T1 · Representation
O-ring with retries: fragility requirement f = φ + (h + ln(1/ε))/(1+ℓ), success converges to a Gumbel kernel, and the scalar quality ladder is exactly the case of homogeneous forgiveness.
In Brief (no math)
Why do cheap AI models catch up to the frontier in months, while frontier labs spend more every year? Because AI tasks differ on two margins — how long a chain of actions they need, and how much they forgive mistakes — and copying is easy on one margin but hard on the other. A clever plan is revealed the first time anyone sees it work. Proving that a system almost never fails is different: to show it fails less than once in ten thousand tries you need on the order of ten thousand real tries; the unforgiving tasks where that proof matters generate few tries; and the tries come from being deployed, which requires the proof you don't yet have. So competition erases the value of anything that can be demonstrated and spares the value of reliability — a capability lead is a bond that stops paying the day it is copied. As followers close in, the frontier race turns from "can it do longer tasks?" to "can it be trusted with unforgiving ones?", and rival labs pile onto that same hard-to-copy frontier. One policy punchline: liability rules and public safety-testing infrastructure both buy reliable AI, but the first concentrates the market and the second opens it.
Three Blind Referee Rounds
The paper has been through three adversarial rounds — each time a blind, neutral-prompt review by two frontier-model referees (a theory referee judged against Econometrica, an IO/innovation referee with live literature search) plus an adversarial self-review with numerical counterexamples. Every fatal objection was then fixed with derived machinery, not wording, and re-verified. The trajectory of top-5 R&R verdicts: June draft: certain reject → v3.0: ~5–10% → v3.1: ~25–35% (first MAJOR REVISION verdicts; RAND judged Accept/Minor) → v3.2: round-three objections addressed. Full reports: v3·GPT, v3·Gemini, v3.1·GPT, v3.1·Gemini.
v3.2's additions close the third round's convergent objections: a data-moat proposition (the sample-complexity floor becomes an imitation lag exactly when trial flow is gated by certified deployment; a simulator good enough to certify the tail must itself be certified against the tail — the real-trial floor is route-independent), an endogenous-intensity proposition (herding survives endogenous breakthrough rates; cheap continuous progress restores differentiation), first-order separability for the direction game, and the cyclical rotation conjecture: herding cannot be absorbing, so the race should alternate between agency and reliability phases.
L2 · Profits are gaps
Bertrand in tasks: frontier profit equals the boundary-value integral over the capability gap to the best imitator. Saturation is an equilibrium state — served, valuable, and rentless.
T2 + P1 · The tail cannot be distilled; the data moat
Certifying failure rate ε needs Ω(1/ε) ground-truth trials — and the floor binds invention too. It becomes a lag because trial flow is gated by certified deployment: lag ratio = flow ratio, a data flywheel. Synthetic data rescales the wall; it cannot remove it.
T3 + T4 · Annuities and rotation
An increment earns its boundary value for exactly its imitation lag: appropriable share = 1−e^(−rτ), exact over a finite look-ahead window. In the depletion phase B_F/B_H rises, so investment rotates to the reliability boundary — and faster imitation brings the rotation earlier. Imitation steers.
T5 + P2 · Herding on the moat
Head-to-head racing replaces r by r+ν inside the annuity — the prize split is derived, not assumed. Herding iff the moat's duo annuity beats the horizon's sole annuity (cutoff ≈ 1.2 at calibrated lags); cheap continuous progress restores differentiation, even with endogenous intensity.
T6 · Wedge & policy reversal
The direction wedge equals (1−e^(−rτ_H))/(1−e^(−rτ_F)) ≈ τ_H/τ_F, signed by observable imitation lags. Liability raises reliability investment and concentration; public verification raises reliability diffusion and lowers concentration.
Revision Log & Verification
Aggregate discipline (public data only)
- Fact 1: 50% time-horizons double every ~7 months; 80% horizons are ~5× shorter (METR).
- Fact 2: open-weight lag is 3–4 months on capability indices but 6–12+ months on autonomy/reliability margins.
- Fact 3: price at fixed capability falls 9–900× per year (median ~50×) while frontier spend rises.
- Honest test: constant hazard predicts a parameter-free 50/80 horizon ratio of 3.11; the data say ~5 — rejecting homogeneity, though scalar mixtures also inflate the ratio (the paper says so itself). The distinctive evidence is the lag structure.
- Appropriable shares: 3.9% (horizon) vs 13.1% (reliability); direction wedge ≈ 0.30 ≈ τ_H/τ_F.
Rounds 95–97: 2026-07-04/05
- R95: referee round on the June draft (certain reject: identities, assumed payoffs) → full rebuild with five derived theorems.
- R96: second round found an HJB inconsistency and an invalid test → certification-delay reformulation (exact finite-window annuities), derived prize split, self-administered mixture counterexample.
- R97: third round returned the first top-5 MAJOR REVISION verdicts → data-moat and endogenous-intensity propositions close the remaining convergent objections.
- Every round: new math pre-verified before writing; suite now 84 checks, 0 failures; compile fully clean.
- Earlier draft archived: June 2026 version (19 pp).
Open full learning log (Rounds 1–100)
Synced from the project-level learning.md; Rounds 95–100 document the three referee rounds, the rebuild, and every verification.
Task Regions
Forgiving (retries, rollback)
Unforgiving (one-shot, costly errors)
Short horizon
Recommendation, ad copy, search help. Saturated interior: served, commoditizing, rentless.
Payments control, medication dosing, driving maneuvers. Reliability margin: slow to imitate, self-protecting rents.
Long horizon
Coding, debugging, research assistance. Horizon margin: plans and scaffolds copy fast, rents are transient.
Autonomous science, critical infrastructure, agentic organizations. Corner where both bind: the direction the race rotates toward.
Paper
The compiled PDF, LaTeX source, design spec, and math checker on this page are synchronized with v3.5. Every numbered claim in the paper has a corresponding check in math_check_v3.py (95 pass, 0 fail).