2D→3D Breach — Supercritical Instability (A–D)

Allen, James Johan Sebastian

2D→3D Breach — Supercritical Instability

Not a “controlled rupture.” A supercritical instability in the 2‑D sheet—set by its size and by tension + potential—triggers a bamboo‑like, staccato chain reaction that lifts into 3‑D.

A) Supercritical instability (not “controlled rupture”)

When the 2‑D sheet grows large enough and the stored tension + potential cross a threshold, long‑wave modes go unstable together — a bamboo‑like, staccato chain reaction that lifts into 3‑D. A minimal free‑energy captures the onset:

\[ \mathcal{F}=\int \Big[\tfrac{\kappa}{2}(\nabla^2 h)^2+\tfrac{\sigma}{2}\lvert\nabla h\rvert^2-\tfrac{\alpha}{2}h^2\Big]\,dA \] \[ \text{Instability when}\quad \kappa k^{4}+\sigma k^{2}-\alpha<0 \quad\Rightarrow\quad 0

Once L > L_crit, many modes ignite at once → chain reaction. Rim segments then circularize under tension, so π emerges at Emergence and governs the rims (\(\oint\kappa\,ds=2\pi\)).

B) Breach instability criterion (details)

Let \(h(x,y)\) be the out‑of‑plane displacement of the coherent 2‑D sheet. With bending rigidity \(\kappa\), surface tension \(\sigma\), and effective potential \(\alpha>0\) from stored energy, linear modes \(h_{\mathbf k}\) grow if

\[ \frac{d}{dt} h_{\mathbf k} \propto \big(\alpha - \sigma k^2 - \kappa k^4\big) h_{\mathbf k}\,. \] \[ \Rightarrow\quad 0<k<k_c,\qquad k_c^2=\frac{-\sigma+\sqrt{\sigma^2+4\kappa\alpha}}{2\kappa}\,. \] \[ \text{Finite domain: } k_{\min}\approx \frac{2\pi}{L}\,;\quad \text{supercritical when } L > L_{\text{crit}}=\frac{2\pi}{k_c}\,. \]

After nucleation, rim tension drives curvature toward a constant value, making the loop π‑locked: \(\oint \kappa\,ds = 2\pi\).

C) Front‑page callout (drop‑in)

Not a controlled rupture: the 2‑D sheet crosses a size–tension–potential threshold and undergoes a supercritical instability — a bamboo‑like, staccato chain reaction into 3‑D. See instability criterion.

D) Breach instability — testable signatures

Size scaling: onset frequency vs domain size jumps at \(L=L_{\text{crit}}\).
Rim circularization: rapid drop in curvature variance; \(R_\pi=\big|\sum\kappa_i\Delta s_i-2\pi\big|\to 0\).
Mode band: initial rupture wavelengths confined to \(0<k<k_c\); spectral histogram cuts off near \(k_c\).

Pattern Field Theory