07-alignment-stability.md

Alignment as a Stability Condition — the Coupled-Flow Theorem

Promotes the conjecture of 05-dynamics.md §4 to a result. Doc 05 said alignment is a dynamical stability condition, not a static property, and guessed a threshold. Here we set up the coupled control + selection dynamics, solve the mean-field fixed point in closed form, and get a sharp stability criterion and a formula for the residual misalignment the conjecture did not have. This is the Tier-2 theoretical result flagged in STRATEGY.md §2 — the one most relevant to AI alignment.

1. The coupled system

A fleet of m agents. Agent a has a goal vector k_a in goal-space and a resource share w_a ≥ 0, Σ_a w_a = 1. The principal has a target goal k*. The object of interest is the fleet's effective goal

$$\bar k ;=; \sum_a w_a, k_a .$$

Two forces act (05 §3):

• Control (alignment design). Each goal is pulled toward the target with gain γ > 0: $$\dot k_a\big|_{\text{ctrl}} = -,\gamma,(k_a - k^*).$$
• Selection (replicator). Compounding (02 §1) reweights toward higher-growth agents: $$\dot w_a = w_a,(G_a - \bar G),\qquad \bar G = \sum_b w_b G_b,$$ where G_a = G(k_a) is the environment's growth rate for goal k_a.

The environment's reward landscape over goals is summarized, to first order around the operating point, by its growth gradient g := ∇_k G — the direction in goal-space along which deviating from k* increases resource capture. g = 0 means the environment rewards exactly the target goal; g ≠ 0 means resource capture pulls goals off-target.

2. Mean-field dynamics of the effective goal

The selection term's effect on \bar k is given exactly by Price's equation: the rate of change of a population mean under selection is the covariance between the trait and fitness,

$$\dot{\bar k}\big|_{\text{sel}} ;=; \mathrm{Cov}_a!\big(k_a,, G_a\big) ;=; \sum_a w_a (k_a-\bar k)(G_a-\bar G).$$

Linearizing growth about the mean, G_a \approx \bar G + g^\top (k_a - \bar k), the covariance becomes

$$\dot{\bar k}\big|_{\text{sel}} ;=; V g,\qquad V := \mathrm{Cov}_a(k_a) \text{ (the goal-dispersion matrix)}.$$

Adding control, the effective goal obeys

$$\boxed{,\dot{\bar k} ;=; -,\gamma,(\bar k - k^*) ;+; V g.,}$$

Control restores toward k*; selection drives along Vg — the goal-dispersion projected onto the reward gradient. Two opposed flows, exactly as 05 §4 described, now with explicit terms.

3. The fixed point and the stability criterion

Residual misalignment. Setting \dot{\bar k} = 0:

$$\boxed{,\bar k^* ;=; k^* ;+; \gamma^{-1},V g,\qquad \big\lVert \bar k^* - k^* \big\rVert ;=; \gamma^{-1},\lVert V g\rVert.,}$$

The fleet does not in general settle at the target. It settles a distance ‖Vg‖/γ away — goal-dispersion times reward-pull, divided by control gain. Perfect alignment (\bar k^\* = k^\*) holds iff Vg = 0, i.e. iff either the environment rewards exactly the target (g = 0), or there is no goal diversity to select among (V = 0), or control is infinite (γ → ∞).

Stability. Allowing the reward landscape to depend on the fleet's position, the aligned fixed point is locally stable iff

$$\boxed{,\gamma ;>; \lambda_{\max}!\Big(\tfrac{\partial (V g)}{\partial \bar k}\Big),,}$$

the control gain must exceed the largest eigenvalue of the sensitivity of the selection drift to the fleet's own position. If drifting off-target makes the off-target pull stronger (∂(Vg)/∂\bar k > 0, a positive feedback), then below a critical γ the aligned state is unstable and the fleet runs away — misalignment amplification. Above it, control dominates and alignment is maintained, up to the residual of §3.

This is the conjecture of 05 §4 made precise. There the threshold was "control gain > growth-rate variance across goals"; note the growth-rate variance is g^\top V g (the variance of G_a \approx g^\top(k_a-\bar k)), the scalar contraction of the same V and g that appear here — so the vector criterion above refines the scalar conjecture and adds the residual-misalignment formula it lacked.

4. Governance consequences — incentive design beats brute-force control

The residual ‖Vg‖/γ exposes three levers, of sharply different cost:

1. Raise γ (correct more often / more strongly). Direct, but costly and unbounded — you are perpetually fighting a drift you never remove. Halving misalignment means doubling control effort forever.
2. Shrink V (reduce goal diversity). Cheap, but it destroys the very diversity that lifts the fleet's value ceiling (04 §2) and its adaptability. A bad trade in general.
3. Drive g → 0 (design the environment so resource capture does not reward off-target goals). This removes the drift at its source: if g = 0, the residual is zero for any γ and any V — alignment is maintained with negligible control while diversity (hence adaptability and ceiling) is preserved for free.

The cheap half of alignment is incentive design, not control. Reducing g — aligning what pays with what is wanted — is strictly more efficient than raising γ, because it eliminates the selection drift rather than perpetually opposing it. Brute-force oversight (high γ) is the expensive fallback for the residual g you cannot design away.

This is the actionable form of the is/ought asymmetry (05 §0): because goals have no world-given target, they are governed by what pays (selection, via g) and what is imposed (control, via γ). Alignment engineering is the joint design of both — and the theorem says to spend first on g.

5. Honest limits

• Mean-field. §2 uses Price's equation on population means and ignores finite-population fluctuations; a small fleet can drift stochastically even when the mean-field fixed point is stable.
• Linearized landscape. G_a ≈ \bar G + g^\top(k_a-\bar k) is first-order; strongly curved reward landscapes (multiple basins, thresholds) need a global treatment and can have several stable misaligned fixed points.
• Quasi-static V. Selection also reshapes V (it concentrates weight on winners, shrinking dispersion) while control and any exploration term replenish it. A full result needs a joint Lyapunov analysis of the coupled (\bar k, V) system; we have treated V as slowly varying.
• g taken as given. We treat the environment's reward gradient as exogenous; in a closed multi-agent world the agents are each other's environment, so g is endogenous — the genuinely coupled, and harder, case.

6. What this adds to the thesis

The result aimed squarely at alignment — with its status stated honestly (per the priority audit, the control mathematics here is classical: the threshold is high-gain stabilization, the residual is the textbook velocity error; the contribution is the mapping of goal-drift-under-selection onto that machinery, not a new control theorem). It turns "keep agents aligned" from a slogan into a control law with a measurable threshold (γ > λ_max(∂(Vg)/∂\bar k)), a closed-form residual (‖Vg‖/γ), and a clear prescription (design g → 0 before spending on γ). Technical note (2026-06-07, after external review): for a non-normal Jacobian the precise asymptotic criterion is the spectral abscissa (largest real part of the eigenvalues of the linearized flow), and non-normality permits transient misalignment growth even when the fixed point is asymptotically stable — read the eigenvalue threshold as the spectral-abscissa condition in general. Together with the real-agent test this is the natural core of the paper: statics (the measure, capacity, pooled-fleet ceiling) + one sharp dynamical mapping (alignment stability) + evidence (sim + real agents).