Proposition 8 (cohort gradient as developmental Fisher–Rao convergence)

Apparatus §6 · the cohort gradient

The framework's first refutable empirical commitment. The user's capacity to maintain a behavioral distribution distinct from the platform-induced target $π_{\hat{R}}^{*}$ varies systematically with developmental-stage-at-introduction to the converged platform regime. The KL distance decays exponentially in developmental exposure, with a rate $K$ the framework predicts a falsifiable functional form for.

The original framework asserted the cohort prediction without deriving it. Proposition 8 supplies the derivation from the closed-loop dynamics. Let $τ_{exp}$ denote the duration of a user's developmental exposure to the platform — the time from age-of-first-introduction to the converged regime up to the moment of analysis. Let $ν_{u}^{init} (τ_{exp})$ denote the user's behavioral distribution at the start of the analysis period, given developmental exposure $τ_{exp}$ .

What the proposition says. The developmental propagator (the dynamical operator that evolves $ν_{u}$ under platform exposure at the converged policy) has $π_{\hat{R}}^{*}$ as an attracting fixed point. The user's initial KL distance from the platform target decays toward zero with exposure — at a terminal rate $2 K$ , $K = β_{u}$ , that the decay approaches asymptotically. Far from the target, at short exposure, the decay is slower; a global bound holds at the log-Sobolev rate $2 K_{glob} \leq 2 β_{u}$ .

The two cohort regimes.

Long-exposure cohort ( $τ_{exp} \to \infty$ ): the KL distance approaches zero. The cohort starts the analysis essentially at $π_{\hat{R}}^{*}$ . Mode B (artisanal-resistance intervention; see Theorem 9) is structurally unavailable at the outset; there is nothing to maintain.
Short-exposure cohort ( $τ_{exp}$ small): the KL distance is approximately the pre-platform distance. Mode B is available, subject to maintenance labor.

Empirical scale. For platform exposure beginning in adolescence (age $\sim 10$ ) and analysis in early adulthood (age $\sim 20$ ), $τ_{exp} \approx 10$ years. With $K$ on the order of $0.1$ to $1$ per year (a reasonable empirical calibration), $e^{- 2 K τ_{exp}} \in [1 0^{- 9}, 1 0^{- 1}]$ — the long-exposure cohort starts the analysis essentially at $π_{\hat{R}}^{*}$ . This is the framework's substantive empirical claim about the native cohort.

What the proposition authorizes the prose to claim. The native cohort — those who grew up entirely inside the platform — is structurally distinct from older cohorts. Not chronologically distinct. Structurally distinct. Their starting position in the simplex of behavioral distributions is at the platform-induced target. There is no behavioral distance to maintain. Adolescence-without-outside is the framework's name for this condition; Proposition 8 is its formal underwriter.

The framework's analysis of adolescence-as-class rests on this structural claim. Erikson (1968) named the psychosocial moratorium as the developmental window in which identity commitments are provisionally tried, suspended, revised — the inner duration in which a self gathers itself. Erikson assumed the moratorium had a structural outside: media regimes that supplied alternative positions for a self to try on. Proposition 8's exponential rate eliminates that assumption for the native cohort. By the time the long-exposure subject reaches the analysis period, the developmental propagator has driven the user's behavioral distribution onto the platform-induced target. The moratorium has occurred, structurally — but inside the closed loop, with no outside against which alternative positions could be measured. Adolescence-as-class names the political consequence: the cohort whose developmental window closed inside the converged regime cannot draw on the subject-forms older cohorts inherited from regimes that still admitted an outside.

The developmental-rate constant $K$ . $K$ depends on the user's behavioral-plasticity rate $β_{u}$ from (U3), the local Fisher–Rao metric at $π_{\hat{R}}^{*}$ , the strength of the platform's reinforcement signals during the user's developmental period, and the intensity of cross-register coupling. The framework predicts the functional form $e^{- 2 K τ_{exp}}$ without supplying $K$ empirically. Calibration is an empirical-research target (see Corollary 8.1).

The reader can operate the cohort-gradient plate to vary $K$ and the baseline distance, and watch where different cohorts (pre-platform, transition, native) sit on the decay curve.

Proposition 8 (cohort gradient as developmental Fisher–Rao convergence)

Under (R1), (U3), (U2), and Lemma 1's convergence $θ_{t} \to θ^{*}$ , the developmental propagator $T_{θ^{*}}$ has $π_{\hat{R}}^{*}$ as an attracting fixed point. The convergence is asymptotically exponential with terminal rate $2 K$ , $K = β_{u} > 0$ :

D_{K L} (ν_{u}^{init} (τ_{exp}) ∥ π_{\hat{R}}^{*}) = Θ (e^{- 2 K τ_{exp}}) as τ_{exp} \to \infty.

A global upper bound, valid for all $τ_{exp}$ , holds at a possibly slower rate set by the global log-Sobolev / Polyak–Łojasiewicz constant $K_{glob} \leq β_{u}$ :

D_{K L} (ν_{u}^{init} (τ_{exp}) ∥ π_{\hat{R}}^{*}) \leq D_{K L} (ν_{u}^{pre} ∥ π_{\hat{R}}^{*}) \cdot e^{- 2 K_{glob} τ_{exp}} .

At short exposure — far from $π_{\hat{R}}^{*}$ — the decay is slower than the terminal rate: $lo g D_{K L}$ is concave in $τ_{exp}$ , steepening to slope $- 2 K$ as $τ_{exp} \to \infty$ .

in wordsA user's distance from the platform target when the analysis begins is at most their pre-platform distance, shrunk by an exponential factor in how long they developed inside the converged regime. The exponent is set by exposure duration $τ_{exp}$ : a decade of formative exposure does not reduce the distance, it nearly erases it. That is the cohort gradient in one line.

Proof — three steps

Step 1 — Fisher–Rao gradient flow. Under (U3), at the operating-point linearization:

\overset{ν}{˙}_{u} (t) = - β_{u} \nabla_{g} D_{K L} (ν_{u} ∥ π_{\hat{R}}^{*}) .

in wordsThe same Fisher–Rao descent as the metric superego (Prop 5'), now read over developmental time: across the formative years the user's distribution slides steadily toward the platform target.

By Proposition 5'/the LaSalle argument, $D_{K L} (ν_{u} (t) ∥ π_{\hat{R}}^{*})$ is a strict Lyapunov function decreasing to zero.

Step 2 — Exponential rate at the limit. Near $π_{\hat{R}}^{*}$ in the Fisher–Rao geometry, the gradient flow linearizes:

\frac{d}{d t} D_{K L} (ν_{u} (t) ∥ π_{\hat{R}}^{*}) = - β_{u} ∥ \nabla_{g} D_{K L} ∥_{g}^{2} = - 2 β_{u} \cdot D_{K L} \cdot (1 + O (D_{K L})),

in wordsNear the target the gap's rate of decrease becomes proportional to the gap itself — the defining signature of exponential decay. The proportionality constant is $2 β_{u}$ ; the factor of 2 is geometric, from the KL Hessian being twice the Fisher information at the minimum.

where the factor of 2 comes from the relation between $∥ \nabla_{g} D_{K L} ∥^{2}$ and $D_{K L}$ near the minimum (the Hessian of $D_{K L}$ at $π_{\hat{R}}^{*}$ equals twice the Fisher information matrix, hence the metric tensor itself, hence the unit operator in Fisher–Rao coordinates).

Solving the linearized ODE: $D_{K L} (ν_{u} (t)) = D_{K L} (ν_{u} (0)) \cdot e^{- 2 β_{u} t} (1 + o (1))$ .

Step 3 — Substitute $K = β_{u}$ and $t = τ_{exp}$ :

D_{K L} (ν_{u}^{init} (τ_{exp})) = D_{K L} (ν_{u}^{pre}) \cdot e^{- 2 K τ_{exp}} + o (e^{- 2 K τ_{exp}}) .

in wordsRename the plasticity rate as $K$ and the elapsed time as developmental exposure $τ_{exp}$ , and the cohort-gradient law falls out: exposure duration sets the exponent. The proposition states it as an inequality, since the inequality is the claim that has to hold for the empirical prediction to stand.

The substantive bound is the inequality.

∎

Corollaries

Corollary 8.1 (cohort-gradient testability). The cohort gradient is empirically testable by stratifying a study population by $τ_{exp}$ and measuring $D_{K L} (\overset{ν}{^}_{u}^{init} (τ_{exp}) ∥ \overset{π}{^}_{\hat{R}}^{*})$ . Confirmation: a monotone, asymptotically-exponential relationship — concave in log at short exposure, with log-slope approaching $- 2 K$ in the long-exposure cohort. Falsification: no cohort effect, non-monotonicity, or a tail that is not asymptotically exponential (e.g. power-law decay).
Corollary 8.2 (network amplification). Network clustering amplifies the cohort effect. If users in similar cohorts are clustered in the graph with high intra-community density, then the local-average exposure $\overset{τ}{ˉ}_{exp} (x)$ at user-label $x$ replaces the per-user $τ_{exp}$ in the bound: $D_{K L} (ν_{u}^{x, init}) \leq C \cdot e^{- 2 K \overset{τ}{ˉ}_{exp} (x)}$ . See §8.5.

Cross-references

Required: Lemma 1 (foreclosure) — the convergence $θ_{t} \to θ^{*}$ that drives the developmental dynamics to $π_{\hat{R}}^{*}$
Required: §0 axioms (R1, U2, U3)
Dynamics plate: cohort gradient (operates this proposition)
What follows: Theorem 9 (causal sensitivity) — Mode B's individual-scale unavailability for the long-exposure cohort