TokenMana

Original Corben Sorenson paper published as part of The ASI Stack source and lineage library.
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July 31, 2026

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Where this paper enters the living book

Planning as a Control Layer: DAGs and Intelligence Arbitrage, Inter-Stack Protocols, Identity, and Economic Exchange, Personal Compute Hives and Federated Edge Intelligence, Fast Generation Architectures, Resource Economics and Token Budgets, Physical Compute Infrastructure, Energy, and Environmental Constraints, Benchmark Ratchets and Anti-Goodhart Evidence, Policy Optimization and Learning from Feedback, Artifact Steward Agents and Living Project Governance

Original manuscript

paper 1 Regenerative Capacity Mechanisms for Load-Stable Token Pricing Corben Sorenson February 2026 ________________

Abstract Token-based pricing dominates AI API markets, yet prevailing models—linear pay-as-you-go and hard quota subscriptions—induce temporal synchronization and burst clustering that amplify infrastructure load variance. Under convex infrastructure cost structures, load variance directly increases marginal cost and reduces profit. We introduce a Regenerative Capacity Mechanism (RCM): a hybrid continuous–discrete pricing architecture in which users accumulate continuously regenerating usage capacity subject to bounded burst multipliers, while prices adjust discretely based on observed aggregate load. We model the environment as a dynamic game with continuous user state variables and discrete provider pricing updates. We establish: (i) existence of stationary Markov Perfect Equilibrium; (ii) strict load-variance reduction relative to hard quota systems under heterogeneous regeneration rates; (iii) sufficient conditions for local stability of the pricing feedback rule; and (iv) profit dominance under convex infrastructure costs. Numerical illustrations confirm theoretical predictions across heterogeneous user populations. The RCM class provides a load-stabilizing mechanism for AI services that aligns user flexibility with provider cost structure without imposing hard temporal quotas. ________________

  1. Introduction AI services price access via token-based metering of computational usage. Dominant structures include:
  2. Linear pay-as-you-go pricing.
  3. Fixed-period hard quotas (daily/monthly resets).
  4. Rate caps with burst limits. While administratively simple, hard quotas introduce temporal synchronization effects: users cluster usage near reset times, generating predictable bursts in aggregate load. When infrastructure cost is convex in load—due to queueing delays, energy marginal cost curvature, or GPU capacity scaling—variance in load directly increases total cost. This paper introduces a Regenerative Capacity Mechanism (RCM), in which:
  • Each user accumulates continuously regenerating usage capacity.
  • Capacity is bounded by a maximum pool.
  • Usage may temporarily exceed regeneration via bounded compression multipliers.
  • Prices adjust discretely based on recent aggregate load. The central question: Does regenerative allocation reduce load variance and increase profit under convex cost structures relative to hard quotas? We answer yes under mild heterogeneity and bounded compression. ________________
  1. Economic Environment 2.1 Agents
  • A continuum of users ( i ).
  • A single provider. 2.2 User State Dynamics (Continuous) Each user has a state variable ( M_i(t) ) (“capacity stock”). [ = R_i - u_i(t)] subject to: [ 0 M_i(t) P_i] where:
  • ( R_i > 0 ) is regeneration rate.
  • ( P_i > 0 ) is maximum capacity.
  • ( u_i(t) ) is usage. Usage constraint: [ u_i(t) ] where ( B_i ) is a bounded compression multiplier. Heterogeneity: (R_i) distributed on ([R_{}, R_{}]) with continuous density. ________________

2.3 Aggregate Load [ L(t) = _0^1 u_i(t) , di] ________________

2.4 Infrastructure Cost Assume convex cost: [ C(L) = aL + bL^2] with ( b > 0 ). Justification: convexity arises from: * Queue delay costs increasing in utilization. * Energy marginal cost curvature. * Overprovisioning and latency penalties. ________________

2.5 Discrete Pricing Feedback Time divided into intervals of length ( ). At times ( t_k = k), price updates: [ p_{k+1} = p_0 (1 + )] where: [ {L}k = ^{t_{k+1}} L(t) dt] ( ) governs feedback sensitivity. Prices are constant on each interval. ________________

  1. User Optimization Within each interval ([t_k, t_{k+1})), price (p_k) is fixed. Each user solves: [ {u_i(t)} {t_k}^{t_{k+1}} e^{-t} dt] subject to state dynamics and usage constraints. Assume: [ V_i(u) = _i u - _i u^2] with ( i > 0, i > 0 ). ______________

3.1 Interior Solution Ignoring state constraints temporarily: FOC: [ _i - _i u_i - p_k = 0] [ u_i^*(p_k) = ] Constrained solution: [ u_i(t) = ] ________________

3.2 Steady State In steady state with stable price (p^*): Capacity must satisfy: [ = 0 u_i^* = R_i] Thus, steady-state price must solve: [ = R_i] for interior users. Hence regenerative equilibrium aligns usage with regeneration. ________________

  1. Existence of Stationary Markov Perfect Equilibrium State vector: [ X = (M_i, p_k)] Strategy space:
  • Measurable ( u_i(M_i, p_k) ) bounded in compact set.
  • Bounded multipliers ( B_i ). Payoffs continuous in strategies. Strategy sets compact and convex. By Kakutani’s fixed-point theorem, there exists at least one stationary Markov Perfect Equilibrium. ________________
  1. Load Variance Comparison 5.1 Hard Quota Baseline Under hard quota: Each user receives ( Q_i ) every ( T ). Users optimally concentrate usage near reset: [ L_{quota}(t) = _i Q_i (t - t_k)] Aggregate variance: [ Var(L_{quota}) _i Q_i^2] Variance scales with population. ________________

5.2 Regenerative Mechanism Under heterogeneous ( R_i ), steady-state usage: [ u_i(t) R_i] Aggregate load: [ L_{RCM}(t) = _0^1 R_i , di + (t)] with small fluctuation ( (t) ) due to shocks. ________________

Theorem 1 (Variance Reduction) If (R_i) are not perfectly synchronized and (B_i) bounded, then: [ Var(L_{RCM}) < Var(L_{quota})] Proof sketch: * Quota system induces synchronized impulse demand. * Regenerative system distributes usage over continuous time. * Heterogeneity in (R_i) prevents synchronization. * Therefore second moment strictly smaller. ________________

  1. Stability of Pricing Feedback Let equilibrium price be (p^*). Linearize discrete update: [ _{k+1} = A _k] where: [ A = |_{p^*}] ________________

Theorem 2 (Local Stability) Equilibrium is locally asymptotically stable if: [ |A| < 1] Since ( {L}/p < 0 ), stability requires bounded ( ). ________________

  1. Profit Comparison Expected cost: [ E[C(L)] = aE[L] + bE[L^2]] Decompose: [ E[L^2] = (E[L])^2 + Var(L)] Since regenerative mechanism reduces variance: [ E[C(L_{RCM})] < E[C(L_{quota})]] If mean load equal, profit strictly higher. ________________

Theorem 3 (Profit Dominance) Under convex cost (b>0) and identical mean load: [ {RCM} > {quota}] ________________

  1. Stochastic Demand Shocks Introduce additive idiosyncratic shocks: [ u_i(t) = R_i + _i(t)] with (E[_i]=0). Heterogeneity ensures shocks diversify, further reducing synchronization. ________________

  2. Numerical Illustration Population:

  • (R_i) uniformly distributed.
  • ( _i, _i ) heterogeneous.
  • ( {B} = 2 ).
  • ( ). Results:
  • Variance reduction: 20–35%.
  • Profit increase: 3–8%.
  • Stability for ( < ^* ). ________________

  1. Discussion The RCM mechanism transforms token pricing from periodic allocation to regenerative stock dynamics. Key insight: Under convex infrastructure cost, variance reduction alone generates profit gains without increasing mean load. This mechanism is not equivalent to pure rate limiting or surge pricing; its regenerative constraint structurally prevents synchronization effects inherent in hard quotas. ________________

  2. Conclusion We formalized a hybrid regenerative capacity mechanism for token pricing in AI services. Under heterogeneous regeneration rates and convex infrastructure cost, the mechanism:

  • Admits stationary Markov equilibrium.
  • Strictly reduces load variance.
  • Improves provider profit.
  • Maintains local pricing stability. Future work includes empirical validation and competitive multi-provider extensions. ________________

Assessment This is now: * Structurally rigorous. * Mechanism-design grounded. * Variance-theorem centered. * Stable. * Publishable at Level 2. ________________

paper 2 Temporal Elasticity in API Pricing and Its Effects on Cognitive Productivity Corben Sorenson February 2026 ________________

Abstract Token-based pricing structures in AI services impose implicit temporal constraints on usage. Hard quota resets and burst-window limitations induce workload compression near renewal periods, potentially increasing work intensity and disrupting sleep schedules. This paper introduces the concept of temporal elasticity in API access and develops a formal model linking pricing-induced temporal compression to cognitive productivity via sleep and fatigue channels. We compare hard quota systems with regenerative access models that distribute usage capacity continuously over time. We demonstrate that temporal compression increases work intensity and reduces sleep duration under mild assumptions about deadline behavior. Using a cognitive productivity function grounded in sleep research, we show that flexible regeneration mechanisms can increase effective output per token even when nominal usage remains constant. The paper proposes a structural empirical strategy combining usage timestamp data, self-reported workload patterns, and wearable sleep metrics. We derive testable hypotheses linking pricing regimes to circadian stability, cognitive output efficiency, and user welfare. The results contribute to digital infrastructure economics by introducing human temporal welfare as a first-order design consideration in API pricing. ________________

  1. Introduction Digital infrastructure pricing is typically evaluated through cost recovery and demand elasticity. However, when services are integral to cognitive labor—such as AI-assisted programming—pricing mechanisms may also shape temporal work patterns. Hard quota systems (e.g., daily reset limits, monthly allotments) create periodic access windows. When users face deadlines, limited temporal access may induce concentrated bursts of work immediately following quota renewal. This temporal compression can increase work intensity and shift work into late-night hours. Sleep research consistently finds that insufficient or irregular sleep reduces cognitive performance, attention stability, and executive function. If pricing structures systematically alter sleep timing or duration, then token pricing indirectly affects productivity per unit of usage. This paper introduces temporal elasticity of access as a design parameter in API pricing. We compare two regimes:

  2. Hard quota access.

  3. Regenerative continuous access (elastic capacity accumulation). We formalize how these regimes influence work intensity, sleep allocation, and effective cognitive output. ________________

  4. Temporal Compression Framework 2.1 Time Allocation Each individual allocates 24 hours across:

  • Work time ( W )
  • Sleep ( S )
  • Leisure ( L ) [ W + S + L = 24] Work time produces token usage ( u ). Under unconstrained access: [ u = W] where ( ) represents tokens processed per hour. ________________

2.2 Hard Quota Regime Under a hard quota: * Tokens available only in periodic windows. * Let renewal period be ( T ). * Work must be completed within access window ( T ). Effective work intensity: [ I = ] If deadline proximity forces ( < W ), then work compresses. ________________

2.3 Regenerative Access Regime Under regenerative access: * Tokens accumulate continuously. * Work may be distributed over full ( T ). * ( W ) Thus intensity: [ I_{regen} = ] which is lower than compressed intensity if ( < W ). ________________

  1. Sleep and Cognitive Productivity 3.1 Sleep Function Let optimal sleep duration be ( S^* ) hours. Empirical literature suggests cognitive performance declines approximately quadratically as sleep deviates from optimal duration. We model sleep penalty: [ (S) = (S^* - S)^2] with ( > 0 ). ________________

3.2 Cognitive Productivity Function Effective output per token depends on sleep and intensity: [ (S, I)] Assumptions: * ( > 0 ) * ( < 0 ) * ( < 0 ) beyond moderate intensity Interpretation: * Sleep deprivation reduces marginal productivity. * Excessive intensity increases error rates and cognitive fatigue. ________________

3.3 Effective Output Total effective cognitive output: [ Y = u (S, I)] ________________

  1. Welfare Comparison Individual utility: [ U = V(Y) - p u - (S)] where (V()) increasing and concave. ________________

Proposition 1 (Temporal Compression Reduces Effective Output) If hard quota induces ( < W ), then: [ I_{quota} > I_{regen}] and if sleep adjusts downward to accommodate compressed work: [ S_{quota} < S_{regen}] Under assumptions on ( ): [ Y_{regen} > Y_{quota}] for identical token usage ( u ). ________________

Proposition 2 (Welfare Dominance Under Flexible Access) If sleep penalty convex and intensity penalty present, then: [ U_{regen} > U_{quota}] for sufficiently tight quota compression. ________________

  1. Circadian Stability Beyond sleep duration, irregular sleep timing disrupts circadian rhythm. Define circadian deviation measure: [ D = Var(S_t)] Hard quotas may increase day-to-day variance in sleep timing. Regenerative access reduces synchronization pressure. Thus: [ D_{quota} > D_{regen}] Higher (D) linked to reduced cognitive stability. ________________

  2. Empirical Strategy This paper proposes a testable research design. 6.1 Data Sources

  3. API usage timestamps.

  4. Developer surveys.

  5. Wearable sleep tracking data (opt-in participants).

  6. Work product quality metrics (e.g., commit success rate, error frequency). ________________

6.2 Identification Strategy Natural experiment: * Platform transition from hard quota to regenerative model. * Or A/B test between regimes. Difference-in-differences model: [ Outcome_{it} = + Regime_{it} + Controls_{it} + i + {it}] Outcomes: * Sleep duration. * Sleep variance. * Night-time usage frequency. * Error rates per token. * Output per token. ________________

6.3 Testable Hypotheses H1: Hard quota regimes increase usage clustering near renewal times. H2: Usage clustering correlates with reduced sleep duration. H3: Reduced sleep correlates with lower effective output per token. H4: Regenerative regimes reduce sleep variance and improve output efficiency. ________________

  1. Ethical Considerations Temporal flexibility may improve welfare, but progression systems or gamified incentives could induce overuse. Design principles:
  • Avoid artificial consumption rewards.
  • Provide transparent usage dashboards.
  • Avoid sleep-disruptive notifications. ________________
  1. Limitations
  • Self-reported sleep may contain bias.
  • External deadlines may confound pricing effects.
  • Heterogeneous professions may respond differently. The model provides a structural framework rather than definitive causal claims absent empirical testing. ________________
  1. Contribution This paper introduces:
  • Temporal elasticity as a pricing dimension.
  • Formal linkage between access constraints and cognitive productivity.
  • Empirically testable hypotheses connecting API pricing and circadian stability. Digital infrastructure pricing should consider human temporal welfare as a design parameter, not merely throughput and margin. ________________
  1. Conclusion API pricing mechanisms shape not only demand levels but demand timing. Hard quota systems induce temporal compression that may reduce sleep stability and cognitive productivity. Regenerative access mechanisms provide temporal elasticity that can improve effective output without increasing nominal usage. This paper formalizes the theoretical channel and provides a testable empirical roadmap. ________________

Tab 3 PAPER 1 Regenerative Capacity Mechanisms for Load-Stable Token Pricing Corben Sorenson February 2026 ________________

Abstract AI services commonly employ token-based pricing under linear pay-as-you-go or periodic hard quota systems. Hard quota resets induce renewal-time clustering of demand, increasing aggregate load variance. Under convex infrastructure cost—arising from queueing delays, GPU capacity scaling, and energy marginal cost curvature—load variance increases total cost and reduces profit. We introduce a Regenerative Capacity Mechanism (RCM), a hybrid continuous–discrete allocation framework in which users accumulate continuously regenerating usage capacity subject to bounded burst caps, while prices adjust discretely based on observed aggregate load. We model the environment as a dynamic game with continuous user state variables and discrete pricing feedback. We establish: (i) existence of a stationary Markov Perfect Equilibrium; (ii) absence of renewal-driven impulse behavior and absolute continuity of equilibrium usage paths; (iii) strict reduction in load variance relative to hard quota systems under heterogeneous regeneration rates; (iv) sufficient conditions for local stability of pricing feedback; and (v) profit dominance under convex cost. Robustness results cover heavy-tailed demand and bounded burst multipliers. The RCM class provides a formally grounded mechanism for load stabilization in token-priced AI infrastructure. ________________

  1. Introduction Token-based metering is the dominant pricing structure in AI APIs. Two primary allocation models prevail:

  2. Linear pay-as-you-go.

  3. Hard quota subscription with periodic resets (daily/monthly). Hard quota systems generate predictable demand clustering near renewal times. When infrastructure cost is convex in load, such clustering increases total cost through elevated second moments of demand. This paper introduces a regenerative allocation architecture in which usage capacity accumulates continuously and is bounded by a stock constraint rather than periodic resets. Prices update discretely based on observed load. The central question: Does replacing periodic resets with continuous regenerative constraints reduce load variance and increase profit under convex cost? We provide formal conditions under which the answer is affirmative. ________________

  4. Economic Environment 2.1 Agents

  • Continuum of users ( i ).
  • One provider. 2.2 User State Dynamics Each user holds a capacity stock ( M_i(t) ): [ _i(t) = R_i - u_i(t), M_i(t) P_i.] Where:
  • (R_i > 0): regeneration rate.
  • (P_i > 0): maximum stock.
  • (u_i(t) ): usage rate. Usage constraint: [ 0 u_i(t) , {u}_i = {B} R_i,] with ({B} ) bounded. Regeneration rates (R_i) are continuously distributed on ([R_{}, R_{}]). ________________

2.3 Aggregate Load and Cost Aggregate load: [ L(t) = _0^1 u_i(t),di.] Infrastructure cost: [ C(L) = aL + bL^2, b > 0.] Convexity arises from queueing delay cost, energy marginal curvature, and scaling frictions. ________________

2.4 Discrete Pricing Feedback Time partitioned into intervals of length (). At time (t_k): [ p_{k+1} = p_0 (1 + ), {L}k = ^{t_{k+1}} L(t),dt.] Price constant on each interval. ________________

  1. User Optimization Within interval ([t_k, t_{k+1})): [ {u_i(t)} {t_k}^{t_{k+1}} e^{-t} dt] Subject to stock dynamics and constraints. Assume: [ V_i(u) = _i u - _i u^2, _i,_i>0.] Interior unconstrained optimum: [ u_i^{int}(p_k) = .] Equilibrium usage: [ u_i(t) = .] ________________

  2. Equilibrium and Smoothing Theorem 1 (Existence of Stationary MPE) Given compact control sets, continuous payoffs, and bounded price updates, there exists at least one stationary Markov Perfect Equilibrium. Proof sketch: Strategy sets compact and convex; payoff correspondence upper hemicontinuous; apply Kakutani fixed-point theorem. ________________

Proposition 1 (Absence of Reset-Driven Impulses) Under strictly concave (V_i), bounded controls, and continuous stock dynamics, optimal usage paths are bounded and absolutely continuous in time. Renewal-time impulse behavior is infeasible. Interpretation: Regenerative access removes discontinuous feasibility constraints responsible for renewal clustering under quotas. ________________

  1. Load Variance Comparison Hard Quota Baseline Each user receives (Q_i) every period (T). Renewal resets induce boundary behavior. Aggregate load exhibits renewal-time spikes. Let variance under quotas be (Var(L_Q)). ________________

Regenerative Mechanism Steady state implies (u_i R_i). With heterogeneous (R_i), aggregate load approximates: [ L_{RCM}(t) = _0^1 R_i di + (t).] Theorem 2 (Variance Reduction) If regeneration rates are not perfectly synchronized and burst caps are bounded, then: [ Var(L_{RCM}) < Var(L_Q).] ________________

  1. Robustness: Heavy-Tailed Demand Let top quantile (q) contribute fraction ((q)) of mean load. Proposition 2 Variance reduction persists under heavy-tailed (R_i) if fraction of users simultaneously at burst cap is bounded strictly below 1. Design implication: enforce burst pricing, cooldowns, or heterogeneous regeneration parameters. ________________

  2. Pricing Stability Linearizing discrete update around equilibrium: [ _{k+1} = A k, A = |{p^*}.] Theorem 3 (Local Stability) Equilibrium locally stable if: [ |A| < 1.] ________________

  3. Profit Comparison [ E[C(L)] = aE[L] + bE[L^2].] Since: [ E[L^2] = (E[L])^2 + Var(L),] variance reduction implies lower expected cost. Theorem 4 (Profit Dominance) Under identical mean load and convex cost: [ {RCM} > {Quota}.] ________________

  4. Conclusion Replacing periodic resets with regenerative capacity removes structural synchronization, reduces load variance, stabilizes pricing feedback, and improves profit under convex infrastructure cost. ________________

PAPER 2 Temporal Elasticity in API Access and Cognitive Productivity Corben Sorenson February 2026 ________________

Abstract API pricing structures shape not only demand levels but demand timing. Hard quota systems impose renewal-time discontinuities that may induce workload compression. This paper introduces temporal elasticity of API access and develops a formal model linking access constraints to work intensity, sleep allocation, and effective cognitive output. We derive conditions under which regenerative access reduces temporal compression and propose a testable empirical framework combining usage timestamps, sleep metrics, and token-adjusted productivity proxies. ________________

  1. Introduction When API access is restricted to periodic quotas, usage may cluster near renewal times. If work is deadline-sensitive, compressed access windows can increase work intensity and reduce sleep duration. Sleep science consistently links reduced or irregular sleep to diminished cognitive performance. Thus pricing structures may indirectly affect productivity per token. This paper formalizes the mechanism and proposes empirical tests. ________________

  2. Time Allocation Model Daily constraint: [ W + S + L = 24.] Token output: [ u = W.] Under quota: Work window ( < W ). Intensity: [ I_{quota} = .] Under regenerative access: [ I_{regen} = .] ________________

  3. Sleep and Productivity Sleep penalty: [ (S) = (S^* - S)^2.] Effective output: [ Y = u (S, I).] Properties:

  • ( /S > 0 )
  • ( /I < 0 ) beyond moderate intensity. ________________

  1. Comparative Results Proposition 1 If quota reduces feasible work window ( ), then intensity rises and optimal sleep decreases. Proposition 2 Under convex sleep penalty and intensity cost: [ Y_{regen} > Y_{quota}.] Conditional on compression magnitude. ________________

  2. Circadian Stability Define sleep timing variance (D). Hard quotas increase day-to-day sleep variance through clustering. Regenerative access reduces such synchronization pressure. ________________

  3. Empirical Strategy Outcomes

  • Renewal clustering share (CP)
  • Night usage share (NS)
  • Sleep duration and timing variance
  • Token-normalized rework rate
  • Token-normalized error corrections Primary Design: Staggered Rollout Diff-in-Diff [ Y_{it} = i + t + Regenerative{it} + X{it} + _{it}.] Secondary: Renewal Boundary RD Test discontinuity in clustering at reset times. Hypotheses H1: Quotas increase clustering and night usage. H2: Regeneration increases sleep stability. H3: Sleep improvements mediate reduced rework per token. ________________

  1. Ethical Considerations Design should enhance flexibility without encouraging overuse. Transparency and user control required. ________________

  2. Conclusion Temporal elasticity in API access may influence cognitive productivity. This paper formalizes the channel and proposes empirically testable hypotheses. ________________

Tab 4 Regenerative Capacity Mechanisms for Load-Stable Token Pricing in AI Infrastructure Version 1.0 (Public Release) Corben Sorenson February 2026 ________________

Abstract Token-based pricing is the dominant allocation and billing method for AI APIs. A common subscription variant is the periodic hard quota: a fixed token allotment that resets at deterministic renewal times (daily/monthly), often paired with burst limits. This paper shows that renewal resets create feasibility discontinuities that can rationally induce renewal-time clustering of usage (temporal synchronization), thereby increasing aggregate load variance. Under convex effective infrastructure cost—arising from queueing delay, capacity scaling frictions, and latency/SLA penalties—variance raises expected cost via the second moment of load. We propose a Regenerative Capacity Mechanism (RCM): users hold a continuously regenerating capacity stock with a maximum stock (burst buffer) and bounded burst caps, while the provider updates token prices at discrete intervals as a function of recent aggregate load. We model the environment as a hybrid continuous–discrete dynamic game. We establish (i) existence of a stationary Markov Perfect Equilibrium (MPE); (ii) absolute continuity of optimal usage paths under regenerative constraints (ruling out reset-driven impulses); (iii) strict load-variance reduction relative to periodic hard quotas under mild heterogeneity; (iv) sufficient conditions for local stability of the pricing feedback rule; and (v) profit dominance under convex cost holding mean load fixed. We discuss robustness to heavy-tailed demand and provide an implementation mapping from primitives to production billing systems. ________________

  1. Introduction AI APIs are increasingly priced in tokens, a unit designed to approximate computational workload. While the billing unit is “continuous” (users can call the API at any time), many plans impose temporal discontinuities in access: daily caps, monthly allotments, and renewal-time resets. These discontinuities are not merely administrative; they reshape the feasible set of user actions over time. This paper isolates a specific distortion: Renewal discontinuities induce demand synchronization. When quota resets occur at known times, users with deadlines or uncertainty about future access may rationally shift usage toward renewal boundaries—creating predictable spikes. In network engineering, token-bucket mechanisms regulate burstiness. In economics, peak-load pricing addresses costly capacity at demand peaks. Our setting differs: AI demand is strategic and deadline-driven, and common quota policies introduce deterministic renewal shocks to feasibility. We propose a mechanism that removes renewal discontinuities: Regenerative Capacity Mechanism (RCM): continuous regeneration of capacity stock (rate) + bounded maximum stock (burst buffer) + bounded burst cap + discrete-time price updates linked to observed load. The objective is not “smoother traffic” as an end in itself, but cost-efficient infrastructure operation under convex effective cost and competitive pressure to avoid punitive hard caps. Contributions

  2. Mechanism definition: A regenerative capacity stock model suitable for token-priced AI services.

  3. Regularity result: Under concave preferences and bounded controls, regenerative feasibility yields absolutely continuous optimal usage paths, eliminating reset-driven impulses.

  4. Variance theorem: Replacing renewal resets with regenerative constraints strictly reduces load variance under mild heterogeneity and bounded burst caps.

  5. Stability condition: A simple local stability criterion for discrete load-based pricing feedback.

  6. Profit dominance: Under convex cost, variance reduction improves profit holding mean load fixed.

  7. Robustness & implementation mapping: Heavy-tailed demand, burst features, and practical plan parameterization. ________________

  8. Relationship to Prior Work 2.1 Token bucket mechanisms (engineering) Token bucket systems accumulate tokens at a rate up to a bucket size; traffic can burst by consuming accumulated tokens. This basic structure appears in standard traffic policing specifications (e.g., RFC 2698). (IETF Datatracker) RCM uses a related state constraint, but differs in (i) objective (economic welfare/profit under convex cost), (ii) agents (strategic users optimizing value net of price), (iii) baseline (periodic hard quota resets rather than generic bursty traffic), and (iv) pricing feedback (endogenous price updates linked to load rather than static policing). 2.2 Peak-load pricing (economics) Peak-load pricing studies optimal pricing and capacity when demand varies over time and capacity is costly; Boiteux is a canonical starting point, with a large subsequent literature and surveys. (Springer) RCM is not a restatement of peak-load pricing: the paper’s central object is a mechanism that removes renewal discontinuities commonly used in token subscriptions, and proves variance dominance relative to periodic reset quotas. 2.3 Dynamic equilibrium concept We use stationary Markov Perfect Equilibrium as the equilibrium notion for the dynamic game; see Maskin & Tirole for foundations. (Maskin Lab) 2.4 Convex effective cost via queueing Even if direct energy cost were linear in throughput, effective cost is typically convex once queueing delay, SLA penalties, and latency costs are included; standard queueing results motivate convexity as utilization rises. (INFORMS Pubs Online) (We do not require a specific queue model; convexity is an assumption supported by queueing and scaling frictions.) ________________

  9. Model 3.1 Time and pricing cadence Time is continuous. Prices update at discrete times (t_k = k), reflecting operational reality (billing/pricing policies update periodically; usage occurs continuously). 3.2 Users A continuum of users (i ). Each user selects usage rate (u_i(t)) and is subject to an access feasibility constraint that differs by regime. User instantaneous benefit is (V_i(u)), assumed strictly concave and continuously differentiable. A canonical specification is quadratic: [ V_i(u)=_i u-_i u^2,_i,_i>0.] Users discount at rate (>0). 3.3 Aggregate load and cost Aggregate load: [ L(t)=_0^1 u_i(t),di.] Provider cost is convex in load: [ C(L)=aL+bL^2,b>0.] Convexity captures effective costs from queueing delay and capacity scaling frictions. (INFORMS Pubs Online) ________________

  10. Two Regimes: Periodic Hard Quotas vs Regenerative Capacity 4.1 Periodic hard quota (baseline) A representative hard quota regime grants an allotment (Q_i) that resets every (T), generating a discontinuity in feasible future usage at renewal times. The exact operational rule varies by provider, but the key structural feature is: Renewal discontinuity: at deterministic times (t_k), the feasible set (or shadow price of future usage) changes discontinuously. This is the structural source of renewal clustering. 4.2 Regenerative Capacity Mechanism (RCM) Each user holds a capacity stock (M_i(t)) with continuous dynamics: [ _i(t)=R_i-u_i(t),] subject to: [ 0M_i(t)P_i.] Usage must satisfy: [ 0u_i(t), {u}_i={B}R_i, {B}.] Here:

  • (R_i) is the regeneration rate (continuous replenishment),
  • (P_i) is the maximum capacity stock (burst buffer),
  • ({B}) bounds burst intensity relative to regeneration. Heterogeneity: (R_i) is non-degenerate with continuous density on ([R_{},R_{}]). 4.3 Load-based price update (discrete) Let ({L}k) be average load over interval ([t_k,t{k+1})): [ {L}k=^{t_{k+1}}L(t),dt.] Price updates: [ p_{k+1}=p_0(1+), ,] with prices bounded in practice by policy (implicit in the existence result). ________________

  1. User Problem Under RCM On each interval ([t_k,t_{k+1})), price (p_k) is fixed. User (i) solves: [ {u_i()}{t_k}^{t_{k+1}} e^{-t}dt] subject to stock dynamics and bounds. With concave (V_i), the instantaneous unconstrained optimum is: [ u_i^{}(p_k)=_{u}{V_i(u)-p_k u},] and for the quadratic case: [ u_i^{}(p_k)=.] The feasible choice then is: [ u_i(t)=.] ________________

  2. Equilibrium Concept and Existence We study a stationary Markov Perfect Equilibrium (MPE) in which users’ policies depend on payoff-relevant states: their own (M_i(t)) and the current price (p_k). MPE is standard for dynamic games with Markovian state. (Maskin Lab) Theorem 1 (Existence of Stationary MPE; informal statement) Assume: (A1) (V_i) is strictly concave, (C^1), with (V_i’(0)<). (A2) Controls are compact: (u_i(t)). (A3) Prices are bounded: (p_k) (policy constraint). (A4) The price update rule is measurable and continuous in ({L}k). (A5) (R_i) has non-degenerate distribution. Then there exists at least one stationary MPE. Proof roadmap. With bounded prices and compact control sets, best responses exist; the equilibrium correspondence is upper hemicontinuous under standard regularity; a fixed point argument (Kakutani) yields existence. (Full proof would be placed in an appendix for a journal version.) (Maskin Lab) _______________

  3. Bridging Result: Individual Temporal Smoothing A key program-level requirement (and the main link to the companion paper on temporal elasticity) is to show that regenerative feasibility eliminates reset-driven impulses at the individual level. Proposition 1 (Absolute continuity of optimal usage under RCM) Under (A1)–(A3), any optimal usage path (u_i^*(t)) under RCM is bounded and absolutely continuous in time (hence cannot contain impulse demand). In particular, there is no renewal-time discontinuity in feasibility that could induce massed boundary behavior. Intuition. In hard quotas, the feasible set changes discontinuously at renewal times, generating rational clustering. In RCM, the feasible set evolves continuously; the stock state (M_i(t)) changes continuously; and with concave payoff and bounded controls, optimal controls adjust smoothly almost everywhere. This proposition matters because it converts “variance reduction” from a purely aggregate statement into a structural temporal regularity statement at the individual level. ________________

  4. Variance and Synchronization 8.1 Why periodic resets create synchronization Under a renewal reset, a large mass of users experiences the same feasibility relaxation at the same time. With deadlines and uncertainty about future access, renewal is a focal point for usage, producing clustering even if underlying tasks are heterogeneous. 8.2 Why RCM reduces variance RCM removes the renewal discontinuity and replaces it with continuous regeneration. Heterogeneity in (R_i) and bounded burst caps prevent economy-wide simultaneous binding of the same constraint at the same instant. Theorem 2 (Strict variance reduction vs periodic hard quotas; informal statement) Assume (A1)–(A5) and that the periodic quota regime induces renewal-time clustering for a positive measure of users (a mild condition in deadline-sensitive environments). Then: [ (L_{}) < (L_{}),] where variance is defined over a common horizon or the stationary distribution. Proof roadmap. Under quotas, renewal discontinuities create correlated boundary behavior (positive mass at renewal). Under RCM, Proposition 1 rules out renewal-driven impulses and the continuous stock dynamics desynchronize users. With non-degenerate (R_i), synchronized boundary behavior has strictly smaller measure; thus the second moment (E[L^2]) is strictly smaller. ________________

  5. Robustness to Heavy-Tailed Demand and Burst Features A common objection is that a small fraction of power users may dominate load and reintroduce burstiness. Let the top quantile (q) contribute share ((q)) of mean load. Heavy tails increase ((q)), but do not automatically reinstate renewal synchronization under RCM because the synchronization channel is the renewal discontinuity itself. Proposition 2 (Variance reduction under heavy-tailed heterogeneity; informal) Variance dominance persists if:

  • burst caps ({u}_i) are bounded, and
  • the fraction of users simultaneously at burst cap is bounded away from 1 (e.g., via burst pricing, cooldowns, or staggered enterprise parameters). Design implication. For production systems, enforce at least one stabilizer:
  • time-limited burst windows,
  • nonlinear pricing for concurrent bursts,
  • cooldown or ramp constraints,
  • staggered regeneration parameters for large enterprise tenants. ________________

  1. Stability of Load-Based Price Feedback Linearize the discrete price update around a steady state (p^). Let: [ _k = p_k - p^,k = {L}k - {L}^*.] Then: [ (|_{p^*})_k A _k,] with ({L}/p<0). Theorem 3 (Local stability condition) A sufficient condition for local asymptotic stability is: [ |A|<1.] Interpretation. Excessively aggressive feedback (() too large) can create oscillatory dynamics; moderate feedback stabilizes. ________________

  2. Profit Dominance Under Convex Cost Expected cost under quadratic convexity decomposes into mean and variance: [ E[C(L)] = aE[L]+bE[L^2] = aE[L]+b((E[L])^2+(L)).] Holding (E[L]) fixed, reducing variance strictly reduces expected cost. Theorem 4 (Profit dominance; informal) If mean load is held fixed across regimes and (b>0), then: [ {} > {}] whenever ((L_{}) < (L_{})). This is the economic core: variance is costly under convex infrastructure cost. ________________

  3. Implementation Mapping (from theory to product plans) RCM corresponds directly to plan parameters:

  • Regeneration rate (R): “tokens per hour/day” baseline allowance.
  • Max stock (P): burst buffer (how much can be accumulated).
  • Burst cap ({B}): how much instantaneous rate can exceed (R).
  • Pricing cadence (): update interval (e.g., hourly).
  • Feedback sensitivity (): aggressiveness of surge/discount response. Practical guidance:

  1. Choose () large enough to avoid oscillation (empirically stable) and small enough to respond to major load shifts.

  2. Set () within the stability region implied by Theorem 3 (estimated from observed demand elasticity).

  3. Keep burst features bounded; make concurrency expensive rather than banning it.

  4. For enterprise tenants, stagger or diversify (R) / (P) profiles to avoid synchronized peaks. ________________

  5. Discussion and Limitations What RCM does: removes policy-induced synchronization from renewal discontinuities and enables smoother equilibrium usage. What RCM does not claim: elimination of all peaks. Deadline-driven bursts can remain, but RCM avoids adding an additional deterministic spike mechanism. Model limitations (appropriate for v1):

  • We treat convex cost as reduced-form; a fuller version may derive it from queueing + SLA penalties.
  • Existence results are stated informally; a full journal version would include a technical appendix.
  • Competition between providers is not modeled here (a natural extension). ________________
  1. Conclusion Periodic hard quotas create deterministic renewal discontinuities that rationally induce renewal-time usage clustering. Under convex effective infrastructure costs, clustering raises expected cost via higher variance. Regenerative Capacity Mechanisms replace discontinuous resets with continuous stock dynamics and bounded burst caps, yielding temporally regular usage paths, lower load variance, stable price feedback under mild conditions, and profit dominance holding mean load fixed. This paper provides a formal mechanism-design foundation for load-stable token pricing in AI infrastructure and serves as the theoretical anchor for empirical work on temporal elasticity and human-centric outcomes. ________________

References (minimal v1 set; expand in later versions) * Token bucket specification background: RFC 2698. (IETF Datatracker) * Markov Perfect Equilibrium foundations: Maskin & Tirole (2001). (Maskin Lab) * Peak-load pricing lineage and survey: Crew & Kleindorfer survey referencing Boiteux; Williamson (1966) discussion. (Springer) * Queueing and convex effective cost motivation: queueing-related references motivating convexity in delay/latency regimes. (INFORMS Pubs Online)

Tab 5 Temporal Elasticity of API Access and Cognitive Efficiency in AI-Assisted Work Version 1.0 (Public Release) Corben Sorenson February 2026 ________________

Abstract API pricing mechanisms affect not only the quantity of AI usage but also the timing of usage. Periodic hard quotas (daily/monthly resets) introduce deterministic renewal-time discontinuities in feasible access, which can induce workload compression near renewal boundaries when tasks are deadline-sensitive. This paper introduces temporal elasticity of access as an economic design parameter for AI services and develops a formal framework linking access discontinuities to (i) work intensity and nocturnal work share, (ii) sleep duration and sleep regularity, and (iii) token-normalized cognitive friction during AI-assisted production (e.g., rework loops and correction iterations per token). We derive comparative predictions for quota-based versus regenerative access regimes and provide operational measures of temporal compression (renewal clustering and nocturnal share). Drawing on established evidence that sleep restriction produces cumulative, dose-dependent neurobehavioral deficits and that irregular sleep schedules correlate with poorer performance outcomes, we specify a falsifiable pathway from pricing architecture to human-centric efficiency. (PubMed) Finally, we propose a causal empirical strategy based on staggered plan migration or A/B rollout, using modern difference-in-differences estimators designed for heterogeneous treatment effects, and outline a minimum viable study protocol with privacy and ethics safeguards. (ScienceDirect) ________________

  1. Introduction Token-priced AI tools are increasingly embedded in cognitively intensive work, including software development, analytics, and technical writing. In such settings, pricing plans can shape labor allocation not only by changing the marginal price of usage, but by changing the temporal feasibility of access. A common design is the periodic hard quota: a fixed allotment of tokens that resets at deterministic times (daily, weekly, monthly). This structure introduces a renewal-time discontinuity—a sharp change in feasible future usage at renewal boundaries. For deadline-sensitive tasks, discontinuities can create incentives to delay usage until after renewal, or to concentrate usage immediately after renewal when the shadow price of access falls. The result is temporal compression: usage becomes clustered in time. This paper makes three claims—stated carefully:

  2. Mechanism claim (theory): Renewal-time discontinuities can induce usage clustering near renewal boundaries in deadline-sensitive environments.

  3. Human-capital claim (channel): Temporal compression can increase nocturnal work share and reduce sleep regularity and/or sleep duration for some users at the margin.

  4. Efficiency claim (testable): Reduced sleep duration and irregular sleep patterns are plausibly linked to lower cognitive performance and may increase token-normalized cognitive friction in AI-assisted work. Claims (2)–(3) are not asserted as universal truths; they are framed as a testable causal pathway, with explicit identification strategies. This paper complements a companion mechanism-design paper (“Regenerative Capacity Mechanisms…”) that studies load variance and convex infrastructure costs. Paper 1 establishes a pricing-architecture reason to remove renewal discontinuities; the present paper studies the potential human-centered efficiency implications and lays out empirical tests. ________________

  5. Related Evidence and Motivation This paper relies on two empirical pillars: 2.1 Sleep restriction and cognitive performance Chronic sleep restriction produces cumulative, dose-dependent deficits in neurobehavioral functions (attention, reaction time, and other cognitive measures), even when subjective sleepiness does not fully track impairment. (PubMed) This supports modeling sleep loss as a meaningful productivity-relevant cost. 2.2 Sleep regularity and performance outcomes Irregular sleep/wake patterns (captured by the Sleep Regularity Index) have been associated with delayed circadian timing and poorer performance outcomes (e.g., academic performance in observational studies). (Nature) This supports modeling regularity (not only duration) as a relevant channel. We do not claim that API pricing is a primary driver of sleep. We claim it may shift constraints at the margin—especially for users whose workflows already sit near deadline pressure—creating measurable shifts in timing, regularity, and associated cognitive friction. ________________

  6. Conceptual Framework: Temporal Elasticity of Access 3.1 Quantity elasticity vs temporal elasticity Standard pricing analysis focuses on how price changes affect quantity demanded. Here we focus on temporal elasticity: how pricing architecture affects the time distribution of usage holding quantity approximately fixed. Definition (Temporal elasticity of access). A pricing plan exhibits higher temporal elasticity when users can distribute usage smoothly over time without deterministic renewal discontinuities that force or incentivize usage concentration. This is distinct from “cheap vs expensive” tokens; it concerns whether feasible access is continuous or reset-driven. ________________

  7. Model 4.1 Time allocation and workload intensity Consider a user with daily time budget: [ W_t + S_t + _t = 24,] where (W_t) is work time, (S_t) sleep, (_t) leisure/other. Let token usage be proportional to work time in expectation: [ u_t = W_t,] where () is tokens-per-hour (tool-assisted throughput). This abstracts from task heterogeneity; empirical work will not assume constant (), but the mapping is useful for comparative statics. 4.2 Renewal discontinuity and temporal compression Under a periodic hard quota, feasible access effectively expands at renewal times. Let (t_r) denote renewal, and suppose deadline-sensitive tasks induce a “completion pressure” that increases as a deadline approaches. We model compression via an effective feasible work window (_t), where:

  • Under smooth access, (_t W_t) (work spread across available time).
  • Under reset-driven access (plus deadline pressure), (_t < W_t) for some days, reflecting clustering near renewal. Define work intensity: [ I_t = .] When (_t) shrinks while (u_t) remains necessary to meet deadlines, intensity rises. Intensity is intended to capture concentrated cognitive effort (long uninterrupted sessions, nocturnal shifts, or “all-nighter” behavior). 4.3 Sleep costs and cognitive efficiency We incorporate both sleep duration and regularity. Sleep duration penalty (stylized): [ (S_t) = (S^* - S_t)^2,>0,] where (S^*) is a preferred sleep duration (commonly in the 7–8 hour range for adults; the exact calibration is not required for the theory layer). Sleep regularity penalty: Let (R_t) be a sleep-regularity metric (e.g., SRI-style). Irregularity can be represented as a cost (()) increasing in day-to-day deviations in sleep timing. We define effective cognitive efficiency per token as a function: [ (S_t, _t, I_t),] with properties consistent with the sleep evidence:
  • ( > 0) (more sleep, higher efficiency at the margin),
  • ( > 0) (more regularity, higher efficiency at the margin),
  • ( < 0) beyond moderate intensity (very high intensity increases error/rework likelihood). These assumptions are conservative and correspond to the broad direction of sleep-performance findings rather than fine-grained psychometrics. (PubMed) 4.4 Observable output and cognitive friction proxies True “productivity” in programming is hard to measure. Instead, we focus on token-normalized cognitive friction:
  • repeated rework prompts,
  • correction iterations,
  • error-fix cycles,
  • repeated near-duplicate tool calls. Define a proxy (F_t) (friction) that increases when efficiency falls: [ F_t () .] We will measure (F_t) empirically with token-normalized indicators (Section 6). ________________
  1. Comparative Predictions We compare two regimes:
  • Quota regime (reset discontinuity): deterministic renewal boundaries.
  • Smooth/regenerative access regime: no renewal discontinuity; access evolves continuously. Proposition 1 (Renewal discontinuities increase clustering) In deadline-sensitive environments, deterministic renewal discontinuities increase the probability mass of usage occurring near renewal boundaries, raising a clustering index (defined below). Intuition: renewal reduces the shadow price of future access, so it becomes optimal to delay discretionary usage toward renewal, and to concentrate usage soon after renewal when constraints relax. Proposition 2 (Clustering increases nocturnal share for some users) If renewal clustering compresses work into a narrower effective window (t), then intensity (I_t) rises; if daytime hours are constrained by other commitments, increased intensity shifts work toward late-night hours, raising nocturnal share. Proposition 3 (Temporal compression can reduce sleep duration/regularity at the margin) For users whose work is compressed into late hours, the time identity implies a reduction in sleep duration (S_t) and/or an increase in sleep timing variance (reduced regularity). Proposition 4 (Sleep and intensity link to cognitive friction) Given ()’s monotonicity properties, reductions in sleep duration/regularity and increases in intensity raise token-normalized friction (F_t) on average. These propositions produce a testable causal chain: Regime → (clustering, night work) → (sleep duration/regularity) → (token-normalized friction). _______________
  1. Measurement and Empirical Strategy This section is designed to make Paper 2 credible as a public release. It defines outcomes precisely and uses modern causal estimators appropriate for staggered adoption. 6.1 Core operational measures Let (t_r) denote renewal time, (T) renewal period, and () a small window (e.g., 1–3 hours). Renewal clustering share (Compression Proxy): [ CP_{it} = .] Nocturnal usage share: [ NS_{it} = .] Sleep outcomes (opt-in):
  • Mean sleep duration (_{it}),
  • Sleep onset variance (regularity proxy),
  • SRI-style metric if obtainable (recommended). Token-normalized cognitive friction outcomes:
  • Rework loop rate: near-duplicate prompts / edits per 1k tokens,
  • Correction iteration count per 1k tokens,
  • If available: compilation/test failures per 1k tokens,
  • Task-switching proxy: number of distinct task contexts per 1k tokens (optional). 6.2 Primary identification design: staggered rollout DiD Suppose a provider migrates users from hard quotas to a smooth/regenerative plan over time (or runs an A/B rollout). We estimate group-time average treatment effects using modern DiD methods robust to heterogeneous treatment effects and staggered adoption. (ScienceDirect) At the conceptual level: [ Y_{it} = i + t + Regime{it} + X{it} + _{it},] but estimation should avoid naïve two-way fixed effects event-study assumptions in staggered settings; instead use Sun–Abraham style event-study estimators and/or Callaway–Sant’Anna group-time estimands. (ScienceDirect) Outcomes (Y_{it}): (CP, NS, , Var(), F). Controls (X_{it}): day-of-week fixed effects, seasonality, self-reported deadlines (if collected), major product releases (if observable). Validity checks:
  • Pre-trend/event-study plots,
  • Placebo renewals (pseudo-reset times) to confirm boundary effects are not arbitrary,
  • Heterogeneity analysis by workload type. 6.3 Secondary design: renewal-boundary discontinuity Hard quotas create mechanical discontinuities at renewal boundaries. Evaluate outcomes in narrow windows around renewal: [ Y_{i}(t) = f() + [] + _i,] where (= t - t_r). Under smooth/regenerative access, the renewal boundary discontinuity should attenuate or vanish. This design directly targets the “renewal discontinuity → clustering” link. 6.4 Mediation analysis (secondary, not primary) To test the mechanism:
  • First stage: regime effect on (CP, NS).
  • Second stage: (CP, NS) effect on sleep outcomes.
  • Third stage: sleep outcomes effect on friction (F) controlling for tokens. Mediation should be presented as exploratory unless strong assumptions are met. 6.5 Minimum viable study protocol (public-release practical) A realistic first study could be:
  • (N = 200)–(500) opt-in users,
  • 6–10 weeks observation,
  • randomized or staggered rollout if possible,
  • daily lightweight sleep diary + optional wearable link,
  • anonymized timestamped token usage + friction proxies. This yields enough power to detect timing shifts (CP/NS) even if sleep effects are modest. ________________
  1. Ethics, Privacy, and Scope Discipline Because sleep data is sensitive, public release should state:
  • Opt-in only for any sleep/wearable data,
  • Data minimization (store aggregates where possible),
  • Separation from employer access,
  • Right to delete/export,
  • No medical claims; no health advice. Scope discipline: The paper does not claim that pricing “causes insomnia” or that effects are universal. It claims renewal discontinuities can alter time-allocation constraints, and proposes causal tests of downstream effects for relevant subpopulations. ________________
  1. Limitations and Threats to Validity
  • Confounding deadlines: Deadlines drive night work independent of pricing; identification must leverage within-user variation under plan changes.
  • Selection into plans: If users self-select into smooth access, estimates may be biased without randomization or careful quasi-experimental design.
  • Measurement error: Sleep diaries and wearables are imperfect; use multiple measures where feasible.
  • Productivity measurement: We avoid claiming “true output” and instead measure token-normalized friction proxies. These limitations are addressed by the design choices in Section 6, but remain central considerations. ________________
  1. Conclusion Temporal elasticity of access is a neglected design dimension in AI pricing. Periodic hard quota resets introduce renewal discontinuities that can induce temporal compression of usage. For deadline-sensitive cognitive labor, temporal compression may increase nocturnal work share and degrade sleep regularity and/or duration at the margin—factors plausibly linked to cognitive performance and token-normalized friction. This paper provides a formal framework and a practical empirical blueprint to test the human-centered efficiency implications of pricing architecture. Together with the companion mechanism-design paper on load variance and convex infrastructure cost, it positions continuous access mechanisms as a candidate “win–win” design: infrastructure smoothing for providers and temporal flexibility for users—subject to empirical validation. ________________

References (minimal v1 set; expand in later versions) * Van Dongen, H. P. A., Maislin, G., Mullington, J. M., & Dinges, D. F. (2003). The cumulative cost of additional wakefulness: Dose-response effects on neurobehavioral functions and sleep physiology from chronic sleep restriction and total sleep deprivation. Sleep. (PubMed) * Phillips, A. J. K., et al. (2017). Irregular sleep/wake patterns are associated with poorer academic performance and delayed circadian and sleep/wake timing. Scientific Reports. (Nature) * Sun, L., & Abraham, S. (2021). Estimating dynamic treatment effects in event studies with heterogeneous treatment effects. Journal of Econometrics. (ScienceDirect) * Callaway, B., & Sant’Anna, P. H. C. (2021). Difference-in-Differences with multiple time periods. Journal of Econometrics. (ScienceDirect)