T14 — Gradient Flow Convergence (Łojasiewicz)

Hero · W4 Capstone group · Cat A. Source: C-0014 / P-0014. Verification: E-0020:E-0022. Canonical version: CV-1.0. Full proof: Canonical Spec — Part 5 · §13.

Statement#

The projected gradient flow on the constraint manifold ${\Sigma }_m$,

starting from any $u_0\in {\Sigma }_m$, converges to a critical point of $\mathcal{E}{|}_{{\Sigma }_m}$.

When the energy is analytic — equivalently, when the distinction parameter $b_D=0$ (or $\epsilon$-smoothed) so that the absolute value $|\cdot |$ in the gradient indicator $g_t$ is replaced by an analytic surrogate — convergence is exponential, with rate determined by the Łojasiewicz–Simon inequality.

Proof idea#

Convergence to critical point. Three standard ingredients:

1. Bounded below. $\mathcal{E}$ is bounded below on the compact manifold ${\Sigma }_m$ (by T1).
2. Monotone decrease. Gradient flow decreases energy: $\frac{d}{dt}\mathcal{E}(u(t))=-\Vert {\Pi }_{{\Sigma }_m}\nabla \mathcal{E}(u(t)){\Vert }^2\le 0$.
3. LaSalle / accumulation. Compactness of ${\Sigma }_m$ ensures accumulation points; monotone decrease forces the gradient to vanish at any accumulation point; therefore accumulation points are critical points of $\mathcal{E}{|}_{{\Sigma }_m}$.

Exponential rate via Łojasiewicz. For analytic functions on compact semi-algebraic sets (which includes ${\Sigma }_m$), the Łojasiewicz–Simon inequality holds: for some $\theta \in (0,1/2]$ and $C>0$,

near any critical point $u^{\ast }$. Combined with the gradient flow equation, this gives:

• If $\theta =1/2$: exponential convergence, $|u(t)-u^{\ast }|\le C{e}^{-\lambda t}$.
• If $\theta <1/2$: polynomial convergence $|u(t)-u^{\ast }|\le C{t}^{-\theta /(1-2\theta )}$.

The analyticity requirement is non-trivial: the absolute value in the original $g_t$ definition breaks analyticity. Setting $b_D=0$ in the distinction operator (or smoothing) is required for the Łojasiewicz machinery, which is why $b_D=0$ is a fixed commitment of the theory (CN13 / Part 4 §11.1 #13).

$\square$

Why this is a hero#

T14 is the dynamical existence theorem. T1 secures static existence of minimizers; T14 secures that the minimizers are reachable by gradient descent. Without T14, the variational framework would be a description of fixed points without a constructive route — minimizers might exist but be inaccessible.

The Łojasiewicz inequality is the workhorse: it guarantees that gradient flow on analytic energies on compact semi-algebraic sets does not get stuck in slow drifts or exhibit pathological convergence. This is far stronger than what generic smooth functions provide.

The analyticity requirement is also load-bearing for SCC's design choices. The decision to set $b_D=0$ (the gradient term in the distinction operator) was driven specifically by T14: keeping the explicit gradient term breaks analyticity, breaks Łojasiewicz, breaks T14, and breaks dynamic accessibility of minimizers. Boundary sensitivity (axiom D-Ax3) is preserved through the spatial structure of $P_t(1-u)$ instead — a clean architectural trade-off.

Logical dependencies#

• Builds on: T1 (energy bounded below on compact ${\Sigma }_m$), analyticity of energy ($b_D=0$ commitment), Łojasiewicz–Simon for analytic functions on compact semi-algebraic sets.
• Builds into: T-Persist-1(b) (basin escape uses gradient flow + Łojasiewicz), T-PreObj-1 (gradient flow attracts to multi-peak F≥2 — explicit dynamic statement requiring T14 to make sense), every implementation that runs scc/optimizer.py find_formation (relies on T14 guaranteeing convergence).

See also#

• Full proof in canonical: Canonical Spec §13 T14
• $b_D=0$ commitment and why analyticity matters: Canonical Spec Part 4 §11.1 #13
• T-PreObj-1 (the W4 result that uses gradient flow attraction): T-PreObj-1 hero page