T11 — Sharp-Interface Γ-Convergence

Hero · Phase + Stability group · Cat A. Source: C-0011 / P-0011. Verification: E-0014. Canonical version: CV-1.0. Full proof: Canonical Spec — Part 5 · §13.

Statement#

Let $\epsilon =\alpha /\beta$ be the smoothness-to-double-well ratio. As $\epsilon \to 0$, the rescaled boundary-morphology energy

Γ-converges to a perimeter functional

defined on characteristic functions ${\chi }_E$ of subsets $E\subseteq X$, where $c_W={\int }_0^1\sqrt{2W(s)}ds$ is the surface-tension constant and ${\mathrm{Per}}_G(E)$ is the graph perimeter (number of edges crossing $\partial E$).

Minimizers of ${\mathcal{E}}_{\mathrm{bd}}^{\epsilon }$ converge (up to subsequence) to characteristic functions of sets minimizing the graph perimeter subject to the volume constraint $|E|=m$. Self-referential correction terms (closure, separation) modify the effective surface tension at higher order in $\epsilon$.

Proof idea#

Standard Modica-Mortola structure. The Allen-Cahn energy on a graph has the form $\epsilon \cdot (\text{Dirichlet})+{\epsilon }^{-1}\cdot (\text{double-well})$. As $\epsilon \to 0$, configurations are forced to either side of the double-well wells (i.e., $u\approx 0$ or $u\approx 1$ everywhere except on a thin transition layer); the energy concentrates on the transition layer with surface-tension density $c_W$.

Lim sup (recovery sequence). For any $E\subseteq X$, construct $u^{\epsilon }$ that approximates ${\chi }_E$ with smooth transition of width $O(\epsilon )$ across $\partial E$; verify ${\mathcal{E}}_{\mathrm{bd}}^{\epsilon }(u^{\epsilon })\to c_W\cdot {\mathrm{Per}}_G(E)$.

Lim inf (compactness + lower bound). Sequences with bounded energy are pre-compact in $L^1$ via the standard equicoercivity argument; any limit is a characteristic function (the double-well term forces this); the lim inf inequality follows from the slicing lemma.

Self-referential corrections. The closure (${\mathcal{E}}_{\mathrm{cl}}$) and separation (${\mathcal{E}}_{\mathrm{sep}}$) terms are perturbations of the Allen-Cahn substrate. Their effect on Γ-convergence is via modified surface tension $c_W\to c_W+O({\lambda }_{\mathrm{cl}}+{\lambda }_{\mathrm{sep}})$ — handled by perturbation analysis. $\square$

Why this is a hero#

T11 is the soft-to-crisp bridge. SCC commits to the primacy of the soft cohesion field $u:X\to [0,1]$, but it must explain how object-like crisp entities are recovered. T11 provides the mathematical mechanism: in the sharp-interface limit, soft minimizers automatically approach characteristic functions of crisp sets minimizing perimeter.

This connects SCC to:

• Geometric measure theory — perimeter minimization is the classical problem; SCC inherits its tools (slicing, BV functions, isoperimetric inequalities).
• Allen-Cahn / Cahn-Hilliard theory — the variational substrate is shared, but SCC adds self-referential corrections that modify the effective surface tension.
• Crisp recovery protocol (open problem) — the asymptotic limit is rigorous, but the threshold rule for finite $\epsilon$ remains under-specified. T11 establishes that some crisp recovery is mathematically inevitable; the protocol for any specific $\epsilon$ is downstream work.

The substantive content of T11 for SCC specifically is that the four-term energy is consistent with a sharp-interface limit. The deliberate non-Allen-Cahn structure of SCC (closure self-reference, distinction asymmetry) does not destroy the limit — it perturbs the surface tension. This is non-trivial: a more aggressive non-locality could have broken Γ-convergence entirely.

Logical dependencies#

• Builds on: T1 (minimizer existence), T8-Core (non-trivial minimizers exist for $\beta /\alpha$ above threshold), discrete isoperimetric structure of the graph.
• Builds into: Deep Core Dominance 2b (uses isoperimetric inequality directly), T-Merge (b) (perimeter ordering on connected graphs is the limit of Γ-convergence), formation birth analysis at ${\beta }_{\mathrm{crit}}$.

See also#

• Full proof in canonical: Canonical Spec §13 T11
• Why crisp recovery is "the soft-to-crisp bridge, not the foundation": Canonical Spec Part 1 §4
• T-Merge (b) (the canonical perimeter-ordering theorem on connected graphs): Canonical Spec §13 T-Merge entry