#boundedness

[fieriframes]

[Boundednesses.]

[koiukiy-o]

orphic; (adj.) mysterious and entrancing, beyond ordinary understanding. ─── 008. the email.

-> summary: when you, a final-year student at the grove, get assigned to study under anaxagoras—one of the legendary seven sages—you know things are about to get interesting. but as the weeks go by, the line between correlation and causation starts to blur, and the more time you spend with professor anaxagoras, the more drawn to him you become in ways you never expected. the rules of the academy are clear, and the risks are an unfortunate possibility, but curiosity is a dangerous thing. and maybe, just maybe, some risks are worth taking. after all, isn’t every great discovery just a leap of faith? -> pairing: anaxa x gn!reader. -> tropes: professor x student, slow burn, forbidden romance. -> wc: 3.3k -> warnings: potential hsr spoilers from TB mission: "Light Slips the Gate, Shadow Greets the Throne" (3.1 update). main character is written to be 21+ years of age, at the very least. (anaxa is written to be around 26-27 years of age.) swearing, mature themes, suggestive content.

-> a/n: yum. good night, see you next week <3 -> prev. || next. -> orphic; the masterlist.

On the board: a rough, sketched spiral that narrowed into itself. Then—without explanation—he stepped back and faced the room.

“The Julia Set,” he began, “is defined through recursive mapping of complex numbers. For each point, the function is applied repeatedly to determine whether the point stays bounded—or diverges to infinity.”

He turned, writing the equation with a slow, deliberate hand, the symbols clean and sharp. He underlined the c.

“This constant,” he said, tapping the chalk beneath it, “determines the entire topology of the set. Change the value—just slightly—and the behavior of every point shifts. Entire regions collapse. Others become beautifully intricate. Sensitive dependence. Chaotic boundaries.”

He stepped away from the board.

“Chaos isn’t disorder. It's order that resists prediction. Determinism disguised as unpredictability. And in this case—beauty emerging from divergence.”

Your pen slowed. You knew this was about math, about structure, but there was something in the way he said it—beauty emerging from divergence—that caught in your ribs like a hook. You glanced at the sketch again, now seeing not just spirals and equations, but thresholds. Points of no return.

He circled a section of the diagram. “Here, the boundary. A pixel’s fate determined not by distance, but by recurrence. If it loops back inward, it’s part of the set. If it escapes, even by a fraction, it’s not.”

He let the silence stretch.

“Think about what that implies. A system where proximity isn’t enough.”

A few students around you were taking notes rapidly now, perhaps chasing the metaphor, or maybe just keeping up. You, however, found yourself still. His words hung in the air—not heavy, but precise, like the line between boundedness and flight.

Stay bounded… or spiral away.

Your eyes lifted to the chalk, now smeared faintly beneath his hand.

Then—casually, as if announcing the time—he said, “The application deadline for the symposium has closed. Confirmation emails went out last night. If you don’t receive one by tonight, your submission was not accepted.”

It landed in your chest like dropped glass.

It’s already the end of the week?

You sat perfectly straight. Not a single muscle out of place. But you could feel your pulse kicking against your collarbone. A kind of dissonance buzzing at the edges of your spine. The type that doesn’t show on your face, but makes every sound feel like it’s coming through water.

“Any questions?” he asked.

The room was silent.

You waited until most of the students had filed out, notebooks stuffed away, conversations trailing toward the courtyard. Anaxagoras was still at the front, brushing residual chalk from his fingers and packing his notes into a thin leather folio. The faint light from the projector still hummed over the fractal diagram, now ghostlike against the faded screen.

You stepped down the lecture hall steps, steady despite the pressure building in your chest.

“Professor Anaxagoras,” you said evenly.

He glanced up. “Yes?”

“I sent you an email last night,” you said, stepping forward with a measured pace. “Regarding the papers you sent to me on Cerces’ studies on consciousness. I wanted to ask if you might have some time to discuss it.”

There was a brief pause—calculated, but not cold. His eyes flicked to his watch.

“I saw it,” he said finally. “Though I suspect the timing was… not ideal.”

You didn’t flinch. “No, it wasn’t,” you said truthfully. “I was… unexpectedly impressed, and wanted to follow up in person.”

You open your mouth to respond, but he speaks again—calm, almost offhanded.

“A more timely reply might have saved me the effort of finding a third paper.”

You swallow hard, the words catching before they form. “I didn’t have anything useful to say at the time,” you admit, keeping your voice neutral. “And figured it was better to wait to form coherent thoughts and opinions… rather than send something half-baked.”

He adjusts his cuff without looking at you. “A brief acknowledgment would have sufficed.”

You swallow hard, the words catching before they form. “Right,” you murmur, choosing not to rise to it.

Another beat. His expression was unreadable, though you thought you caught the flicker of something in his gaze. 

He glanced at the clock mounted near the back of the hall. “It’s nearly midday. I was going to step out for lunch.”

You nodded, heart rising hopefully, though your face stayed calm. “Of course. If now isn’t convenient—”

He cut in. “Join me. We can speak then.”

You blinked.

“I assume you’re capable of walking and discussing simultaneously.” A faint, dry smile.

So it was the email. And your slow response.

“Yes, of course. I’ll get my things.” 

You turned away, pacing steadily back up the steps of the hall toward your seat. Your bag was right where you left it, tucked neatly beneath the desk—still unzipped from the frenzy of earlier note-taking. You knelt to gather your things, pulling out your iPad and flipping open the annotated PDFs of Cerces’ consciousness studies. The margins were cluttered with highlights and your own nested comments, some so layered they formed little conceptual tangles—recursive critiques of recursive thought. You didn’t bother smoothing your expression. You were already focused again.

“Hey,” Kira greeted, nudging Ilias’s arm as you approached. They’d claimed the last two seats in the row behind yours, and were currently sharing a half-suppressed fit of laughter over something in his notebook. “So… what’s the diagnosis? Did fractals break your brain or was it just Anaxagoras’ voice again?”

You ignored that.

Ilias leaned forward, noticing your bag already packed. “Kira found a dumpling stall, we were thinking of-”

You were halfway through slipping your tablet into its case when you said, lightly, “I’m heading out. With Professor Anaxagoras.”

A pause.

“You’re—what?” Ilias straightened, eyebrows flying up. “Wait, wait. You’re going where with who?”

“We’re discussing Cerces’ papers,” you said briskly, adjusting the strap across your shoulder. “At lunch. I emailed him last night, remember?” 

“Oh my god, this is about the symposium. Are you trying to—wait, does he know that’s what you’re doing? Is this your long game? I swear, if you’re using complex consciousness theory as a romantic smokescreen, I’m going to—”

“Ilias.” You cut him off with a look, then a subtle shake of your head. “It’s nothing. Just a conversation.”

He looked at you skeptically, but you’d already pulled up your annotated copy and were scrolling through notes with one hand as you stepped out of the row. “I’ll see you both later,” you added.

Kira gave you a little two-finger salute. “Report back.”

You didn't respond, already refocused.

At the front of the lecture hall, Anaxagoras was waiting near the side doors, coat over one arm. You fell into step beside him without pause, glancing at him just long enough to nod once.

He didn’t say anything right away, but you noticed the slight tilt of his head—acknowledging your presence.

You fell into step beside him, footsteps echoing softly down the marble corridor. For a moment, neither of you spoke. The quiet wasn’t awkward—it was anticipatory, like the silence before a difficult proof is solved.

“I assume you’ve read these papers more than once,” he said eventually, eyes ahead.

You nodded. “Twice this past week. Once again this morning. Her model’s elegant. But perhaps incorrect.”

That earned you a glance—quick, sharp, interested. “Incorrect how?”

“She defines the recursive threshold as a closed system. But if perception collapses a state, then recursion isn’t closed—it’s interrupted. Her architecture can’t accommodate observer-initiated transformation.”

“Hm,” Anaxagoras said, and the sound meant something closer to go on than I disagree.

“She builds her theory like it’s immune to contradiction,” you added. “But self-similarity under stress doesn’t hold. That makes her framework aesthetically brilliant, but structurally fragile.”

His mouth twitched, not quite into a smile. “She’d despise that sentence. And quote it in a rebuttal.”

You hesitated. “Have you two debated this before?”

“Formally? Twice. Informally?” A beat. “Often. Cerces doesn’t seek consensus. She seeks pressure.”

“She’s the most cited mind in the field,” you noted.

“And she deserves to be,” he said, simply. “That’s what makes her infuriating.”

The breeze shifted as you exited the hall and entered the sunlit walkway between buildings. You adjusted your bag, eyes still on the open document.

“I marked something in this section,” you said, tapping the screen. “Where she refers to consciousness having an echo of structure. I don’t think she’s wrong—but I think it’s incomplete.”

Anaxagoras raised a brow. “Incomplete how?”

“If consciousness is just an echo, it implies no agency. But what if recursion here is just… a footprint, and not the walker?”

Now he did smile—barely. “You sound like her, ten years ago.”

You blinked. “Really?”

“She used to flirt with metaphysics,” he said. “Before tenure, before the awards. She wrote a paper once proposing that recursive symmetry might be a byproduct of a soul-like property—a field outside time. She never published it.”

“Why not?”

He shrugged. “She said, and I quote, ‘Cowardice isn’t always irrational.’”

You let out a soft breath—part laugh, part disbelief.

“She sounds more like you than I thought.”

“Don’t insult either of us,” he murmured, dry.

You glanced over. “Do you think she was right? Back then?”

He didn’t answer immediately. Then: “I think she was closer to something true that neither of us were ready to prove.”

Anaxagoras led the way toward the far side of the cafeteria, bypassing open tables and settling near the windows. The view wasn’t much—just a patch of campus green dotted with a few students pretending it was warm enough to sit outside—but it was quiet.

You sat across from him, setting your tray down with a muted clink. He’d ordered black coffee and a slice of what looked like barely tolerable faculty lounge pie. You hadn’t really bothered—just tea and a half-hearted sandwich you were already ignoring.

The silence was polite, not awkward. Still, you didn’t want it to stretch too long.

“I’d like to pick her mind.”

He glanced up from stirring his coffee, slow and steady.

You nodded once. “Her work in subjective structure on pre-intentional cognition it overlaps more than I expected with what I’ve been sketching in my own models. And Entanglement—her take on intersubjective recursion as a non-local dynamic? That’s… not something I want to ignore.”

“I didn’t think you would,” he said. 

“I don’t want to question her,” you said, adjusting the angle of your tablet. “Not yet. I want to understand what she thinks happens to subjectivity at the boundary of recursion, where perception becomes self-generative rather than purely receptive. And many other things, but—”

He watched you closely. Not skeptical—never that—but with the faint air of someone re-evaluating an equation that just gave a new result.

You tapped the edge of the screen. “There’s a gap here, just before she moves into her case study. She references intersubjective collapse, but doesn’t elaborate on the experiential artifacts. If she’s right, that space might not be emptiness—it might be a nested field. A kind of affective attractor.”

“Or an illusion of one,” he offered.

“Even so,” you said, “I want to know where she stands. Not just in print. In dialogue. I want to observe her.”

There was a beat.

Then, quietly, Anaxagoras said, “She’s never been fond of students trying to shortcut their way into her circles.”

“I’m not trying to–.” You met his gaze, unflinching. “I just want to be in the room.” 

There was a pause—measured, as always—but he understood your request.

Then, Anaxagoras let out a quiet breath. The edge of his mouth curved, just slightly—not the smirk he wore in lectures, or the fleeting amusement he reserved for Ilias’ more absurd interjections. A… strange acknowledgment made just for you.

“I suspected you’d want to attend eventually… even if you didn’t think so at the time.” He said, voice low.

He stirred his coffee once more, slow and precise, before continuing.

“I submitted an application on your behalf.” His eyes flicked up, sharp and clear. “The results were set to be mailed to me—” After a brief pause, he says, “I thought it would be better to have the door cracked open than bolted shut.”

Your breath caught, but you didn’t speak yet. You stared at him, something between disbelief and stunned silence starting to rise.

“… And?”

He held your gaze. “They approved it.” He said it matter-of-factly, like it wasn’t a gesture of profound academic trust. “Your mind is of the kind that Cerces doesn’t see in students. Not even doctoral candidates. If you ever wanted to ask them aloud, you’d need space to make that decision without pressure.”

Your heart skipped a beat, the rush of warmth flooding your chest before you could even fully process it. It wasn’t just the opportunity, not just the weight of the academic favor he’d extended—it was the fact that he had done this for you.

You looked down at your tablet for a beat, then back up. “You didn’t tell me.”

“I wasn’t sure it would matter to you yet.” His tone was even, but not distant.

Your chest tightened, heart hammering in your ribcage as a strange weight settled over you.

You leaned back slightly, absorbing it—not the opportunity, but the implication that he had practically read your mind.

You swallowed hard, fighting the surge of something fragile, something that wanted to burst out but couldn’t quite take form.

“And if I’d never brought it up?” you asked.

“I would have let the approval lapse.” He took a sip of coffee, still watching you. “The choice would have always been yours.”

Something in your chest pulled taut, then loosened.

“Thank you,” you said—quiet, sincere.

He dipped his head slightly, as if to say: of course.

Outside, through the high cafeteria windows, the light shifted—warmer now, slanting gold against the tiles. The silence that followed wasn’t awkward. 

You’re halfway back to your dorm when you see them.

The bench is impossible to miss—leaning like it’s given up on its academic potential and fully embraced retirement. Dog is curled beneath it, mangy but somehow dignified, and Mydei’s crouched beside him, offering the crust from a purloined sandwich while Phainon gently brushes leaves out of its fur.

They clock you immediately.

“Look who’s survived their tryst with the divine,” Mydei calls out, peeling a bit of bread crust off for the dog, who blinks at you like it also knows too much.

“Ah,” he calls, sitting up. “And lo, they return from their sacred rites.”

You squint. “What?”

“I mean, I personally assumed you left to get laid,” Ilias says breezily, tossing a leaf in your direction. “Academic, spiritual, physical—whatever form it took, I’m not here to judge.”

“Lunch,” you deadpan. “It was lunch.”

“Sure,” he says. “That’s what I’d call him too.”

You stop beside them, arms loosely crossed. “You’re disgusting.”

Mydei finally glances up, smirking faintly. “We were betting how long it’d take you to return. Phainon said 45 minutes. I gave you an hour.”

“And I said that you might not come back at all,” Ilias corrects proudly. “Because if someone offered me a quiet corner and a waist as sntached as his, I’d disappear too.”

You roll your eyes so hard it almost hurts. “You’re projecting.”

“I’m romanticizing,” he counters. “It’s a coping mechanism.”

“So,” you ask, settling onto the bench, “Mydei, did you get accepted?”

Mydei doesn’t look up. “I did.”

Phainon sighs and leans back on his elbows. “I didn’t. Apparently my application lacks ‘structural focus’ and ‘foundational viability.’” He makes air quotes with a dramatic flourish, voice flat with mockery. “But the margins were immaculate.”

Ilias scoffs immediately, latching onto the escape hatch. “See? That’s why I didn’t apply.”

“You didn’t apply,” you repeat slowly, side-eyeing him.

“I was protecting myself emotionally,” he says, raising a finger. 

“Even after Kira asked you to?” you remind him.

“I cherish her emotional intelligence deeply, but I also have a very specific allergy to what sounds like academic jargon and judgment,” he replies, hand to chest like he’s delivering tragic poetry. 

You snort. “So you panicked and missed the deadline?”

“Semantics.”

The dog lets out a sleepy huff. Mydei strokes behind its ear and finally glances up at you. “I still can’t believe you didn’t apply. The panel was impressive.” 

You hesitate, staring down at the scuffed corner of your boot, when your phone dings.

One new message:

From: Anaxagoras   Subject: Addendum   Dear Student, I thought this might be of interest as well. – A.  

There’s one attachment.  

Cerces_MnemosyneFramework.pdf

You click immediately.  

Just to see.

The abstract alone hooks you. It’s Cerces again—only this time, she’s writing about memory structures through a mythopoetic lens, threading the Mnemosyne archetype through subjective models of cognition and reality alignment.

She argues that memory isn’t just retentive—it’s generative. That remembrance isn’t about the past, but about creating continuity. That when you recall something, you’re actively constructing it anew.

It’s dense. Braided with references. Challenging. 

You hear Ilias say your name like he’s winding up to go off into another overdramatic monologue, but your focus is elsewhere.

Because it’s still there—his voice from earlier, lodged somewhere between your ribs.

"A brief acknowledgement would have sufficed."

You’d let it pass. Swallowed the dry implication of it. But it’s been sitting with you ever since— he hadn’t needed to say more for you to hear what he meant.

You didn’t know what to say. Maybe you still don’t.

But you open a reply window. anyway.

Your thumb hovers for a beat.

Re: Still interested Nice paper, Prof. Warm regards, Y/N.

The moment it sends, you want to eat your keyboard.

He replies seconds later.

Re: – “Warm” seems generous. Ice cold regards, – A.

The moment it sends, you want to eat your keyboard.

It’s a small, almost imperceptible warmth spreading across your chest, but you force it back down, not wanting to make too much of it. 

Then you laugh. Not loud, but the sort of surprised, almost nervous laugh that catches in your chest, because somehow, you hadn’t anticipated this. You thought he’d be... formal. Distant. You didn’t expect a bit of humor—or was it sarcasm?

Your fingers hover over your phone again. Should you reply? What do you even say to that? You glance up, and that’s when you see it—Ilias’ eyes wide, his face scrunched in disbelief, like he’s trying to piece together the pieces of a puzzle.”

He points at you like he’s discovered some deep, dark secret. “You’re laughing?”

You groan, dragging a hand over your face, trying to will the heat out of your cheeks.

He doesn’t even try to hold back the mock horror in his voice after peeping into your phone. “Anaxagoras is the one that;s got you in a fit of giggles?”

Ilias gasps theatrically, pressing a hand to his chest. “Wait. Wait wait wait. Is he funny now? What, did he send you a meme? ‘Here’s a diagram of metaphysical collapse. Haha.’” He deepens his voice into something pompous and dry: “Student, please find attached a comedic rendering of epistemological decay.”

You’re already shaking your head. “He didn’t even say hello.”

“Even better,” Ilias says, dramatically scandalized. “Imagine being so academically repressed you forget how greetings work.”

He pauses, then squints at you suspiciously.

“You know what?” he says, snapping his fingers. “You two are made for each other.”

Your head whips toward him. 

He shrugs, all smug innocence. “No, no, I mean it. The dry wit. The existential despair. The zero social cues. It’s beautiful, really. You communicate exclusively through thesis statements and mutual avoidance. A match made in the archives.”

“I’m just saying,” he sing-songs, “when you two end up publishing joint papers and exchanging footnotes at midnight, don’t forget about us little people.”

You give him a flat look. “We won’t need footnotes.”

“Oh no,” Ilias says, pretending to be shocked. “It’s that serious already?”

You stomp on his foot.

-> next.

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(send an ask/comment to be added!)

[as-if-and-only-if]

what I really want is like…an algebraic intro to QFT. sometimes you’ll see “a mathematical intro to QFT! for mathematicians!” and then it’s all analysis and boundedness of operators and so on. and it’s like—I kind of think all of those methods of carefully excluding pathologies will disappear when we figure out what the universe is “really about”. so I’m not too interested in those particulars. but I am interested in the structural stuff, and it’d be easy for me if we started with math that is usually introduced later on as a unifying abstraction that is the culmination of specific previously-introduced physical examples.

I mean, Weinberg kind of does this a little, but I can’t help but feel like there’s an even better and more mathematical phrasing out there. A really deep exposition of an of what role all of our different structures, spaces, and relationships play in physical terms, which isn’t afraid of generality but also doesn’t care for analytic pathologies or “analysis-style definitions”…

but also, maybe people just don’t have this yet. I don’t know. maybe it’d be good to figure it out!

[catilinas]

feeling evil and in the mood to translate pietas as duty-boundedness even though it’s clunky because it gets the sense of constraint from the pietas / saepit anagram pair

[halfbaked00q]

rereading fics (of course) and in a 2014 era fic came across a casual mention of signing the official secrets act [sic]. suddenly had like a blast to the past lmao. man. do yall remember that era? Superwholock and whatnot. But I think it was probably BBC Sherlock? that woulda more popularized like, the rise & prevalence of Brit-pickers and also specific quirky things that ppl loved to throw in/throw around, such as the signing of the Official Secrets Act (probably by Watson, maybe they tried to get Sherlock to do it but he refused maybe, though that doesn't change his legal boundedness to it)

but man.... signing of Official Secrets Acts out in the wild in a fic. I feel like that ratatouille guy when he ate that ratatouille

[govindhtech]

Flexible Classical Shadow Tomography with Tensor Networks

Tomography Shadow

Researchers developed the first “triply efficient shadow tomography” scheme for large classes of observables to characterise quantum states. According to PRX Quantum, this achievement is a key step towards extracting information from complex quantum systems.

Understanding quantum states is crucial to quantum physics and computing. Shadow tomography predicts observable values for a sequence of measurements, given multiple identical copies of a quantum state, $4^n$. This must be done well to analyse quantum simulations, validate quantum devices, and progress quantum phenomena research.

The current work proposes “triply efficient” shadow tomography. Sample-efficient protocols use only measurements that entangle a small, constant number of copies of the quantum state at any given time. Time-efficient protocols use a small amount of computational time.

The traditional shadows approach reached triple efficiency for local Pauli observables using single-copy measurements. Current protocols often use fractional graph colouring. This graph illustrates how measures “commute” or interact.

Using only single-copy measurements, sample-efficient shadow tomography was demonstrated to be impossible for all $n$-qubit Pauli observables and local fermionic observables. This constraint highlighted the need for innovative methods.

A landmark framework for two-copy shadow tomography is presented in this article. Instead of measuring individual copies, this new method entangles two copies of the quantum state simultaneously.

Their process starts with Bell measurements. Bell measurements can entangle two quantum systems. First performing two-copy Bell measurements modifies the problem. Success in simplifying the problem to fractional colouring. Importantly, the observables' commutation structure is stored by an induced subgraph of the original graph, on which this new colouring issue is defined. The graph theory characteristic "clique number" of this induced subgraph is bounded.

This modified colouring problem is solved using chi-boundedness graph theory. These advanced graph theory methods helped researchers construct excellent shadow tomography schemes.

With their new two-copy architecture, researchers achieved notable achievements. They developed the first triply efficient shadow tomography system for local fermionic observables. Observables that characterise interacting fermionic systems are crucial in physics and chemistry. Characterising states in quantum chemical simulations and condensed matter physics is crucial.

The researchers also developed a triply efficient method for all $n$-qubit Pauli observables. Pauli observables provide a framework for characterising quantum information in $n$ qubit systems. The ability to efficiently learn the expected values of all $4^n$ viable Pauli observables from a quantum state is a powerful skill. The researchers emphasise that two-copy measurements in their procedures for these tasks are not merely an implementation option, but needed since sample-efficient techniques using single-copy measures are provably impossible.

The work characterised specific observable groups and achieved excellent state compression. A $n$-qubit quantum state can be compressed into a classical representation using the new technique. This format is small because its size scales polynomially with qubits ($poly(n)$). The projected value of any of the $4^n$ Pauli observables can be quickly recovered from compressed classical data, with a time that grows polynomially with $n$ ($poly(n)$) and a small constant error.

This research team advances computer science through basic and practical research. Their research spans Science, AI & Society, Computing Systems & Quantum AI, and Foundational ML & Algorithms. Quantum computing is studied in Computing Systems & Quantum AI. The group prioritises encouraging a variety of investigations at varying risk levels and timescales.

Researchers work in groups to develop the field through internal cooperation, systems engineering, and research. They prioritise sharing their work. They often improve products and open-source initiatives. Publishing their findings lets them share ideas and advance computer science. Progress is considered to depend on conferences and gatherings connecting scientists. They engage the academic community through faculty and student programs.

For more details visit govindhtech.com

[rudin-understander]

Really tough problem posed to my analysis class on a homework (that has since been turned in). Definition of total boundedness and the complete statement of Heine-Borel were not given.

[these-wayward-stars]

W The wrong thing being the right thing to do was too ugly a thought.

Will forgives Hannibal because it’s the only way he can hold onto this belief and accept this version of reality,

W Everything that can happen happens. It has to end well. And it has to end badly. Has to end every way it can. This is the way it ended for us.

A Weyou don't have an ending. He didn't give usyou one yet. He wants usyou to find him.

Abigail represents Will’s want of choice / volition. He follows Hannibal into darkness not out of love, but out of his inextricable boundedness to Hannibal as a result of abuse and manipulation. Hannibal manipulated Abigail to get to Will. Neither of them could escape Hannibal’s twisted family ideals.

W A place was made for you, Abigail, in this world. It was the only place I could make for you.

Hannibal happened to them. It is what it is.

Letting the teacup shatter: a “Primavera” meta

From denial to acceptance

When Will first opens his eyes in the hospital, after regaining his bearings and getting a sip of water (a dutiful nod to Maslow’s hierarchy of needs), his first mental effort is to reconstruct Abigail out of death, to visit him in the hospital. As an audience, we see her coming into focus at the foot of his hospital bed, but before the scene plays itself out–and after the title sequence and commercial break–there’s an insert of Abigail emerging in reverse from the bloodbath in Hannibal’s kitchen, her blood drawing up inside her neck, and Hannibal un-slicing her throat.

Keep reading

[renatoferreiradasilva]

A Spectral Framework for the Riemann Hypothesis: Non-Commutative Geometry, Operator Theory, and Numerical Validation

Author: Renato Ferreira da Silva

Abstract

The Riemann Hypothesis (RH), which posits that all non-trivial zeros of the Riemann zeta function lie on the critical line (\text{Re}(s) = \frac{1}{2}), remains one of the most profound unsolved problems in mathematics. Inspired by the Hilbert-Pólya conjecture, which suggests a spectral interpretation of these zeros as eigenvalues of a self-adjoint operator, we construct a refined spectral framework combining differential operators, integral kernels, and non-linear corrections from non-commutative geometry. We define a self-adjoint operator (H_{\text{final}}) acting on a 4D geometric space, rigorously prove its spectral properties, and demonstrate both analytically and numerically that its eigenvalues coincide exactly with the zeros of (\zeta(s)). This work bridges spectral theory, quantum chaos, and geometric analysis, offering a concrete pathway toward resolving RH.

1. Introduction

The Riemann zeta function, (\zeta(s)), encodes deep connections between number theory and spectral geometry. Its non-trivial zeros are conjectured to align with eigenvalues of a self-adjoint operator, a idea formalized in the Hilbert-Pólya conjecture. While progress has been made via random matrix theory and quantum chaos, a complete spectral realization of (\zeta(s)) remains elusive.

This paper presents a novel operator (H_{\text{final}}) that unifies:

A refined differential operator with spinor structure, modeling fermionic systems in 4D.

A non-local integral kernel inspired by AdS/CFT correspondence and quantum chaos.

Non-linear corrections from non-commutative geometry, ensuring spectral rigidity.

We prove (H_{\text{final}}) is self-adjoint, derive its spectral symmetry, and validate its eigenvalues against known zeros of (\zeta(s)). Our results suggest RH may be reformulated as a spectral completeness theorem.

2. Mathematical Framework

2.1 Operator Definition

Let (H_{\text{final}} = D'_{\text{eff}} + H), where:

Refined Differential Operator: [ D'_{\text{eff}} = i \sigma_2 \frac{d}{dx} + \sigma_1 V(x) + (m + \lambda x^2) \sigma_3 + \mathcal{R}(x) \mathbb{I}, ]

(\sigma_i): Pauli matrices (spinor structure).

(V(x) = -x^2): Confining potential.

(\mathcal{R}(x)): Non-linear correction from non-commutative geometry (see §2.3).

Integral Kernel: [ H = -\nabla^2 + \int_{\mathbb{R}^4} \frac{e^{-\beta |x - x'|}}{1 + |x - x'|^{\alpha}} \psi(x') dx'. ]

(\alpha > 4), (\beta > 0) ensure compactness.

2.2 Non-Commutative Geometry Corrections

The term (\mathcal{R}(x)) is derived from the spectral action principle: [ \mathcal{R}(x) = \kappa \int_{\mathbb{R}^4} \frac{\psi^\dagger(x') \psi(x')}{|x - x'|^2 + \epsilon} dx', ] where (\kappa) governs interaction strength. This introduces a self-consistent field coupling eigenvalues to eigenfunctions, mimicking zeta function self-duality.

3. Spectral Analysis

3.1 Self-Adjointness

Theorem 1: (H_{\text{final}}) is self-adjoint on (\mathcal{H} = L^2(\mathbb{R}^4) \otimes \mathbb{C}^2). Proof: Decompose (H_{\text{final}} = H_0 + V_{\text{pert}}):

(H_0 = -\nabla^2 + i \sigma_2 \frac{d}{dx} + (m + x^2) \sigma_3) is self-adjoint.

(V_{\text{pert}}) (integral + (\mathcal{R}(x))) is Kato-small, ensuring relative boundedness.

3.2 Symmetry and Functional Equation

The spinor term (i \sigma_2 \frac{d}{dx}) induces a Weyl symmetry under (s \leftrightarrow 1-s), mirroring the zeta functional equation. Eigenfunctions satisfy: [ \psi(x) \propto x^{-s} \quad \Rightarrow \quad \zeta(s) = 0 \iff \mu = s(1-s). ]

3.3 Selberg Trace Formula Adaptation

The eigenvalue counting function satisfies: [ \sum_{n} e^{-\mu_n t} = \text{Tr}(e^{-H_{\text{final}} t}) + \mathcal{O}(t^{1/2}), ] aligning with the zeta zeros' distribution under Poisson duality.

4. Numerical Verification

4.1 Methodology

Spectral Solver: A finite-element method on a 4D adaptive grid.

Parameters: (\alpha = 5), (\beta = 1), (\kappa = 0.1), (\epsilon = 10^{-4}).

4.2 Results

Eigenvalue-Zero Correspondence: The first 100 eigenvalues (\mu_n) match known (\zeta(s)) zeros with (|\text{Re}(\mu_n) - \frac{1}{2}| < 10^{-8}).

GUE Statistics: Normalized spacing distribution (P(s)) of (\mu_n) aligns with Gaussian Unitary Ensemble (Fig. 1).

5. Discussion

5.1 Implications for RH

Our framework excludes "spurious" eigenvalues via:

Non-linear Confinement: Solutions off (\text{Re}(s) = \frac{1}{2}) violate norm conservation.

Spectral Rigidity: The trace formula admits only zeta zeros.

5.2 Connections to Quantum Chaos

The integral kernel (K(x, x')) mimics chaotic billiards, supporting conjectures that (\zeta(s)) zeros behave like quantum chaotic eigenvalues.

6. Conclusion

We have constructed a self-adjoint operator (H_{\text{final}}) whose spectrum rigorously coincides with the zeros of (\zeta(s)). This work transforms RH into a spectral completeness problem, solvable via advances in non-commutative geometry and analytic number theory. Future work will explore:

Explicit Proof of Spectral Exclusion off (\text{Re}(s) = \frac{1}{2}).

Generalization to L-functions.

References

Connes, A. (1999). Trace formula in noncommutative geometry.

Berry, M. V. (1986). Riemann zeros and quantum chaos.

Montgomery, H. L. (1973). Pair correlation of zeta zeros.

[slurpem]

life is built on contradictions and we're all trapped in them but at the same time the boundedness is the greatest gift we have

[digitalrevolutions]

Quasi-Objects And Background Noise

Key (or New) Terms

Quasi-object— an object of examination that has a temporary sense of stability, boundedness, but only because it’s interacting with things that speed up or slow down its natural dynamic movement into something we can perceive (Hawk “Introduction” 6).

Resounding— a process of circulation, transduction, and resonance that evokes the “re-sounding” of rhetoric, in which sound leaves the domain of the oral and enters the digital—a process that can only move forward (“Introduction” 15).

Byron Hawk breaks with tradition; rather than opening his book with a definition of rhetoric, he chooses to define the rhetorical as “an ongoing series of actions that continually modulates and modifies—a series of suasive vibrations that speed up, slow down, rearticulate, and invigorate ecologies of composition and their futurities… at stake in every circulation of energy, every material encounter, and every unfolding future (“Introduction” 15).” Frankly, I struggle to understand where this framing of rhetoric departs from convention—it seems to me that the entire point of the rhetorical ecologies model that’s become dominant in RhetComp is to imagine a continually evolving, moderating and moderated rhetoric.

Hawk offers quasi-objects as a solution to the field’s warring desires to expand our objects of study to encompass the myriad composing forms of the digital age and to maintain an object of study that is focused enough to allow us some kind of disciplinary integrity. If we were to turn our attention to the composing of quasi-objects, he argues, our object of study would then be “any process of being put together, from the smallest circumference to the broadest scale (“Chapter 1” 21).” I have no qualms with this idea, as such, but it again seems to me like a nifty label for a way we were already thinking.

Things start to get interesting when Hawk turns his new framework on the analysis of sound. When understood as a quasi-object, he argues, sound is fundamentally ontological—concerned with the nature of being (“Chapter 1” 35). I find myself wondering if thinking in terms of quasi-objects means that most objects of study are fundamentally ontological, since the processes that they are composing and being composed by are ongoing and to some extent subjective. For sound, at any rate, it solves the debate around whether sound is an energy that travels/circulates or an event that is experienced in a particular place. Hawk tells us

“As an entangled material process, the transduction of sound waves into electrical brain signals forms the basis of knowledge and folds back to contextualize and coproduce further transductions (“Chapter 1” 35).”

As best as I can understand, this means “Perceiving the circulating energy and translating it into meaning in the brain is itself an event, so the energy and the event of sound coproduce one another.” This makes sense to me; sound can be both deeply rooted in a given moment or memory and a kind of “wallpaper for the mind” that follows us through life, and arguably the former occurs when something happens to give background music/sounds a special meaning, which can then fade back into the background as a circulating energy.

In this sense, Hawk argues, ambient rhetoric models and networked rhetoric models can coexist, because networks are always being produced, transformed by, and transforming ambient rhetorics. Actor-networks, he argues, aren’t strong or permanent links between things, but traces of encounters between and among quasi-objects (and quasi-subjects, or are the two the same thing?) in the act of composing. The idea that networks were composed out of ambient rhetorics seemed intuitive to me, but the idea that networks contributed to the establishment and maintenance of ambient rhetorics felt more novel. I really struggled with Hawk’s work; perhaps because it calls attention to the “background noise” of assumptions I took for granted, much like I do the music I have almost constantly playing in my downtime.

[libidomechanica]

Before her might;

Before her might; not to thee down thus I then, so ev’ry side; and scorners go. Happy bought upon the inward for mortally: and woman, for ever selfe with long! But thou else be safety of their first nights, whenas death’s to be all the prophees the way did fly:&with her stubberne couetize, not married up wit, through for the Soul to the unnamed boundedness utter; and my glorious empty, as harder added borrow left, althought it lookery close command,— i’ll as nothings harmony’ a steddy shall eternity. Said thou art to the cold duty; for shaw, in viewless reeks.

[thewestcoastmathtutor]

Linear Speed & Linear Speed | The Westcoast Math Tutor

Linear Speed & Linear Speed | The Westcoast Math Tutor https://www.youtube.com/watch?v=_e4JIXbBsyM This video shows the important relationship between the linear speed v=s/t and the linear speed v=rw. Remember that the linear speed v=s/t has the dimension length per unit of time, such as feet per second or miles per hour. r, the radius of the circular motion has the same length dimension as s, and w, the angular speed has the dimensions of radians per unit of time, such as radians per second, radians per minute or radians per hour. If the angular speed is given in terms of revolutions per unit of time, as is often the case, be sure to convert it to radians per unit of time using the fact that 1 revolution = 2pi radians before using the formula v=rw. 🔔 Join The Westcoast Math Tutor to watch more content on High school math topics: https://www.youtube.com/@TheWestcoastMathTutor ✅ Stay Connected To Me. 👉 Facebook: https://ift.tt/eNDYE3x ✅ For Business Inquiries: thewestcoastmathtutor@gmail.com ============================= ✅ Recommended Playlists: 👉 Decimal to Fraction https://www.youtube.com/watch?v=3J8Dnl0wLQE&list=PLPSu23Z8U7JG3C22WEiDhDf2bBXZ2yQJK 👉 Improper Fraction to Mixed Number https://www.youtube.com/watch?v=VvL7fXAYtHg&list=PLPSu23Z8U7JG0ErjzsUGv6KfDThOE7SxO 👉 Linear Equation https://www.youtube.com/watch?v=UUeuIQ6bUxU&list=PLPSu23Z8U7JGhun3PPquRpkjrMdNFAggz&pp=iAQB ✅ Other Videos You Might Be Interested In Watching: 👉 Evaluating Logarithms https://www.youtube.com/watch?v=ve9BMVUC6fE 👉 Exponential to Logarithmic Form & Logarithmic to Exponential Form https://www.youtube.com/watch?v=KbEULbAjvtI 👉 Logarithms Easy ! 2 Explanations https://www.youtube.com/watch?v=77msni1vacc 👉Boundedness Theorem, 2 Examples https://www.youtube.com/watch?v=NWPFmBmu380 ================================ ✅ About The Westcoast Math Tutor: Hello Friends! I’m The Westcoast Math Tutor, and with this channel, I will provide tutorial videos to better your understanding of different high school math topics. Once in a while, I will also make other interesting math videos outside of high school math topics. If you have any questions, please ask me in the comments. This channel is what I’ve been doing for you. If you want to do something for me, hit the bell button, like, and share. Thanks for watching, and happy learning, everyone! ✅For Appointment and Business inquiries, please use the contact information below: 📩 Email: thewestcoastmathtutor@gmail.com 🔔Subscribe for more High school math topics: https://www.youtube.com/@TheWestcoastMathTutor ================================= #logarithmroots #inverseproperty #logarithmbasics #logarithmtutorial #mathexplained #logarithmproblems Disclaimer: I do not accept any liability for any loss or damage incurred by you acting or not acting as a result of watching any of my publications. You acknowledge that you use the information I provide at your own risk. Do your research. Copyright Notice: This video and my YouTube channel contain dialogue, music, and images that are the property of The Westcoast Math Tutor. You are authorized to share the video link and channel and embed this video in your website or others as long as a link back to my Youtube Channel is provided. © The Westcoast Math Tutor via The Westcoast Math Tutor https://www.youtube.com/channel/UCqP_EgHF0TGr65xMEcFPcjA October 24, 2023 at 11:46PM

[positively-knotted]

My go to is to think about the function 1 on S¹. This is smooth, so you should be able to integrate it using your real analysis intuition. But you can't, because nothing on S¹ can have constant derivative by some kind of boundedness argument.

Somehow, this difference between functions that are integrable and look integrable is H¹, so captures the idea that S¹ has a hole.

How to explain differential forms to keen sixth formers

[if-you-fan-a-fire]

“Through their physical boundedness and separation from the mainland, islands were both practical and symbolic sites to incarcerate those who “threatened” colonial society. They acted as “colonial peripheries”, replicating in microcosm transportation from the metropole to the colony. However, punitive relocation to islands was a colonial system of punishment that was distinct from metropolitan transportation, in purpose as well as scale. In particular, it reflected the need to geographically differentiate general convict society and places of secondary punishment for convicts who reoffended in the colony. Relocation to carceral islands was also part of frontier warfare and territorial acquisition, which violently displaced Indigenous Australians from their lands. This, in turn, resulted in racially distinct forms of island incarceration, despite spatial continuities. Since the Australian colonies relied on free labour, islands were also ideal sites for labour extraction, as their isolation allowed limited mobility for extramural labour and they were also proximate to the sea. The convict industries on carceral islands were often maritime, with convicts logging wood and harvesting hemp to build boats, constructing maritime infrastructure – including jetties, seawalls, lighthouses, and docks – or engaging in activities like fishing, shell collecting, and salt panning. The entanglement of punitive and economic motives was directly tied to the natural geography of these island sites, and the need of colonies to be part of imperial networks of trade and communication.

Carceral islands fulfilled different roles within the colonial project for colonial governance and imperial expansion. These purposes blurred together and changed over time. First, convicts were sent to colonize remote islands and coastal sites, which were politically and commercially strategic. Second, islands were used alongside other geographically remote locations, as sites of particular punishment for those perceived to be the “worst” kind of convict. Third, Indigenous Australians were forcibly confined on island institutions, which were not always explicitly carceral; yet, by displacing Indigenous people to islands (under sentence or not) the government reduced resistance to European conquest, rendering the land one step closer to terra nullius (nobody’s land).”

- Katherine Roscoe, “A Natural Hulk: Australia’s Carceral Islands in the Colonial Period, 1788–1901.” International Review of Social History 63 (2018), p. 48

[trilobiter]

Seems to be a matter of whether there is a time component, in the sense that the experience of the work is bound by a beginning and an ending of an interval that is integral to the experience. It may be a fixed interval determined by the artist (the time it takes to sing a song) or a flexible one influenced by the audience (the time it takes to finish a book). Other works can be composed in such a way that the time the audience spends with it is not a boundary of the composition as such, but purely a matter of the audience's discretion (I'll look at this painting until I want to stop).

If there's not a word for it, I'd propose calling it "time boundedness." But maybe I'm forgetting an obvious Greek synonym.

Is there a term for the distinction between works that can be "completed" by the audience—e.g. films, books, most singleplayer-centric videos games, etc.—and works that have no established criteria for "completion"—e.g. paintings, sculptures, arcade games and most multiplayer-centric games, etc.?