Reliability, Compute Scarcity, and Frontier AI Races

Discrete dynamic-game paper PDF, step-by-step derivation, and equilibrium diagnostic.
Open PDF Math audit Verification note

What Changed

Current package: 62-page discrete dynamic-game paper, reader-facing PDF view, math audit, verification note, selected equilibrium diagnostics, calibration bridge, and empirical source roadmap. Detailed artifacts are collapsed below.

Replication artifacts and audit files Collapsed by default to keep the paper view readable. The files are still served with the site.

Core Documents

Paper PDF Step-by-step math audit Verification README

Equilibrium Diagnostics

Stationary diagnostic JSON Selection diagnostic envelope Selection correspondence envelope Selected Nash-deviation audit Policy-evaluation audit

Calibration

Calibration bridge JSON Task-reliability scaling Source-freshness audit Rental-comparability audit Capacity prediction envelope

Empirical Evidence

Data-source roadmap Claim-threshold audit Parameter-status ledger Panel-estimand map Empirical-readiness audit Acceptance scorecard Synthetic mechanism-pattern audit

Paper Reframe

The title now names the mechanism directly: reliability, compute scarcity, and frontier AI races.

The paper now leads with the economic mechanism: consumer AI quality can saturate while producer value keeps compounding, and frontier training bids against saturated serving for scarce compute. The abstract foregrounds selected Markov-chain moments, the \(S_n=p^n\) calibration bridge, and matched hardware cells instead of closing on the continuous-time appendix caveat.

The introduction now clarifies that the two-laboratory game is a frontier-pair reduction inside a matched hardware and task-horizon cell, not an industry-wide two-firm assumption.

The introduction now adds a concentration-scope map: pair-level selected dominance, cell-level compute pressure, cross-cell aggregation, \(N\)-firm industry dynamics, and welfare or antitrust claims are separate objects with separate evidence requirements.

The introduction now states the scaling-law notation bridge: the common \(1-q_r^n\) task-failure notation is written as \(1-p^n\) because \(q\) is reserved for capability in the model.

The reliability section now derives the task-horizon gap bridge: at a common success threshold, \(H_{\bar S}(q)=(-\log\bar S)e^q\), so log horizon ratios equal capability gaps.

The calibration section now reports frontier-value curvature sensitivity: the same 1440-minute horizon is 10.51x the 137-minute baseline at \(a=1\), 34.08x at \(a=1.5\), and 110.48x at \(a=2\).

The introduction now states early that public task-horizon, data-center, GPU, and rental-price numbers discipline units and conditional prediction envelopes, but do not estimate private training allocation, race intensity, or equilibrium selection without matched hardware-cell evidence.

The calibration section now adds a source-freshness audit: task-horizon evidence and cloud list prices are volatile, and every public anchor must be rechecked before being used as current empirical evidence rather than as a scenario input.

The introduction also states that finite-horizon and stationary computations verify computability and selected-transition diagnostics, not uniqueness, full robustness, or private behavior.

The introduction now adds a formal-result dependency matrix: five formal rows require no public calibration input, and only the tipping-diagnostic row requires equilibrium selection before producing point predictions.

The claim-status table is now generated by a discrete-vs-continuous result-transfer audit: six major result blocks, zero unique-global-equilibrium claims, zero universal-threshold claims, zero global continuous-time claims, and only the tipping/concentration block requiring equilibrium selection for point objects.

The empirical section now adds a claim-threshold audit: formal theory, selected diagnostics, calibration identities, and public scenarios are current-package outputs, while full empirical support, causal estimates, and welfare or policy rankings require additional evidence or primitives.

The empirical section now adds a data-source roadmap: task horizons, training allocation, comparable rental prices, cold floors, supplier bottlenecks, access outcomes, consumer-serving saturation, selection, and the joined mechanism cell are mapped to source classes and remaining evidence gaps.

The appendix now aggregates the selected stationary diagnostics into one envelope: Markov-chain row accounting is stable across 29 rows, while convergence and signed parity drift remain selection/path dependent.

The appendix now also adds a selected Nash-deviation audit: state-level continuation games have maximum unilateral deviation gains at machine precision, while Bellman residuals still determine which selected paths are stationary fixed-point claims.

The appendix now adds a stationary policy-evaluation audit: freezing selected stationary policies and solving \((I-\beta M_\sigma)V_i=r_i(\sigma)\) gives machine-precision linear residuals for the small stationary sweep.

The appendix now turns the same selected diagnostics into a moment correspondence: the clean converged Bellman-Nash pool spans both parity persistence and rapid exit, so empirical tests should compare data to a pre-specified selection set rather than to the benchmark row alone.

The stationary-equilibrium theorem now separates finite Bellman-Nash representation from algorithmic convergence: value-iteration, policy-iteration, and homotopy routines are residual-checked computational paths, not implications of the existence theorem.

The result-status comparison now also explains why the major results survive the discrete switch: static reliability, local parity instability, finite-horizon continuation games, stationary finite-game existence, selected-\(M_\sigma\) Markov-chain diagnostics, and compute-rent accounting are separate links in the same mechanism.

The empirical section now states the mechanism's boundary conditions: serving demand must be locally saturated, training and serving must draw on substitutable accelerator capacity, and frontier stakes must rise with task horizon or capability gaps in the same hardware cell.

The empirical section now also orders implementation: construct the hardware-cell panel and boundary checks, transform raw observations into model objects, then compare joint sign restrictions under an explicit selection environment.

The empirical section now states the selection-correspondence discipline: without a pre-specified selection environment, selected-equilibrium moments are a set-valued prediction rather than a point prediction.

The empirical section now adds a generated acceptance/rejection scorecard: the public pilot partially supports unit/scenario gates but does not test the full mechanism because training allocation, repeated gap dynamics, and the full joint evidence package remain missing.

The empirical section now adds a synthetic mechanism-pattern audit: one constructed mechanism-consistent panel is accepted, while rent-only, mean-reverting-gap, and ex-post-selection false positives are rejected by the stated gates.

The empirical section now adds a panel estimand map: six descriptive matched-cell contrasts say how to operationalize serving saturation, rental pressure, scarcity validation, gap transitions, access leakage, and the full mechanism pattern without pretending to identify structural parameters from the current public pilot.

The empirical section now adds a generated parameter-status ledger so reliability units, primitives, finite-grid domains, selected-equilibrium outputs, public scenario inputs, conditional rental inversions, and missing empirical targets are not treated as the same kind of evidence.

The empirical section now aligns panel timing with the discrete game: start from \(g_t\), measure within-period training and rental pressure, and compare to the next gap transition \(g_{t+1}\).

The empirical section now separates pattern evidence from causality: joint sign restrictions are necessary fingerprints, while causal claims require variation in training pressure or capacity within comparable hardware cells.

The calibration section now separates selected-equilibrium moments, public unit anchors, external rental comparators, and derived scarcity inversions before combining them in the moment bridge.

The selected-equilibrium moment bridge now separates the average-demand rental index \(R(\mathbb E[T])\) from the action-weighted transaction rental \(\mathbb E[R(T)]\); the latter is weakly larger under mixed actions by Jensen's inequality.

The calibration section now also separates the exact task-horizon identity from conditional capex-to-capacity and published-price-to-scarcity conversions, so the capacity envelope is not read as observed GPU deployment or private training contracts.

The calibration section now audits rental comparability row by row: CoreWeave GB200 is the closest public scenario input, H100/H200 rows are cross-hardware sensitivity checks, and no public row is treated as a matched private frontier-training contract.

The hardware-cost bridge now states the power-boundary/PUE convention: if public MW capacity is facility power and rack kW is IT load, a factor \(\phi\) scales the cold rental floor and dollar schedules without changing the Bellman-Nash game; the calibration JSON records the baseline \(\phi=1\).

The calibration section now reports an elasticity audit: a one-percent driver change that moves \(r_0\) one-for-one changes the CoreWeave-GB200 \(\hat\tau\) scenario by 0.28 percentage points under Stargate and 0.07 percentage points under Abilene.

The capacity envelope now states its deployment-timing convention: reported accelerator counts and annual accelerator-hours set average in-service fraction \(\zeta=1\), so staggered deployment scales the physical stock and hour columns without changing the equilibrium model.

The introduction now also states that the paper is positive rather than a welfare theorem: it does not rank concentration, prescribe antitrust remedies, or decide whether compute rents are transfers or distortions without extra welfare primitives.

The tipping proposition now also separates existence of invariant distributions from uniqueness: dominance mass is reported under a specified initial-state long-run/Cesaro convention or a specified invariant distribution unless extra irreducibility and aperiodicity conditions are imposed.

Modeling Choice

The main model is discrete time because the important objects are the finite-state Bellman-Nash system, the kink at parity, and equilibrium selection across finite normal-form games. Continuous-time HJB formulas remain in the appendix as local marginal approximations; they are not used as global equilibrium or threshold claims.

Discrete vs Continuous Result Transfer

Result block Discrete claim Continuous-time role Status
Two-faced reliability Household value saturates; producer frontier value grows as \(A(q)=Ke^{aq}\). Same algebra; no equilibrium step. Retained.
Local racing at parity Inactive parity is not a one-period Nash equilibrium under continuous actions or a fine grid. Local marginal intuition. Retained locally.
Finite dynamic equilibrium Finite horizons use backward induction; admissible finite discounted games have stationary mixed Markov equilibria. HJB equations are local approximations. No uniqueness claim.
Tipping, dominance, compute concentration A selected \(\sigma\) induces \(M_\sigma\), drift, first exit, dominance, and training-demand moments. No universal threshold theorem. Selection-conditional.
Compute-market discipline \(R(T)=\bar W/(\bar Q-T)\) and \(E_T=\bar WT/(\bar Q-T)\). Same scarcity accounting. Retained.
Scaling-law calibration \(S_n=p^n\), \(1-p^n\), cold rental floors, and \(\hat\tau=1-r_0/R^m\) put the model in real units. No independent role. Unit bridge, not estimator.

Generated by verification/discrete_result_transfer_audit.py. The JSON records zero unique-global-equilibrium claims, zero universal-threshold claims, and zero global continuous-time claims.

Concentration Scope

Layer Inside current model Stronger claim requires Not claimed
Matched frontier pair \(g=q_1-q_2\) tracks leader-challenger dynamics in one cell. Cell definition, benchmark timing, and challenger rule. Industry concentration or HHI theorem.
Selected pair dominance \(M_\sigma,D_\sigma\), first exits, and mass on \(|g|\ge c\). Pre-specified selection and Markov-chain convention. Unique dominance or universal threshold.
Compute pressure \(\mathcal T_\sigma(g)\), \(T/\bar Q\), \(R(T)\), and rental wedges. Observed allocation, comparable contracts, capacity, and bottlenecks. Consumer share or downstream markup result.
Cross-cell aggregation \(\sum_c w_c\mathbf{1}\{|g_{c,t}|\ge b\}\) after cell moments are measured. Weights, common definitions, selection environments, and aggregation rule. Aggregate theorem from the two-lab game.
\(N\)-firm dynamics Outside the maintained state space. Vector gaps, entry/exit, larger games, and new computation. \(N\)-firm equilibrium or tipping result.
Welfare and antitrust Positive mechanism only. Welfare weights, spillovers, rent incidence, and policy instruments. Welfare ranking or optimal remedy.

Generated by verification/concentration_scope_map.py. The artifact has six rows, adds zero model primitives, and adds zero industry-concentration theorems.

Formal Result Dependencies

Formal statement Uses Does not use
Two faces Reliability primitive, finite household task measure, \(B(n)=bn^a\). Compute market, equilibrium selection, finite grid, or public data.
One-/finite-period mechanics \(\omega,r_0,\pi(g)\), fine/admissible grid, finite continuation games. Stationary uniqueness, continuous-time HJB, or public rental benchmarks.
Compute-market discipline \(\bar W,\bar Q,T<\bar Q\), residual-capacity buffer. Solved dynamic game, selected equilibrium, or observed private training allocation.
Finite-state dynamic theorem Finite \(G,X\), admissibility, \(\beta\), bounded payoffs, row-stochastic \(P\). Closed-form HJB, uniqueness, algorithmic convergence, or public calibration.
Tipping diagnostics Fixed \(\sigma\), finite \(G\), row-stochastic \(P\), Markov-chain convention. Selection-free point prediction or universal closed-form threshold.
Composition/access Logistic hazard, Bertrand repeatable services, access/leakage benchmark. Welfare ranking, access-price panel, or private training allocation.

Generated by verification/result_dependency_matrix.py. The full matrix has six rows and is a theorem-scope audit, not a new theorem.

Claim Thresholds

Claim level Current status Not claimed without more evidence
Formal discrete theory Supported Unique global equilibrium, welfare ranking, or industry-wide concentration theorem
Selected-equilibrium computation Supported with selection caveat Selection-free point prediction or universal tipping threshold
Calibration identity Supported Structural estimates of private training allocation or race intensity
Public scenario prediction Supported as scenario Observed GPU spot prices, private training contracts, or deployed training stock
Empirical mechanism support Missing critical evidence Full mechanism support from public list prices or selected simulations alone
Causal estimate Outside current package Causal effects from cross-sectional correlations or one selected transition matrix
Welfare or policy ranking Outside current model Normative conclusions from the positive race mechanism alone

Generated by verification/claim_threshold_audit.py. The audit has seven rows: four current-package or scenario rows, one missing-critical empirical row, and two rows outside the current package or model.

Empirical Data-Source Roadmap

Evidence block Current status Remaining gap
Task horizons and gap transitions Public partial Hardware-cell match, training allocation, and pre-specified selection
Training allocation bounds Nonpublic required Public prices or capex do not reveal private training share
Comparable rental or lease prices Public scenario partial Price alone does not identify \(T/\bar Q\), race intensity, or selection
Cold floor and capacity boundary Public partial In-service stock and controlled-capacity denominator by cell
Supplier bottlenecks and margins Public aggregate partial Cell-level bottleneck or margin evidence matched to laboratories
Access outcomes and leakage exposure Nonpublic required Gap-linked access requests and refusal reasons
Consumer-serving saturation Public partial; private needed Fixed product family and separation from training-class pressure
Selection environment Computational partial Ex post selection is fit, not a test
Full mechanism cell Missing critical evidence No single public anchor, selected simulation, or list price substitutes for the joined cell

Generated by verification/empirical_data_source_roadmap.py. The roadmap has nine rows, five public or public-partial source classes, four rows requiring nonpublic/private or missing joined evidence, nine rows requiring matched cells, and zero structural parameters estimated.

Main Theoretical Changes

  • Rebuilt the primitives around task reliability, with a household menu face that saturates and a producer frontier face that grows as \(A(q)=Ke^{aq}\).
  • Changed the cheap-race claim: commoditization alone is not the driver; capacity buildout lowers the cold rental price \(r_0=\bar W/\bar Q\), which can make frontier training cheaper.
  • Rebuilt the main model in discrete time: one-period mechanics first, then finite-horizon games, then the discounted finite-state dynamic game.
  • Characterized finite-horizon Markov-perfect equilibria by backward induction on finite normal-form games.
  • Made finite action-grid admissibility explicit: a residual-capacity buffer \(\eta>0\) keeps \(R(T)\) and one-period payoffs bounded.
  • Characterized the infinite discounted grid model as a finite stochastic game with stationary Markov equilibria on admissible grids, while separating existence from convergence of any selected numerical algorithm.
  • Added the equilibrium-induced transition matrix \(M_\sigma\) as the discrete object for parity, tipping, dominance, and compute concentration.
  • Moved continuous-time HJB logic to the appendix as a small-period approximation rather than the source of global claims.
  • Kept compute-market accounting, composition/kink logic, escalation through input scarcity, and access-pricing implications.
  • Defined the empirical objects needed to take the model to data: training-class compute, capacity pre-commitments, supplier margins, rental prices, task-horizon gaps, and access refusals.
  • Clarified the literature position: the paper changes the prize, the input cost, and the tipping object relative to canonical race models.
  • Added a calibration bridge from task-horizon success/failure probabilities to capability units and from public infrastructure-capex anchors to \(\bar Q\), \(r_0\), and rental-price schedules.
  • Added a calibration object ledger separating endogenous selected-equilibrium moments, public unit anchors, external price comparators, and derived scarcity inversions.
  • Added a calibration accounting-boundary clarification separating exact reliability identities from conditional capex-to-\(r_0\) and comparable-price-to-\(\hat\tau\) conversions.
  • Added a rental-comparability audit showing that two public rows are GB200-family scenario inputs, six rows are cross-hardware sensitivity references, and zero rows observe a matched private training contract or \(T/\bar Q\).
  • Added a power-boundary/PUE convention: when public MW is facility power but rack kW is IT load, \(k\) is replaced by \(\phi k\), scaling \(r_0\) and dollar rental schedules without changing the equilibrium model; the verification output now records baseline \(\phi=1\).
  • Added a mechanism-preservation paragraph explaining why the discrete model keeps the major economic results while dropping unique closed-form equilibrium and universal-threshold claims.
  • Added a two-laboratory scope clarification: the gap state is a frontier-pair reduction in a matched hardware/task cell, while industry-wide concentration claims need aggregation or an explicit \(N\)-firm extension.
  • Added empirical boundary conditions clarifying when the mechanism is distinctive rather than a generic AI-demand or hardware-scarcity story.
  • Added an empirical implementation sequence separating panel construction, object transformations, and joint sign tests under explicit equilibrium selection.
  • Added a selection-correspondence discipline: absent a pre-specified selection rule, the empirical object is the set of selected-equilibrium moments, not an ex post point prediction.
  • Added empirical timing discipline so training reservations, capacity commitments, and rental premia are interpreted as period-\(t\) inputs for the next gap transition, not contemporaneous sorting evidence.
  • Added a causal-design guardrail separating necessary joint-pattern evidence from causal estimates of input rents or gap drift.
  • Added an early scaling-law notation bridge: the common \(1-q_r^n\) failure form is the paper's \(1-p^n\), because \(q\) denotes capability.
  • Added an explicit task-horizon gap derivation showing that same-threshold horizon ratios identify \(g=q_i-q_j\).
  • Added a task-reliability scaling table showing how 50-percent task horizons map to \(p\), per-step failure, model \(q\), gap relative to 137 minutes, and the \(a=1\) frontier-value ratio.
  • Added a frontier-value curvature sensitivity table showing how \((n/137)^a\) changes the real-number implication of task-horizon improvements.
  • Added a public-source provenance ledger documenting the raw public numbers, transformations, model uses, and limitations behind the calibration bridge.
  • Added a calibration source-freshness audit recording six public anchors, three high-volatility rows, two public list-price rows, and six rows requiring recheck before empirical use.
  • Added a calibration-target residual showing that the illustrative selected stationary benchmark does not match the public GB200 rental-implied scarcity targets.
  • Added a capacity-unit prediction envelope translating public power-capacity anchors into GB200-equivalent accelerators and annual training accelerator-hours.
  • Added a prediction-versus-forecast guardrail so the capacity envelope is read as a conditional model implication, not an unconditional forecast of actual private GPU use.
  • Added a deployment-timing stock-flow convention: the capacity envelope records average in-service fraction \(\zeta=1\), and staggered deployment scales physical stock and accelerator-hour predictions by \(\zeta\).
  • Reframed the abstract around the economic mechanism, selected Markov-chain moments, the \(S_n=p^n\) calibration bridge, and matched hardware-cell evidence.
  • Added an early empirical-status paragraph separating public calibration unit discipline from evidence on private training allocation, race intensity, and equilibrium selection.
  • Added an early computational-status paragraph separating finite-grid diagnostic evidence from uniqueness, robustness, and private-behavior claims.
  • Added an early welfare-scope paragraph separating positive concentration-pressure results from welfare rankings and policy prescriptions.
  • Added a Markov-chain convention for dominance mass: selected \(M_\sigma\) always has invariant distributions, but long-run dominance is tied to an initial-state/Cesaro convention or a specified invariant distribution unless extra chain-uniqueness conditions are imposed.
  • Added a calibration-sensitivity sweep varying discount rate, asset life, and utilization around the public capex anchors.
  • Added a calibration-elasticity audit differentiating the existing \(r_0\), \(\hat\tau\), and GB200-equivalent capacity-unit formulas.
  • Added a selected-equilibrium moment bridge translating the benchmark \(M_\sigma\) into drift, exit, dominance, selected training share, average-demand rental multiplier, action-weighted transaction rental multiplier, and dollar rental moments under the public capex anchors.
  • Added an identification map separating what is measured, conditionally identified, inverted from prices, disciplined by selected transition moments, and not identified from a single observed series.
  • Added a generated parameter-status ledger recording eleven object classes and zero structural parameters estimated from public data.
  • Added a generated claim-threshold audit separating current formal, computational, calibration, and scenario claims from missing empirical mechanism support, causal claims, and welfare or policy ranking.
  • Added a generated empirical data-source roadmap mapping nine evidence blocks to source classes, current status, remaining gaps, and the matched-cell requirement.
  • Added a generated formal-result dependency matrix separating theorem assumptions from calibration inputs, selected simulations, and continuous-time approximations.
  • Added a generated discrete-vs-continuous result-transfer audit showing six major result blocks, no unique global equilibrium claim, no universal tipping-threshold claim, and no global continuous-time theorem claim.
  • Added a generated concentration-scope map separating matched-pair dominance, compute pressure, cross-cell aggregation, \(N\)-firm extensions, and welfare or antitrust claims.
  • Added a generated panel estimand map translating existing model objects into six descriptive matched-cell contrasts, with explicit nonidentification warnings for each contrast.
  • Added a model-implied rental schedule table reporting dollars per accelerator-hour from the public capex anchors at training shares \(T/\bar Q=0,0.25,0.50,0.75,0.90\).
  • Added the rental-benchmark inversion \(\hat\tau=1-r_0/R^m\), so published cloud rental prices can be read as conditional scarcity-share scenarios.
  • Added an empirical discipline and falsification table tying consumer margins, training-class rental premia, scarcity shares, gap dynamics, access refusals, and selection rules to joint predictions and rejection tests.
  • Added a generated empirical acceptance/rejection scorecard separating support conditions, scope failures, rejection tests, and current public-pilot status across eight mechanism gates.
  • Added a synthetic mechanism-pattern audit showing that the empirical gates accept one mechanism-consistent constructed panel and reject rent-only, mean-reverting-gap, and ex-post-selection false positives.
  • Added a competing-explanations table separating the mechanism from generic AI demand, hardware supply shocks, benchmark noise, broad access policy, and pure selection artifacts.
  • Added a result-status hierarchy separating results retained by the discrete model from tipping, dominance, and compute-concentration claims that are conditional on equilibrium selection.
  • Added a claim-status comparison table separating exact discrete results, selected-equilibrium diagnostics, calibration identities, and continuous-time local approximation.
  • Added a result-preservation clarification: the discrete model keeps the major economic results while dropping unique closed-form global equilibrium and universal tipping-threshold claims.
  • Added a maintained-primitives ledger collecting the reliability, frontier-task, compute-market, finite-game, selection, and calibration objects used below.
  • Tightened the introduction's result ladder so unconditional formal claims, selected-equilibrium Markov-chain diagnostics, and extra refinement assumptions are separated without correction-memo framing.
  • Added a stochastic-tremble sensitivity table for the benchmark selected stationary diagnostic, keeping it explicitly outside the equilibrium theorem.
  • Added a selected Nash-deviation audit separating state-level continuation-game Nash accuracy from stationary Bellman fixed-point convergence.
  • Added a stationary policy-evaluation audit that freezes selected policies, forms \(M_\sigma\), and solves the induced linear Bellman evaluation equations exactly.
  • Added a damping-path sensitivity table for the benchmark selected stationary diagnostic, keeping computational path dependence separate from equilibrium existence.
  • Added a homotopy-path sensitivity table for the benchmark selected stationary diagnostic, showing stable row-stochastic accounting, parity training, exit, and dominance but path-sensitive signed drift and nonconverged warm-start rows.
  • Added an initial-value sensitivity table for the benchmark selected stationary diagnostic, changing only starting value functions and showing stable accounting with path-sensitive signed drift.
  • Added a state-noise sensitivity table that perturbs the transition kernel inside the stationary Bellman-Nash computation before resolving the selected fixed point.
  • Added a selected-diagnostic envelope summarizing 29 stationary diagnostic rows across selection, grid, damping, homotopy, initial-value, state-noise, and tremble checks.
  • Added a selected moment-correspondence envelope showing that the clean converged Bellman-Nash pool spans both parity persistence and rapid exit, so selection-free empirical predictions are set-valued.
  • Added a standalone conclusion summarizing the mechanism, retained formal results, selection-conditional dynamic objects, calibration scenarios, and empirical burden.
  • Added a claim ladder separating formal theory, selected-equilibrium computation, calibration-unit conversions, and top-level empirical mechanism evidence.
  • Added a minimum evidence package requiring matched hardware cells, controlled capacity, training-allocation bounds, comparable prices, cold floors, dated task-horizon gaps, supplier bottlenecks, and access outcomes before the mechanism is treated as empirically tested.
  • Added an empirical sufficient-statistics checklist tying task-horizon transitions, training-allocation bounds, rental wedges, supplier bottlenecks, access outcomes, and selection environments to the model objects they discipline.
  • Added a public-pilot missingness disclosure: current public rows do not populate training-allocation bounds, so \(\hat\tau\) remains a rental-implied scarcity scenario rather than a measured training share.
  • Added a public-pilot coverage table showing that the public panel has six hardware rows, five rental rows, two capex/capacity rows, one task-horizon row, and zero training-allocation rows.
  • Added an empirical-readiness audit showing which public-pilot gates pass for unit/scenario work and which gates remain missing for the full mechanism test.
  • Added a laboratory-hardware-quarter measurement protocol for accelerator/rack hours, training reservations, rental prices, power and data-center commitments, task-horizon gaps, supplier bottlenecks, and access restrictions.
  • Added data-appendix design rules for sample construction, hardware-class comparability, controlled capacity, training-allocation bounds, rental-price conversion, validation checks, and access-outcome coding.
  • Added an identification hierarchy that separates access-price evidence, rental-price evidence, scarcity-share evidence, selected-transition evidence, and public-anchor unit checks.
  • Added a pilot data appendix template listing variables, source classes, transformations, and model uses for the minimum empirical panel.
  • Added a machine-readable blank pilot CSV and transformation script that validate the same fields and compute \(T/\bar Q\), \(p\), \(\lambda\), model \(q\), log horizon gaps, \(r_0\), and \(\hat\tau\).
  • Added a six-row public-anchor pilot panel using listed cloud prices, public infrastructure commitments, NVIDIA rack specifications, and METR's task-horizon example.

Verification Brought Alongside It

  • The derivation tab now walks through the discrete-time mathematical claims without skipping intermediate algebraic steps.
  • The finite-horizon diagnostic computes a three-period grid game by backward induction.
  • The finite-grid feasibility diagnostic checks residual-capacity admissibility, bounded one-period payoffs, and row-stochastic interpolation.
  • The stationary diagnostic iterates selected continuation games and reports the Bellman residual, transition row sums, drift, first-exit probabilities, and compute demand.
  • The stationary theorem gives existence and a finite Bellman-Nash representation; convergence of selected value-iteration, policy-iteration, or homotopy paths is reported through residual and path diagnostics.
  • The paper now reports a compact stationary sweep table with Bellman residuals, row-stochastic transition checks, parity training, first-exit probabilities, and dominance mass.
  • The selected stationary runs use the largest-joint-continuation-payoff selection rule; stronger computational claims require alternative selection rules, larger grids, stochastic perturbations, and path checks.
  • The selection-rule stress test reruns the benchmark grid under alternative computational selection rules; only the largest-joint rule converges under the same damped iteration, so the other rows are sensitivity diagnostics.
  • The grid-refinement audit holds the benchmark economics fixed and varies only the grid, showing row-stochastic transition accounting but grid-sensitive convergence and tipping diagnostics.
  • The damping-path audit holds the benchmark grid and selection rule fixed while varying value-iteration damping and iteration caps; nonconverged rows are path diagnostics.
  • The homotopy-path audit holds the benchmark grid, final \(\omega=0.25\), damping, and selection rule fixed while varying only the \(\omega\)-continuation warm start.
  • The initial-value sweep holds the benchmark grid, \(\omega=0.25\), damping, economics, and selection rule fixed while varying only the starting value functions.
  • The state-noise sweep perturbs the transition kernel inside the selected stationary Bellman-Nash problem and resolves the perturbed game under the same max-joint selection rule.
  • The stochastic-tremble sweep mixes the benchmark selected policies with uniform action trembles and recomputes \(M_\sigma\)-style diagnostics without treating the trembled rows as equilibria.
  • The selected-diagnostic envelope CSV reports 29 diagnostic rows, 18 converged or base-converged rows, 11 nonconverged sensitivity-path rows, 26 rapid-exit rows, 26 high-dominance rows, and maximum row-sum error 4.44e-16.
  • The selected moment-correspondence CSV reports 14 clean converged Bellman-Nash rows, 25 stationary diagnostic paths, 4 policy-tremble matrices, and maximum row-sum error 4.44e-16.
  • The stationary policy-evaluation audit CSV reports three converged rows, maximum policy-evaluation gap 5.44e-4, maximum linear residual 6.38e-16, and maximum row-sum error 2.22e-16.
  • The concentration-scope map CSV reports six scope layers, zero model primitives added, zero industry-concentration theorems added, and explicit aggregation, \(N\)-firm, and welfare-primitives requirements.
  • The panel-estimand map CSV reports six descriptive matched-cell contrasts, zero structural parameters estimated, and explicit matched-cell and pre-specified-selection requirements.
  • The synthetic mechanism-pattern audit CSV reports four constructed panels, 24 synthetic rows, one accepted scenario, three rejected scenarios, and maximum rental identity error 3.33e-16.
  • The stationary sweep records low-, benchmark-, and high-intensity selected dynamics without claiming uniqueness or calibration.
  • The calibration JSON records the \(1-p^n\) task-reliability map, representing the common \(1-q_r^n\) scaling-law notation without overloading capability \(q\), plus public GB200/Stargate/Abilene cost scenarios and published rental benchmark comparisons without treating them as equilibrium estimates.
  • The task-scaling CSV reports 30-, 137-, 480-, and 1440-minute 50-percent horizons with \(q=3.77, 5.29, 6.54, 7.64\) and \(a=1\) frontier-value ratios 0.22, 1.00, 3.50, and 10.51 relative to 137 minutes.
  • The curvature sensitivity CSV reports the same horizon ratios under \(a=0.5,1.0,1.5,2.0\), with the 1440-minute row equal to 3.24, 10.51, 34.08, and 110.48.
  • The source-provenance CSV records the public quantities behind the calibration bridge and checks \(42.00/4=10.50\) and \(49.24/8=6.155\).
  • The rental-comparability audit CSV records two hardware-matched CoreWeave GB200 rows, six cross-hardware H100/H200 rows, zero observed comparable private-contract rows, and a maximum rental-inversion identity error of 4.44e-16.
  • The target-residual CSV reports \(\tau_\sigma(0)=0.168\), public GB200 rental-implied targets 0.719 and 0.930, and the current grid maximum training share 0.324.
  • The capacity-envelope CSV reports 6.00m GB200-equivalent accelerators for the 10 GW anchor, 0.72m for the 1.2 GW anchor, the average in-service fraction \(\zeta=1\), and the implied selected and rental-target training accelerator-hours.
  • The calibration sensitivity CSV shows how \(r_0\), \(R(0.75\bar Q)\), and \(\hat\tau\) move under low-, baseline-, and high-floor annualization assumptions.
  • The calibration elasticity CSV reports 22 scenario-driver rows; a one-percent driver change that moves \(r_0\) one-for-one changes \(\hat\tau\) by 0.28 percentage points under Stargate and 0.07 under Abilene.
  • The equilibrium moment bridge CSV combines the benchmark selected transition matrix with the calibration bridge, showing \(\mathcal T_\sigma(0)/\bar Q=0.168\), average-demand rental multiplier 1.20245, transaction-rental multiplier 1.20292, exit 10 equal to 1.00, and dollar translations under the public capex anchors.
  • The identification map CSV records the required normalizations and nonidentification warnings for capability gaps, \(a\), \(r_0\), \(\tau\), \(\mathcal T_\sigma\), \(M_\sigma\), and selection claims.
  • The parameter-status ledger CSV classifies eleven objects and records zero structural parameters estimated from public data.
  • The empirical data-source roadmap CSV reports nine source-class rows, five public or public-partial rows, four rows requiring nonpublic/private or missing joined evidence, nine matched-cell requirements, and zero structural parameters estimated.
  • The result-dependency matrix CSV covers six formal statements, with five rows requiring no public calibration input and only the tipping row requiring selection for point predictions.
  • The rental benchmark CSV exposes the \(\hat\tau\) calculations behind the paper's calibration table.
  • The pilot transform summary reports that the blank template has zero empirical rows and passes built-in arithmetic checks for \(S_n=p^n\), \(1-p^n\), and \((1-\hat\tau)R^m=r_0\).
  • The public pilot panel recovers \(r_0=2.95\), \(r_0=0.74\), \(\hat\tau=0.72\), \(\hat\tau=0.93\), and maps the 137-minute, 50-percent horizon to model \(q=5.29\).
  • The empirical readiness CSV marks validation identities, task-reliability units, capex cold-floor units, and rental-wedge scenarios as pass; training allocation, access outcomes, and a joint mechanism cell are missing.
  • The diagnostic JSON records the selected equilibrium at parity for each date.
  • The horizon comparison graph and table compare one-, two-, and three-period computed equilibria and flag the multi-period model as selection-dependent.

Task-Reliability Scaling

Horizon \(n\) \(p=0.5^{1/n}\) \(1-p\) \(q\) \(q-q_{137}\) \(A/A_{137}\), \(a=1\)
30 0.97716 0.02284 3.77 -1.52 0.22
137 0.99495 0.00505 5.29 0.00 1.00
480 0.99856 0.00144 6.54 1.25 3.50
1440 0.99952 0.00048 7.64 2.35 10.51

Rows use 50 percent success at a common task-length unit. The last column is \(e^{q-q_{137}}\), so for general \(a\) the frontier-value ratio is \((n/137)^a\). This is a scaling-law unit conversion, not an equilibrium estimate.

Frontier-Value Curvature Sensitivity

Horizon \(n\) \(n/137\) \(a=0.5\) \(a=1.0\) \(a=1.5\) \(a=2.0\)
30 0.22 0.47 0.22 0.10 0.05
137 1.00 1.00 1.00 1.00 1.00
480 3.50 1.87 3.50 6.56 12.28
1440 10.51 3.24 10.51 34.08 110.48

Generated by verification/frontier_value_curvature_sensitivity.py. Entries are \((n/137)^a\), not estimates of \(a\) or equilibrium outcomes.

Model-Implied Rental Schedule

Scenario \(T/\bar Q=0\) 0.25 0.50 0.75 0.90
Stargate full commitment $2.95 $3.94 $5.90 $11.81 $29.52
Abilene joint venture $0.74 $0.98 $1.48 $2.95 $7.38

Entries are dollars per accelerator-hour from \(R(T)=r_0/(1-T/\bar Q)\). They are conditional schedules from public capex anchors, not observed transaction prices.

Calibration Elasticity Audit

Driver \(r_0\) elasticity Stock/hours elasticity \(\Delta\hat\tau\), pp
Investment \(I\), MW fixed +1 0 / 0 -0.28 / -0.07
Capacity \(M\), investment fixed -1 +1 / +1 +0.28 / +0.07
Rack kW \(k\) or \(\phi\) +1 -1 / -1 -0.28 / -0.07
Accelerators/rack \(G_R\) -1 +1 / +1 +0.28 / +0.07
Annuity \(\alpha\) +1 0 / 0 -0.28 / -0.07
Utilization \(u\) -1 0 / +1 +0.28 / +0.07
Rental \(R^m\) 0 0 / 0 +0.28 / +0.07
In-service \(\zeta\) 0 +1 / +1 0.00 / 0.00

Generated by verification/calibration_elasticity_audit.py. The last column uses the CoreWeave GB200 comparator and reports Stargate / Abilene percentage-point effects for a one-percent driver increase.

Public Calibration Source Provenance

Source Raw public quantity Transformation Limitation
METR time horizons 2h17 at 50% success \(p=0.5^{1/137}\), \(q=5.29\) One public example
NVIDIA GB200 NVL72 72 GPUs, about 120 kW per rack \(G_R=72\), \(k=120\) kW GB200-equivalent assumption
Stargate $500b and 10 GW $50.0m/MW; \(r_0^{gpu}\) = $2.95 Aggregate commitment
Abilene $15b and 1.2 GW $12.5m/MW; \(r_0^{gpu}\) = $0.74 Project financing anchor
CoreWeave GB200 $42.00/hour for 4 GPUs $10.50/GPU-hour for \(R^m\) On-demand list price
CoreWeave H100 $49.24/hour for 8 GPUs $6.155/GPU-hour comparison Different hardware class
Crusoe H100/H200 $3.90 and $4.29/GPU-hour Direct comparison benchmarks Different cloud product

The provenance ledger has a June 13, 2026 source-check date and validates the simple per-accelerator conversions. It does not estimate private training allocation.

Calibration Source Freshness

Source Timing Volatility Current use
METR task horizons Updated 2026-05-08; checked 2026-06-13 High Unit anchor; recheck for latest horizon
NVIDIA GB200 NVL72 Checked 2026-06-13 Medium Hardware normalization; recheck SKU/power boundary
Stargate 2025-01-21 / 2025-09-23; checked 2026-06-13 Medium Scenario anchor; recheck buildout/capex timing
Abilene 2025-05-21 body; page metadata 2026-04-25; checked 2026-06-13 Medium Scenario anchor; recheck deployed stock
CoreWeave prices Checked 2026-06-13 High Price comparator; recheck each calibration
Crusoe prices Checked 2026-06-13 High Cross-hardware sensitivity; recheck each calibration

Generated by verification/calibration_source_freshness_audit.py. The audit reports six rows, three high-volatility rows, two public list-price rows, and six rows requiring recheck before empirical use.

Rental Benchmark Comparability

Benchmark class Hardware gate Missing gates Use
CoreWeave GB200 NVL72 Pass: same GB200 NVL72 family Private training contract, matched region, utilization/power boundary, observed \(T/\bar Q\) Closest public scenario; not observed evidence
CoreWeave/Crusoe H100 and H200 Fail: different accelerator generation Same contract, region, utilization, and allocation gaps Cross-hardware sensitivity only
All public rental rows Identity check passes Comparable private-contract rows: 0 of 8 \(\hat\tau\) remains conditional

Generated by verification/rental_comparability_audit.py. The audit reports two GB200-family rows, six cross-hardware rows, and zero observed comparable private training-contract rows.

Selected-Equilibrium Moment Bridge

Scenario \(\tau_\sigma(0)\) \(|D_\sigma(0)|\) \(R_\sigma^A(0)\) \(R_\sigma^A/R^m\) \(\hat\tau^m\) Exit 10 Dominance
Stargate full commitment 0.17 1.38 $3.55 0.34 0.72 1.00 1.00
Abilene joint venture 0.17 1.38 $0.89 0.08 0.93 1.00 1.00

The displayed rental column is the average-demand index \(R_\sigma^A=R(\mathbb E[T])\). The moment-bridge JSON also reports the action-weighted transaction rental \(\mathbb E[R(T)]\), which is weakly larger under mixed actions. This is a moment bridge, not a fit to the published rental benchmark.

Calibration Target Residual

Scenario \(\tau_\sigma(0)\) \(\hat\tau^m\) Gap Max grid \(\tau\) \(R^m/R_\sigma^A\) Bound needed
Stargate full commitment 0.17 0.72 0.55 0.32 2.96 2.68
Abilene joint venture 0.17 0.93 0.76 0.32 11.83 3.05

The rental ratio uses the average-demand index \(R_\sigma^A=R(\mathbb E[T])\), not the action-weighted transaction rental reported in the moment-bridge JSON. Both rental-implied targets exceed the current grid's maximum training share. This is a calibration limitation diagnostic, not a re-estimation.

Capacity-Unit Prediction Envelope

Scenario Total acc. Selected train acc. Selected train hrs Rental-implied acc. Rental-implied hrs
Stargate full commitment 6.00m 1.01m 7.52b 4.31m 32.11b
Abilene joint venture 0.72m 0.12m 0.90b 0.67m 4.98b

Counts assume all announced power capacity is GB200-NVL72-equivalent at 120 kW per 72-accelerator rack, 85% utilization, matched power boundary, and average in-service fraction \(\zeta=1\). If public capacity is facility MW and rack kW is IT load, the PUE factor rescales the dollar floor. Staggered deployment scales the physical stock and hour columns by \(\zeta\). Prediction here means a conditional model implication, not an unconditional forecast of actual private GPU use.

Calibration Sensitivity

Scenario Case \(r,L,u\) \(r_0^{gpu}\) \(R(0.75\bar Q)\) \(\hat\tau\)
Stargate Low 5%, 7, 0.95 $1.73 $6.92 0.84
Stargate Base 10%, 5, 0.85 $2.95 $11.81 0.72
Stargate High 15%, 3, 0.60 $6.94 $27.78 0.34
Abilene Low 5%, 7, 0.95 $0.43 $1.73 0.96
Abilene Base 10%, 5, 0.85 $0.74 $2.95 0.93
Abilene High 15%, 3, 0.60 $1.74 $6.94 0.83

The table varies only annualization and utilization assumptions. It is a scenario sensitivity around public capex anchors, not an estimate of private allocation.

Selected Nash-Deviation Audit

Case Converged Bellman residual Max deviation gain Row error \(\mathcal T_\sigma(0)\) Exit 10
Benchmark max-joint Yes 9.3e-7 6.7e-16 2.2e-16 1.68 1.00
Selection min-joint No 1.8e-1 2.2e-16 1.1e-16 1.01 1.00
State noise 0.05 Yes 9.6e-7 4.4e-16 2.2e-16 1.69 1.00
Coarse grid No 5.3e-2 1.1e-16 1.1e-16 0.36 0.00

Generated by verification/selected_nash_deviation_audit.py. Max deviation gain checks finite continuation-game Nash accuracy; Bellman residual still governs stationary fixed-point status.

Stationary Policy-Evaluation Audit

Case Converged Bellman residual Eval. gap Linear residual Row error
Low intensity coarse Yes 2.9e-5 5.4e-4 6.9e-18 1.1e-16
Benchmark Yes 9.3e-7 1.8e-5 5.0e-16 2.2e-16
High intensity coarse Yes 2.9e-5 5.4e-4 6.4e-16 2.2e-16

Generated by verification/stationary_policy_evaluation_audit.py. Frozen selected policies solve the induced linear Bellman systems to machine precision; this is a fixed-policy accounting audit, not a new equilibrium selection rule.

Grid-Refinement Audit

Grid Converged Residual Row error \(\mathcal T_\sigma(0)\) Exit 10 Dominance
15 x 4 No 5.3e-2 1.1e-16 0.36 0.00 0.00
21 x 5 Yes 9.3e-7 2.2e-16 1.68 1.00 1.00
25 x 5 No 1.5e-2 2.2e-16 1.68 1.00 1.00

Rows fix the benchmark economics and max-joint selection rule, changing only the grid. Nonconverged rows are sensitivity diagnostics, not stationary-equilibrium claims.

Damping-Path Sensitivity

Damping Iter. Converged Residual Row error \(D(0)\) \(\mathcal T(0)\) Exit 10 Dominance
0.50 220 No 1.0e-1 0.0 0.00 0.81 0.00 0.00
0.70 325 Yes 9.3e-7 2.2e-16 -1.38 1.68 1.00 1.00
0.90 220 No 3.2e-5 2.2e-16 -1.38 1.68 1.00 1.00

Rows fix the benchmark grid, economics, and max-joint selection rule, changing only value-iteration damping and the iteration cap. Nonconverged rows are computational-path diagnostics, not stationary-equilibrium claims.

Homotopy-Path Sensitivity

Path Iter. Converged Residual Row error \(D(0)\) \(\mathcal T(0)\) Exit 10 Dominance
Direct 0.25 325 Yes 9.3e-7 2.2e-16 -1.38 1.68 1.00 1.00
Low 0.10 -> 0.25 220 No 5.3e-5 2.2e-16 1.37 1.68 1.00 1.00
High 0.40 -> 0.25 220 No 1.9e-5 2.2e-16 1.37 1.68 1.00 1.00

Rows hold the benchmark grid, final \(\omega=0.25\), damping, and max-joint selection rule fixed. Only the \(\omega\)-continuation warm start changes; nonconverged rows are path diagnostics, not stationary-equilibrium claims.

Initial-Value Sensitivity

Start Iter. Converged Residual Row error \(D(0)\) \(\mathcal T(0)\) Exit 10 Dominance
Zero 325 Yes 9.3e-7 2.2e-16 -1.38 1.68 1.00 1.00
Leader tilt 324 Yes 9.6e-7 1.1e-16 -1.38 1.68 1.00 1.00
Follower tilt 350 No 7.5e-5 2.2e-16 1.35 1.69 1.00 1.00
Gap bonus 330 Yes 9.5e-7 1.1e-16 1.35 1.68 1.00 1.00

Rows hold the benchmark grid, \(\omega=0.25\), damping, economics, and max-joint selection rule fixed. Only starting value functions change; nonconverged rows are path diagnostics, not stationary-equilibrium claims.

State-Noise Sensitivity

\(\epsilon\) Iter. Residual Row error \(D_\epsilon(0)\) \(\mathcal T_\epsilon(0)\) Exit 10 Dominance
0.00 325 9.3e-7 2.2e-16 -1.38 1.68 1.00 1.00
0.01 330 9.3e-7 2.2e-16 -1.38 1.69 1.00 1.00
0.05 324 9.6e-7 2.2e-16 -1.36 1.69 1.00 1.00
0.10 324 9.6e-7 2.2e-16 -1.37 1.73 1.00 1.00

Rows perturb the transition kernel inside the Bellman-Nash continuation game before resolving the selected stationary problem. The maintained benchmark is \(\epsilon=0\); positive rows are sensitivity diagnostics, not new model primitives.

Stochastic-Tremble Sensitivity

\(\epsilon\) Row error \(D_\epsilon(0)\) \(\mathcal T_\epsilon(0)\) Exit 10 Dominance
0.00 2.2e-16 -1.38 1.68 1.00 1.00
0.01 2.2e-16 -1.36 1.68 1.00 1.00
0.05 3.3e-16 -1.31 1.66 1.00 0.98
0.10 4.4e-16 -1.24 1.64 1.00 0.96

Rows with \(\epsilon>0\) mix the selected max-joint stationary policies with uniform action trembles. They are transition-matrix sensitivity checks, not equilibria of a perturbed game.

Selected-Diagnostic Envelope

Family Rows, conv. Max row err. Exit/Dominance range Takeaway
Race intensity 3, 3 2.2e-16 0.00-1.00 / 0.00-1.00 Regime variation; low intensity can persist at parity.
Selection rule 5, 1 2.2e-16 1.00-1.00 / 1.00-1.00 Accounting holds, but convergence depends on the rule.
Grid refinement 3, 1 2.2e-16 0.00-1.00 / 0.00-1.00 Coarse nonconverged row displays parity persistence.
Damping path 3, 1 2.2e-16 0.00-1.00 / 0.00-1.00 Low damping has not converged and remains near parity.
Homotopy path 3, 1 2.2e-16 1.00-1.00 / 1.00-1.00 Exit and dominance persist, but drift sign flips.
Initial values 4, 3 2.2e-16 1.00-1.00 / 1.00-1.00 Exit and dominance persist, but drift sign flips.
State noise 4, 4 2.2e-16 1.00-1.00 / 1.00-1.00 Small transition-kernel noise preserves the benchmark pattern.
Policy tremble 4, 4 4.4e-16 1.00-1.00 / 0.96-1.00 Exit survives; dominance declines with trembles.

Generated by verification/selection_diagnostic_envelope.py. Across 29 rows, 26 show rapid exit and high dominance, but 11 are nonconverged sensitivity-path rows; the envelope is not a uniqueness or robustness theorem.

Selected Moment Correspondence

Pool Rows, conv. Max resid. \(D_\sigma(0)\) \(\mathcal T_\sigma(0)\) Exit / dominance
Converged stationary correspondence 14, 14 2.9e-5 [-1.38, 1.35] [0.00, 1.81] [0.00, 1.00] / [0.00, 1.00]
All stationary diagnostic paths 25, 14 1.9e-1 [-1.40, 1.37] [0.00, 1.81] [0.00, 1.00] / [0.00, 1.00]
Policy-tremble induced matrices 4, 4 n/a [-1.38, -1.24] [1.64, 1.68] [1.00, 1.00] / [0.96, 1.00]

Generated by verification/selection_correspondence_envelope.py. The table is a set-valued empirical discipline: ranges are not frequency weights, and the benchmark selected row is one point inside the correspondence.

Panel Estimand Map

Estimand Panel contrast Support pattern Not claimed
Serving saturation \(m_{j,t}=\alpha_j+\delta_t+b_C H_{j,t}+u_{j,t}\) \(b_C\approx0\) while frontier horizons improve. Welfare or demand estimation.
Rental pressure \(\log R^m_{c,t}=\alpha_c+\delta_t+b_R T_{c,t}/\bar Q_{c,t}+u_{c,t}\) \(b_R>0\) in matched cells. Race intensity or selection.
Scarcity validation \(T_{c,t}/\bar Q_{c,t}-\hat\tau_{c,t}\), \(\hat\tau=1-r_0/R^m\) Small gap or clear comovement. Allocation from price alone.
Gap transition \(\mathbf{1}\{|g_{t+1}|>|g_t|\}=\alpha_b+\delta_t+b_D P_{c,t}+u_{c,t}\) \(b_D>0\) under pre-specified selection. Unique equilibrium or causality.
Access leakage \(A_{c,t}=\alpha_c+\delta_t+b_A\mathbf{1}\{|g_t|\le b\}L_{c,t}+u_{c,t}\) \(b_A>0\), with relaxation at large leads. Welfare ranking.
Full pattern \(J_{c,t}=\mathbf{1}\{\text{critical gates pass}\}\) Critical evidence appears in the same cell. One-variable test.

Generated by verification/panel_estimand_map.py. The artifact has six rows, estimates zero structural parameters, and requires matched hardware cells plus a pre-specified selection environment.

Empirical Sufficient Statistics

Statistic Disciplines Failure mode
Task-horizon gaps and transitions \(g\), \(M_\sigma\), \(D_\sigma\) Gaps move independently of measured training pressure.
Training-allocation bounds \(\mathcal T_\sigma(g)\), \(T/\bar Q\) Rental premia appear without reservations, cluster use, or power commitments.
Comparable rental wedge \(R(T)\), \(\hat\tau=1-r_0/R^m\) The wedge disappears after hardware, bundle, region, utilization, and term matching.
Supplier bottlenecks Scarcity channel Prices rise without delivery delays, supplier margins, financing constraints, or reserved capacity.
Access outcomes by gap \(E(1-e^{-ag})\ge \ell(E)\Delta V(g)\) Refusals are unrelated to gaps, leakage exposure, or competitive-use restrictions.
Selection environment Selected \(\sigma\) and \(M_\sigma\) Tipping appears only under an arbitrary computational selection.

This is a data checklist for the existing model objects, not a new estimator or equilibrium refinement.

Identification Map

Object Disciplined by Required restriction Not identified alone
Capability gap \(g\) Task-horizon ratios \(H_i/H_j\) Common suite, success threshold, date, and unit. Absolute capability or race intensity.
Frontier elasticity \(a\) \(\log P_0/\log(H_L/H_F)\) Observed access prices and no or separately priced leakage. Refused, bundled, rationed, or learning-confounded access.
Cold floor \(r_0\) Capex, power, accelerators, annuity, utilization. Common hardware, matched power-boundary/PUE convention, and financing assumptions. Private training demand or selection.
Scarcity share \(\tau\) \(\hat\tau=1-r_0/R^m\) Matched rental product and \(R^m\ge r_0\). Observed training allocation without usage or reservation data.
\(\mathcal T_\sigma(g)\) Training-allocation bounds. Controlled-capacity denominator and training-use bounds. Gap dynamics or dominance.
\(M_\sigma\) Capability-gap transition frequencies. State bins, benchmark-date comparability, and selection environment. A unique finite-game equilibrium.
Race intensity and selection Joint fit of drift, exit, dominance, training, rents, bottlenecks, access. Matched panel and explicit selection interpretation. A structural claim from one price, gap, or selected simulation.

The map is an identification discipline over existing objects. It is not a new estimator or model restriction.

Parameter Status Ledger

Object Claim status Paper use Not claimed
\(p,q,g\) Normalization and measurement index. Task-horizon units and gap states. Welfare quality, race intensity, or industry-wide concentration without aggregation.
\(a,\omega,s,\theta,\mu,\beta\) Primitive or conditionally identifiable parameter. Frontier curvature, race intensity, composition, imitation, and discounting. Structural estimates from current public rental or capex rows.
\(G,X,\eta,P\) Computational domain. Finite and bounded Bellman-Nash games. Continuous-state convergence, uniqueness, or an empirical estimate.
\(\sigma,M_\sigma,D_\sigma,\mathcal T_\sigma\) Selected-equilibrium object and endogenous outputs. Drift, first exit, dominance, selected training, and rental moments. Unique equilibrium, closed-form threshold, or robustness across selections.
\(r_0,\bar Q,R(T),\phi,\zeta\) Public-scenario input and accounting convention. Dollar rental schedules and capacity-unit scenarios. Observed private training allocation, equilibrium selection, or an unconditional forecast.
\(R^m,\hat\tau\), training allocation, access, bottlenecks, gaps Conditional inversion or missing empirical target. Minimum matched-hardware evidence for the mechanism. Observed \(T/\bar Q\) or full mechanism validation from public prices alone.

Generated by verification/parameter_status_ledger.py. The full ledger has eleven rows and reports zero structural parameters estimated from public data.

Competing Explanations

Nearby story Overlap Discriminating evidence
Generic AI demand High compute prices and large capacity commitments. Training-class wedges rise while consumer-serving margins can remain flat.
Hardware supply shock Rental prices rise for the same accelerator generation. Pressure is stronger in training-matched cells than comparable serving cells.
Benchmark noise Task-horizon gaps move across dates. Gap transitions persist and covary with measured training pressure.
Access policy Frontier access is refused, delayed, or throttled. Restrictions are tied to gaps, leakage exposure, or competitive-use terms.
Selection artifact A computed selected matrix exits parity quickly. Drift, first-exit, and dominance patterns survive plausible selection environments or are explicitly conditioned.

This table uses the existing model objects as discriminating evidence; it does not add a new primitive.

Conclusion Takeaway

The conclusion closes the main text with the paper's disciplined claim: the model explains why upstream training pressure can rise while consumer-facing margins look mature; finite-grid equilibria and selected transition matrices provide the dynamic objects; calibration gives real-number scenarios; empirical evidence must jointly match training allocation, prices, bottlenecks, access outcomes, and gap transitions.

Public-Pilot Coverage

Field group Populated rows Interpretation
Panel identifier and hardware class 6 Public rows can be organized into laboratory-hardware-quarter cells.
Listed rental price 5 Prices discipline rental-unit conversions, not private training allocation.
Public capex and controlled capacity 2 Scenario anchors for \(r_0\), \(\bar Q\), and implied cold floors.
Task horizon and success probability 1 Unit check for \(p\), \(\lambda\), and model \(q\).
Training-allocation lower/upper bounds 0 Missing object needed to turn \(\hat\tau\) into measured training pressure.

Empirical Readiness Audit

Gate Status Interpretation
Validation and identities Pass Six rows, no validation warnings, max identity residual 2.6e-15.
Task-reliability units Pass One task-horizon row maps a 50% horizon into \(p\), \(\lambda\), and \(q\).
Capex cold-floor units Pass Two public capex rows compute scenario cold rental floors.
Rental-wedge scenarios Pass Five rental rows and two \(\hat\tau\) rows support comparable-price scenarios.
Training allocation bounds Missing No row has controlled capacity plus training lower and upper bounds.
Supplier bottleneck evidence Partial Public infrastructure proxies exist, but matched supplier-margin evidence is absent.
Access outcomes and gap transitions Missing No access outcomes and no repeated task-horizon gaps; no row joins the minimum mechanism objects.

Generated by verification/empirical_readiness_audit.py. This is a readiness diagnostic, not an estimator or equilibrium result.

Acceptance and Rejection Scorecard

Gate Support condition Rejection condition Public pilot
Unit and hardware-cell comparability Common task unit, threshold, hardware/service bundle, capacity boundary, and identities. Unit identities fail or incompatible hardware, utilization, region, or contract terms are mixed. Partial support
Household-serving saturation Consumer-serving margins are flat while frontier task horizons improve. Frontier pressure appears mainly as higher consumer-serving margins. Missing
Training allocation and rental wedge Observed training bounds move with comparable rental premia and scarcity shares. Rental-implied scarcity is high while observed training allocation is low and no bottleneck is present. Missing critical
Supplier bottlenecks and upstream rents Training pressure is accompanied by rental premia, pre-commitments, bottlenecks, or supplier margins. Training pressure rises while comparable premia and bottleneck indicators stay flat. Missing
Gap dynamics and tipping High race pressure is followed by outward drift, first exit, or dominance under pre-specified selection. Gaps mean-revert despite high measured training pressure and comparable inputs. Missing critical
Access and leakage Refusals or throttling concentrate near parity or high-leakage environments and relax at large leads. Restrictions are unrelated to gaps or leakage exposure. Missing
Selection environment A rule, refinement, or public-correlation interpretation is specified before the data comparison. Tipping appears only by ex post selection and disappears under plausible alternatives. Partial support
Full mechanism test The same matched cells jointly show saturated margins, high training allocation, wedges, bottlenecks, access patterns, and selected-model gap drift. High measured race pressure does not produce upstream rents, access discipline, or gap movement. Not tested

Generated by verification/empirical_acceptance_scorecard.py. The JSON records eight rejection gates and no public-pilot support for the full mechanism test.

Synthetic Mechanism Pattern Audit

Synthetic panel Avg. \(T/\bar Q\) / \(\hat\tau\) Outward gap rate Failed gate Verdict
Mechanism-consistent 0.45 / 0.45 1.00 None Accepted
Rental-only false positive 0.08 / 0.73 0.00 Training-rental alignment; gap dynamics; access-leakage discipline Rejected
Mean-reverting gaps 0.61 / 0.61 0.00 Gap dynamics Rejected
Ex-post selection fit 0.49 / 0.49 1.00 Selection pre-specification Rejected

Generated by verification/synthetic_mechanism_pattern_audit.py. These are constructed panels, not data; the audit checks that the empirical gates reject single-variable false positives.

Current Status

The website now serves the discrete dynamic-game paper PDF as rendered page images, links to the source PDF, exposes the current step-by-step derivation, reports the selected-equilibrium diagnostics, links the finite-grid feasibility check and calibration bridge, and records the conclusion, concentration-scope map, panel-estimand map, empirical sufficient-statistics, competing-explanations, falsification, acceptance-scorecard, synthetic-pattern, identification-map, parameter-status-ledger, claim-threshold, data-source-roadmap, result-dependency-matrix, measurement, data-design, pilot-template, pilot-transformation, stationary policy-evaluation, grid-refinement, damping-path, homotopy-path, initial-value, state-noise, stochastic-tremble, selection-envelope, source-provenance, source-freshness, curvature-sensitivity, rental-comparability, calibration-target-residual, calibration-sensitivity, calibration-elasticity, selected-equilibrium moment-bridge, model-implied rental-schedule, public-anchor, public-pilot coverage, and empirical-readiness layers.

Discrete Equilibrium and Tipping Objects

This tab states the finite-state Bellman-Nash object and the induced Markov chain used to discuss tipping. It does not claim a closed-form or unique equilibrium.

Equilibrium object. The characterization is the Bellman-Nash fixed point together with the support equal-payoff conditions at every gap state. Tipping is read from the transition matrix induced by the selected equilibrium, not from a universal continuous-time threshold formula.

1. Exact Equilibrium Object

The state is the capability gap \(g\). Each firm chooses training \(x_i\). Training demand is \(T_i=x_i^2/2\), total training is \(T=(x_1^2+x_2^2)/2\), and the compute rental price is \(R(T)=\bar W/(\bar Q-T)\).

The finite action grid is admissible: for some residual-capacity buffer \(\eta>0\), every pair of grid actions satisfies \(T\le \bar Q-\eta\). This keeps \(R(T)\le \bar W/\eta\), so the finite normal-form games have bounded payoff entries.

The maintained primitives are fixed before any equilibrium computation: reliability \(\lambda(q)=e^{-q}\), task success \(e^{-\lambda(q)n}\), frontier value \(B(n)=bn^a\), scarce compute \((\bar Q,\bar W)\), finite grids \((G,X)\), discounting \(\beta\), imitation \(\mu\), transition kernel \(P\), and the selected equilibrium \(\sigma\) used to form \(M_\sigma\).

\[ \tilde g=(1-\mu)g+x_1-x_2, \qquad P(g'\mid g,x_1,x_2) \text{ maps } \tilde g \text{ back to } G. \]

The frontier payoff from the post-training gap is

\[ \pi(g)= \omega\left[ s\left(L(ag)-\frac12\right) +(1-s)\left[2\sinh(ag/2)\right]^+ \right], \qquad L(z)=\frac{1}{1+e^{-z}}. \]

Stage payoffs are

\[ u_1(g,x_1,x_2)=\pi(\tilde g)-R(T)\frac{x_1^2}{2}, \qquad u_2(g,x_1,x_2)=\pi(-\tilde g)-R(T)\frac{x_2^2}{2}. \]

2. Bellman-Nash Characterization

A stationary Markov equilibrium is a pair of mixed rules \(g\mapsto\sigma_i(g)\) and values \(V_i(g)\). Given the rival's rule and the value function, the payoff from choosing a pure action \(x_i\) at state \(g\) is

\[ W_i(g,x_i;\sigma_j,V_i) = \sum_{x_j\in X}\sigma_j(x_j\mid g) \left[ u_i(g,x_i,x_j) +\beta\sum_{g'\in G}P(g'\mid g,x_i,x_j)V_i(g') \right]. \]

The exact equilibrium conditions are the support conditions:

\[ W_i(g,x_i;\sigma_j,V_i)=V_i(g) \quad\text{if}\quad \sigma_i(x_i\mid g)>0, \]
\[ W_i(g,x_i;\sigma_j,V_i)\le V_i(g) \quad\text{if}\quad \sigma_i(x_i\mid g)=0. \]

Values are the equilibrium expected payoffs:

\[ V_i(g)= \sum_{x_i\in X}\sigma_i(x_i\mid g) W_i(g,x_i;\sigma_j,V_i). \]

Symmetry lets the computation impose

\[ V_2(g)=V_1(-g), \qquad \sigma_2(x\mid g)=\sigma_1(x\mid -g). \]
This is the actual finite-state equilibrium characterization. A pure-strategy equilibrium is the special case where each support has one action; then each chosen action is an argmax of \(W_i\).

3. Finite-Horizon Characterization

For \(H\) periods, set \(V_{i,0}(g)=0\). With \(h\) periods left, solve the finite matrix game with payoff

\[ U_{i,h}(g,x_1,x_2) = u_i(g,x_1,x_2) + \beta\sum_{g'\in G}P(g'\mid g,x_1,x_2)V_{i,h-1}(g'). \]

The Nash equilibrium of this matrix game gives \(\sigma_{i,h}(g)\) and \(V_{i,h}(g)\). This is exact backward induction, not an approximation.

4. Equilibrium-Induced Tipping

Once an equilibrium selection \(\sigma=(\sigma_1,\sigma_2)\) is fixed, it induces a Markov transition matrix over gap states:

\[ M_\sigma(g,g'') = \sum_{x_1,x_2\in X} \sigma_1(x_1\mid g)\sigma_2(x_2\mid g) P(g''\mid g,x_1,x_2). \]

The expected drift and expected training demand are

\[ D_\sigma(g)= \sum_{g''\in G}(g''-g)M_\sigma(g,g''), \qquad \mathcal T_\sigma(g)= \sum_{x_1,x_2\in X} \sigma_1(x_1\mid g)\sigma_2(x_2\mid g) \frac{x_1^2+x_2^2}{2}. \]

Parity is locally attracting when \(D_\sigma(g)\) points toward zero on a neighborhood of zero. It is locally repelling when \(D_\sigma(g)\) points away from zero. Dominance is measured by invariant or long-run mass on large \(|g|\), with the initial-state/Cesaro or invariant-distribution convention stated explicitly when the selected chain is not known to have a unique invariant distribution. First-exit probabilities from a parity band are computed from the band-restricted submatrix of \(M_\sigma\).

This is the discrete tipping structure: selected equilibrium \(\rightarrow\) transition matrix \(\rightarrow\) drift, first exit, convention-specific long-run mass, and compute concentration.

The same selected policy gives the empirical training share and rental moment once capacity units are fixed:

\[ \tau_\sigma(g)=\frac{\mathcal T_\sigma(g)}{\bar Q}, \qquad R_\sigma^A(g)=\frac{r_0}{1-\tau_\sigma(g)}, \qquad \bar R_\sigma(g)=\mathbb E_\sigma[R(T)]. \]

For the benchmark selected stationary diagnostic at parity, \(\tau_\sigma(0)=0.168\), the average-demand rental multiplier is 1.20245, the transaction-rental multiplier is 1.20292, exit by ten periods is 1.00, and dominance mass is 1.00. Those are predicted moments of the selected numerical equilibrium, not estimates of observed private training pressure.

5. Discrete versus Continuous Claim Status

Claim Discrete-time status Continuous-time role
Reliability and frontier value Exact static proposition: household value saturates and producer value grows as \(A(q)=Ke^{aq}\). Same primitive; no dynamic equilibrium issue.
Dynamic equilibrium Finite horizons are computed by backward induction; the admissible finite discounted game has stationary mixed Markov equilibria and a finite Bellman-Nash representation. Uniqueness and algorithmic convergence are not claimed. Smooth HJB equations are local approximations, not the global equilibrium object.
Tipping and dominance Defined after selecting an equilibrium: \(M_\sigma\), \(D_\sigma(g)\), first exit, convention-specific invariant or Cesaro mass, and \(\mathcal T_\sigma(g)\). No universal continuous-time threshold is used as a theorem.
Compute rents and calibration \(R(T)\), \(E_T\), \(S_n=p^n\), and \(\hat\tau=1-r_0/R^m\) are accounting and unit-conversion results. No additional equilibrium characterization.

6. Main Results in the Discrete Paper

Result Characterization Status in discrete time
Two faces Household value saturates, while producer value is \(A(q)=Ke^{aq}\). Relative producer value is \(e^{ag}\). Exact.
One-period mechanics At parity, a small unilateral training action has first-order frontier benefit and second-order compute cost. Exact for continuous actions; grid-dependent for finite action grids.
Finite horizon With \(H\) periods, each date-state problem is a finite continuation matrix game solved by backward induction. Exact after choosing finite state grids, admissible action grids, and an equilibrium selection at each matrix game.
Dynamic equilibrium The equilibrium is the Bellman-Nash support system above. It returns state-contingent training, compute demand, values, and transition dynamics. Exact finite-state characterization on admissible grids.
Tipping diagnostics A selected equilibrium induces \(M_\sigma\), drift \(D_\sigma(g)\), invariant mass, escape probabilities, and expected compute demand \(\mathcal T_\sigma(g)\). Exact conditional on equilibrium selection.
Compute-market discipline \(R(T)=\bar W/(\bar Q-T)\) and \(E_T=\bar WT/(\bar Q-T)\). Escalation shows up as compute rents when \(T\) approaches capacity. Exact accounting.
Calibration bridge Observed success \(S_n=p^n\) gives \(p=S_n^{1/n}\), \(\lambda=-\log p\), and \(q=-\log\lambda\). Public capex/capacity anchors set \(\bar Q\) and \(r_0\), with a matched power-boundary/PUE convention, and observed rental \(R^m\) maps to \(\hat\tau=1-r_0/R^m\). Exact unit conversion; not an equilibrium or price-estimation result.
Moment bridge A selected \(M_\sigma\) plus \(r_0\) gives \(D_\sigma\), exit, dominance, \(\mathcal T_\sigma/\bar Q\), and \(R_\sigma=r_0/(1-\mathcal T_\sigma/\bar Q)\). Exact conditional translation of a selected equilibrium; not a calibration fit.
Composition Priority tasks are discouraging when behind; repeatable Bertrand tasks give comeback incentives and a kink at parity. Exact payoff objects; smooth \(s^*=1/(1+\theta^3)\) is only the old local smooth projection.
Access market The static no-learning limit price is \(P_0(g)=e^{ag}\). With leakage, sell only when access rents cover the dynamic value lost to the rival. Exact static benchmark plus dynamic profitability condition.

7. Appendix Approximation Only

The continuous-time limit is retained only to show the local marginal training formula that comes out of the discrete Bellman equation as the period length shrinks:

\[ x(g)=\frac{[V'(g)]^+}{r}. \]

This formula is useful intuition for why higher continuation value raises training and higher compute price lowers training. It is not the global equilibrium characterization, and it is not the paper's tipping result.

8. What the Equilibrium Returns

  • Policy functions: training distributions \(\sigma_i(g)\) at each gap.
  • Values: \(V_i(g)\), so we know the value of leading or lagging.
  • Compute demand: \(T(g)\) and rental price \(R(T(g))\).
  • Gap dynamics: \(M_\sigma\), \(D_\sigma(g)\), invariant mass, and finite-horizon first-exit probabilities.
  • Access rule: whether the leader sells or forecloses at each gap once leakage is specified.

9. What Must Be Computed

  • The one-, two-, and three-period games are exact by backward induction.
  • The infinite-horizon game is exact after choosing a finite state grid, admissible action grid, and equilibrium-selection rule.
  • The stationary diagnostic JSON reports one selected solution path for this system; the selection-rule, grid-refinement, damping-path, homotopy-path, state-noise, and tremble sweeps expose sensitivity rather than proving robustness.
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