Morning sunlight filtered through the formu-aligned windows of the Grand Formu Library, casting precise geometric patterns across the marble floor. Azaril had returned early, drawn by the discoveries of the previous day and eager to continue his research. This time, he came alone—Silvius had mentioned other business in the city that required his attention.

  The familiar weight of Magistra Illumina's tokens rested in Azaril's pocket as he navigated the concentric circles of shelves, heading toward the same restricted section they had explored yesterday. As he rounded a corner between towering stacks of mathematical treatises, something small and lithe darted across his path.

  A cat—or what appeared to be one at first gnce. Looking closer, Azaril noticed distinctive markings across its silver-gray fur, subtle patterns that shifted like living equations. Its eyes, an unusual amber-gold, held an intelligence that seemed to evaluate him with deliberate calcution.

  "Ah, you've met Theorema," came Magistra Illumina's voice from behind him.

  Azaril turned to find his mentor approaching, her robes today featuring complex formu patterns in silver thread against deep purple fabric.

  "Calcution Cat," she expined, nodding toward the feline that had now settled on a nearby reading table, tail curled in a perfect logarithmic spiral. "Quite rare, and notoriously selective about their human companions. This one has resided in the library for nearly twelve years."

  The cat's gaze remained fixed on Azaril, its head tilted slightly as if solving a complex equation.

  "Do they serve a particur purpose?" Azaril asked, recalling simir companion creatures in the demon realm that were bred for specific magical functions.

  "They're drawn to mathematical precision," Magistra Illumina expined, moving to stand beside him. "Some schors believe they can perceive formus directly—see the underlying patterns in reality that our most advanced mathematics can only approximate." She smiled slightly. "More practically, they possess an uncanny ability to locate specific texts, often guiding researchers to references they didn't know they needed."

  As if demonstrating this talent, Theorema suddenly leapt from the table and trotted purposefully down one of the narrow aisles. After a few steps, the cat paused, looking back at them expectantly.

  "It seems you've piqued her interest," Magistra Illumina observed. "A rare occurrence for a newcomer to the library."

  Intrigued, Azaril followed the cat, with Magistra Illumina a few steps behind. Theorema led them through a series of turns, deeper into the stacks than Azaril had ventured the previous day. They arrived at a section beled with formu symbols indicating historical chronologies.

  "Curious," Magistra Illumina murmured as Theorema sat pointedly before a specific shelf. "This contains pre-Codification records of mathematical anomalies. Not typically of interest to new schors."

  Theorema stood on her hind legs, batting at a particur volume bound in faded blue leather. Magistra Illumina carefully removed it, examining its spine with raised eyebrows.

  "'Observational Divergences in Cross-Realm Formu Applications,'" she read, handing the book to Azaril. "An obscure work documenting cases where standard formus produced unexpected results near realm boundaries."

  Azaril accepted the volume, noting the complex library seal indicating restricted circution. "Is this normally avaible to assistants?"

  "It would require special authorization," she admitted. "But since Theorema has selected it specifically for you..." She gnced at the cat, who had settled into a self-satisfied posture. "Let us call it an intuitive recommendation."

  The three of them relocated to a small alcove with a reading table. As Azaril opened the book, Theorema immediately jumped onto the table and positioned herself beside the text, occasionally batting a paw at specific passages as if highlighting them for attention.

  "Remarkable," Magistra Illumina observed. "In all my years, I've never seen her so engaged with a new library visitor."

  Azaril studied the text, which documented formu experiments conducted near the borders between the Human Empire and other realms. The researchers had discovered that certain mathematical principles behaved differently at these junctures—formus that worked fwlessly within imperial territories produced unpredictable results or failed entirely when tested too close to boundaries.

  "These findings contradict the fundamental assumption that mathematical ws are universal," Azaril noted, carefully phrasing his observation to avoid revealing his non-human perspective. "They suggest that each realm might operate under slightly different natural principles."

  "A controversial idea," Magistra Illumina agreed. "The orthodox position maintains that formu inconsistencies near boundaries result from contaminating influences rather than fundamental differences in mathematical reality."

  Theorema purred loudly, as if approving of this discussion, and nudged another passage with her paw. This section described experiments with "hybrid formus" that incorporated elements from multiple realm traditions, reportedly achieving effects impossible through standard imperial mathematics alone.

  "Most interesting," Magistra Illumina said, leaning closer to examine the text. "This research was discontinued after the Great Codification, when mixing formu traditions was formally prohibited as potentially destabilizing."

  As they continued studying the text, Theorema remained attentive, occasionally stretching across pages to indicate connections between seemingly unreted sections. The cat's behavior exhibited a pattern-recognition ability that seemed almost supernatural, revealing retionships in the material that might have taken days to discover otherwise.

  "She truly does understand the mathematics," Azaril remarked, watching Theorema precisely arrange herself to highlight a particur diagram.

  "Indeed." Magistra Illumina watched the cat with thoughtful eyes. "And she seems to have developed a particur interest in you."

  Throughout the morning, Theorema continued guiding their research, leading them to additional references reted to boundary phenomena and formu variations. The cat's selections followed a coherent intellectual thread, building an understanding of how mathematical principles might differ across realms—information directly relevant to Azaril's unique perspective but carefully disguised as theoretical academic interest.

  When midday arrived, bringing the gentle chiming of formu-tuned bells from the Schor's Tower, Magistra Illumina gathered the materials they had been examining.

  "I have teaching obligations this afternoon," she expined, carefully returning books to their proper pces. "But I believe Theorema would be content to continue assisting your research." She regarded the cat with an appraising look. "In fact, given her unprecedented interest, I wonder if you might consider a more permanent arrangement."

  "Permanent?" Azaril asked, surprised.

  "Calcution Cats choose their companions, not the reverse," Magistra Illumina expined. "They're not pets in any conventional sense. They form intellectual partnerships with those whose mathematical thinking resonates with their own perception of patterns." She smiled slightly. "Theorema has resided in the library since before my appointment as Magistra, showing occasional interest in schors but never attaching herself to anyone specifically. Until now, it seems."

  Theorema, as if understanding the conversation, rose and stretched deliberately before walking to Azaril and pressing against his leg, her fur momentarily dispying complex formu patterns.

  "I would be honored," Azaril replied carefully, "but surely such a valuable library resident—"

  "Would be wasted confined to these walls if she's found a mind she wishes to accompany." Magistra Illumina completed his thought. "The purpose of Calcution Cats is to further mathematical understanding. If Theorema believes your work warrants her assistance, who am I to interfere with such a selective creature's judgment?"

  The matter settled with surprising ease. After completing the necessary documentation—a simple formu contract acknowledging the cat's voluntary partnership—Azaril found himself with an unexpected companion. Theorema showed no hesitation about leaving the library, trotting alongside him as he navigated the Schor's District, occasionally darting ahead to indicate turns as if she knew exactly where they were headed.

  They arrived at the Schor's Quill just as afternoon shadows began to lengthen across the cobblestone streets. Marta, the innkeeper, raised her eyebrows at the sight of the distinctive feline.

  "A Calcution Cat? Those are rarer than honest tax collectors," she remarked. "And they normally never leave the Academy grounds."

  "She seems to have adopted me," Azaril expined, still somewhat bewildered by the day's development.

  "Well, she's welcome here," Marta replied. "Though I'd keep your door closed at night. Some would pay handsomely for such a companion, willingness notwithstanding."

  Upstairs, Azaril found Silvius waiting in their shared sitting room, examining a complex star chart spread across the table. He looked up as they entered, his silver eyes widening slightly at the sight of Theorema.

  "You've acquired an illustrious companion," he observed, setting aside his work. "Calcution Cats are highly selective. This is... quite exceptional."

  Theorema approached Silvius with deliberate steps, her tail held high. She stopped before him, her amber-gold eyes fixed on his silver ones in what appeared to be mutual assessment. After a long moment, the cat did something Azaril had not yet seen—she bowed her head slightly, a gesture almost like deference.

  "Even more interesting," Silvius murmured, gently extending a hand. When Theorema pressed her head against his fingers, he added, "She recognizes something in me as well."

  "Magistra Illumina said they perceive mathematical patterns directly," Azaril expined, watching the interaction with curiosity. "She guided our research with remarkable precision today, leading us to texts about formu variations near realm boundaries."

  "Did she indeed?" Silvius stroked Theorema's head, a thoughtful expression crossing his face. "These creatures often understand more than we realize. Their perception transcends the rigid categories humans use to organize knowledge."

  Theorema settled comfortably between them, occasionally gncing from Azaril to Silvius as if following their conversation.

  "Magistra Illumina seemed surprised by her interest in me," Azaril noted. "Apparently, Theorema has resided in the library for years without choosing a specific companion."

  "Perhaps she recognizes a mind unconstrained by traditional formu teachings," Silvius suggested. "Your unique perspective might resonate with her pattern-perception."

  The evening passed pleasantly as Azaril shared his research discoveries. Theorema remained attentive, occasionally repositioning herself to be closer to whatever text or diagram they discussed. When dinner arrived, the cat accepted a small portion with dignified appreciation, her table manners as precise as any imperial noble's.

  Later, as Azaril prepared for sleep, Theorema arranged herself at the foot of his bed, her form creating a perfect arc against the bnket. Just before extinguishing the formu-light, he noticed the cat's fur momentarily dispying patterns that resembled the demon script from his homend—a coincidence surely, but unsettling nonetheless.

  The following morning, Azaril woke to find Theorema sitting at his window, her silhouette outlined by early sunlight. As he stirred, she turned to regard him with those intelligent amber-gold eyes, then jumped down and moved to the door, clearly indicating her desire to continue their library work.

  Silvius joined them for breakfast, watching with amusement as Theorema selected morsels from a small pte provided by an impressed Marta.

  "I have matters to attend to at the Administrative Quarter today," Silvius said. "Perhaps you and your new companion can continue yesterday's research path?"

  "I'd pnned to," Azaril replied. "Theorema seems to have a specific direction in mind."

  Silvius nodded, his silver eyes briefly meeting the cat's amber-gold gaze in what almost seemed like communication. "I suspect she does. Calcution Cats don't make choices without purpose."

  When they parted ways after breakfast, Azaril heading toward the library with Theorema at his side, he reflected on how quickly his circumstances had evolved. From outcast prince to imperial assistant, now with a rare magical companion that somehow recognized something special in him—the pattern suggested possibilities he couldn't yet fully comprehend.

  At the library, they were granted immediate access to the restricted sections, Theorema's presence apparently serving as additional credentials. The cat led him confidently through the stacks, bypassing the previous day's materials and heading toward even older collections.

  By midmorning, Azaril found himself surrounded by ancient texts describing early interactions between human formu masters and practitioners of other magical traditions. Theorema had guided him to historical accounts that suggested the rigid separation between magical systems was retively recent—that for centuries after the formation of discrete realms, practitioners had continued to exchange knowledge and techniques.

  This research continued for several days, with Theorema consistently leading him to materials that challenged the orthodox imperial view of formu magic as an isoted, self-contained system. Each evening, he shared his findings with Silvius, whose insightful comments often connected disparate facts into compelling patterns.

  Throughout the week, Azaril and Theorema established a comfortable routine. Each morning they would arrive at the library, and the cat would systematically guide him through increasingly specialized sections, always seeming to know exactly which texts would most benefit his understanding.

  On several occasions, Theorema's guidance proved almost prescient. When Azaril was puzzling over a reference to ancient diplomatic exchanges between early human kingdoms and demon cns, the cat would disappear momentarily only to return and lead him to a previously unknown supplementary text that provided crucial context.

  "It's as if she understands not just what I'm reading, but what questions I'm forming in my mind," Azaril remarked to Magistra Illumina during one of her visits to check on his progress.

  "Calcution Cats perceive patterns beyond our comprehension," she replied, watching Theorema delicately turn a page with her paw. "Some schors believe they exist partially in a mathematical realm beyond physical reality, seeing connections we can only theorize."

  The cat's assistance extended beyond merely finding texts. When Azaril struggled with a particurly complex formu concept, Theorema would often arrange herself on the page in positions that highlighted retionships between symbols, sometimes creating living examples of the patterns being described through the subtle shifting markings in her fur.

  During a particurly intensive study session on cross-realm mathematical anomalies, Junior Guard Atticus passed by their alcove, giving a respectful nod to Azaril and a curious gnce at Theorema.

  "Remarkable creature," he commented. "I've worked here five years and never seen her take such interest in a schor. You must be doing something extraordinary to earn such attention."

  Azaril smiled politely. "Perhaps she simply appreciates a fresh perspective."

  "Perhaps," Atticus agreed, though his tone suggested mild skepticism. "Though the Chief Librarian has noted your unusually rapid progress through the restricted sections. Quite impressive for someone so new to the Imperial Academy."

  Theorema simply blinked at him, but her amber-gold eyes seemed to hold a knowing intelligence that transcended ordinary animal awareness.

  Later that afternoon, when Silvius returned to their quarters, Azaril described his day's research, expining the unusual guidance Theorema had provided. The cat sat between them, occasionally adding what seemed like deliberate emphasis to certain points through subtle movements or gentle paws on specific words in Azaril's notes.

  "She's proven remarkably helpful," Silvius commented, his silver eyes thoughtful. "Though perhaps not entirely surprising. Calcution Cats have always served as bridges between conventional knowledge and deeper patterns."

  "The texts she guided me to suggest potential connections between magical systems that modern schors consider entirely separate," Azaril expined. "Historical accounts of formu masters colborating with practitioners from other realms."

  Silvius nodded slowly. "Such knowledge challenges the comfortable isotion each kingdom prefers to maintain."

  Theorema chose that moment to stretch deliberately, her fur briefly dispying patterns that resembled mathematical formus—complex symbols that seemed to dance across her coat before settling back into normal feline markings.

  "She's been extraordinarily helpful," Azaril observed. "Almost as if she's deliberately guiding me toward specific discoveries."

  "Calcution Cats perceive patterns humans cannot," Silvius agreed. "Perhaps she recognizes in you someone capable of understanding connections others miss." His silver eyes met Azaril's with unusual intensity. "Someone who might appreciate knowledge that transcends the artificial boundaries between realms."

  That evening, as twilight settled over Aurelium, Azaril studied Theorema as she arranged herself precisely on the windowsill, her silhouette outlined against the deepening purple sky. In just a few days, this remarkable creature had led him to discoveries that challenged everything he'd been taught about the separation of realms and the isotion of different magical traditions.

  Like him, she seemed to exist somewhat outside the rigid categories others took for granted. Perhaps that expined her unprecedented interest in a disguised demon prince navigating the complexities of human society—she recognized a kindred spirit who didn't quite belong in the neat cssifications that ordered the Human Empire.

  As if sensing his thoughts, Theorema turned to regard him with those intelligent amber-gold eyes, then deliberately blinked once, slowly—a gesture of trust and connection that transcended species boundaries, just as his journey was teaching him to transcend the boundaries between realms.