# Stage 2 Derivation: Certification, Entry, and the Reliability Lag This note develops the replacement for the reduced-form data-moat proposition. It is not manuscript prose. The protected mechanism is directional imitation; the purpose of the game is to make the reliability lag, entry margin, and public-verification comparative statics equilibrium objects. ## 1. Certificate technology A certificate at failure tolerance `epsilon` requires `N(epsilon)` independent, usable observations. The testing theorem imposes `N(epsilon) >= 4(1-2 delta)^2 / epsilon`. Firm `j` inherits a fraction `gamma_j in [0,1)` of the certificate through evidence that transfers from related tasks. Its residual evidence requirement is `R_j = (1-gamma_j) N(epsilon)`. This parameter makes the scope of the statistical claim explicit. If `gamma_j` is bounded away from one, the residual target remains `Omega(1/epsilon)`. Complete formal or cross-task transfer (`gamma_j=1`) removes the task-specific sampling wall. Usable evidence arrives at baseline rate `b_j`. For the incumbent `L` and entrant `E`, `b_L = D_pub + d M`, `b_E = D_pub + omega d M`, where `D_pub >= 0` is public verification, `d M >= 0` is telemetry from the incumbent's certified installed base, and `omega in [0,1]` is the share of that telemetry usable by the entrant. The proprietary-data benchmark is `omega=0`. There is no circularity: installed-base evidence from the already certified generation informs certification of the next reliability increment. Firm `j` can buy additional verification throughput `x_j >= 0` at capacity cost `kappa_j x_j^p / p`, with `p >= 2`. Independent trials then accumulate at rate `b_j+x_j`, and certification time is `T_j(x_j) = R_j/(b_j+x_j)`. The firm incurs a time-to-market cost `w_j > 0` per unit of delay. Conditional on seeking a certificate, it minimizes `K_j(x) = kappa_j x^p/p + w_j R_j/(b_j+x)`. This is an exact project-cost objective: the first term is purchased evaluation capacity and the second is pre-certification burn or opportunity cost. Discounted post-certification profits enter the separate entry decision below. ## 2. Verification equilibrium For `R,w,kappa>0`, `K` is strictly convex: `K_xx = kappa(p-1)x^(p-2) + 2wR/(b+x)^3 > 0`. The unique optimum is strictly positive and solves `kappa x^(p-1)(b+x)^2 = wR`. (CE1) Let `A = (p-1)b + (p+1)x`. Implicit differentiation of (CE1) gives `x_b = -2x/A < 0`, `x_R = x(b+x)/(A R) > 0`, `x_w = x(b+x)/(A w) > 0`, `x_kappa = -x(b+x)/(A kappa) < 0`. Public evidence crowds out paid testing, while a larger residual sample target or higher time value induces more testing. Total throughput nevertheless rises in the baseline flow because `d(b+x)/db = (p-1)(b+x)/A > 0`. At the equilibrium effort, `T=R/(b+x)` satisfies `T_b = -(p-1)T/A < 0`, `T_R = ((p-1)b+p x)/(A(b+x)) > 0`, `T_w = -T x/(A w) < 0`, `T_kappa = T x/(A kappa) > 0`. (CE2) Thus public verification reduces certification time despite crowding out private effort. Cross-task transfer reduces `R` and therefore time. More expensive testing lengthens time; a higher cost of delay shortens it. For `p>=2`, `T(b)` is convex. Direct differentiation gives `T_bb = 2(p-1) T{(p-1)^2 b + (p+1)(p-2)x}/A^3 >= 0`. (CE3) This curvature is what makes common public infrastructure narrow a fixed installed-base lag advantage. ## 3. Incumbent advantage and public verification Suppose the firms share `(w,kappa,p)` and the entrant inherits weakly less evidence: `R_E >= R_L`. Because `b_E <= b_L`, (CE2) implies `T_E >= T_L`, strictly if either inequality is strict. With common residual requirements, `b_Ex_L` but `b_E+x_E0` be the incumbent's baseline-flow advantage and write `T_E=T(D_pub+omega dM)` and `T_L=T(D_pub+dM)=T(b_E+z)`. Convexity gives `d(T_E-T_L)/dD_pub = T_b(b_E)-T_b(b_E+z) <= 0`. (CE4) Both lags fall, their gap shrinks, and the gap converges to zero as public flow grows. Disclosure (`omega` rising) lowers the entrant lag directly. The minimized certification project cost `K*` obeys the envelope formulas `K*_b = -wR/(b+x)^2 < 0`, `K*_R = w/(b+x) > 0`, `K*_w = T > 0`, `K*_kappa = x^p/p > 0`. (CE5) Public evidence and transferable evidence therefore make entry cheaper even though they reduce privately purchased testing capacity. ## 4. Entry and waiting Let `V_j` be the post-certificate value of serving the task class and `F_j` a fixed entry cost. The certification-and-entry subgame has the equilibrium rule `enter_j iff V_j >= F_j + K_j*` (CE6) (with a stated tie-breaking convention). If the entrant does not enter, its absorption time is infinite. Public verification can therefore have an extensive margin: by (CE5), it can induce entry, not merely shorten the lag of an entrant that would enter anyway. With continuously divisible testing and `w>0`, an entering firm always buys some capacity. Pure waiting is the limiting case `w=0`, in which `x*=0` and public evidence alone completes the certificate. If activating a private testing program costs a fixed setup fee `s>0`, the firm compares `wR/b` with `s + kappa x_hat^p/p + wR/(b+x_hat)`, where `x_hat` solves (CE1). It waits exactly when the first expression is smaller. The setup-cost extension creates an explicit waiting/testing region; the baseline sets `s=0` so public evidence unambiguously shortens equilibrium certification time. ## 5. When reliability is slower than plans Let `tau_H` be the plan-access lag. At the interior verification optimum, `T_E > tau_H` if and only if `R_E/tau_H > b_E` and `kappa_E (R_E/tau_H-b_E)^(p-1)(R_E/tau_H)^2 > w_E R_E`. (CE7) Proof: `T_E>tau_H` is equivalent to `x_E < R_E/tau_H-b_E`. The left side of (CE1) is strictly increasing in `x`, so substitution of the proposed upper bound gives exactly (CE7). This replaces the old unqualified assertion `tau_F>tau_H` with a primitive necessary-and-sufficient condition. ## 6. The equilibrium exponential wall For bounded `b`, incomplete evidence transfer, and an interior optimum, (CE1) implies `x ~ (wR/kappa)^(1/(p+1))`, `T ~ (kappa/w)^(1/(p+1)) R^(p/(p+1))`. (CE8) Since the statistical target at the marginal task obeys `N(q_F) = Omega(exp((1+ell)(q_F-phi))/n)`, and `R=(1-gamma)N`, any `gamma` bounded away from one yields `T_E(q_F) = Omega(exp[p(1+ell)(q_F-phi)/(p+1)] / n^(p/(p+1)))`. Endogenous paid testing reduces the exponent from one to `p/(p+1)`; it does not turn the wall into a bounded lag. If testing throughput is capped at `x_bar`, the cap eventually binds and the original full-order result returns: `T ~ R/(b+x_bar) = Omega(N)`. The wall disappears only if transferable evidence approaches completeness quickly enough (`1-gamma` offsets `N`) or if verification capacity scales comparably with the sample target. These are explicit diffusion technologies, not hidden exceptions. ## 7. Innovation versus diffusion Let a frontier reliability increment earn local boundary flow `B_F` until entrant certification. Its deterministic-lag annuity is `A_F(T_E)=B_F(1-exp(-rT_E))/r`. Because `A_F` is increasing in `T_E`, (CE2) implies that more public verification reduces the increment's private appropriability. At the same time, (CE5)--(CE6) show that it reduces the cost of certification and can induce entry. Public verification therefore has three distinct effects: 1. it directly lowers the leader's own next-generation certification cost; 2. it accelerates entrant certification and can induce entry; 3. by shortening the entrant lag, it reduces the ex ante rent from creating a reliability increment. The first two promote deployment and diffusion; the third weakens frontier innovation incentives. The net effect on future frontier reliability is not signed without an innovation model. This distinction must replace the manuscript's unqualified claim that public verification always produces safer future AI.