The Mech Touch Chapter 3737: Cubed Root Relationship

Chapter 3737: Cubed Root Relationship

Suffice to say, the Golden Skull Alliance and more specifically the Larkinson Clan experienced a long and exciting day.
No one expected that after the Minerva demonstrated its prowess, the brother to its pilot would break through under circumstances that had never taken place before.
An incredible amount of coincidences had to take place at once to make such an improbable event happen.
If Casella and Imon worked for any other organization, it would have been impossible for the Sentinel Commander to connect to her brother with an empowering bond at the time of his breakthrough.
It was clear to numerous people who possessed an understanding of expert pilots that a unique and possibly legendary breakthrough had occurred.
There was no way that Commander Casella remained unchanged after getting involved in her brother's transformation.
It was also unlikely that Venerable Imon Ingvar had broken through to become a normal expert pilot.
Right now, the Larkinson Clan's highest priority was checking up on the health of the two power siblings.
Both brother and sister had been carted off to the Dragon's Den where the most knowledgeable and experienced Lifer medical specialists examined their new patients.
Commander Casella Ingvar's condition was relatively mild. She had exhausted her willpower during the field test. Though she received an infusion of energy from her brother's breakthrough, she had exerted herself far too much during the last test.
"How is she, doc?" Ves asked as Lucky clung to his shoulder.
"Casella is doing okay." Director Ranya told her as they stood behind a transparent screen. "Aside from her mental exhaustion and signs of stress, there should be nothing wrong with her. She will require several days of rest in order to return to her peak."
Ves looked relieved. "That is great to hear. Have you also detected any foreign presences, significant changes in her brain structure, elevated energy signals or other anomalies?"
Ranya narrowed her eyes. "Are you suspecting that Casella has experienced greater changes?"
It was hard for Ves to miss the spiritual bond that had recently formed between brother and sister. The connection emerging from Casella's head went right through the bulkhead and off into the direction where Imon was currently being treated!
"I suggest you keep the Sentinel Commander under observation for a few more days." He mildly said. "I think it would be good for them both if you can put the Ingvars together. This breakthrough event has produced changes that we have never seen before and cannot fully understand."
Ranya looked thoughtful and nodded. "Hm, you are correct in that, though I believe that Casella will not experience any detrimental side effects from the incident. Her brother on the other hand is in a much more serious condition."
The two moved over to observing the treatment room where Imon's unconscious body was being subjected to a lot of intensive scanning.
"He looks in bad shape alright." Ves commented. "His breakthrough event was quite a doozy."
"It shouldn't be more violent than a typical breakthrough in battle, but…" Ranya hesitated. "The current scans show that his brain signals are more disordered than usual. What that exactly means, we do not know. Usually, expert pilots gain more physical resilience due to their changes that always helps with recovering from a stressful transformation. The concern we have right now is whether this normal healing process is enough for a newly-advanced expert pilot who has channeled substantially more power than usual."
Everything had a price. Commander Casella's act of empowering her brother during his breakthrough created both positive and negative repercussions.
Ves predicted that the former outweighed the latter, but that did not mean he should disregard the cost.
He briefly extended his spiritual senses to the unconscious expert pilot.
Imon's force of will was more frazzled and messy than normal. Though expert pilots ought to be strong enough to remedy this damage, it might take weeks or even months before Imon consolidated his new state!
"I think… we should pay more attention to healing and rehabilitating Venerable Imon." He told Dr. Ranya. "Take it slow with regards to his recovery. There is no hurry. It will take many months before we can develop an expert mech for him so he does not need to push himself to pilot mechs anytime soon."
"Very well. I will take that into consideration."
Ves waited for a time in order to see whether Imon was able to wake up soon. When Ranya indicated that he would remain out cold for at least several days, he gave up and headed back to the room where Casella was recovering.
"Meow."
Lucky flew to the tired-looking Sentinel Commander and tried to cheer her up by acting cute.
Casella slowly reached out and petted Lucky, but it was clear her heart was not in it. Her mind was a jumble as she thought about the implications of her brother's successful apotheosis.
"Well. We had a big day today." Ves spoke in a light tone. "Let's start with your expert mech. Now that you have finished a session with it and have time to reflect on your experiences, what do you think about your Minerva?"
"I do not have much to say beyond the feedback that I have already given before and during the field test. The Minerva is both powerful and fully capable of amplifying my strengths. I cannot wish for anything better, truly." She answered.
"How well do you think you can improve your Command Field going forward?"
"Hmm. I already have a sense that I can increase the range of my Command Field by growing my resonance strength. When I stretched it all the way out to 8 kilometers from my Minerva, I already felt that I was being limited by my lack of growth. I cannot tell you how much more powerful I have to become to extend that field to 16 kilometers, but it will not happen soon."
Ves nodded. "I wonder whether your growth in resonance strength will expand the range of your Command Field at a linear rate or progress in a cubed root relationship."
Casella frowned as she failed to understand this point. "What do you mean by that?"
"There are two possible models for range extension that I can think of. If the range of your Command Field only looks at distance from your expert mech, then growing twice as strong as an expert pilot will also multiply the radius by two. If you grow four times as powerful as now, the maximum range will be four times as much as now. Understood?"
"That is simple enough to understand. Does this mean that my Command Field can cover entire battlefields in space if I grow strong enough? If I can encompass enemy mechs in my range, maybe I can affect them in a detrimental way."
Casella already started to think about what she could do to debilitate any hostile mech that fell into her spider web!
"Let's not get too excited, commander. Your Command Field was fairly strong back then because all of those mech pilots voluntarily embraced your intervention. If they considered you their enemy, then you wouldn't be able to affect them that much with the dispersal of your strength over such a massive area."
"I see. Well, I will see for myself once we fight our next battle. What about that square root relationship you mentioned?"
"Ah, that's another possible model that describes the growth of your Command Field. If it is not based on range but rather on the volume of space that you can influence with your expert mech, then the increase in radius will not be as exaggerated. You can think of it this way."
Ves activated his comm and used it to draw a simple projected circle in the air. "Imagine this is a standard 25 cm diameter pizza."
He drew a larger circle around the first one.
"Imagine this is a 35 cm diameter pizza. Now, you don't need to be a mathematician to figure out that the bigger pizza exceeds the smaller one by 5 cm on both halves. A 10 cm difference in diameter does not sound like a massive difference, but the bigger pizza is actually twice as big as the smaller pizza!"
Casella became more attentive as she studied the two circles.
"In other words, the 35 cm pizza offers as much food as two 25 cm pizzas?"
Ves smiled. "Correct. Next time, don't get fooled by the sizes of the pizzas that you order in the future. Even a 5 cm difference in diameter can dump a lot more joules of energy in your stomach. Now, I didn't present you this story to teach you about pizzas. What I really wanted to do was to put you in the right perspective."
"What does this have to do with the range of my Command Field, then?"
"Your Command Field is a big pizza." Ves simply stated. "No, it's more than that. It's a big ball-shaped loaf of bread. Since we are working in three dimensions rather than two, you need to describe the expansion in volume in a cubed relationship."
Casella frowned and rubbed the side of her head with a pained expression.
"Meow?" Lucky looked concerned.
"I had a long day today. I am not like this normally, but can you please give me the short version of your explanation?"
Ves let out an exasperated breath. "Okay, imagine a balloon that is 5 cm across. If you want to increase its radius by 1 cm, you only need to pump in a small breath of air. Now consider a balloon that is 5 meters across. If you want to increase this big boy's radius by 1 cm, how much air do you think you need to blow inside?"
"…I am guessing a lot more air than with the smaller balloon."
"Correct. It is so much more that at some point the cost becomes too exorbitant. I am afraid that this will be the case with your Command Field as well. Instead of air, the substance that ‘inflates' your field is the true resonance that you are able to generate with your expert mech. There is only a finite amount of energy that you can generate, and the larger your Command Field, the more difficult it becomes to expand its radius by just 1 meter. This is because you need to blow a huge amount of ‘air' just to fill up all of the extra volume!"
Although the analogies made her feel woozy, Casella roughly understood the gist of his message. She would have been able to figure this out easily enough on her own if she was in her prime, but she should have been asleep a while ago considering how much she had exerted herself today!
"Let's talk about another aspect of your Command Field." Ves moved on. "Have you figured out how you can affect more mechs with your Commandeering ability? During the testing session, we noticed that you struggled to Commandeer more than a couple of hundred mechs. It would be nice if you can raise this number over time, but we aren't sure what is needed for that to happen."
Casella smiled. "That is easier. While I think I can do better when my resonance strength has grown, I already had a thought back then that it is a matter of exercise and proficiency. The more I become familiar with this operation and the more I exercise my mind and willpower with the Commandeering ability, I can probably affect more mechs over time, especially if I push my numbers limit on a regular basis. I would have to practice alongside many friendly mechs in reality for that to happen."
"Ah, that sounds good. I suspect that we can also improve this aspect of your ability by augmenting your multitasking capabilities. This is a risky operation, though. A botched implantation procedure can ruin your genetic aptitude or your qualifications as an expert pilot. The risks are far too great."
The sentinel Commander already understood this truth. "I will exercise the quantity limit of my Command Field the old fashioned way if that is the case. This should be the most natural and healthy way for me to strengthen my Command Field."
"I hope it will grow to encompass the entire Larkinson Army in the future."