# Paper v3 Design Spec — "The AI Task Frontier" rebuild

**Date:** 2026-07-04. **Approved architecture:** Facts → 5 theorems → aggregate quantification
(user-approved; theoretical work with macro/aggregate data only, no micro empirics).

## One-line pitch
A theory of the direction of innovation when imitation is directionally asymmetric:
frontier AI competition races along the moat (what cannot be copied), not along value —
with the two-dimensional task frontier (horizon × forgiveness) derived, not assumed.

## Headline objects
- Primitive: O-ring-with-retries task technology. Task = n serial stages, per-stage retry
  allowance ℓ, tolerated failure ε.
- Derived kernel (Lemma 1): task success → Gumbel CDF exp(−e^{−(1+ℓ)(q_F−f)}), scale 1/(1+ℓ).
- Derived threshold: f = φ + (h + ln(1/ε))/(1+ℓ), h = ln n. Slope 1/(1+ℓ): retries divide
  the compounding tax.

## Theorems (all pre-verified in math_check_v3.py, 44/44)
- **T1 Representation:** task economy admits a scalar-ladder representation iff reduced
  requirements (h,f) form a chain (⇔ homogeneous forgiveness). Heterogeneous ℓ ⇒ irreducibly 2-D.
- **L2 Bertrand-in-tasks:** per-task asymmetric Bertrand ⇒ allocation by delivered net value
  (upgrades old Prop 2 identity into equilibrium rule); leader profit = W(q^L)−W(q^m) =
  path integral of boundary values over the gap; saturation = fringe convergence (derived).
- **T2 Tail cannot be distilled:** a plan is copied from one demonstration; a failure rate
  needs Θ(1/ε) samples (Pinsker: N ≥ 2(1−2δ)²/KL, KL ≤ 0.27ε). Required ε on marginal fragile
  tasks ~ n·e^{−(1+ℓ)(q_F−φ)} ⇒ imitation rate λ_F ↓ exponentially in fragility frontier;
  λ_H bounded below (plans observable). Appropriability derived, directional.
- **T3 Rotation:** leader–fringe dynamics, linear-flow HJB ⇒ shadow value μ_i = B_i/(r+λ_i);
  bang-bang direction rule by persistence-adjusted boundary value per dollar; along an H-push
  B_F/B_H rises (∂B_F/∂q_H = C > 0 unlock mechanism + H-depletion) ⇒ finite rotation time T*;
  ∂T*/∂λ_H < 0 (imitation steers, doesn't just deter).
- **T4 Direction game:** two labs choose boundaries; herding on the moat direction iff
  σ ≥ V_H/V_F (dominance), differentiation equilibria otherwise, mixed p* increasing in V_F/V_H.
- **T5 Wedge + policy reversal:** planner ranks by B_i/r, market by B_i/(r+λ_i); relative
  direction wedge = (r+λ_F)/(r+λ_H), signed by imitation asymmetry (observable). Liability θ
  raises x_F and steady-state moat Δ*_F (concentration ↑); public verification raises λ_F,
  thins the moat (concentration ↓). Both "pro-safety," opposite market structure.

## Facts (sources verified 2026-07-04)
- F1: 50%-horizon doubles ≈7 months (212d); 80%-horizon ≈5× shorter, same doubling (~213d).
  [METR / Kwa et al. 2025; metr.org/time-horizons]
- F2: imitation-lag asymmetry: open-weight lag ≈3–4 months on Epoch capability index /
  ~7 months FrontierMath, but mid-2025 open models ≈ late-2024 frontier on METR autonomy
  suites; long-horizon agentic reliability persistently favors closed frontier.
  [Epoch open-vs-closed insights; METR DeepSeek/Qwen report]
- F3: price to reach fixed benchmark score falls 9×–900×/yr, median ≈50×/yr, while frontier
  training spend rises. [Epoch inference price trends; Epoch trends]
- Quantification hook: constant-hazard scalar benchmark (Ord 2025) predicts 50/80 horizon
  ratio = ln.5/ln.8 = 3.11, parameter-free; observed ≈5 ⇒ excess reliability tax ⇒ second
  dimension quantitatively needed.

## Section map
1 Intro · 2 Facts · 3 Task technology + T1 · 4 Market (L2) · 5 Imitation (T2) ·
6 Dynamics (T3) · 7 Strategic competition (T4) · 8 Welfare & policy (T5) ·
9 Quantitative discipline · 10 Conclusion · App A proofs · App B verification note.

## Porting table
- horizon_forgiveness.tex: 2-D primitive & sufficient statistics → §3–4 (rewritten on derived kernel).
- agentic_racing.tex: decomposition/welfare-wedge machinery → §8 (2-D, derived payoffs).
- task_frontier_race.tex: prose style, examples, old Prop 2 → upgraded into L2; old Props 3–4 deleted.
- paper_v2 verification infra: calibration spirit → §9 (aggregate only).

## Deletions (from current canonical paper)
Static 2×2 "two-period race" as-is; exogenous α_N, π_S(δ_S), σ as primitives; Ω_R prize
shortcut; rank-2 matrix "test" (demoted to remark); "preemption/dynamic" mislabels.

## Positioning (must-engage)
Kremer 1993 (O-ring ancestor); Bryan–Lemus 2017 (direction of innovation — closest competitor);
Acemoglu 2002/2007 (DTC); Acemoglu–Autor 2011, Acemoglu–Restrepo 2018 (task models);
Aghion et al. 2001 (escape competition); Gilbert–Newbery 1982, Reinganum 1983 (preemption);
Arrow 1962, Anton–Yao 1994/2002, Teece 1986 (appropriability); Mussa–Rosen, Shaked–Sutton
(multidim quality); Bresnahan–Trajtenberg 1995 (GPT); BJVW 2020 (aggregate discipline format);
Ord 2025 (constant-hazard scalar benchmark); Kwa et al. 2025 (METR); Hinton et al. 2015 (distillation).