News & Updates
The latest news and updates from companies in the WLTH portfolio.
SpaceX Sets Its $1.75tn Valuation with Fixed $135 Share Price in Bold, Unconventional IPO Push
In a highly unusual move that underscores Elon Musk's willingness to rewrite the traditional IPO playbook, SpaceX is planning to price its shares at a fixed $135 each, aiming to raise a record $75 billion and achieve a $1.75 trillion valuation, according to sources cited by Reuters. The rocket and satellite company intends to sell 555.6 million shares in what would be one of the largest public offerings in history. The roadshow begins on Thursday, with the debut tentatively scheduled for June 12 on the Nasdaq under the ticker "SPCX". Goldman Sachs, Morgan Stanley, BofA Securities, Citigroup, and J.P. Morgan are leading the underwriting syndicate. Unlike conventional IPOs, where companies set a price range and adjust based on investor feedback during bookbuilding, SpaceX is taking a "take-it-or-leave-it" approach. This fixed-price strategy reflects Musk's confidence in strong demand and the company's unique position in the market. "Musk is simply taking a 'take-it-or-leave-it' approach which works for his followers and is also sensible given the market conditions and the lack of comparables," Weiheng Chen, a senior partner at Wilson Sonsini, noted. Breaking Tradition on Multiple Fronts SpaceX is deviating from norms in several other ways. The offering is structured as all-primary, meaning all proceeds go directly to the company rather than allowing existing shareholders to sell shares. Musk himself will face a 366-day lock-up period on his holdings, signaling a long-term commitment to investors. The company is also planning to allocate up to 30% of the shares to retail investors -- an unusually large portion designed to tap into Musk's dedicated following. Proceeds will fund expansion of AI computing resources and the Starlink satellite constellation, two areas central to Musk's vision of building an interconnected future that spans Earth and beyond. At a $1.75 trillion valuation, SpaceX would trade at approximately 93.7 times its 2025 revenue of $18.67 billion. This is rich even by high-growth tech standards. For comparison, Rocket Lab trades at around 118 times revenue, Palantir at 81 times, and Tesla at roughly 17 times. Morningstar recently valued SpaceX at $780 billion, well below its current private-market valuation, with most of the value attributed to the profitable Starlink business. The company's broader pitch to investors, however, rests heavily on futuristic bets: Mars colonization missions, space-based AI data centers powered by solar energy, and other technologies that do not yet exist at commercial scale. SpaceX has tied a significant portion of its growth narrative to a potential $28.5 trillion addressable market in these emerging areas. Financially, the picture is mixed. Starlink remains the clear cash cow, driving most revenue, profits, and growth. However, the launch services and other segments continue to burn cash. In the first quarter, revenue rose to $4.69 billion from $4.07 billion a year earlier, but losses widened. For the full year 2025, SpaceX swung to a net loss of $4.94 billion from a profit the prior year. The governance structure is designed to preserve Musk's control. As with Tesla, SpaceX is implementing a dual-class share structure that will concentrate voting power in the hands of Musk and a small group of insiders. While this ensures strategic continuity, it may give some institutional investors pause regarding corporate governance and long-term accountability. A Catalyst for the Next Wave of Mega IPOs SpaceX's listing is expected to kick off a wave of massive public debuts. Together with anticipated IPOs from OpenAI and Anthropic, these three companies alone could add nearly $4 trillion in market capitalization to public markets, intensifying competition for investor capital in an already crowded tech sector. The offering comes after years of subdued large-cap IPO activity. Strong demand is widely anticipated, fueled by Musk's track record and retail enthusiasm, but execution risks remain high. Two of SpaceX's three main businesses are still unprofitable, and much of the valuation depends on unproven future technologies. Still, business leaders don't seem to be backing out. "When you're the most anticipated IPO ever, you can ask investors to adapt to your process rather than the other way around," former Bank of America executive Craig Coben observed. Increasingly, analysts are seeing SpaceX's bold approach, fixed pricing, heavy retail allocation, and strong founder control as a reflection of Musk's signature style: high conviction, minimal compromise, and a long-term horizon that extends far beyond traditional Wall Street timelines.
Dow Jones Top Company Headlines at 9 PM ET: Terms Revealed for SpaceX's Unconventional $75 Billion IPO | SoftBank ...
Terms Revealed for SpaceX's Unconventional $75 Billion IPO The company plans to sell shares at $135 apiece, eschewing the norm of setting a price range and incorporating investor feedback. ---- SoftBank CEO's Bad Bets Left Him in Despair. An AI Spree Has Him Back on Top. Masayoshi Son said that his Tokyo-based technology conglomerate would unleash at least $52 billion of investment in French data centers. ---- CrowdStrike Lifts Outlook After Swinging to First-Quarter Profit The cybersecurity company said it now expects annual recurring revenue of $6.53 billion to $6.56 billion for all of fiscal 2027, up from its previous target of $6.47 billion to $6.52 billion. ---- Quantinuum Prices IPO at $60 a Share. It's Slated to Go Public Thursday. Bullish investors see Quantinuum as the next superpower in quantum computing. Here's why. ---- Broadcom Shares Slide Despite Jump in Revenue on AI Chip Demand The semiconductor and software maker's shares slid after-hours as the report and guidance failed to live up to expectations among some investors. ---- Bill Ackman's Pershing Square Set to Make $600 Million on Universal Stake The hedge-fund firm first invested in the company in 2021 and has failed to clinch two proposed deals. ---- Citi Names New Head of Strategy as Jane Fraser Steers Bank Into 'Next Phase' Margo Pilic will be Citi's new head of strategy, M&A, and investor relations. ---- Real-Estate Giant Compass Under Antitrust Investigation in New York The state attorney general is probing a $1.6 billion acquisition of a rival brokerage firm. ---- PVH Cuts Outlook Citing Iran War Impact on Sales The owner of Tommy Hilfiger and Calvin Klein said it now expects full-year revenue to stay approximately flat, compared to previous guidance of a slight increase. ---- Hasbro's New AI Studio Looks to Bring Its Iconic Characters to Next-Generation Experiences Hasbro launched Sixth Wall, a new artificial-intelligence studio aiming to bring the toymaker's cast of characters into the new technological era. ---- Silicon Valley Stalwart Benchmark Breaks From Past, Embraces Mature Startups After a late-stage bet on Cerebras delivered big returns, Benchmark decided to raise its first-ever growth fund. ---- Sleep Number Prepares for Bankruptcy Filing to Address Debt Load The mattress maker is preparing to file for bankruptcy as it contends with a high debt burden and deteriorating financial performance, people familiar with the matter said. ---- Eli Lilly, Ascidian Sign $1.9 Billion Kidney-Disease Treatment Deal Eli Lilly signed a collaboration and licensing agreement worth up to $1.9 billion with Ascidian Therapeutics to research and develop kidney-disease treatments. ---- Medtronic Posts Higher Profit, Sales Medtronic said its profit and sales rose during its fiscal fourth quarter, reflecting steps the company has taken to improve business performance and drive growth. ---- Shoppers Are Spending More at Macy's as Turnaround Continues Macy's and its sister brands Bloomingdale's and Bluemercury have added higher-end products. Customers are snapping them up. ---- Zara Owner Shrugs Off Middle East Turmoil to Post Faster Sales Growth Sales at the Spanish fashion retailer-which houses Zara, Massimo Dutti and Bershka-rose 11.5% compared with the prior year. (END) Dow Jones Newswires June 03, 2026 21:15 ET (01:15 GMT) Copyright (c) 2026 Dow Jones & Company, Inc.
Anthropic Confidentially Files for IPO, Beating OpenAI
AI giant Anthropic has confidentially filed for a U.S. initial public offering, the company said on Monday, edging ahead of rival OpenAI in a closely watched race to reach public markets. The move sets up an early test of whether investor appetite for artificial intelligence, which has fueled lofty private valuations and talk of potential trillion-dollar listings, will hold up under public scrutiny, and which company gets to set the template for how the fast-growing sector is valued. Anthropic, which makes agentic coding assistant Claude Code, did not disclose the size or the terms of the offering. It last raised $65 billion at a post-money valuation of $965 billion in late May, putting it ahead of OpenAI. The listing would represent one of the most consequential stock market debuts in years, potentially reshaping benchmark indexes, investor flows and the broader narrative driving U.S. equities. Reuters reported in May that OpenAI was also preparing to confidentially file for a U.S. IPO in the coming weeks. That follows SpaceX's mega-IPO filing, which is on course to rewrite the record books as the Elon Musk-led company pursues a $75 billion offering at a $1.75 trillion valuation and could begin trading within two weeks. Confidential submissions let companies advance IPO preparations while shielding sensitive financial details from rivals and the public. "Filing shortly after SpaceX allows Anthropic to capitalize on strong investor interest in AI and growth stocks while the window remains favorable," Kat Liu, vice president at IPO research firm IPOX, said. "Anthropic's valuation ambitions appear far less aggressive in comparison (to SpaceX) than they might have looked in isolation." RACE FOR AI DOMINANCE OpenAI and Anthropic have become the face of the AI boom that has redrawn corporate strategies, sparked a global arms race for computing power and talent, and turned AI-linked companies into some of the market's most richly valued firms. "For OpenAI, the conventional read is that Anthropic just seized the narrative advantage by filing first," said Harrison Rolfes, a senior analyst at PitchBook. "The unconventional read is that OpenAI got the better end of this: Anthropic just volunteered to absorb all the disclosure risk first, and OpenAI now has a free option to watch how institutional investors react to audited frontier AI financials before committing to its own price." On prediction markets, where traders wager on the outcome of future events, most participants had expected, opens new tab OpenAI to file for an IPO before Anthropic. OpenAI CEO Sam Altman said in a CNBC interview that he is not focused on the timing of a potential initial public offering for the ChatGPT maker, following news of Anthropic's confidential filing. He added that the company will go public when it makes sense to do so. Anthropic's valuation has more than doubled from $380 billion in February, when it raised $30 billion in a funding round. The company's rapid rise in early 2026 rattled markets, triggering sharp selloffs in software and IT stocks as investors worried its increasingly autonomous AI tools could upend traditional business models and accelerate disruption across industries. Its latest funding round drew backing from a mix of Silicon Valley and Wall Street investors, including Blackstone, Brookfield, D1 Capital Partners, GIC, General Catalyst and Insight Partners. A MARKET MILESTONE As a slew of blockbuster listings races toward public markets, companies from SpaceX to AI giants are competing for a finite pool of investor capital. Analysts, including D.A. Davidson's Gil Luria, said the two companies were racing to go public before capital on Wall Street ran out, and to set the agenda for how a frontier AI model - one that pushes the boundaries of what machines can do in language, reasoning, or coding - reports financials in a way that is favorable to their financial model. "The combined demand for capital from SpaceX, OpenAI and Anthropic will be so considerable that it is likely to create disruptions in the capital markets, so going early will be a great advantage," Luria said. At a valuation of close to $1 trillion, Anthropic would vault to the top tier of the S&P 500, alongside a handful of elite companies that dominate global equity markets. The IPO market has regained momentum in recent weeks, with companies raising $87.5 billion through May 26, the highest year-to-date global total since 2021, according to Dealogic data. Several sizable U.S. IPOs are also set to hit the market later this week, including Honeywell-backed quantum computing firm Quantinuum, Blackstone-backed Liftoff and gas engine manufacturer Innio.
Weatherford Announces Definitive Agreement to Acquire NCS Multistage, Expanding Completions Portfolio and Unconventional Resource Exposure
HOUSTON, June 01, 2026 (GLOBE NEWSWIRE) -- Weatherford International plc (NASDAQ: WFRD) ("Weatherford" or the "Company") and NCS Multistage Holdings, Inc. (NASDAQ: NCSM) ("NCS Multistage") today announced that Weatherford has entered into a definitive agreement to acquire NCS Multistage. Under the terms of the agreement, NCS Multistage stockholders have an election to receive either Weatherford common stock or a combination of Weatherford common stock and cash. On a blended basis, this is expected to be the equivalent of 0.463 shares of Weatherford common stock for each NCS Multistage share with up to 19.99% of this payable in cash. Annual cost synergies are expected to be at least $15 million and be realized within 18 months of closing. The deal is expected to be immediately accretive to adjusted Free Cash Flow per share. NCS Multistage brings a complementary technology portfolio aimed at supporting the optimization of oil and gas well completions and field development strategies. Its solutions are designed to enhance reliability and performance in complex well environments and are widely recognized for engineering rigor and capital-efficient deployment. Compelling Strategic Benefits The acquisition is expected to complement and enhance Weatherford's portfolio by: * Expanding offerings in the well completions segment, while deepening Weatherford's capabilities in the unconventional space. * Supporting the delivery of differentiated, technology-enabled solutions that help customers improve operational and production outcomes. * Providing an avenue for further growth of NCS Multistage's portfolio by leveraging Weatherford's international footprint. Girish Saligram, Weatherford's President and Chief Executive Officer, commented, "The acquisition of NCS Multistage is a natural complement to our completions strategy and enhances the application fit of our well construction products portfolio. NCS Multistage's technology is expected to enhance our ability to serve customers across the completion lifecycle, from well design through production optimization and late-life interventions, while deepening our exposure to the growing unconventional resource market. We expect to realize at least $15 million in annual run-rate cost synergies over a period of 18 months. Additionally, we see a meaningful opportunity to create additional value by bringing this technology to our global customer base, and we look forward to welcoming NCS Multistage into Weatherford." Ryan Hummer, NCS Multistage's Chief Executive Officer, commented, "This is a significant step for NCS Multistage that we believe positions our business -- and the talented people who built it -- for the next phase of growth as part of a leading global energy services company. I am proud of the company that our team at NCS Multistage has built, and it is clear from our interactions that Weatherford recognizes the strength of our technology, the quality of our operations, and the commitment of our people. This combination creates an opportunity for our products, technology, and people to reach a broader set of customers and markets faster than we could on our own, supported by Weatherford's financial strength and international footprint, providing long-term opportunity and value for our stakeholders." Transaction Details and Approvals The transaction has been approved by the Board of Directors of Weatherford, the Board of Directors of NCS Multistage, and the controlling stockholder of NCS Multistage that owns more than 50% of NCS Multistage's outstanding common stock. The transaction is subject to certain customary closing conditions, including regulatory approvals, and is expected to close in the second half of 2026. Until the transaction closes, Weatherford and NCS Multistage will continue to operate as separate, independent companies. Under the terms of the agreement, NCS Multistage stockholders can elect to receive either 0.554 shares of Weatherford common stock at closing, or a combination of 0.239 shares of Weatherford common stock and a cash amount equal to 0.137 shares of Weatherford common stock at closing, subject to proration and certain limitations and adjustments. On a blended basis, this is expected to be the equivalent of 0.463 shares of Weatherford common stock with up to 19.99% of the total equity consideration payable in cash. Advisors King & Spalding LLP is acting as legal counsel to Weatherford and Baker Botts L.L.P. is acting as legal counsel to NCS Multistage. Piper Sandler & Co. is serving as financial advisor to NCS Multistage. About Weatherford Weatherford is a global energy services company that helps customers drill smarter, complete wells more effectively, and maximize production across the entire well lifecycle. With a differentiated portfolio of market-leading solutions, integrated technologies, and a broad global customer footprint across six continents, we blend advanced engineering, digital intelligence, and world-class field expertise to reduce risk, improve performance, and maximize the value of customer assets. Together, we elevate every operation, delivering stronger wells, sharper decisions, and better energy for the world. Visit weatherford.com for more information and connect with us on social media. About NCS Multistage NCS Multistage is a leading provider of highly engineered products and support services that facilitate the optimization of oil and natural gas well construction, well completion and field development strategies. NCS Multistage provides products and services primarily to exploration and production companies for use in onshore and offshore wells, predominantly those that have been drilled with horizontal laterals in both unconventional and conventional oil and natural gas formations. NCS Multistage's products and services are utilized in oil and natural gas basins throughout North America and in selected international markets, including the North Sea, the Middle East and Argentina. Visit ncsmultistage.com for more information. Forward-Looking Statements This communication includes statements, which, to the extent they are not statements of historical or present fact, constitute "forward-looking statements" within the meaning of the U.S. Private Securities Litigation Reform Act of 1995. Forward-looking statements, and any related oral statements, can be identified by the use of terms such as "believe," "project," "expect," "anticipate," "estimate," "outlook," "budget," "intend," "strategy," "plan," "guidance," "may," "should," "could," "will," "would," "will be," "will continue," "will likely result," and similar expressions, although not all forward-looking statements contain these identifying words. These statements include, but are not limited to, statements about the expected timing and completion of the proposed transaction between Weatherford and NCS Multistage, the anticipated benefits of the proposed transaction, and plans and expectations for the new combined company after the completion of the proposed transaction. Such statements are based upon the current beliefs of Weatherford's and NCS Multistage's management and are subject to significant risks, assumptions, and uncertainties. Should one or more of these risks or uncertainties materialize, or underlying assumptions prove incorrect, actual results may vary materially from those indicated in our forward-looking statements. Readers are cautioned that forward-looking statements are only estimates and may differ materially from actual future events or results, based on factors including but not limited to the ability to complete the proposed transaction on the timeframe or on the terms currently anticipated or at all, including due to a failure to obtain requisite regulatory approvals; risks related to difficulties, inabilities or delays in integrating the parties' businesses; the ability to realize the anticipated benefits of the proposed transaction, including estimated synergies; the occurrence of any event, change or other circumstance that could give rise to the right of either or both parties to terminate the Merger Agreement; the potential impact of the announcement or consummation of the proposed transaction on the parties' stock price and on their respective business, contractual and operational relationships; risks related to business disruptions from the proposed transaction that may harm the business or current plans and operations of either or both parties, including diversion of management time from ongoing business operations; the risk that the proposed transaction and its announcement could have an adverse effect on the ability of either or both parties to hire and retain key personnel; the outcome of any legal proceedings that may be instituted against Weatherford or NCS Multistage, or their respective directors; the possibility that the proposed transaction may be more expensive to complete than anticipated, including as a result of unexpected factors or events, or unforeseen or unknown liabilities; Weatherford's ability to receive, in a timely manner and on satisfactory terms, required shareholder and court approval, and to satisfy the other conditions to the proposed redomestication within the expected timeframe or at all; our ability to realize the expected benefits from the proposed redomestication; the occurrence of difficulties in connection with the redomestication, including any costs related thereto; the risk that the proposed redomestication disrupts current plans and operations; global political, economic and market conditions, political disturbances, war or other global conflicts, terrorist attacks, public health issues such as pandemics, changes in global trade policies, tariffs and sanctions, weak local economic conditions and international currency fluctuations; general global economic repercussions related to U.S. and global inflationary pressures and potential recessionary concerns; as well as the factors and risks described in Weatherford's Annual Report on Form 10-K for the year ended December 31, 2025 and NCS Multistage's Annual Report on Form 10-K for the year ended December 31, 2025, and, in each case, in subsequent filings with the U.S. Securities and Exchange Commission. Other unpredictable factors not discussed in this communication could also have material adverse effects on forward-looking statements. You should not place undue reliance on any of Weatherford's or NCS Multistage's forward-looking statements. Any forward-looking statement speaks only as of the date on which such statement is made, and Weatherford and NCS Multistage undertake no obligation to correct or update any forward-looking statement, whether as a result of new information, future events or otherwise, except as required by applicable law, and we caution you not to rely on them unduly. No Offer or Solicitation This communication is not intended to and shall not constitute an offer to sell or the solicitation of an offer to sell or the solicitation of an offer to buy any securities, or a solicitation of any vote or approval, nor shall there be any sale of securities in any jurisdiction in which such offer, solicitation or sale would be unlawful prior to registration or qualification under the securities laws of any such jurisdiction. No offer of securities shall be made except by means of a prospectus meeting the requirements of Section 10 of the Securities Act of 1933, as amended (the "Securities Act"), or in a transaction exempt from the registration requirements of the Securities Act. Additional Information About the Transaction and Where to Find It In connection with the proposed transaction, Weatherford intends to file a registration statement on Form S-4 (the "Form S-4") that also constitutes a prospectus of Weatherford with respect to the shares of Weatherford to be issued in the proposed transaction (the "prospectus") and NCS Multistage intends to file an information statement on Schedule 14C, with the Securities and Exchange Commission (the "SEC"). Each of Weatherford and NCS Multistage may also file other relevant documents with the SEC regarding the proposed transaction. This document is not a substitute for the Form S-4 or prospectus or any other document that Weatherford or NCS Multistage may file with the SEC. INVESTORS AND SECURITY HOLDERS ARE URGED TO READ THE REGISTRATION STATEMENT, THE INFORMATION STATEMENT/PROSPECTUS AND ANY OTHER RELEVANT DOCUMENTS THAT MAY BE FILED WITH THE SEC, AS WELL AS ANY AMENDMENTS OR SUPPLEMENTS TO THESE DOCUMENTS, CAREFULLY AND IN THEIR ENTIRETY IF AND WHEN THEY BECOME AVAILABLE BECAUSE THEY CONTAIN OR WILL CONTAIN IMPORTANT INFORMATION ABOUT THE PROPOSED TRANSACTION. Investors and security holders will be able to obtain free copies of the Form S-4 and the information statement/prospectus (if and when available) and other documents containing important information about Weatherford, NCS Multistage and the proposed transaction, once such documents are filed with the SEC through the website maintained by the SEC at http://www.sec.gov. Copies of the documents filed with, or furnished to, the SEC by Weatherford will be available free of charge on Weatherford's website at https://weatherford.com/investor-relations/home. Copies of the documents filed with, or furnished to, the SEC by NCS Multistage will be available free of charge on NCS Multistage's website at https://ir.ncsmultistage.com. The information included on, or accessible through, Weatherford's or NCS Multistage's website is not incorporated by reference into this communication. For Investors: Luke Lemoine Weatherford Investor Relations +1 713-836-7777 [email protected] Mike Morrison NCS Multistage Holdings Chief Financial Officer and Treasurer +1 281-453-2222 [email protected] For Media: Kelley Hughes Weatherford Corporate Communications, Marketing & Sustainability [email protected]
Fractional-order systems for neuromorphic computing: software and hardware opportunities and challenges - npj Unconventional Computing
Applying a fractional derivative transforms these inputs into signals that approximate white noise, thereby maximizing entropy and improving the efficiency of spike coding. This implies that fractional-order dynamics are not only biologically plausible but may confer functional advantages by aligning neural dynamics with the statistics of natural sensory inputs. The introduction of fractional-order dynamics into RC provides control over a fundamental trade-off in how the reservoir processes temporal information. By replacing the standard Leaky Integrate-and-Fire (LIF) with an Fractional-order Leaky Integrate-and-Fire (FLIF), α may be used to control a balance between retaining a rich history of past inputs versus maintaining sensitivity to new environmental signals. Fractional calculus generalizes the concepts of differentiation and integration to non-integer orders. The most classical definition is the Riemann-Liouville form, which expresses a fractional derivative as an integral operator with a power-law kernel. For a function x(t), the Riemann-Liouville derivative of order α (0 < α < 1) is defined as where Γ(⋅) is the Gamma function. This formulation makes explicit the non-local nature of fractional differentiation: the present state depends on the entire history of the system, weighted by a power law. Unlike integer-order derivatives, which depend only on instantaneous changes, the Riemann-Liouville derivative encodes memory of all past dynamics. Two alternative formulations are commonly used in applications. The Grünwald-Letnikov (GL) derivative is defined in terms of finite differences: where are generalized binomial coefficients. The GL form is particularly useful for numerical simulations, as it lends itself to discrete-time implementations. Accordingly, the GL form is utilized throughout this work. The Caputo derivative modifies the Riemann-Liouville form by moving differentiation inside the integral: where denotes the first derivative of x(τ). This definition is often favored in physical modeling because it allows for initial conditions specified in terms of integer-order derivatives, making it compatible with experimentally accessible quantities such as voltages and currents at t = 0. In practice, the GL and Caputo forms yield very similar dynamical behavior, and both converge to the Riemann-Liouville derivative in the appropriate limits. All three definitions emphasize the same fundamental property: fractional differentiation couples the present dynamics of a system to its entire history, with memory decaying according to a power law. To connect fractional calculus with RC, the standard LIF neuron is extended to a FLIF neuron model. The classical LIF neuron describes the membrane potential V(t) as a first-order differential equation with a leak toward the resting potential and input-driven currents. In extending this model to the FLIF model, the first-order derivative is replaced by a fractional derivative of order α, yielding dynamics with long-tailed memory. A common biophysical form is written in terms of the effective capacitance C and resistance R: where V is the resting potential and I(t) is the input current. An equivalent form, which we use throughout this work and in our simulations, rewrites the leak term in terms of the membrane time constant τ = RC: where b is an optional bias current. This representation is more convenient for numerical implementations, since τ is typically treated as a tunable parameter. When V(t) reaches a threshold V, the neuron emits a spike and the potential is reset to V. The introduction of the fractional derivative fundamentally changes the dynamics. For α = 1, the model reduces to the classical LIF neuron with exponential decay. For α < 1, the neuron exhibits a power-law memory kernel, integrating its input with a heavy-tailed weighting of past activity. This produces dynamics consistent with experimental observations of spike-rate adaptation, anomalous diffusion of ions, and the whitening of power-law input signals. The parameter α thus controls the balance between sensitivity to recent inputs and the retention of long-term history, offering a tunable mechanism for matching neural dynamics to the statistics of input stimuli. The FLIF neuron provides a compact and biologically grounded way to introduce scale-free memory into artificial neural reservoirs. It also serves as a bridge between theoretical models of fractional dynamics and practical implementations in both software and hardware, making it a natural building block for FOR computing. In the sections that follow, we adopt the τ-based form in Eq. (5), as it directly corresponds to our software implementation using the Grünwald-Letnikov discretization. An important consequence of fractional dynamics is a shift in the effective rheobase of the neuron. For α < 1, the fractional derivative acts as an additional dampening term on the membrane potential, reducing the rate at which inputs accumulate toward threshold. This means that lower values of α require stronger input to elicit a spike, effectively raising the rheobase. Accordingly, to compare across different α values, one must account for this difference in excitability, either by adjusting bias currents or scaling input gain. Without such normalization, reservoirs at lower α may appear artificially quiescent as they are under-driven relative to higher-α systems. Algorithm 1 summarizes the complete discrete-time update procedure for a single FLIF neuron using the GL discretization; Fig. 1 shows the corresponding signal-flow block diagram. Algorithm 1 GL-FLIF neuron update Require: α, Δt, L, τ, b, V, V, V 1: Precompute for k = 1, ..., L 2: Initialize circular buffer , V ← V 3: for each timestep n do 4: Read input current I 5: ⊳ GL history term 6: 7: if V≥V then 8: Emit spike; V ← V 9: end if 10: Push V into circular buffer 11: end for Information-theoretic analysis This section establishes the information-theoretic foundation for understanding FORs. We show that the fractional order α acts as a control parameter that simultaneously governs three key computational properties: memory capacity (the ability to store past inputs), AIS (predictability of future states from present states), and information transfer (sensitivity to external inputs). Critically, these properties do not all peak at the same value of α, implying that different tasks will demand different optimal settings depending on their computational requirements. We begin by formalizing the FOR as an extension of the standard reservoir computing framework, then systematically characterize how α shapes the reservoir's information processing capabilities. This analysis provides the theoretical foundation for understanding why FORs outperform integer-order baselines on certain tasks, and offers principled guidance for selecting α based on task structure. Formally, a reservoir can be described as a collection of N neurons, each with its own state x(t) and update rule. A general form is where is the input, W and are the input and recurrent weight matrices, and f(⋅) denotes the neuron model (e.g., standard LIF, fractional LIF). The readout is trained as a linear combination of reservoir states, with W learned, for example, by regression or another learning rule. In this work, the nonlinear update rules f are implemented as fractional-order neuron models, introducing the order α as a tunable dynamical parameter. Replacing standard integer-order neurons with fractional-order neurons introduces power-law memory kernels into the reservoir dynamics. The fractional-order α acts as a control parameter, continuously tuning the balance between responsiveness to new inputs and retention of long-term history. To understand the role of α, we analyze the reservoir's information-theoretic and dynamical properties. A standard measure of reservoir performance is the Memory Capacity (MC), which quantifies how well past inputs can be reconstructed from the present reservoir state. Two forms are relevant here. The linear memory capacity measures the recovery of delayed inputs themselves. For a scalar input sequence {u}, the linear MC at lag k is defined as where is the optimal linear reconstruction of u from the reservoir state X. The nonlinear memory capacity extends this definition by replacing u with a nonlinear function of the delayed input. A common choice is to use Legendre polynomials P(u), yielding The total nonlinear memory capacity is the sum of across lags k and polynomial orders m. The cumulative memory capacity reported here is the sum of both linear and nonlinear contributions: This combined measure captures both the ability of the reservoir to store direct input histories and its ability to encode nonlinear transformations of them. As shown in Fig. 2, empirically, the total (i.e., cumulative) memory capacity decreases with increasing α in an approximately sigmoidal fashion: for low α, the reservoir integrates over long histories and achieves high memory capacity; as α increases, memory decays more rapidly, and at α ≈ 1 the reservoir retains only short-term input traces. This decreasing sigmoid relationship highlights a smooth but sharp transition from history-rich to history-poor dynamics. Interestingly, correlations between neuron activities peak in the transition region of this curve, suggesting that reservoir units are maximally coupled when the system is shifting between long-memory and short-memory regimes. This correlation peak may reflect the presence of critical-like dynamics where the reservoir is most sensitive to both its own state and its external input. The AIS quantifies how much information the reservoir's present state holds about its immediate future. Formally, where is the reservoir state vector. For small α, the reservoir's long memory produces an inertial, slow-moving state dominated by the past reservoir states. New inputs act as strong perturbations, and X is therefore not highly predictable from X alone, yielding low AIS. As α increases, the reservoir becomes more predictably responsive to inputs as the dynamics become dominated by the recurrent structure, thus AIS rises. In other words, higher α values produce reservoirs with more deterministic internal evolution. The complementary quantity is information transfer, which measures how much of the reservoir's current state reflects the history of the external input. Formally, where U denotes a delay-embedded representation of the input's past. For low α, the reservoir excels at integrating the input signal, storing substantial information about its external environment. For high α, the reservoir becomes dominated by the dynamics of the recurrent structure of the reservoir, and information transfer from the input decreases. With reference to Fig. 3, together these measures reveal a fundamental trade-off. Low α values configure the reservoir as a sensitive sensor: highly responsive to external inputs, capable of storing long histories, but internally less predictable. High α values configure the reservoir as a stable internal model: self-determined, predictable, but less permeable to input. At intermediate α, both information transfer and AIS are significant, producing a balance where the reservoir both listens to the input and stabilizes its own dynamics. This intermediate regime, which coincides with the transition in memory capacity and the peak in neuron-to-neuron correlations, represents the most computationally powerful setting. FORs therefore generalize the standard framework by introducing a tunable continuum of memory and dynamical regimes. The fractional-order α interpolates between input-driven and self-driven dynamics, between long-memory storage and short-memory responsiveness. The balance achieved at intermediate α suggests a critical-like operating point where reservoirs are maximally expressive: capable of capturing external structure, internally stable, and richly interactive. This positions fractional reservoirs as a principled extension of reservoir computing, with information-theoretic properties that can be tuned continuously through α. Fractional-order neuron implementations The realization of fractional-order neurons requires translating continuous-time dynamics into discrete or physical forms while preserving the long-memory and power-law characteristics that define fractional systems. Because fractional differentiation is inherently non-local in time, each implementation involves trade-offs among accuracy, computational efficiency, and resource usage. We organize implementations into two primary domains: software-based numerical realizations and hardware-based digital or analog realizations. In software, fractional dynamics are typically approximated by discrete convolution over past membrane potentials, capturing history-dependent effects through weighted sums with power-law coefficients. This approach allows precise control of parameters such as fractional order α, timestep Δt, and history length L, making it suitable for systematic experimentation and benchmarking. The Grünwald-Letnikov (GL) formulation is particularly advantageous due to its direct correspondence with the continuous fractional derivative and its straightforward implementation as a running sum over stored voltages. In hardware, fractional-order dynamics can be approximated through digital fixed-point implementations or realized physically through analog elements with inherent fractional characteristics, such as memcapacitive and memristive devices. Digital realizations, particularly those targeting field-programmable gate array (FPGA) or application-specific integrated circuit (ASIC) platforms, enable parallel execution and exploration of scaling behaviors, while analog approaches offer the possibility of directly computing with the physics of fractional-order components. Together, these implementations establish the foundation for practical fractional-order neuromorphic systems, spanning from precise numerical simulation to physical embodiment of fractional dynamics. Fractional-order software implementations Simulating fractional-order neurons requires discretizing the continuous-time dynamics. Here we describe the Grünwald-Letnikov (GL) formulation and its direct update rule. For comparison, we briefly note the role of Euler methods commonly used for integer-order LIF models. The GL form expresses the fractional derivative as a weighted sum over past values: with timestep Δt, history window length L, and coefficients Substituting this into the fractional LIF equation with explicit evaluation of the leak term at V yields the practical update rule: When V≥V, the neuron emits a spike and is reset to V. This direct update aligns with our implementation: a circular buffer maintains past voltages, GL coefficients are precomputed, and the scaling factor Δt ensures consistency across timesteps. For integer-order LIF neurons (α = 1), the forward Euler method gives the familiar update while the implicit Euler scheme evaluates the leak at V instead of V, improving stability for stiff systems. The fractional GL update in Eq. (15) can be viewed as a natural generalization: the instantaneous update is still explicit, but past states enter through the convolutional history term with power-law weights. The principal computational cost of the GL method arises from the history term in Eq. (15). At each timestep, evaluating the convolution requires O(L) operations, where L is the history window length. For long simulations or large networks, this scaling can dominate runtime. The memory requirement is also O(L) per neuron, since each must maintain a buffer of its past voltages. In practice, L is chosen to balance accuracy and efficiency. Smaller L truncates the power-law kernel, reducing memory effects but lowering cost. Larger L captures the full long-tail dynamics more accurately at the expense of higher runtime and memory. For a reservoir of N neurons simulated over T timesteps, the total cost of the GL convolution is O(N⋅L⋅T), and the memory requirement is O(N⋅L) for the history buffers plus O(L) for the shared coefficient array. For typical values of L used in our benchmarks, this cost is tractable on modern CPUs and GPUs, but becomes a bottleneck when scaling to millions of neurons. To characterize the accuracy-efficiency trade-off empirically, we swept L from 5 to 250 at α = 0.5 on the FSDD spoken-digit task across 10 trials. Results are shown in Fig. 4. Accuracy follows a sigmoid-like curve with L: performance is essentially degenerate below L = 30, rises steeply through L ∈ [50, 100], and saturates empirically beyond L ≈ 150. The saturation point is task-dependent, determined by the temporal scale of the relevant history in the input signal; tasks with longer-range dependencies would be expected to saturate at larger L. In the case of FSDD there is a hard upper bound at L = 250, imposed by the sample duration: each sample spans 250 micro-steps (25 macro-steps × Δt = 0.1), so L > 250 cannot add further within-sample history. One practical design choice for FSDD is L = 100, which achieves 93.3% accuracy at the knee of the saturation curve, as beyond L = 100 gains of less than 1% come at linearly increasing compute cost. Several optimizations exist to mitigate this cost, such as using recursive coefficient updates, Fast Fourier Transforms for long convolutions, or rational transfer-function (IIR) approximations to the power-law kernel. We adopt the direct truncated GL form in this work for clarity and reproducibility. In hardware realizations of fractional-order neurons, the fractional order α is set by physical device parameters that are subject to manufacturing variation, temperature sensitivity, and long-term drift. We evaluate robustness to two non-ideality scenarios: drift in the realized value of α, and additive noise on the input signal. We simulated device-level variation in α by training at the nominal α = 0.5 and testing at drifted values ranging from -10% to +10%. As shown in Fig. 5, classification accuracy on the FSDD task is nearly invariant to drift across this range, changing by less than 2 percentage points at ±10% and approximately 0.5 percentage points at ±5%. The near-symmetric response about α = 0.5 is consistent with operation near a local optimum of the accuracy-α curve. We evaluated robustness to additive i.i.d. Gaussian noise on the input features (σ = 0 to 0.5), simulating sensor noise or signal corruption. Figure 6 compares the fractional reservoir (α = 0.5) against the integer-order baseline (α = 1.0). Performance at α = 0.5 remains stable up to σ = 0.01 and degrades gracefully beyond, while maintaining a consistent accuracy advantage over the integer-order baseline across all tested noise levels. Notably, the integer-order reservoir shows comparatively less sensitivity to increasing noise, with the performance gap between α = 0.5 and α = 1.0 narrowing at higher σ. Two effects likely contribute to this. The standard LIF acts as a low-pass filter over its membrane time constant, attenuating high-frequency noise components before they influence spiking. In contrast, the fractional LIF incorporates input noise into its power-law memory trace, where it persists across timesteps and is recurrently recirculated through the reservoir dynamics. Fractional-order hardware implementations Implementation of fractional-order systems directly in hardware has the potential to improve performance over a software-only implementation, given the opportunity for increased parallelism and the speed advantages of dedicated hardware. A digital hardware implementation may serve as a bridge, allowing for the evaluation of fractional dynamics applied across a range of problems while analog components with fractional characteristics are still in development. We explore existing work below, including review articles on digital fractional-order systems, as well as two articles describing fractional-order neurons targeting an FPGA execution environment. We then offer an examination of relevant design considerations for digital fractional-order neurons, an overview of a fractional LIF neuron implemented in SystemVerilog, and a discussion of implications for achieving fractional dynamics in a digital context. Regarding a general approach to the implementation of fractional-order calculus in digital systems, review articles and ref. contain mathematical grounding through an overview of fractional-order definitions and options for numerical approximations. Clemente-López et al. address fractional-order systems on both FPGAs and embedded hardware, while Ali et al. focus on FPGAs and their unique architectural features. Both perform tradeoff analysis between different numerical approximation methods and discuss topics most relevant to digital systems, such as discretization techniques and usage of floating-point versus fixed-point representations. Clemente-López et al. highlight the Adomian Decomposition Method as well as the Grünwald-Letnikov (GL) method as the two primary approximation methods used in digital implementations. They further note that the GL method is generally preferred due to its simplicity and flexibility. In ref. , Ali et al. additionally describe the usage of the popular Tustin method for discretization. The above review articles are in general agreement that the GL method for fractional-order approximation is well-suited for digital platforms. In addition, they both cite the choice to use fixed-point as an appropriate design decision for numerical representation. In ref. , Malik and Mir present a Hindmarsh Rose (HR) neuron with fractional-order update dynamics. Their work spans single neuron behavior as well as the combined behavior resulting from a coupling of two neurons, with the execution of neuron activity on FPGA. The authors include waveforms showing neuronal dynamics and a range of spiking behavior. Details on representation in hardware include FPGA resource utilization and notes describing the use of a 32-bit fixed-point format. Similarly, Tolba et al. outline an approach for fractional-order update dynamics in an Izhikevich neuron model executing on an FPGA. They also produced waveforms showing the resulting spiking behaviors, as well as block-level hardware diagrams. Though their work includes greater depth regarding details such as bit widths and routing between high-level components, the focus of the article tends towards the comparison between the integer-order and fractional-order implementations, and the information on engineering tradeoffs and their implications for the fractional-order alone is limited. As discussed above, selection of α (alpha) determines a system's history retention as well as the relative weighting of recent historical values. This is due to the binomial coefficient magnitudes resulting from a particular value of α at a given timestep, as seen in Eq (13). Figure 7 shows the relative coefficient magnitude decay with increasing timestep for different values of α. Lower values of α provide greater history retention, as seen by the minimal decay in coefficient magnitude, while higher values of α decay much more quickly. In each case, the coefficient magnitude of the historical value at timestep 1 is equal to the value of α itself, so that higher values of α more heavily weight values in recent history. In the context of digital hardware representation, the memory capacity of a system is dependent on the precision afforded to the binomial coefficient values. If the bit width of the binomial coefficients is too small, the memory capacity will shrink to match the time step with the coefficient magnitude matching the smallest value representable, given the number of bits. Table 1 shows the maximum history length, or value of k, across α values in 3 different fixed-point formats: UQ0.8, UQ0.16, and UQ0.32, where each format is unsigned with zero bits representing the integer component, and 8, 16, or 32 bits, respectively, representing the fractional component. The required fixed-point representation for binomial coefficients should be determined by the chosen α value for a given application, along with the desired history length. Figure 8 shows the maximum history length for a range of α values in fixed-point formats UQ0.8, UQ0.16, and UQ0.32. An unsigned fixed-point format utilizing 64 bits for the fractional component would allow for a history length of nearly 4 billion, e.g., when using α = 0.9. As such, a fixed-point format of UQ0.64 is likely larger than necessary for most applications, and has the potential to produce time- and space-complexity issues in execution. Conversely, UQ0.16 becomes insufficient at higher α values where rapid coefficient decay limits history to fewer than 100 steps; UQ0.32 is the practical minimum for applications requiring both high α and extended history. We chose to implement a FLIF neuron using the hardware description language SystemVerilog (SV). Simulation was performed using cocotb, a Python framework targeting chip verification, along with tools from the oss-cad-suite toolchain, such as verilator and gtkwave. Our goal in simulation was to verify the spike timing adaptation described in ref. . We created a preliminary proof-of-concept implementation using a coefficient bit width of 8 and a history length of 8. We iteratively doubled the history length in simulation until the target behaviors were achieved, with a history length of 256 and a coefficient bit width of 16. The final implementation uses Q8.0 fixed-point for input current and membrane potential, and Q0.16 for the GL coefficients. Figure 9 shows the hardware datapath of the implementation, and Fig. 10 summarizes the development pipeline. We verified that our implementation produced an increasing spike frequency in the presence of a constant input current, as well as a higher recovery spike frequency after a dropout of a constant input current. These findings are consistent with the behavior described by Teka et al. in ref. . Figure 11 shows the increasing spike frequency over time due to the fractional-order system memory. Similar memory effects are seen in Fig. 12, which shows a higher recovery spike frequency than the initial frequency after a brief period of zero input current. Primary implications of our findings when considering the execution of our design on an FPGA include the identification of available board resources. In particular, the mathematical operations associated with the multiplication and accumulation of coefficients and history terms are ideally mapped to efficient resources such as DSP slices. As the history length grows, the need for DSP slices increases linearly. Similarly, pre-computed binomial coefficient values require storage proportionate to history length. Intrinsically fractional-order hardware Analog realizations of fractional dynamics exploit physical devices whose impedance exhibits constant-phase ("fractance") behavior. A promising approach is the fractional-order memcapacitor, whose state-dependent capacitance produces a power-law memory kernel at the circuit level. Recent work has shown that such memcapacitive elements can reproduce neuronal behaviors, including spike-rate adaptation, long-lived responses, and signatures consistent with critical dynamics, positioning them as efficient and biologically plausible neuromorphic primitives. A practical fractional leaky integrate-and-fire (FLIF) neuron can be realized using four functional components: (i) a leak pathway, (ii) a memcapacitive integrator acting as the fractional memory core, (iii) a Schmitt trigger for robust thresholding, and (iv) a reset switch. In this configuration, the membrane node integrates input current through the memcapacitive branch, producing a slow fractional ramp whose rate depends on the fractional order α. The Schmitt trigger converts this ramp into discrete spikes with hysteresis, while a MOSFET or transmission gate provides the reset path, rapidly discharging the memcapacitor after each spike. Fractional memory effectively increases the rheobase current, so bias and gain normalization across α values are often required to maintain consistent firing thresholds. Schmitt trigger trip points should be chosen so the fractional ramp reliably crosses both thresholds under the expected input range, ensuring stable oscillation and noise immunity. By embedding the memory kernel directly into the physical capacitance rather than into algorithmic state variables, these designs achieve intrinsic fractional behavior with minimal computational overhead and high analog bandwidth. When physical fractional-order memcapacitors are unavailable, several well-established approximation methods can emulate their behavior over a target frequency band (ω, ω) corresponding to the neuron's operating spectrum. The Oustaloup method approximates s (0 < γ < 1) by a stable, minimum-phase IIR filter composed of first-order sections: where poles and zeros are geometrically distributed between ω and ω, and N controls the approximation order. Larger N broadens the constant-phase region but increases circuit complexity and power consumption; shallow approximations suffice for narrowband operation. ORA maps directly to active-RC or Gm-C biquad sections, and can be implemented in switched-capacitor or mixed-signal IIR topologies. A finite Foster or Cauer ladder of resistors and capacitors -- referred to here as an R-C network to distinguish it from the reservoir computing (RC) abbreviation used throughout this paper -- approximates a constant-phase element (CPE) whose impedance scales as Z(s) ∝ s over a target frequency band (ω, ω). The pole-zero structure of the ladder, with poles and zeros interlaced along the negative real axis, produces a nearly flat phase response of approximately -πα/2 across the band -- a defining property of a fractional-order element. This frequency-domain characterization has a direct time-domain counterpart: the impedance Z(s) ∝ s corresponds to convolution with the power-law kernel t/Γ(α), which is precisely the memory kernel underlying the FLIF dynamics described in the Fractional leaky integrate-and-fire neuron subsection. The Grünwald-Letnikov discretization of this kernel (Eq. (13)) approximates the continuous convolution via the weighted sum , where the GL coefficients c decay as k for large k. The R-C ladder thus physically realizes the same power-law memory that the GL algorithm implements in software, providing a direct analog instantiation of the fractional-order neuron's temporal dynamics. These networks are fully passive, compact, and compatible with SPICE-level modeling. The trade-off between accuracy and area is governed by ladder depth: deeper ladders yield better constant-phase fidelity but at higher component cost. Together, these methods -- memcapacitive and analog -- demonstrate several design strategies for realizing fractional-order neuronal dynamics. To illustrate how the theoretical trade-offs identified in the FOR computing subsection manifest in concrete applications, we consider a diverse set of benchmark tasks spanning classification, control, and prediction. These case studies are chosen to highlight complementary aspects of reservoir computation: (i) long-term memory and complex prediction, (ii) efficient representation of natural input signals, (iii) real-time control and stability, and (iv) classification and prediction capabilities. Specifically, we study: * spoken digit classification, * control of the cart-pole balancing problem, and * classification of blood glucose dynamics for diabetes prediction. Each task is implemented using a FOR computing framework or a fractional-order spiking neural network (FOSNN), with fractional order α varied systematically. Together, these examples demonstrate that FORs consistently outperform their integer-order counterparts, and that the benefits align with the theoretical trade-offs in memory, information storage, and sensitivity to natural statistics developed in earlier sections. Time-series classification Time-series prediction and classification is a canonical benchmark for reservoir computing, as it simultaneously probes a network's nonlinear transformation capacity and its ability to retain and process temporal dependencies. Here, we evaluate the performance of FORs on the Free Spoken Digit Dataset (FSDD), a widely used corpus for speech-based classification tasks. FSDD consists of short audio recordings of spoken digits (0-9) from multiple speakers, offering natural temporal variability. The task tests the reservoir's ability to process natural auditory signals that exhibit temporal correlations and spectral regularities common to many real-world sensory signals. As described in the biological motivation discussion above, fractional dynamics may be advantageous in such contexts, as fractional differentiation can act as a flexible signal whitening filter for 1/f-type power spectra. As shown in Fig. 13, FORs outperform their integer-order counterparts (α = 1.0) on the FSDD task. A complete sweep of α values shows a characteristic peak of accuracy at intermediate fractional orders (α ≈ 0.3-0.5), consistent with the theoretical prediction that optimal information processing occurs where the trade-off between memory retention and whitening balance is achieved. Lower α values yield strong memory but reduced temporal responsiveness, while higher α values approach the Markovian dynamics of standard reservoirs, losing long-range dependencies. The results above use offline ridge regression as the readout. To confirm that the effectiveness of α is not specific to this choice, we repeated the full sweep using Recursive Least Squares (RLS), an online readout that updates readout weights incrementally as each sample is processed and leaves the reservoir weights unchanged. Figure 14 shows classification accuracy for both methods across α = 0.1-1.0. The two readouts track closely throughout, both peaking at α = 0.5, and the fractional-order advantage over the integer-order baseline is preserved under both. The α optimum and the qualitative shape of the accuracy curve are thus readout-independent, confirming that α controls an intrinsic property of the reservoir dynamics rather than an artifact of the training procedure. Cart-pole balancing In this case study, a FOSNN is investigated to test if fractional memory dynamics improve performance and robustness on the cart-pole control task, compared to a standard SNN and multi-layer perceptron (MLP) across a range of α values. A comparison of model performance on the CartPole-v1 task is shown in Table 2. The MLP displayed the most significant performance change with the introduction of noise, as shown in Fig. 15. The average solve episode and standard deviation increased by 35.35% and 48.18%, respectively. Additionally, the MLP's completion rate decreased by 17%. The standard SNN offered moderate improvements, achieving a slightly higher stability than the MLP in noisy conditions, but it demonstrated a lower completion rate (7.9% loss) and overall performance in comparison to the FOSNN. In contrast, the mid-α FOSNN (α = 0.5-0.7) models displayed similar combinations of higher resilience and absolute performance. The baseline and noisy performances of an α = 0.6 FOSNN is shown in Fig. 16. In general, the mid-α FOSNN group maintained the highest completion rate (98%) while completing the environment the fastest on average (1411 episodes, noisy). Notably, FOSNN (α = 0.7) was the only value to show an absolute improvement in its completion rate (from 95% to 98%), and FOSNN (α = 0.6) improved its standard deviation by 4.32% (from 102.40 to 97.98). When tested under progressive node removal, the MLP showed the greatest overall robustness, achieving a normalized area under curve (AUC) of 0.540, a lower standard deviation of 43.47, and a critical damage threshold (CDT) of 70%. In comparison, while most FOSNN models were less robust than the MLP architecture, the FOSNN (α = 0.8) showed one notable advantage with the lowest magnitude of initial performance loss, with an initial degradation rate (IDR) of -4.04, as shown in Fig. 17. Diabetes prediction The diabetes prediction task evaluates the reservoir's ability to identify diagnostic patterns from physiological time-series data. Figure 18 shows the effect of varying the fractional order α on performance. Both accuracy and F1 score decreased monotonically as α increased, revealing that reservoirs with stronger memory effects (lower α) achieve superior predictive accuracy on physiological time-series data. The best performance was achieved near α = 0.1, corresponding to a long memory kernel and slower voltage decay, which effectively integrates temporal correlations across patient features. At higher α values, the network dynamics approach the Markovian regime of standard LIF neurons, leading to diminished temporal representation and reduced discriminative power. A comparative evaluation of all four network architectures is shown in Fig. 19 and Table 3. The FOR achieved the highest mean accuracy and F1 score, outperforming both the standard reservoir and feedforward models. While the MLP and SNN exhibited similar performance, the inclusion of recurrent dynamics in both reservoir variants yielded substantial gains, confirming that temporal integration -- especially when extended by fractional memory -- plays a key role in representing slow-varying physiological processes.
Unconventional orbital currents and torques due to ferro-rotational orbital textures - npj Spintronics
In this work, we demonstrate the electrical generation of unconventional orbital currents in FR systems that preserve both and symmetries. Symmetry arguments reveal that these -even rotation-induced orbital currents -- analogous to -odd magnetization-induced spin currents in FM systems -- manifest as (i) longitudinal orbital currents polarized along the FR axis and (ii) unconventional orbital Hall currents with polarization collinear with either the charge or orbital current [e.g., see (Fig. 1c). Tight-binding calculations show that these effects are driven by an electric hexadecapole (16-pole) moment arising from the FR order, through an intrinsic and nonrelativistic mechanism. To corroborate these findings, we perform first-principles calculations for the FR material TiAu4. Finally, we explore the experimental implications of rotation-induced orbital currents by studying an FR/FM bilayer within a tight-binding framework, demonstrating current-induced orbital accumulation in the FR layer as well as orbital torque in the FM layer. We further suggest that this unconventional orbital torque can enable deterministic, field-free switching of the FM order, pointing to a promising route for orbitronics research based on novel ferroic orders and higher-order electric multipoles. In the linear-response regime, the orbital current J (or spin current J) generated by an electric field E is expressed as , where X = L or S. Here, α and γ denote the orbital (spin) current flow and polarization directions, respectively. The rank-3 orbital (spin) conductivity tensor σ (σ) can generally be decomposed into -even and -odd contributions, with their nonzero components dictated by the system's symmetry. For example, in a nonmagnetic cubic system with point group O, only the -even conventional Hall components of , where α, β, and γ are mutually orthogonal, are symmetrically allowed. Symmetry breaking due to ferroic orders can induce additional nonzero components of σ. Here, we focus on ferroic orders that preserve symmetry, classified into two types: -odd FM order and -even FR order. In a cubic system, the FR and FM orders aligned along the z direction reduce the symmetry, leading to the point group 4/m and the magnetic point group , respectively. For both cases, the nonzero components of the total σ are given by refs. (see Supplementary Note 1 and Supplementary Table 1 for all nonmagnetic crystallographic point groups): In addition to the -even conventional Hall components (, , and ), the components induced by ferroic orders can be categorized into two groups: (i) diagonal components ( and ), describing longitudinal currents polarized along the order parameter (pink arrows in Fig. 1c), and (ii) off-diagonal components ( and ), representing unconventional Hall currents, where the polarization is collinear with either E or J (green arrows in Fig. 1c). It is worth noting that, in the presence of the first-type longitudinal components , the second-type Hall components take the form . This implies a conversion of a primary current into a secondary current (or of into ) for α ≠ β, corresponding to spin swapping or orbital swapping. These ferroic-order-induced currents inherit the -parity of the associated order parameters. In -odd FM metals, -odd longitudinal spin currents are electrically generated due to the nonrelativistic spin-polarized band structure. Additionally, -odd unconventional spin Hall currents -- also known as the magnetic spin Hall effect or spin swapping -- arise from SOC. These magnetization-induced spin currents in FM metals can accompany the relativistic -odd orbital currents via SOC, e.g., the magnetic OHE. In contrast, in -even FR systems, the rotation-induced orbital currents can be generated, including the longitudinal orbital currents and unconventional orbital Hall (or orbital swapping) currents. Importantly and distinctively, they require neither broken nor SOC, as will be demonstrated. Rotation-induced longitudinal orbital current To see how the orbital current can be generated in FR systems, we introduce a minimal tight-binding model with a relevant order parameter. The OAM dynamics can be driven by multipole degrees of freedom. Although the FR order is often described by an axial vector, such as the electric toroidal moment, we focus here on another emergent multipole in FR systems: the electric hexadecapole moment (rank-4), H ∝ xy(x - y) (Fig. 2a), which is even under and . The quantum mechanical operator for this can be constructed by replacing r = (x, y, z) with the OAM operators . Accordingly, we define an atomic-site electric hexadecapole moment operator as where and ℏ is the reduced Planck constant. Note that can emerge under the point group 4/m exhibiting the FR order along the z direction. In the atomic d-orbital basis {}, Eq. (2) simplifies to , which implies that hybridizes orbital wave functions, effectively rotating them around the z-axis, as illustrated in Fig. 2b. Let us introduce a two-dimensional square lattice tight-binding model incorporating . We adopt a minimal two-orbital basis {} for describing . Considering only nearest-neighbor hopping, the Hamiltonian is given by where k is the crystal momentum, a is the lattice constant, is the identity matrix, are the pseudospin Pauli matrices, and is determined by the Slater-Koster hopping parameters in units of eV. The term in Eq. (3) accounts for the crystal field that splits d and levels. The effect of the FR order along the z direction is incorporated through , which is equivalent to in the two-orbital basis, with its magnitude set by Δ = 0.1 eV. Figure 2c shows that a gap between d and bands is opened due to the electric hexadecapole moment. Near the gap, the eigenstates , with energies ϵ ≈ ± Δ, yield the expectation values . We note that corresponds to the submatrix of that is defined in the full d-orbital basis, so effectively captures the out-of-plane OAM. Here, we derive an intuitive picture of how an electric field E drives the dynamics of for a single Bloch state near the gap. Under , an electron with charge -e after time δt acquires momentum δk = - eEδt/ℏ, leading to the perturbation . The dynamics of follow the Bloch equation , where B(k) is the effective magnetic field satisfying , with arising from the electric-field-induced crystal field variation. In the vicinity of the band gap, with an initial condition , the solutions for small deviations from equilibrium are given by , , and This result shows that the electric hexadecapole moment undergoes precession due to the intrinsic crystal field that acts as a current-induced effective field, generating the nonequilibrium OAM . This behavior resembles spin dynamics in FM systems under an intrinsic spin-orbit field, although the effect here is nonrelativistic. Note that in Eq. (4) diverges as Δ → 0, but the net value vanishes as the gap closes. Although the net OAM (or ) vanishes upon k-integration, the net orbital current remains finite, leading to a nonzero . The conventional orbital current operator is defined as , where is the velocity operator. Substituting and , the longitudinal orbital current to first order in E is given by Integration of Eq. (5) over k-space yields a finite value, confirming the emergence of a rotation-induced orbital current driven by an intrinsic, nonrelativistic mechanism associated with a higher-order electric multipole. Unconventional orbital Hall current Additional orbital currents can emerge in multi-orbital systems exhibiting richer orbital texture. To explore this, we construct a three-dimensional tight-binding model for an FR system with the point group 4/m (Fig. 3a; see Methods and Supplementary Note 2), which constrains σ as given in Eq. (1). The tetragonal unit cell consists of A atoms with five d orbitals, and B atoms with an s orbital. The FR order along the z-axis arises from a rotational displacement of the four B atoms by an angle ϕ. The hopping pairs included in the model are shown in Fig. 3a. The next-nearest-neighbor hopping between d orbitals gives rise to the momentum-dependent d-orbital texture responsible for the conventional OHE. Its amplitude is assumed proportional to that of the nearest-neighbor hopping, with the proportionality factor η initially set to 0.5. The hopping between s and d orbitals, which depends on ϕ, characterizes the FR order. Figure 3b shows the band structure of this model with ϕ = 20°, which exhibits a nonzero expectation value of [defined in Eq. (2)] in equilibrium. Unlike earlier works, where was manually introduced into the Hamiltonian, in this model, it naturally emerges from structural rotation. It is noteworthy that downfolding our Hamiltonian into the two-dimensional d-orbital subspace yields a term proportional to for small ϕ (see Supplementary Note 3), revealing a direct connection between the electric hexadecapole moment and the FR order. We now proceed to compute the -even part of the orbital conductivity tensor σ using the Kubo formula (see Methods). Figure 3c presents numerical results for the nonzero orbital conductivity components for different values of ϕ, with . The longitudinal () and unconventional orbital Hall () components (e.g., see Fig. 1c), represented by pink circles and green triangles, respectively, vanish at ϕ = 0 and reverse sign under FR-order-reversal (ϕ → - ϕ). In contrast, the conventional orbital Hall components, indicated by blue × and orange + symbols, remain finite at ϕ = 0 and are invariant under FR-order-reversal. These results clearly demonstrate that rotation-induced OHE and conventional OHE have distinct physical origins, while both are -even and nonrelativistic. To further investigate the mechanism behind rotation-induced orbital currents, we compute for different values of η, which controls the next-nearest-neighbor hopping amplitudes, while fixing ϕ = 20° (Fig. 3d). We find that only the longitudinal component remains finite for η = 0, indicating that it arises solely from the FR order, specifically the electric hexadecapole moment, as demonstrated by our two-orbital model. On the other hand, both conventional and unconventional Hall components emerge as η increases, suggesting that the rotation-induced OHE requires not only the FR order but also the orbital texture responsible for the conventional OHE. This phenomenon can be understood in terms of nonrelativistic orbital swapping -- an orbital analog of spin swapping. It has been shown that in FM metals, a spin-polarized current is converted into a swapped spin current through the interplay of the orbital texture and SOC. Similarly, our results show that the longitudinal orbital current , induced by the FR order, is converted into the unconventional orbital Hall current (or into when ) via the orbital texture. Notably, this conversion does not require SOC, in contrast to spin swapping. First-principles calculation for TiAu Next, we investigate the FR material candidate, tetragonal TiAu (space group I4/m) using first-principles calculations (see Methods). The crystal structure exhibits the FR order along the z-axis (Fig. 4a), leading to a nonzero electric hexadecapole moment (Fig. 4b). The orbital conductivity tensor σ takes the same form as Eq. (1), with seven independent nonzero components of , including those for α = x () and α = z (). The rotation-induced orbital currents associated with these components are illustrated in the left and right panels in Fig. 1c, respectively. The components for are related to those for by four-fold rotational symmetry about the z-axis. By evaluating the Kubo formula, the nonzero components are obtained as functions of the chemical potential. For (Fig. 4c), the conventional Hall components exceed 1000(ℏ/e)(Ω cm) at the Fermi level. Additionally, we identify rotation-induced components, including the longitudinal orbital conductivity and the unconventional orbital Hall conductivity . For (Fig. 4d), the rotation-induced components are smaller, with and . The magnitude of the unconventional terms depends on the FR orbital texture (e.g., see Fig. 3), motivating further materials exploration. While the orbital conductivity is fully nonrelativistic, the corresponding nonzero components of the spin conductivity tensor can manifest, too, due to SOC. When SOC is present, not only the electric multipole moments but also the atomic-site electric toroidal moments, defined in the spinful basis, can emerge from the FR order, contributing to the -even spin current generation. A key distinction, however, is that the spin conductivity vanishes in the absence of SOC, whereas the orbital conductivity remains largely unaffected by SOC due to its nonrelativistic origin (see Supplementary Note 4). Furthermore, the -even orbital conductivity arises purely from the interband contribution, which is robust against scattering time (see Supplementary Note 4), while possible extrinsic contributions are not considered. By contrast, the -odd conductivity in FM systems is dominated by the intraband contribution that scales with the scattering time, but it is prohibited here by invariance. Unconventional orbital torque and field-free switching So far, we have discussed rotation-induced orbital currents based on the conventional definition of the orbital current operator, which is not directly measurable. In this section, we show that FR order gives rise not only to orbital currents but also to effects that can be probed experimentally, such as OAM accumulation and orbital torque. To illustrate this, we examine an FR/FM bilayer using a tight-binding model (Fig. 5a; see Methods). In our model, under , the FR layer with generates conventional () and unconventional () orbital Hall currents without spin currents, while the FM layer does not produce orbital or spin Hall currents on its own. This design ensures that the current-induced torque on magnetization M in the FM layer originates solely from the OAM injection by the FR layer. Within linear-response theory, we compute the current-induced non-equilibrium OAM (δL) and spin (δS) in the FR/FM system with , where is the unit vector of M (Methods). Figure 5b shows the layer-resolved δL per applied electric field. Large OAM components along x and y appear near the top and bottom surfaces of the FR layer, demonstrating orbital accumulation from the unconventional (δL) and conventional (δL) orbital Hall currents. The induced OAM is transferred across the FR/FM interface and subsequently interacts with M. This generates δS, which acts as an effective field for the torque . In particular, the spin-orbit precession with δL results in , leading to the damping-like orbital torque . Due to the presence of δL and δL, we obtain , where and are the effective fields for the conventional and unconventional orbital torques, respectively. The effective fields per charge current density J are estimated in our model (see Methods) as and , which fall within the range of reported values for spin-orbit torque devices using heavy metals. To gain further insight into how these orbital torques contribute to magnetization switching, we simulate magnetization dynamics within a macrospin model. The dynamics of is described by the Landau-Lifshitz-Gilbert equation where γ is the gyromagnetic ratio, B is the magnetic anisotropy field, and α is the Gilbert damping parameter. We consider a type-x geometry, in which an easy axis is collinear with the charge current, i.e., . Parameters are set to α = 0.05 and B = 30 mT. The same and obtained above are used here, together with current pulses of J = ± 10 A/m. Figure 5d shows the trajectory of over time t upon a current pulse, illustrated in Fig. 5e. Initially, points along (t = t). The current pulse exerts the damping-like orbital torques on the magnetization. Because of the presence of , is not perfectly aligned with but instead slightly tilted toward (t = t), as shown in Fig. 5f. Hence, after the pulse is turned off, relaxes deterministically to (t = t) without the aid of an external magnetic field. Figure 5e, f illustrates the repetitive switching between under opposite current pulses. These results demonstrate that unconventional orbital torques from FR materials offer a viable route to field-free magnetization switching.
Star City EP Teases The 'Unconventional Love Story' Ahead For Anastasia And Sasha -- Plus, Grade The Spin-Off's Premiere - TVLine
You might not guess it from watching the first two episodes of Apple TV's "Star City," but it seems we're witnessing the beginning of a great love story. "Star City," a spin-off of the streamer's alt-history drama "For All Mankind," launched with two installments on Friday, chronicling the mothership's same story -- what if America didn't win the Space Race? -- but from the Soviet perspective. During the premiere, polar-opposite cosmonauts Anastasia Belikova (Alice Englert) and Sasha Polivanov (Solly McLeod) are forced by the Soviet Union to marry each other, largely because Anastasia -- the first woman to walk on the moon -- is unwed, and the Soviets can't have that. "It's actually inspired by a true story we couldn't believe, like a lot of the stories in 'Star City,'" series co-creator Ben Nedivi shares with TVLine. "The idea of a single woman being a hero to the Soviet Union was not possible. There was a situation where they forced cosmonauts to get married. And it felt like, 'Oh, that's something we have to dip our toes into as writers.'"
SpaceX IPO filing lays bare losses and Musk control as it stakes future on AI
May 20 : SpaceX took the wraps off its IPO filing on Wednesday, laying bare for investors just how much Elon Musk is losing on artificial intelligence while betting the company's future on transforming the rocket maker into an AI powerhouse. Much of its outlook relies on SpaceX dominating technologies and markets that do not yet exist - from Mars missions to AI data centers in space. For many, Musk's record turning Tesla into the most valuable auto company in the world and developing the world's first fully reusable rocket and largest satellite network is enough to justify investment. The filing cements Musk's tight control of SpaceX while giving shareholders little say over his decisions. It shows just how central AI has become following the February purchase of xAI, which drove most of the company's spending and a majority of its losses in the first quarter. The listing could become the first U.S. market debut above $1 trillion and would immediately make SpaceX one of the world's most valuable publicly traded companies. Of SpaceX's three divisions, only the connectivity segment powered by satellite internet unit Starlink was profitable in the first three months of the year. While Starlink generated an operating profit of $1.19 billion, it wasn't enough to prevent the company from booking a total operating loss of $1.94 billion in the first quarter on $4.69 billion in revenue. Its AI division, alone, accounted for $2.47 billion in losses on $818 million in revenue. Musk's purchase of his social media and AI company xAI gave SpaceX new capabilities and opportunities but a staggering amount of spending, accounting for 76 per cent of its $10.1 billion in capital spending in the first quarter, as well as fresh losses. The company's plans rely on technology that's not yet been built for much of its future revenue stream, including operating data centers powered by solar power in space, to reach a potential market of $28.5 trillion, according to the filing. The filing confirmed a series of recent Reuters reports about the IPO. SpaceX has grown into the world's largest space business since its founding in 2002 by launching thousands of Starlink internet satellites. Its pioneering use of reusable rockets has transformed the economics of space, forcing competitors like Jeff Bezos' Blue Origin to play catch-up. A successful share sale could value the company at a record-setting $1.75 trillion, which would put its founder on track to become the first trillionaire in history. Musk will also retain 85.1 per cent of the combined voting power of the company, the filing showed. The company's regulatory disclosure comes during a critical week for the rocket maker, which is preparing to launch a test flight of its next-generation Starship rocket on Thursday. The board has given Musk control over the company, but has tied much of his compensation to audacious targets of establishing a permanent human colony on Mars and building space data centers with compute capacity powered by the equivalent of 100 terawatts, or 100,000 one-gigawatt nuclear reactors, Reuters previously reported. SpaceX is aiming to list its shares as early as June 12, with a roadshow launch targeted for June 4 and the share sale expected as early as June 11, Reuters reported last week. 'HALO EFFECT' Musk's CEO celebrity persona may matter more to some investors than SpaceX's underlying business fundamentals, analysts and academics said, because there are no other comparable companies against which to benchmark its valuation. "There is somewhat of a halo effect around Musk and his unconventional vision," said Reena Aggarwal, a finance professor at Georgetown University. "It is difficult to value companies like this because there is no peer group for comparison." The $1.75 trillion valuation target, if achieved, would eclipse Saudi Aramco's 2019 offering, which set a record for the world's biggest IPO when it debuted on Riyadh's exchange at a value of $1.7 trillion. SpaceX had planned to try to raise more than $75 billion in the offering, Reuters previously reported. SpaceX will use a dual-class share structure that gives Class B shareholders 10 votes each, concentrating control with Musk and a handful of other insiders, while Class A shares sold to public investors will carry one vote each, the prospectus showed. The company has adopted numerous provisions that, taken together, severely limit shareholder rights, forcing legal claims through arbitration, restricting where cases can be filed and protecting Musk from being fired by anyone other than Musk. The scale of the offering has drawn attention to the increasingly interconnected structure of Musk's business empire, often dubbed the "Muskonomy," which includes leading electric vehicle company Tesla, as well as his businesses in AI and brain-chip implants. SpaceX merged with Musk's AI startup xAI in a deal that valued the rocket company at $1 trillion and the developer of the Grok chatbot at $250 billion. Through its AI infrastructure platform, SpaceX inked deals for Anthropic to pay it $1.25 billion a month to use compute capacity from its Colossus and Colossus II data center clusters in Memphis, Tennessee through May 2029, the filing shows. The company disclosed that it had been named as a defendant in multiple lawsuits arising from its AI chatbot Grok's image-generation and editing features. INTENSIFYING SPACE RACE The race to commercialize space has intensified as private companies led by SpaceX and Blue Origin compete to slash launch costs, deploy satellite networks and secure government contracts. SpaceX's revenue is driven by Starlink, the world's largest satellite operator. The network of about 10,000 satellites offers broadband internet to consumers, governments and enterprise customers. But the company's expanding footprint across aviation, maritime and enterprise markets is helping turn capital-intensive space projects into a recurring revenue engine. SpaceX plans to earmark a significant portion of shares for retail investors and will host about 1,500 of them at an event in June, Reuters reported in April. The company is expected to list on the Nasdaq and Nasdaq Texas under the ticker symbol "SPCX." Goldman Sachs, Morgan Stanley, Bank of America, Citigroup and J.P. Morgan are the bookrunners.
SpaceX IPO filing brings Musk's interplanetary ambitions to Wall Street
May 20 : As Elon Musk's SpaceX races toward what could be the largest IPO in history, its filing delivers a rare mix of hard financial data and bold ambitions of exploring the frontiers of space. The filing's references to lunar missions and Mars settlement echo the popular space-age futuristic themes of "The Martian" and "Interstellar," while grounding those ambitions in the more familiar language of commercial space development. The company identified asteroid mining, in-orbit manufacturing and energy production on the moon and Mars as potential future opportunities, even though these ventures appear nowhere near feasibility. The language in the filing at times veered from conventional corporate disclosure to warnings of existential peril. "We do not want humans to have the same fate as dinosaurs," the company said, as it made a case for interplanetary travel. The part-balance-sheet, part-science-fiction nature of the paperwork is just one of several signs that show how unprecedented SpaceX's IPO truly is, in terms of size, ambition and business model. It is also consistent with Musk's public persona. The billionaire founder and CEO of the company has developed a reputation for being unconventional, which has earned him a loyal following but also unsettled investors at times. "Love him or hate him, Musk is definitely not boring, and his capacity to spin business narratives that seem outlandish at first hearing but become conventional wisdom later clearly adds to the allure of SpaceX," said Aswath Damodaran, a finance professor at New York University's Stern School of Business. Projections deemed excessively rosy can backfire, as hard-nosed investors are often more interested in tangible details about a company's business than grandiose visions. SpaceX acknowledged the risk, warning that many of its initiatives depend on technologies that are either nascent or do not yet exist, and may never become commercially viable. Musk's solid track record at Tesla, however, has turned some of his hardened skeptics into believers, with "never bet against Elon" becoming a common saying on Wall Street. Since SpaceX has no obvious public market peer, the projections may be treated as a necessary part of the pitch rather than a flaw. "Very little captures public imagination like space travel, and I think investors will want that in their portfolios. SpaceX will be pitching itself as a generational company, one with a long-term vision for investors to hold onto for 20 or 30+ years," said Matt Kennedy, senior strategist at Renaissance Capital, a provider of IPO-focused research and ETFs. The IPO is expected to raise a record $75 billion at a valuation of roughly $1.75 trillion, placing SpaceX among a handful of the world's elite kilocorn companies and sending ripples across nearly every corner of the equity markets. The rocket and satellite maker accelerated the timeline for its IPO earlier this month. It is now aiming to list its shares as early as June 12 on the Nasdaq, Reuters reported, citing sources.
