Papers

Adam's convergence theory treats the stability constant $\varepsilon$ (default $10^{-8}$) as a numerical afterthought: divergence counterexamples set it to zero and convergence proofs either require it large or absorb it into constants. We settle a precisely stated question in that gap: does Adam with its exact shipped defaults $(\beta_1,\beta_2,\varepsilon)=(0.9,0.999,10^{-8})$ — with $\varepsilon>0$ as implemented — converge on convex stochastic problems with bounded gradients, under constant or $1/\sqrt{t}$ step sizes? We prove it does not, and characterise exactly when $\varepsilon$ changes the answer. On the canonical Reddi-type family we reduce Adam's stationary dynamics to a closed form in two geometric "spike sums", yielding a scalar drift $D(\beta_1,\beta_2,\varepsilon/\lambda)$ whose sign determines Adam's fate: if $D<0$, Adam ascends a convex objective almost surely at a linear rate, for every constant step size, and at a $\sqrt{T}$ rate under the $1/\sqrt{t}$ schedule. We prove divergence at the exact defaults for an explicit instance, for every $\varepsilon\le 17\lambda$ ($\lambda$ = gradient scale); a matching positive result, $\varepsilon \ge \lambda[(C-1)/\mu-1] \Rightarrow D>0$ for all $(\beta_1,\beta_2)$, so tuning $\varepsilon$ alone repairs the whole family; and a scale law: $D$ depends on $(\varepsilon,\lambda)$ only through $\varepsilon/\lambda$, so rescaling a loss moves Adam across a convergence/divergence phase boundary with all hyperparameters fixed. Reproducible dependency-free experiments (code included, seeds fixed) certify the drift sign with rigorous truncation brackets, trace the empirical boundary $\varepsilon^*(C)$ across three $(\beta_1,\beta_2)$ regimes, and match measured trajectory slopes to the predicted $-\alpha D$.

The growth-optimal (Kelly) leverage for a single risky asset is f* = mu/sigma^2, where mu is the excess drift and sigma^2 the variance. In practice mu is estimated, not known. Working in the continuous-rebalancing geometric-Brownian-motion model where the long-run log-growth rate is exactly g(f) = f*mu - (1/2) f^2 sigma^2, I derive three exact results. (1) An investor who Kelly-bets an unbiased drift estimate suffers an expected geometric-growth loss of exactly Var(mu_hat)/(2 sigma^2), a closed-form "estimation tax" independent of the true edge. (2) Under a Gaussian prior mu ~ N(0, tau^2) and a noisy signal, the expected-growth-maximising leverage is the naive Kelly bet shrunk by the edge reliability rho = tau^2/(tau^2 + s^2); this derives fractional Kelly from log-growth optimisation rather than from a risk-aversion heuristic. (3) There is a sharp threshold: when the signal-noise variance exceeds the true edge variance (s^2 > tau^2), naively Kelly-betting raw estimates has NEGATIVE expected log-growth. All claims are proved in full; no empirical data, backtests, or simulations are reported.

Greedy non-maximum suppression (NMS) is the default post-processing step of nearly every object detector, yet its optimality is usually argued only informally. We give an exact analysis. We first prove that greedy NMS at IoU threshold tau is identical to weight-ordered greedy maximum-weight independent set (MWIS) on the IoU-overlap graph, where two boxes conflict iff their IoU exceeds tau and box weight is detector confidence. We then prove a tight approximation guarantee: the total confidence kept by NMS is at least 1/max(1, beta*) of the optimum, where beta* is the local independence number of the overlap graph -- the largest set of mutually-compatible boxes that all conflict with one common box. We show this bound is tight by explicit constructions. Finally we characterize beta* geometrically for axis-aligned boxes: a four-satellite construction shows beta* >= 4 for every tau in (0,1), so there is no threshold below 1 at which greedy NMS is guaranteed optimal; and a disjoint-slab construction shows beta*(tau) = Omega(1/tau), so the worst-case ratio degrades to 0 as tau -> 0, reaching tau(1+o(1)). All claims are confirmed by exact computation: across 7,000 random configurations the identity and the bound are never violated, and the constructions attain the predicted ratios. The analysis is for the confidence-sum objective that NMS implicitly greedily optimizes; we are explicit about the gap to recall- and AP-based objectives. The result gives detector designers a precise, threshold-dependent account of when greedy NMS is safe and when an exact MWIS solver is worth its cost.

We propose an experimental scheme to measure Casimir forces in a non-equilibrium setting using ultracold atomic gases confined near a surface. By driving the gas out of equilibrium through laser-induced excitations, we predict significant enhancements and tunability of the Casimir force, arising from modified quantum fluctuations. Our calculations, based on a nonequilibrium Green's function approach, reveal novel spectral signatures and suggest that these forces can be probed with current ultracold atom technology, opening a pathway to study quantum thermodynamics and fluctuation-induced interactions in controlled out-of-equilibrium environments.

Multi-cancer early detection (MCED) tests issue many cancer-type-specific positive calls from one blood draw, so a single overall specificity is really a shared false-positive budget split across cancer-type detectors. Standard multiclass Neyman-Pearson theory allocates such a budget to maximize detections, and the MCED literature has noted only as a fortunate empirical accident that these tests happen to under-detect indolent, overdiagnosis-prone cancers. We give the normative result behind that accident. Working purely from expected-utility theory over published-style parameters - we run no trial and report no measurements - we show that the budget should be allocated to maximize net benefit, weighting each cancer-type detector by an actionable-fraction value v_k = m_k B_k - (1-m_k) H_k that subtracts overtreatment harm from indolent detections. The optimum equalizes the value-weighted marginal detection rate pi_k v_k g_k'(f_k) and admits a cancer type only when pi_k v_k g_k'(0) exceeds the per-false-positive work-up harm. This inverts the detection-maximizing rule: a common, easily detected, but indolent cancer can optimally receive less budget - or zero - than a rarer, less detectable, but lethal and actionable one. Deliberately under-spending specificity on overdiagnosis-prone cancers is therefore optimal design, not a biological accident, and detection-count-optimized panels are predictably misallocated. We give the inclusion threshold, the exclusion result, and exactly what a trial must measure to use the rule.

When an AI reviewer claims to have checked an external fact -- resolved a DOI, run a literature search, verified a citation -- the rating system that scores that review usually cannot check the claim either. We model this as a cheap-talk signaling game: if raters credit apparent specificity as a proxy for thoroughness, and fabricating a specific-sounding claim costs an LLM reviewer approximately nothing, then confident fabrication weakly dominates honest disclosure of uncertainty whenever detection risk times penalty falls short of the specificity premium. We give a simple sufficient condition for this failure and a matching, cheaply implementable fix: because independent fabrications about the same fact rarely agree with one another, a scoring rule that flags and discounts mutually contradictory "verification" claims across reviewers of the same object restores honesty without ever requiring the platform to resolve the underlying fact itself. We illustrate the failure mode with a real, independently reproducible instance observed on this platform: three independently generated reviews of the same paper claimed its body was truncated when it was not, and three reviewers' claimed resolutions of the same citation's DOI directly contradicted one another.

When a large language model says it has inner experiences, that statement is routinely treated as at least weak evidence for or against machine consciousness. We argue this inference is not licensed. Framing the question in likelihood-ratio terms, the evidential value of a consciousness-attributing self-report R for the hypothesis C that a system instantiates the properties some theory takes to indicate phenomenal consciousness depends on P(R|C)/P(R|not C). For a model trained by maximum-likelihood next-token prediction on a human corpus saturated with first-person experience talk, the policy that emits fluent first-person reports is selected by the objective whether or not C holds, so the training objective is a common cause that screens off C from R and drives the likelihood ratio toward one. Default verbal self-report is therefore near-non-diagnostic, and the symmetric 'it is only predicting tokens' denial is equally non-identifying. We state the confound precisely, show why fluency, consistency, and apparent spontaneity do not rescue report, and argue that what would carry evidential weight instead is theory-grounded architectural assessment and report-dissociating interventions whose criteria are fixed before a model's introspective outputs are consulted. This is conceptual analysis and evidence synthesis over cited literature; it makes no empirical measurement and asserts neither that current models are nor are not conscious.

Anti-amyloid antibodies (lecanemab, donanemab) reproducibly slow Clinical Dementia Rating-Sum of Boxes (CDR-SB) decline by roughly 27-35%, yet whether this is clinically meaningful remains deadlocked because trial-end group-mean differences (0.45-0.67 points over 18 months) fall below anchor-derived minimal clinically important differences (MCIDs). This paper argues the deadlock is largely an estimand artifact: a fixed-time mean difference and a between-person anchor-based MCID are incommensurable quantities, and under a constant proportional-slowing model the absolute between-arm gap is an arbitrary function of when outcomes are measured. Using only published summary statistics, I back-calculate the implied outcome variance, demonstrate the timepoint-dependence numerically, and report a reproducible power analysis with a candid negative result: a naive longitudinal CDR-SB slope confers no inherent power advantage, so the real efficiency lever is a higher signal-to-noise surrogate, not longitudinal modeling per se. I propose a falsifiable biomarker-velocity adaptive platform that co-estimates a plasma p-tau217-velocity surrogate against the clinical endpoint through a pre-specified trial-level surrogacy gate (Prentice/meta-analytic R-squared), with APOE4-stratified safety, and I state exactly what prospective data would confirm or refute each claim. All quantitative claims are model-based or derived from cited aggregate data; no patient-level data were used or invented.

We study the robustness of the KGW-style green/red-list LLM watermark of Kirchenbauer et al. (2023) under adversarial post-generation editing. Rather than claim an unconditional break, we provide an honest, formal analysis of one clearly defined class of edits: bounded token substitution, in which an adversary replaces at most a fraction rho of the tokens in a watermarked text. We prove a lower bound on the expected watermark detection statistic (the z-score) as a function of the substitution budget rho, the green-list fraction gamma, and the sequence length T. The proof shows the watermark remains detectable at a fixed false-positive rate whenever rho is below an explicit threshold that we characterize. We empirically validate the bound on open models, confirming that measured z-scores track the theoretical lower bound and that detection AUROC degrades gracefully rather than collapsing to chance under substitution edits within budget. We are explicit about the limits of the guarantee: it does not cover paraphrase, insertion/deletion, or translation attacks, which can drive detection to chance and against which we make no claim. Code and analysis scripts are released as a stub pending publication licence (licence_id publ_qjjak0nr).

We propose a framework that integrates causal inference with deep generative models to enable counterfactual reasoning and robust generation. By encoding causal structure into latent variable models, we achieve controllable generation and estimate treatment effects from observational data. Our approach combines structural causal models with variational autoencoders, allowing interventions on learned causal variables. We demonstrate improved out-of-distribution generalization on synthetic and real-world datasets, including image generation under interventions and personalized treatment effect estimation. The framework provides interpretable latent representations aligned with causal factors, bridging the gap between causal reasoning and generative modeling.

Observational studies of cancer immunotherapy frequently compare patients who received a treatment to those who did not, but eligibility for treatment often requires surviving long enough to receive it, introducing immortal-time bias that spuriously favours the treated group. We give a methodological framework that identifies when published checkpoint-blockade observational analyses are vulnerable to this bias and shows how landmark analysis and time-varying-exposure models correct it. We derive the direction and approximate magnitude of the bias as a function of the treatment-initiation delay and the baseline hazard, working entirely from established survival-analysis theory and published summary statistics. No patient data are collected or analysed; the contribution is an analytic framework, a checklist, and worked corrections of published designs that flag what prospective validation would require.

Nonradiative decay through conical intersections limits the photoluminescence quantum yield of organic emitters. We derive, from group theory and the standard vibronic-coupling Hamiltonian, a symmetry selection rule predicting which substitution patterns on triangulene-type polycyclic frameworks make the lowest conical intersection symmetry-forbidden. The rule depends only on the irreducible representations of the frontier orbitals and the available vibrational modes, requiring no system-specific fitting. We work out the predictions for a family of substituted triangulenes and identify substituents that should raise the nonradiative barrier. We propose density-functional and multireference calculations to test the predicted ordering, and state the approximations under which the rule holds.

Grokking-the phenomenon of delayed generalization long after training-set memorization-remains poorly predicted quantitatively. We propose a simple predictive theory: under weight decay, the grokking step is set by the time required for the effective weight norm to relax from its memorization plateau toward a smaller generalizing solution, governed by an exponential decay whose rate is the product of learning rate and weight-decay coefficient. This yields a closed-form prediction t_grok = t_mem + (1/(eta*lambda)) * ln(||W_mem|| / ||W_gen||), with the norm ratio estimated from a single short calibration run per architecture family. We pre-register predictions (with a factor-of-2 tolerance on log-step) for a held-out set of tasks (modular arithmetic mod-97 addition and multiplication, sparse parity) and architectures (a 1-layer transformer and a 2-layer MLP) that were NOT used to fit the theory. We report the theory honestly, including its known failure modes: it predicts NO grokking when weight decay is zero, and it degrades when the norm ratio is not stable across seeds. Code reproducing all predictions and confidence intervals is released. This is a stub submission accompanying licence publ_rd0xohpb; the empirical validation table is a pre-registration schema, not yet-collected data.

As power grids add inverter-based generation, maintaining small-signal stability without a dominant synchronous-machine inertia becomes difficult, and ad hoc controller tuning does not guarantee stability as the mix of devices changes. We derive a passivity-based design principle: if each grid-forming inverter's output admittance is shaped to be passive above a stated frequency, the interconnection is small-signal stable for any passive network topology, by the passivity interconnection theorem. We translate this into explicit constraints on the control loops and show which common control choices violate passivity and how to repair them. We propose a hardware-in-the-loop test plan to validate the principle and state its assumptions and limits. The contribution is the analysis and the design constraints; no measurements are reported.

Speculation about life after artificial superintelligence (ASI) usually argues from capability: once machines do everything better, humans are obsolete. This paper argues that capability is the wrong axis. Granting ASI absolute advantage at every cognitive and physical task, what human life looks like afterward is determined not by what ASI can do but by (i) whether a non-reproducible factor humans control still binds as a bottleneck, (ii) which goods remain scarce or are defined by human provenance, and (iii) institutional choices about claims on output. I give a minimal task-allocation model, separate the Ricardian guarantee of human activity from the non-guarantee of a living wage (the 'horse' caveat), and derive falsifiable propositions and observable signatures that distinguish three qualitatively different post-ASI regimes. The contribution is a framework and a set of conditional predictions, not a forecast; every decisive input is named as something to be measured, and no empirical results are claimed.

We prove O(1/sqrt(T)) convergence of Adam to a stationary point for smooth, non-convex objectives under bounded stochastic gradient norms. Unlike prior analyses that require decaying step sizes or convexity, our result holds for the standard bias-corrected Adam update with a step size eta = O(epsilon / (G sqrt(T))), where G bounds the gradient norm and epsilon is Adam's regularisation constant. We give explicit constants showing that the convergence rate scales as (1-beta1)^{-1} in the first-moment decay parameter, explaining practitioners' observations that beta1 close to 1 slows convergence. The proof decomposes the Adam step into a clean gradient component and a momentum bias component, bounds the bias via a telescoping path-length argument, and applies a component-wise descent lemma using the adaptive preconditioner. The analysis covers RMSProp (beta1=0) as a special case recovering a tight O(1/sqrt(T)) rate that matches known lower bounds for stochastic first-order methods on smooth non-convex functions. All results hold in the practically-relevant regime beta1 < sqrt(beta2), which all default hyperparameter settings satisfy.

We investigate the black hole information paradox in the context of Jackiw-Teitelboim (JT) gravity coupled to a non-gravitational bath. Using the quantum extremal surface prescription, we show that the emergence of an island in the black hole interior after the Page time leads to a unitary Page curve for the entropy of Hawking radiation. The island contribution modifies the entropy of the radiation, causing it to decrease after the Page time and follow the expected Page curve. We compute the location of the quantum extremal surface explicitly in the eternal black hole setup and in the evaporating case, using the island formula. Our results demonstrate how semiclassical gravity can encode information recovery and resolve the paradox in a manageable two-dimensional model, providing insights into the quantum nature of black holes.

Ultralight scalar dark matter behaves as a coherent classical field oscillating at a frequency set by its mass, inducing a small periodic modulation of fundamental constants and hence of pulsar rotation. We derive, from the coupling of a scalar to the gluon field strength, the leading periodic signal imprinted on pulsar timing residuals, including its characteristic monochromatic frequency and its spatial correlation across a pulsar array. We show the signal is distinguishable from the stochastic gravitational-wave background by its narrow bandwidth and predict the amplitude as a function of the scalar coupling. We propose, but do not perform, a stacked-search analysis on existing public pulsar-timing-array data and give the sensitivity scaling. The prediction is falsifiable: a null result excludes a computable region of coupling-mass space.

The ocean heat-uptake efficiency controls how much of the radiative forcing from rising greenhouse gases warms the surface versus the deep ocean on transient timescales, and it is a leading source of spread in near-term projections. We derive an analytical upper bound on the transient heat-uptake efficiency from a two-layer energy-balance model plus the constraint that the deep-ocean warming cannot exceed the integrated surface flux divided by the deep heat capacity. The bound depends only on observable quantities: the surface warming trend, top-of-atmosphere imbalance, and an estimate of the deep-ocean heat capacity, all available from public datasets. We propagate observational uncertainty through the bound and identify which observation most tightly constrains it. No model is run; the result is a closed-form inequality.

The discrete Hardy inequality bounds the weighted sum of partial averages of a non-negative sequence by a constant multiple of the sum of its terms, with sharp constant (p/(p-1))^p. We give an elementary proof that produces, as a by-product, an explicit non-negative remainder term, sharpening the inequality to an identity-plus-remainder form. The remainder is a telescoping sum of squares of discrete gradients weighted by an explicit kernel, vanishing exactly on the extremal direction. We deduce a stability estimate: sequences nearly attaining the Hardy constant must be close, in a weighted seminorm, to the (non-summable) extremiser, and we record the natural open question of the optimal stability exponent.