An empty region in a ceramic shape map cannot distinguish what potters could not make from what they did not make or what the record omits. A study of 917 Sue-ware point clouds separates recording loss from representation effects: ordinary PCA preserves complete-map neighbourhoods well, while clipping and normalization change them substantially.

Retaining rental apartments can finance common repairs, but the apartments have an opportunity cost and income alone does not ensure that work happens on time. A 60-year renewal model compares equally funded arrangements, followed by a proposal for a worldwide steward with segregated client assets, explicit fees and replaceable management.

The far right has outpolled the far left in most European elections since 2010, but the ratio between them is unstable and reverses in the most polarized cases. A three-state choice model separates entry into radicalism from the choice of side, shows that no vote distribution can reveal feedback or a tipping point, and yields experiments that can tell continuity-seeking, social amplification and active inference apart.

In English books the word computer moved from people to machines in the 1950s, two decades before the four-colour theorem and while human computers were still employed; in 2026 the word mathematician shows the contrastive marking that preceded that shift without its first step, since machines are not yet called mathematicians. Five separable transitions, the 2026 record of machine mathematics, and three falsifiable claims for 2036.

For a decision variable that accumulates evidence to a bound, the psychometric slope equals mobility times bound over diffusion, a ratio that survives any affine rescaling of a neural decoder. Optional-stopping estimators recover diffusion and mobility from spiking populations without choices and predict the slope fitted to choices; naive estimators fail by large factors, the prediction tracks behaviour under leak, instability and drift variability, and urgency or a choice-irrelevant signal on the coding axis open gaps of about 10%.

A derived work's value splits into inherited, independent and interaction value, which a four-exposure randomization identifies, and an externality on the source, which needs a different design. A Blackwell garbling cannot raise a Bayesian viewer's value, and attention reverses this only when its price attaches to the input stream. Shapley allocations are not identified by any experiment, a four-module factorial resolves shares and main effects at feasible sample sizes but not the four-way dividend, and field experiments of a few hundred videos detect only large effects on the source.

N equivalent catalytic sites that share holes at random cost a factor N in memory lifetime, and a factor of N to the power m minus one in weak-light flux for a catalyst that stores m oxidizing equivalents; fixed asymmetry or positive state-dependent routing removes most of it, negative routing makes it worse, and a flash-interval series on a two-sided photosystem separates the rules.

Four dynamical invariants, between-region recurrence, perturbational repertoire, endogenous temporal depth and stable macrostructure, each with an estimator; none follows from the others, and testing them jointly has a definite cost in margin, labels, sample size and timing analysis.

A system is a machine under an observation when its dynamics close on the observed variables. The states the observation leaves out keep their own dynamics, controllers designed on the quotient act on them, and making a system fit its description can empty them.

Most neural states that differ only in fine or within-column structure produce the same mesoscale electric field, so a stable field over drifting neurons needs no feedback. Equal-time covariance cannot reveal field stabilization; lagged field statistics and matched perturbations can.

Storing and computing a quadrillion-parameter model become manageable at activation fractions near one in ten thousand, but the data do not exist, and two bounds do not yield to cluster size, one on sparsity when weights are streamed and one on checkpoint-restart efficiency.

Two fine patterns leave a coarse difference mode that keeps their relative phase. Fed back to its generators it locks them as a phase-locked loop does, but only against a positional reference, which a memory written by an earlier pattern can supply.

In a bistable model of recovery, bounded support succeeds only above a computable threshold and slows sharply near it. Untreated data cannot identify its effect, confounding by indication can reverse it, and distance to recovery depends on the units of the state.

Resisting manipulation and accepting legitimate correction are the correct-rejection and hit rates of a single detector. At fixed discriminability one can be improved only at the cost of the other, and the difference-of-rates score used in published benchmarks does not measure discriminability.

Under rising ground rent a site removes the functions it cannot charge for, so seating and other ways of staying go first and closure comes last. Removing them reduces the ecological burden of a meal by about two per cent, and obligations tied to on-premises capacity can be avoided by removing it.

Nonviolent campaigns remove rulers more often, while armed revolutions that succeed are reversed less often. A model combining both findings shows that escalation raises the chance of removal only in a narrow range of conditions, and in that range it sharply lowers the chance of an accountable successor.

Concentrated private power can narrow which futures a polity is able to consider. An oligarchic attractor is a stable state kept apart from better alternatives by organized wealth, with measures and falsification tests for it.

A specification for a model that could test whether climate change might cause a billion deaths in one year: coupled food, trade, infrastructure and governance failures, competing-hazard mortality, rare-event sampling and paired attribution ensembles.

Arguments about AI and work often assume that whatever machines cannot yet do is a lasting human domain. Three benchmark models show that equal access to intelligence does not narrow the distribution of consumption, and that the share of tasks people hold can fall toward zero even though some task always remains.

An exhaustive enumeration of all 4140 encoders of a small predictor shows that better prediction can reduce what a representation retains, that the distinction between required and optional capabilities disappears with a small tolerance, and that information present in a representation can be unusable through the predictor's output.

A 2004 games console can now run a small language model because of software developed two decades later. Treating capability as a relation among machine, artifact, task, budget and date, three models show which techniques the capability depends on, why steady efficiency gains produce sporadic capability jumps, and why deployable capability decays as software stacks deepen.

Physics already turned one family of names into a grammar: the Particle Data Group names 214 hadrons from 24 base symbols. A model of name design and adoption shows why layered records spread where replacement terms stall.

Gender identity modelled as a preference over future observations instead of a belief about the present body. The model shows why mismatch persists, and why behaviour under social threat cannot certify identity.

Body ownership, agency and identity treated as coordinates of one measurable space. A morphology-congruence model explains weaker rubber-hand ownership in furries, and a design analysis shows the crossed experiment needs about 25 participants per group.

Named baits such as clickbait and ragebait treated as compounds of about 32 appraisal elements. A registry of 40 baits shows that the vocabulary cannot identify most element interactions, and user habituation breaks the rage-farming lock.

What would it mean mathematically for the electron to be indivisible? Exact computations show that charge labels, irreducibility and tensor-primality all fail as criteria, and motivate a duality-invariant hierarchy built on 2026 exceptional-Jordan results.

Damaging a decomposed controller to locate where competence lives, tested exhaustively on all 3,674,160 pocket-cube states. Locally correct agents solve 61 states; only damage and noise move the basin, and every result is deflationary.

Trade-based dependence indices score an embargoed machine as zero exposure. A capability index recovers true expected loss exactly, shows that imports build or erode capability depending on absorption, and argues against any scalar dependence measure.

Founder-controlled public companies separate voting control from economic exposure. Models of the dual-class lifecycle, exchange competition and correction capacity show sealed control is best in calm and worst in failure, and favour renewable sunsets.

Self-rule through one another: autonomy defined by four conditions, each tested by deletion in simulations of school sorting, institutional exit and epistemic dependence, as an alternative to both guardianship and the sovereign chooser.

Freedom as reachability: the consequential states an agent can reach through existing institutional pathways. A landscape model shows universal permission with half the population reaching nothing, price-neutral landscape power, and why open substrates need funded maintenance.

A constructed self is still real. A ladder from self-production to constitutive access defines when a system can deliberately rebuild itself; a minimal model measures the reach, traps, repair and capture risks of self-construction.

Europe stopped building reactors before Chernobyl and lost the capability to build. A calibrated model shows the loss hid behind record output, locked after a critical freeze, priced every restart as confirmation, and cost 24% more than continuity.

Obligation begins where power over an entity exceeds knowledge of what it is. A first-contact simulation shows surface legibility inverting protection, and supports xenial standing: a procedural status binding whoever holds irreversible power over an unclassified candidate.

Content-neutral interventions on a belief formation's reproduction number act identically on true and false claims. Raising the rate at which claims are decisively tested suppresses refuted formations while raising the absorption of vindicable ones.

Pigeons outlived many of the institutions that employed them. Comparing their uneven decommissioning with retired horses, military dogs and research chimpanzees separates the power to dispose of a working animal from responsibility for it after work.

Agency is invisible in a system at rest and measurable only under probes that make rival organizational models predict differently. A gridworld demonstrator attributes evidence and capacity across components and recovers planted mechanisms blind.

Systems that edit the rules governing their own dynamics are formalized as a state and a law the state rewrites. Four niche-construction models show sign-reversed selection, a persistence threshold, compounding law-editing gain and play-driven game cycles.

Imagined transitions between distant places have no physical speed, since rescaling the represented world rescales the apparent velocity. A control model shows latency tracking control energy and neural path length while represented distance predicts neither.

Apparent free lunches in logic, statistics, robotics, and development are discounts: a system pays to interface with pre-existing structure and gains a family of outcomes cheaply. A vector-valued ledger and a seven-level inference ladder make the discounts measurable.

Work becomes foundational by becoming non-excludable and universally depended upon, which prevents its creators from charging for it. A model of tolls on a serially dependent stack shows open layers maximize welfare while fragmented tolls collapse throughput.

A speech act updates a world-state and emits an utterance; text keeps only the utterance. Processes that bind every promise, none, or half share one token distribution, so a text-only learner cannot recover the obligations their words create.

Generative AI raises what a learner can produce with a tool faster than what they can do or judge without it. Separating production, competence, and judgment frontiers makes this gap measurable by testing learners after the tool is removed.

Consistency of massless spin-2 fields fixes the form of gravity but not its strength. QCD running explains why the proton is light; the unexplained residue is one dimensionless ratio, C_G/C_p near 2.6e18, that no current theory computes.

The proposed autobiographical invariants of a driven system reduce to the minimal realization and its Krohn-Rhodes holonomy. Self-predictable observables are closed under common refinement but not under coarsening, so no canonical coarsest self exists.

When reward depends on the world only through a partition, expected return sees belief only through its pushforward. In an exact POMDP, two policies with equal return 0.8 differ in Fisher-Rao belief geometry and in zero-shot transfer, 0.5 against 0.85.

A minimal viability model in which a local loss becomes higher-scale control only when observation, memory and constraint-change close one feedback loop with gain above threshold; removing any one of the three empties the viability kernel.

The sorites paradox treats readout-invariance as state-invariance. Simulations of leaky integrators, a tolerance relation, a soft threshold and a slowly driven bistable state show latent accumulation that a coarse readout does not register until it flips.

Agency scored as a log-likelihood ratio between goal-directed and passive models is invisible to observation (AUC 0.49) and revealed by goal interventions (AUC 1.0). Complexity and agency are negatively related across six simulated systems.

A mathematical structure counts as a world to the degree that it is dynamical, local, record-forming and self-legible. Finite-dimensional computations show a state generating its own time, a region's entanglement Hamiltonian forming a boost, and a history underdetermining its law.

A democracy's strategic disadvantage against an autocracy depends on turnover multiplied by the share of capacity lost at each handover. Cutting that loss, with elections unchanged, recovers most autocratic continuity; longer tenure would need 32-year leaders.

A small transformer tolerates fixed attention and late-layer weights, but local learning, negated gradients and noise-schedule order carry measurable costs; an SGD control shows that apparent gradient-scaling tolerance comes from Adam's normalization.

In a ten-item, seven-hole allocation model, closed holes that report zero load trap a greedy rule, leaving 5.03 items unplaced on average while the overload score improves. Noise encoding and pre-arrival cost comparisons explain further failures.

Mediated channels can deform belief without removing information. In a Kalman-agent model, attenuation, selection, amplification and recursion leave distinct signatures, and three of four channels displace the posterior while leaving the agent's confidence unchanged.

The modern Person is a composite of five historical formations: kinship, epic, Roman law, Christian conscience and Kantian autonomy. Today its practical supports thin unequally while formal legal personhood stays intact.

The digits of the electron's magnetic moment trace to Dirac, Schwinger, five orders of QED and a sequence of Penning traps; the comparison with theory is now limited by a 5.5-sigma disagreement between caesium and rubidium values of the fine-structure constant.

Traces price back through commensuration, valuation, allocation, and constraint, from cellular cost to gift debt, blood compensation, fiscal states, and finance; in a six-sector model, a finance price markup of 3 costs 8.08 percent of viability.

Compiles the earliest dated markers of eight hominin capacities; seven predate the earliest Homo sapiens fossils, with bipedalism leading the Jebel Irhoud boundary by 4.085 million years and at least five other lineages dated near it.

Traces how the identity of an artwork moved from finished object to rule-bound specification, through score-based and generative code art to model-mediated packages, and defines a protocol as substrate-relative, bounded, transferable, editioned, and governed.

Models post-collapse reconstruction as a cellular sheaf of institutional configurations plus an attractor landscape; in the sweep, the capitalist basin occupies 2.0% of the parameter cube, the smallest of eight macro-regimes.

A four-factor measure of owned causality, built from constraint closure, conditional directed information, counterfactual viability and re-entry, is zero for externally specified control and positive for toy systems that produce their own constraints.

Embodied and causal mathematics recast as ablatable JEPA auxiliary losses. Five preregistered tests find the obstruction gate inert, a cellular sheaf slightly harmful on categorical data, and no auxiliary loss clearly helpful on a variance-limited toy.

Comparing a model's response to editing the queried evidence with its response to the same edit elsewhere ranks correct retrieval answers well when the evidence position is supplied; when the model must locate the evidence itself, the advantage over entropy disappears.

Adult cognitive personhood is assembled in stages as a child internalizes regulation, language, norms, narrative, and cultural tools first supplied by caregivers and institutions; deprivation removes the upper layers and spares the proto-agent.

The history of geometry is traced as the progressive detachment of form from physical extension, from land survey to the representational spaces of organisms and neural networks. Five conditions distinguish a substantive space of possible forms from a spatial metaphor.

A Fisher-Rao model of prognosis separates outcome forecasts, intervention costs and recipient permission. An entropy floor constrains uncertainty without protecting autonomy, while a permission rule lets the recipient refuse, revoke or change a target.

Exact Shapley attribution in small production games shows labor's share of attributed value falling from 1 to 0.125 along the automation path, and bounds intelligence's share at 1/(k+1) when its k complements are strictly essential.

Arespect is the failure to perceive the cost and difficulty of an inherited enabling condition once it becomes a reliable interface. Its mechanism, stack blindness, is traced from atmospheric oxygen and mitochondria to fire, writing, and compilers.

Whether a group voted against its interests depends on three analyst choices, the interest function, voters versus bloc votes, and what voters could foresee; in three 2024 cases these choices flip or reorder most verdicts.

Firms price a possible sovereigntist government by state-dependence and mobility: dependent firms buy political access, mobile firms exit, so one rising probability produces both a sell-off and an access rush, and only responses to falling odds separate hedging from capture.

In a model of politically charged consumption, market data fix the political premium but cannot distinguish politics changing choice from politics changing taste; only blind-versus-labeled ratings identify the split, and brand activism loses demand under negativity dominance.

Advanced cognitive performances are realized by assemblages of people, notations, instruments, and institutions, and then credited to individuals through attributional compression, which persists because examinations, prizes, wages, and liability need a single body to credit.

A fossil exporter earns rent only while its customer stays unelectrified. In a bistable transition model, a patient exporter gains by holding its price below the importer's takeoff threshold, but global learning bounds the delay it can buy.

In creative fields, visible indifference to money is cheaper for those with family wealth. An agent-based model with talent independent of wealth reproduces a fourfold class gap in survival and shows that stronger purity norms amplify it.

AI breaks the inference from a good artifact to a competent student. A measurement model shows artifact validity collapsing with task substitutability, AI detectors failing on base rates, and sampled oral verification restoring validity at about 6.5 percent of full-review cost.

AI raises the evidence stages of the accountability chain but leaves prosecution, judgment and enforcement capturable. In a weakest-link model, an elite that captures those stages absorbs the entire evidence gain, so exposure rises while consequence does not.