Cohort gradient

The framework's first refutable empirical commitment. Proposition 8 derives the cohort gradient from the closed-loop dynamics: the developmental propagator ${\mathcal{T}}_{{\theta }^{\ast }}$ has ${\pi }_{\hat{R}}^{\ast }$as an attracting fixed point. The user's initial behavioral distance decays toward zero asymptotically exponentially — terminal rate $2K$, $K={\beta }_u$ — with a global upper bound at the slower log-Sobolev rate $2K_{\mathrm{glob}}\le 2{\beta }_u$ and $\log D_{KL}$ concave at short exposure. The plate renders the terminal exponential.

The two parameters.

• K (developmental rate, /year).The rate at which behavioral distance decays under platform exposure. Depends on the user's behavioral-plasticity rate ${\beta }_u$ (axiom U3), the local Fisher–Rao metric at ${\pi }_{\hat{R}}^{\ast }$, the platform's reinforcement-signal strength, and cross-register coupling intensity.
• D_KL^pre (baseline distance).The user's pre-platform KL distance from the platform-induced target. The pre-platform cohort starts here.

The operational-foreclosure threshold. The horizontal dashed line at $D_{KL}=10^{-2}$marks where the cohort's behavioral distribution becomes operationally indistinguishable from the platform-induced target. At $K=0.3$ and $D_{KL}^{\mathrm{pre}}=2$ (typical parameter set), the native cohort (τ_exp = 15y) sits well below the threshold.

Half-life. The readout ${\tau }_{1/2}=\log 2/(2K)$ gives the time required for the KL distance to halve. At $K=0.3$, ${\tau }_{1/2}\approx 1.2$ years — every 1.2 years of platform exposure halves the behavioral distance.

The empirical-research target. The framework predicts a falsifiable functional form. To calibrate $K$empirically: stratify a study population by τ_exp, measure each cohort's KL distance from the platform target, fit the curve. Confirmation: a monotone, asymptotically-exponential relationship — concave in log at short exposure, with log-slope approaching $-2K$ in the long-exposure cohort. Falsification: no cohort effect, non-monotonicity, or a tail that is not asymptotically exponential (e.g. power-law decay). The framework's priority-1 empirical study targets this calibration directly.