4: Chapter 4: Anchoring to Reality, Brilliant Logic

Consciousness returned. Zhang Hao sat up abruptly from his dorm bed, heart pounding. The gloom of the Chamber of Oblivion, the tension of solving the puzzles, and especially that mysterious resonance during his Talent Awakening were still clearly etched in his mind.

He immediately looked at his shoulder—it was empty.

As expected, Silver Star remained in the Dream World. However, a sense of unprecedented close connection let him know clearly that he could summon it to his side whenever he wished.

He couldn't wait to pull up his personal panel. Looking at the 233 points of Dream Power and the brand-new [Logical Connection] Talent, he took a deep breath to suppress his excitement.

"It's time."

He focused his mind, guiding the meager but real [Dream Power] within his body toward the invisible "coordinate" bound to Silver Star, just as he had driven his power in the Dream World.

"Manifest!"

The moment the command was issued, he felt the Dream Power in his body drain like a floodgate opening, losing a full 100 points! Simultaneously, a speck of silver light emerged out of thin air in front of him, rapidly expanding and taking shape.

As the light dissipated, a small hamster covered in dark silver metallic scales, its eyes flashing with a calm blue light, stepped out as if from a ripple in water and landed lightly on his desk. It sniffed the air with its small nose, its synthesized electronic voice carrying a hint of imperceptible rawness, as if being started for the first time:

"Real World coordinates confirmed. Host Zhang Hao, [Dream Spirit] Silver Star has officially completed Real-world Anchoring."

It worked!

Zhang Hao excitedly reached out his finger, wanting to touch Silver Star. His fingertip felt a cold and hard metallic texture, along with a faint, electric-like energy pulse. This wasn't an illusion or a projection; it was a real, tangible existence! His guide, his partner, was standing right on his desk in the Real World!

"Recording Real World environmental parameters... Air composition, gravitational constant, physical rules stable... Confirmed differences from the Dream World." Silver Star turned its little head, quickly analyzing its surroundings. "Host, to maintain my basic existence in reality, it costs 1 point of Dream Power per hour. If high-intensity calculations, information scanning, or auxiliary abilities are used, consumption will increase."

One point per hour! Zhang Hao's heart tightened, and he quickly checked his panel: 133 points of Dream Power remaining. This meant he had to constantly enter the Dream World to complete Dream Tasks or find other ways to obtain Dream Power to maintain Silver Star's long-term existence.

Just then, the dormitory door was pushed open, and his roommate rushed in noisily: "Haozi! Hurry, hurry! The Abbess of Extinction is about to take attendance for 'Higher Mathematics II'! If we're late, our participation points will be docked!"

Only then did Zhang Hao realize he had class this morning. He hurriedly stuffed Silver Star into the front pocket of his hoodie (Silver Star cooperatively reduced its presence, looking like a small charm) and grabbed his backpack to run out.

In the classroom, Professor Li, known as "The Abbess of Extinction," stood at the podium with a stern face, explaining the complex content of Fourier Series expansion. Zhang Hao sat in the back row, trying to keep up, but the mental energy spent in the secret chamber last night (or rather, early this morning) hadn't fully recovered. Coupled with the excitement of just manifesting Silver Star, he was a bit distracted.

"...Therefore, for this non-periodic function, we need to perform an Odd Extension, and then..." Professor Li's eyes were like torches, scanning the drowsy students below, eventually fixing on Zhang Hao, whose gaze was a bit vacant.

"Zhang Hao!"

Zhang Hao jolted and stood up.

"Tell me, based on what I just said, how do you determine if a function needs an odd or Even Extension, and how do you perform it? What is the core point?" Professor Li's question carried a clear intent to test him; this was her usual "wake-up" tactic.

Surrounding students cast sympathetic or gloating looks. This material was already difficult, and Zhang Hao was clearly spacing out; he was likely in trouble.

Zhang Hao's mind went blank. Odd Extension? Even Extension? He vaguely remembered the concepts, but the specific criteria and steps eluded him in the moment of tension.

It's over... He could already see the tragic state of his participation points being docked.

Just then, Silver Star, in his hoodie pocket, transmitted a faint, cool stream of energy. Simultaneously, a line of tiny blue text, visible only to him, floated in front of his retina like subtitles:

"[Detected Host encountering difficulty in knowledge retrieval. Triggering auxiliary protocol. Parsing content based on recorded classroom audio...]"

It was Silver Star! It retained its analytical abilities even in reality!

Almost simultaneously, Zhang Hao felt as if a clear spring had been injected into his brain. His [Logical Connection] Talent was passively triggered in the Real World for the first time! Those originally blurred and scattered knowledge points were linked together as if by an invisible thread, rapidly constructing a clear logical chain in his mind.

Inspiration struck, his initial panic instantly subsided, and his eyes regained their calm. He cleared his throat and answered loudly:

"Professor Li, the core point in determining whether a function f(x) defined on [0, L] needs an odd or Even Extension, and how to perform it, lies in what kind of Fourier Series property we want to obtain."

His speaking pace was steady, and his points were clear:

"First, if we wish to expand it into a sine series (meaning the series contains only sine terms), we need to perform an Odd Extension on f(x). The extended function F(x) should be an odd function on [-L, L], satisfying F(-x) = -F(x). At this point, the key to the extension is to ensure the function is 0 at the origin (x=0) and exhibits odd symmetry throughout the symmetric interval."

"Second, if we wish to expand it into a cosine series (meaning the series contains only cosine terms), we need to perform an Even Extension on f(x). The extended function F(x) should be an even function on [-L, L], satisfying F(-x) = F(x). At this point, the key to the extension is to ensure the function's derivative at the origin (if it exists) is 0, or to handle singularities properly, and exhibit even symmetry throughout the symmetric interval."

"Therefore, the core point lies in the choice of the target series type, and constructing a periodic extension function with the corresponding parity on [-L, L] accordingly."

His answer was not only complete but even more refined and essential than the textbook's description. The classroom fell silent; even the students who intended to watch the joke were stunned.

Professor Li pushed up her glasses, a rare hint of surprise appearing on her stern face. She nodded: "A very good answer; you've grasped the key to the problem. Sit down."

Zhang Hao sat down, his heart still beating with excitement. He quietly pressed Silver Star in his hoodie pocket, feeling the cold metal shell and the subtle energy fluctuations.

At that moment, it wasn't just Silver Star providing information prompts; it was the [Logical Connection] Talent that allowed him to instantly understand the internal Logic of that information and organize it into perfect language!

He subconsciously checked his personal panel (he could mentally access it even in reality):

"[Dream Power Reserve]: 132/133 points (Silver Star's basic Real World existence consumption -1 point/hour)"

He had consumed one point of Dream Power, but it had perfectly resolved a crisis and verified the powerful effects of the Talent and Spirit in reality!

This was more than just answering a classroom question. It meant that with Silver Star and [Logical Connection], he would possess an advantage far beyond ordinary people in learning, analysis, and even any brain-taxing problems he might encounter in the future!

The path to transcendence wasn't just found in combat-heavy Dream Tasks. Knowledge, Logic, and wisdom were also Domains that could be strengthened! And all of this stemmed from the tiny metallic hamster he brought into reality and his own awakened Talent related to thought.

Zhang Hao took a deep breath and looked toward the podium. His eyes no longer held confusion or fear, but were filled with a novel desire to explore knowledge. He knew that from this moment on, his real life had completely changed its trajectory. And the starting point of it all was that seemingly ordinary night and the... unknown existence that created worlds in dreams.