T8-Core — Phase Transition (Spectral Universality)

Hero · Phase + Stability group · Cat A. Source: C-0008 / P-0008. Verification: E-0010, E-0011. Canonical version: CV-1.0. Full statement and proof: Canonical Spec — Part 5 · §13.

Statement#

Let $X_t$ be a finite connected graph with Fiedler eigenvalue $\lambda_2 > 0$ (algebraic connectivity). Let $m = cn$ with volume fraction

and let $W(u) = u^2(1-u)^2$ be the double-well potential. If

then the global minimizer of $\mathcal{E}_{\mathrm{bd}}|_{\Sigma_m}$ is non-uniform (i.e., not the constant field $u \equiv c$).

Proof idea#

Second variation analysis at the uniform state. The ordered-pair summation convention (Section 0 of the canonical spec) gives the smoothness functional $2\alpha\, v^T L v$ where $L$ is the graph Laplacian and $v = u_t - c\mathbf{1}$, contributing a Hessian term $4\alpha L$.

The second variation of $\mathcal{E}_{\mathrm{bd}}$ at $u \equiv c$ along the Fiedler eigenvector $\psi_2$ (eigenvalue $\lambda_2$) has eigenvalue

The double-well derivative $W''(u) = 2 - 12u + 12u^2$ satisfies $W''(c) < 0$ on the spinodal interval $\big((3-\sqrt{3})/6,\, (3+\sqrt{3})/6\big)$. Therefore, when

the uniform state is a saddle point of $\mathcal{E}_{\mathrm{bd}}|_{\Sigma_m}$. By T1, a global minimizer exists; since the uniform state is a saddle, the global minimizer must be non-uniform. $\square$

Why this is a hero#

T8-Core is spectral universality. The critical condition $\beta/\alpha > 4\lambda_2/|W''(c)|$ depends only on:

• the spectral gap $\lambda_2$ of the graph (a topological invariant), and
• the volume fraction $c$ (a structural parameter of the theory).

It does not depend on the specific graph — it works on any connected graph with $\lambda_2 > 0$. Formation birth is therefore not an artifact of a particular geometry; it is a universal phenomenon determined by the graph's algebraic connectivity.

This is the analogue of the classical phase transition condition in Allen-Cahn / Cahn-Hilliard theory, but reformulated for the graph setting and with the spectral gap as the universal control parameter. T8-Core is the SCC version of the statement "spinodal decomposition is topology-independent."

Caveat (scaling regime). On graph families where $\lambda_2 \to 0$ as $n \to \infty$ (e.g., $k \times k$ grids with $\lambda_2 \sim \pi^2/k^2$), the threshold vanishes — the criterion is trivially satisfied for any $\beta > 0$. T8-Core then guarantees existence but not a meaningful selection criterion. On graphs with $\lambda_2 = \Omega(1)$ (expanders, bounded-diameter graphs, fixed-size applications), the criterion is genuinely informative. At scale, the diagnostic vector $\mathbf{d} = (\mathsf{Bind}, \mathsf{Sep}, \mathsf{Inside}, \mathsf{Persist})$ takes over — formation quality, not formation existence, becomes the meaningful question. See T-PreObj-1 for the W4 refinement: under full SCC parameters, the F=1 single-disk minimizer is non-critical, so existence shifts from "non-uniform" to "F≥2 multi-peak."

Logical dependencies#

• Builds on: T1 (existence of minimizers), spectral graph theory (Fiedler eigenvalue).
• Builds into: T8-Full (extends to full energy via IFT), T-Birth-Parametric (supercritical pitchfork at $\beta_{\mathrm{crit}}$), T11 (sharp-interface limit of the same energy), T-PreObj-1 (full SCC version where F=1 disk is non-critical).

See also#

• Full proof in canonical: Canonical Spec §13 T8-Core entry including scaling caveats and finite-element rescaling remark
• Volume constraint and admissible $c$ range: Canonical Spec Part 3 §8.0
• T8-Full extension to full energy: Canonical Spec §13 T8-Full entry
• W4 refinement: T-PreObj-1 hero page