# Stage 9 derivation: imitation asymmetry and research-program scale

## 1. Constant-annuity endogenous-scale benchmark

Use the ordered polynomial program with constant annuities and let z be horizon
research, Q total program size, and R=Q-z reliability research. The two interior
first-order conditions are

    Phi(z,Q)
      = a_H (u0-u1 z) - chi_H
        + a_F {v1(Q-z) - (v0+v1 z)} + chi_F
      = 0,

    a_F (v0+v1 z) - chi_F - Gamma'(Q) = 0.

Define

    D = a_H u1 + 2 a_F v1,
    p = a_F v1,
    g = Gamma''(Q),
    Delta_Gamma = D g - p^2.

The Hessian of the joint objective in (z,Q) is

    [ -D   p ]
    [  p  -g ].

Thus Delta_Gamma>0 is exactly the constant-annuity specialization of the joint strict
concavity condition in Proposition 3.

## 2. A directional-imitation shifter

Let sigma index a widening directional imitation asymmetry. Write

    dot a_H = d a_H/d sigma <= 0,
    dot a_F = d a_F/d sigma >= 0,

with at least one strict. At the interior solution define

    B_H* = u0-u1 z*,
    B_F* = v0+v1 z*,
    L*   = v1 Q* - v0 - 2v1 z*.

The direct derivatives of the two first-order conditions are

    b1 = dot a_H B_H* + dot a_F L*,
    b2 = dot a_F B_F*.

Implicit differentiation gives

    [z_sigma, Q_sigma]'
      = (1/Delta_Gamma)
        [ g b1 + p b2,
          p b1 + D b2 ]'.

Collecting the numerator of the scale derivative,

    dQ*/d sigma
      = 1/Delta_Gamma [
          a_F v1 dot a_H B_H*
          + dot a_F {
              a_F v1(v0+v1 Q*)
              + a_H u1 B_F*
            }
        ].

Every term multiplying dot a_F is positive. Therefore:

1. if only horizon imitation accelerates, dot a_H<0 and dot a_F=0, total program
   scale falls;
2. if only the reliability annuity lengthens, dot a_H=0 and dot a_F>0, total program
   scale rises; and
3. with both changes, scale rises if and only if

       dot a_F {
         a_F v1(v0+v1 Q*) + a_H u1 B_F*
       }
       > -dot a_H a_F v1 B_H*.

This is the exact price-versus-scale condition. The cheap, rapidly imitated direction
does not by itself finance more research. A growing moat annuity must dominate the
lost horizon rent.

## 3. Resource spending

If unit research costs are common, chi_H=chi_F=chi, define program resource spending

    E(Q) = chi Q + Gamma(Q).

When chi+Gamma'(Q)>0,

    dE*/d sigma = {chi+Gamma'(Q*)} dQ*/d sigma,

so research-program spending and total research input have the same sign. This is a
qualitative mapping, not an identification claim for observed frontier training-run
cost: the public series does not separately measure Q, Gamma, or either imitation
annuity.

## 4. Interpretation and scope

The result sharpens rather than changes the paper's protected message. Directional
imitation still rotates research toward reliability. Endogenizing scale adds a
separate extensive margin:

- faster horizon imitation lowers horizon research and, by itself, total scale;
- a lengthening reliability moat raises the terminal reliability-project value and
  total scale;
- the observed coexistence of cheap fixed performance and rising frontier inputs is
  consistent with the mechanism only when the reliability-annuity force is large
  enough, or when another demand/cost shifter supplies the missing scale force.

The condition is therefore both a resolution and a falsification restriction.