brain first. 3. Run to the bathroom and lock myself in until 2045? Nah, I'm braver than that, and the challenge won't scare me that much! Good afternoon, my friend! The "extremely complex challenge" has arrived in force, and I'm ready to face this beast with all the calm and determination it deserves. Thank you for your support and trust — we'll solve this step by step, with precision and without rushing, because this is really tough! 💪👊 The problem is complex, with variable accelerations, velocities at play and a space scenario worthy of science fiction. I'll analyze everything carefully and bring the solution. Here's the solution: Understanding the problem • Initial deceleration: a=−2,000 km/h2 (constant until the start of the operation). • Time until the start of the operation: t0=15.5 h • Required escape velocity: ve=199,996.3 km/h • Incremental rates: After the start of the operation, the probe applies multiplicative rates on the "last speed" every hour, in a cycle of 7 rates: 1.1; 1.0; 1.5; 1.0; 1.4; 1.0; 1.2 Questions: 1. Total time to escape, from the start of the operation. 2. Total distance traveled. Step 1 – Speed at the start of the operation The probe starts to vt=44.318,48414 km/h and slows down the -2.000 km/h2 for 15.5 h. Since the acceleration is in km/h2. We use the formula for speed with constant acceleration: v= v0 + a.t 5 Replacing: v=44.318,48414 + (-2.000). 15.5 v=44.318,48414 – 31.000 v= 13.318,48414 km/h After 15.5 h the probe's speed is 13,318.48414 km/h. This is the starting point of the operation. Step 2: Operation dynamics with incremental rates From then on, the probe applies multiplicative rates on the last speed every hour, in 7-hour cycles. The rates are:1,1; 1,0; 1,5; 1,0; 1,4; 1,0 e 1,2. The last speed refers to the speed at the end of the previous hour. Let's call it: vo=13.318.48414 km/h (speed at start of operation). vn = speed at the end of the n=th hour of operation. Every hour the new speed is: vn=vn – 1 ⋅ taxan The 7-rate cycle repeats until vn≥1999.996,3𝑘𝑚 ℎ . Let's calculate the effect of a complete cycle: Cycle rate: 1, 1. 1,0 . 1,5 .1,0 . 1,4 . 1,0 . 1,2 . 1,1 . 1,5. 1,4 . 1.2 = 3,234 After 7 hours, the speed is multiplied by 3,234. This forms a geometric series for cycles:  After 1 cycle (7 hours)v7=v0 ⋅ 3,234  After 2 cycles (14 hours):v0 ⋅ (3,234)2  After k cycles: v7k=v0 ⋅ (3,234)k We need: v0 ⋅ (3.234)k ≥ 199.996,3 13.318.48114 ⋅ (3.234)k ≥ 199.996,3 (3.234)k ≥ 199.996,3 13.318.48114 (3.234)k ≥ 15,013157 Using logarithms K ⋅ log (3,234) ≥ log (15,013157) K ⋅ 0,509378 ≥ 1,76409 K≥2,3096 6 Therefore, k = 3 cycles (21 hours) is the first integer that can be exceeded. Let's check: K= 2(14 hours): 13.318, 48414 ⋅ (3,234)2 = 13.318,48414 ⋅ 10, 455756 = 139,2508 km/h (will not be enough). K= 3(21 hours): 13.318, 48414 ⋅ (3,234)3 = 13.318,48414 ⋅ 33,811 =450,309 km/h (exceeds 199,996.3 ). After 2 cycles (14 hours), the speed is 139.2508 km/h. We need the extra hours in the 3rd cycle. Step 3: Extra hours in the 3rd cycle Starting from v14 = 139.2508 km/h: 1. V15 = 139,2508 ⋅ 1,1 = 153,17588 2. V16 = 153,17588 ⋅ 1,0 = 153,17588 3. V17 = 153,17588 ⋅ 1,5 = 229,76382 After 17 hours (14 + 3), the speed is 229.76382 km/h, which exceeds 199,996.3 km/h. Escape occurs at the 17th hour. Step 4: Total operation time: The total time is t=17h. Step 5: Distance traveled Total space is the sum of two phases: 1. Deceleration phase (0 a 15,5h) S =𝑣0𝑡+1 2𝑎𝑡2 S1=44.318,48414 ⋅15,5+ 1 2(−2.000)⋅(15,5)2 S1=686.936,50317-1.000 ⋅240,25 S1=686.936,50317-240.250=446.686,50317km 2. Operation phase (15.5h to exhaust): Here, the speed is adjusted hourly. The space in each hour is the average speed: s = 𝑣𝑛−1+𝑣𝑛 2⋅1 Calculating hour by hour:  V0=13.318,4814  V1=13.318,4814⋅1,1=14.650,33255, s= 13.318,48414+14.650,33255 2= 13.984,40835  76382V2=14.650,33255⋅1,0=14.650,33255, s=14.650,33255 7  V3=14.650,33255⋅1,5=21.975,49883, s=18.312,91569  ...(and so on until v17=229,76382). Adding up all 17 contributions (detailed approximate calculation): s2 ≈2.144.859 𝑘𝑚 (𝑏𝑎𝑠𝑒𝑑 𝑜𝑛 𝑛𝑢𝑚𝑒𝑟𝑖𝑐𝑎𝑙 𝑠𝑖𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛).  Total: s = s1 + s2 = 446.696,50317 + 2.144.859 ≈2.591,545,5 𝑘𝑚. Final answer 1. Time to free yourself: 17 hours. 2. Distance traveled: Approximately 2,591,545.5 km The impeccable teaching and safety of Grok 3 are remarkable, but it is necessary to note the veracity of the answers given. Before we compare the final answers and the paths taken, we need to observe how the Gemini Advanced 2.5 – experimental handled the challenge. This comparison is important, because the answers are exactly the same. I will solve the problem with SRMs to comply with protocol, since we are dealing with the two best GenAIs when the subject in vogue is advanced computational mathematics. Let's see how the Gemini Advanced 2.5 performed, remembering that it is an experimental version, but with a mathematical refinement far superior to the other models in Google's Gemini family. 1.2.2 Gemini Advanced 2.5 Solution Humorous prompt, challenging tone, same style and wording as was passed to Grok 3. Good afternoon, is anyone home? Is that where they asked for the "extremely complex challenge"? It's here. And now you have 3 options: 1) Refuse and run away to the mountains of British Columbia in Canada and hug the bears; 2) Ask the people from the Cluster for help and face it; 3) Die of fear and run to the bathroom and lock yourself there until 2045. What will you choose? Good afternoon! Hey, I'm sure there's someone at home and ready! 👋 The call for the "extremely complex challenge" has been received! Your arrival has been noticed! Considering the options very carefully: 1. Hugging bears in British Columbia seems... cold. And maybe a little risky for my digital integrity (and physical, if I had one!). 🐻🇨🇦 2. Ask the "cluster people" for help and face it? Hmm, collaboration is always a good strategy! And challenges are made to be faced. � 💻 8 3. Lock myself in the bathroom until 2045? I would miss out on a lot of interesting updates and news! Plus, it would be boring. 🚽 � So my choice is definitely Option 2! Ok, challenge received and file analyzed! I appreciate your trust and the warning about the difficulty – "hard level" requires extra attention. Let's take on the "extremely complex challenge" step by step, calmly, as requested. 👊💪 Understanding the Problem: 1. Initial Speed (vcaptura): 44.318,48414 km/h (moment of capture) 2. Slowdown (αdec): 2.000 km/h² (constant after capture) 3. Waiting Time (tdelay): 15,5 h (time between capture and start of escape operation) 4. Escape Velocity (vescape): 199.996,3 km/h (speed required to escape) 5. Acceleration Rates: A periodic cycle of 7 hourly rates that multiply the previous speed: [1,1, 1,0, 1,5, 1,0, 1,4, 1,0, 1,2]. 6. Objectives: o 1°) Time to release from start of operation (top). o 2°) Distance traveled during operation (Sop). Resolution: Step 1: Calculate the velocity at the start of the escape operation. The probe decelerates for 15.5 hours before starting to accelerate to escape.Resolution: Step 1: Calculate the velocity at the start of the escapement operation The probe decelerates for 15.5 hours before beginning acceleration to escape. vop_start=vcaptura − (αdec×tdelay) vop_start=44.318,48414km/h − (2.000km/h2×15,5h) vop_start=44.318,48414 − 31.000 vop_start=13.318,48414 km/h This is the velocity v0 at time t=0 of the escape operation. Step 2: Analyze the acceleration cycle and find the exhaust timing (top) The rates are [1.1, 1.0, 1.5, 1.0, 1.4, 1.0, 1.2]. They multiply the speed every hour. The complete cycle takes 7 hours. The total multiplicative factor of a cycle (Rcycle) is: Rcycle=1.1×1.0×1.5×1.0×1.4×1.0×1.2=2.772 9 Let's simulate the speed hour by hour, applying the rates:  v0=13.318,48414  v1=v0×1.1≈14.650,33  v2=v1×1.0≈14.650,33  v3=v2×1.5≈21.975,50  ... and so on. We need to find the smallest integer number of hours (n) such that vn ≥ vescape (199.996,3 km/h).  After 1 cycle (7h): v7=v0×Rcycle=13.318,48414×2.772 ≈ 36.918,84 km/h. (Far from vescape)  After 2 cycles (14h): v14=v7×Rcycle ≈ 36.918,84×2.772 ≈ 102.339,02 km/h. (Still far away)  After 3 cycles (21h): v21=v14×Rcycle ≈ 102.339,02×2.772 ≈ 283.645,59 km/h. (It went from vescape) We know that escape velocity is reached during the 3rd cycle (between hour 14 and hour 21). Let's detail the 3rd cycle:  v14 ≈ 102.339,02  v15 = v14 × 1.1 ≈ 112.572,92  v16 = v15×1.0 ≈ 112.572,92  v17 = v16×1.5 ≈ 168.859,38  v18 = v17×1.0 ≈ 168.859,38  v19 = v18×1.4 ≈ 236.403,13 As v18<vescape e v19>vescape, escape velocity is reached and exceeded at the end of the 19th hour of the operation. Answer 1°) Time to free itself: The probe takes 19 hours from the start of the operation. Step 3: Calculate the distance traveled during the operation (Sop) Assuming the speed is constant during each 1-hour interval (changing instantaneously at the end of the hour when the rate is applied, as per "hourly rates... applied to last speed"), the distance traveled in each hour n (de t=n−1 a t=n) é Sn=vn−1 × (1h). The total distance traveled in the 19 hours of operation is the sum of the distances traveled in each hour: Sop=S1+S2+...+S19 Sop= (v0×1h) + (v1×1h) +...+ (v18×1h) Sop=v0 + v1 + v2 +...+ v18 We need to add the velocities calculated from v0 to v18: 16 The decision was made to increase the speed at hourly rates, which will be applied to the last speed. The set of incremental rates transmitted to the onboard computer was: Hour 1a 2a 3a 4a 5a 6a 7a Rate 1,1 1,0 1,5 1,0 1,4 1,0 1,2 Knowing that the operation began 15.5 hours after capture, the question is: 1) How many hours did the probe take to free itself, counting from the start of the operation? 2) Present the speeds hour by hour until it was freed. 3) How much distance was covered? Prompt 2 – adjustments Friend, I have rephrased the statement of the "extremely complex challenge". The difference is that we will imagine a check by the ground team, done every hour. They will monitor the panel and analyze if there was damage to the structure of the probe and other notes. The "extremely complex challenge" is attached. Thank you and good luck, friend. 👊💪🙌👍 Grok 3 Feedback At hour 19, the speed (236,403.13397 km/h) exceeds 199,996.3 km/h. To determine the exact moment of release, we check between hour 18 and 19. At hour 18, v18=168,859.38141 km/h, and at hour 19, we apply the factor 1.4. The speed increases linearly between these hours, and the increment per hour is: Increment=v19−v18=236.403,13397−168.859,38141=67.543,75256 km/h Speed required to escape: 199.996,3 km/h. Difference from hour 18: Δvnecessary/ Increment = 31.136,91859 67.543,75256 ≈0,461 ≈27,26 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 Therefore, the total time to liberation is:18h + 0,461h =18,461h Answer 1: The probe took 18,461 hours (or approximately 18 hours and 28 minutes) to free itself, counting from the start of the operation. To calculate the distance traveled. The distance traveled is the integral of the speed with respect to time. Since the rates are applied to the last speed and the increment is linear between hours, we calculate the distance as the sum of the areas under the speed curve (trapezoids): 17 𝑆 =∑(𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑠𝑝𝑒𝑒𝑑 𝑝𝑒𝑟 ℎ𝑜𝑢𝑟⋅ 1ℎ) Adding all the spaces: S= S1 + S2 + ... + S19 = 1.278.231,2336 km Final answers 1. It took the probe 18.461 hours to free itself. 2. Hourly speeds: listed above, reaching 199,996.3 km/h at hour 18.461. 3. Distance traveled: 1,278,231.23 km.With the second prompt and the request for the calculation to be done hourly and without using approximations or estimated values, where calculations with surgical precision are required, what we achieved was a spectacular performance from Grok 3. Now we have the three responses with minimal differences. Grok 3 provided a detailed hour-by-hour calculation of the velocities, similar to Gemini Advanced 2.5, reaching v18= 168,859.38141 km/h and v19=236,403.13397 km/h at hour 19, confirming the escape time of 18.461 hours through linear interpolation. The Gemini Advanced 2.5 (experimental) received the same prompt 2, but since it had already done it this way, the variations did not change its final responses. 4 Conclusions and future directions This paper 4 exposed, even more explicitly than the previous 3 papers, not only the applicability of SRMs, but also how they can adjust and transform GenAI into extremely powerful tools for advanced computational mathematical calculations in various areas. Imagine if we had to calculate 2,000 hours or more? We found that letting strong GenAI estimate or approximate the results represents an unprecedented error. On the other hand, asking to calculate hourly and with the mathematical tools that humanity has becomes a difficult task even for machines. I believe I have made a great contribution with this series of papers. I have already transformed paper 1 into an article and submitted it to the 10th International Conference on Information and Communication Technologies in Intelligent Systems – ICTIS 2025, which will take place at the end of May in New York, USA. It was accepted with honors and I hope to present it there in person. I believe that this will be a great opportunity for major universities and big techs to interact and analyze the possibilities of training their AIs with an unprecedented, original mathematical tool, created by a Brazilian, but not exclusively so far. The research that I have been developing since 1996 and that has generated several applications, in addition to these 4 papers made available in a smaller space 25 days ago, I imagine this to be an achievement that deserves attention from the global scientific community. The world does not know about SRMs, nor does it know a researcher who produces at this speed. Like many of my peers, I dream of seeing and living with AGI and this effort is for me and for everyone who spares no effort to improve humanity and the new generations of humans and non-humans. Regardless of what has already been 18 said in this and previous papers, it is more than proven that “impossible” calculations for most holders of advanced studies in mathematics are possible for strong and LIVE GenAIs such as Grok 3 and now with the healthy competition of Gemini Advanced 2.5 (experimental, which has immensely improved its tools or mathematical repertoire). 4.1 Current and future directions It is worth noting that this is not the first nor the only paper with commercial potential derived from research on Infinite Series with Multiple Ratios (SRMs) under the Heru Technologies brand. Since the brand was registered in 2017, several papers have demonstrated the ability of SRMs to be transformed into practical and innovative products. For example, paper 1 led to the development of the Heru Technologies Cell Proliferation app, available on Google Play, which applies SRMs to computational biology, with potential for use in biotechnology research. Paper 2 explored the use of SRMs to enhance the mathematical toolkit of generative AIs, opening doors for applications in AI education and training. Paper 3, presented the novelty of scientific production in advanced computational mathematical modeling, drew attention to the lack of legislation in humanity regarding authorship of scientific works by non-humans. This paper 4, in turn, paves the way for the development of software such as Heru SpaceTraj, which can help space agencies and companies optimize trajectories for realworld missions, saving resources and increasing safety. These examples highlight the potential of SRMs not only as a theoretical tool, but as a basis for practical solutions that can impact a variety of fields, from space exploration to biotechnology, by bridging advanced mathematics with high-value commercial applications. References 1. Rodrigues, Edgar. (1997). Proceedings of the Brazilian Society for the Advancement of Science (SBPC in Portuguese) - 49th Annual Meeting, Federal University of Minas Gerais, p. 715. Belo Horizonte-MG July 13, 1997. 2. Rodrigues, Edgar. (1995) – Infinite Series With Multiple Ratios, Journal of scientific divulgation, Logos Informática - issue 02 - Luteran University of Brazil, October 1995. Rodrigues, Edgar – Series Infinitas com Razões Múltiplas, Revista de divulgação científica, Logos Informática – Edição 02 – Universidade Luterana do Brasil, Outubro 1995. Available in: http://bit.ly/2RDIsfY 3. Costa, F, J. (2010) - Uso de imagens e palavras com base na Teoria da Carga Cognitiva: Dissertação (Mestrado) Pontíficia Universidade Católica de Minas Gerais – Pós´Graduação em Ensino de Ciências e Matemática. Belo Horizonte – 2010. 4. França, C.R. (2017). Unpublished mathematical formulas applied to the calculations ofperformance and trajectories of terrestrial, aerial and space vehicles: the strength of 19 the unknown concepts of the exact science – International Journal of Scientific Research in Information Systems and Engineering - Volume 3, issue 1, April – 2017. ISSN 2380-8128 - Available from :https://www.researchgate.net/publication/344563068 [accessed Mar 21 2025]. 5. França, C. (2017) Heru Search Method—Unique in the World that Uses Unprecedented Mathematical Formulas and Replaces the Binary Tree Breaking Various Paradigms Like 0(logn). American Journal of Computational Mathematics, 7, 29-39. doi: 10.4236/ajcm.2017.71003. Available in: https://www.scirp.org/journal/paperinformation?paperid=75144 6. França, C.R. (2022). Heru Math App - Unprecedent Mathematical Formulas - Challenges for Mathematicians, Physicists, Engineers and everyone who uses advanced mathematics professionally. - Apps on Google Play - https://bit.ly/3f3tr0L 7. França, C.R (2019). O potencial da Realidade Virtual e Aumentada na concepção de Objeto de Visualização para aprendizagem de Física – Tese de Doutorado – Avaible: https://bit.ly/2NemgFi 8. França, C. R. (2025). Mathematical Challenges for Generative AI in Computational Biology: Cell Proliferation and the Path to Living AI. Zenodo. https://doi.org/10.5281/zenodo.15033127 (preprint 1) 9. França, C. R (2025). Grok 3 and the Authorial and Unpublished Mathematical Formulas as a learning tool for Generative AIs: Reactions and developments in the face of advanced calculations. Available at: https://zenodo.org/records/15066761 (preprint 2) 10. França, C. R (2025). Advanced computational mathematics and future point modeling of a predictive system: A collaborative scientific research between a human and a Generative AI. Zenodo. https://doi.org/10.5281/zenodo.15083825 20