The Breach Event in Pattern Field Theory™

Date: August 28, 2025

Disambiguation. Not to be confused with “field patterns” in PT‑symmetric time‑varying media (Mattei & Milton, 2017) nor with Grenander/Mumford Pattern Theory. Pattern Field Theory (PFT™) is authored by James Johan Sebastian Allen and uses “pattern” for physically realized field dynamics.
Canonical snapshot (PFT).
  • NULL ≠ 0: no domain, no metric, no numbers.
  • Emergence: geometry turns on; with the first curvature, π emerges as its invariant.
  • Π‑particle: first coherent loop with \(\oint \kappa\,ds=2\pi\); Π‑matrix enforces π‑locking.
  • Breach (2D→3D): a supercritical instability — bamboo‑like, staccato chain reaction. Not a “controlled rupture.”

Overview

The breach is the geometric transition from a coherent 2‑D sheet into 3‑D volume. It is triggered when the sheet’s size–tension–potential crosses a threshold; many long‑wavelength modes ignite at once, creating rims that circularize quickly. Since π emerges at Emergence, rims are π‑locked (\(\oint\kappa\,ds=2\pi\)).

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 > Lcrit, many modes ignite at once → chain reaction. Rim segments circularize under tension, so π governs the rims.

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 \(\alpha>0\), 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 \(k_c\).

Π‑locking metrics (for rims)

  • Closure residual: \(R_\pi=\big|\sum_i \kappa_i \Delta s_i - 2\pi\big|\).
  • Curvature variance: length‑weighted \( \mathrm{var}_w(\kappa) \).
  • Π‑matrix stability: smallest eigenvalue \( \lambda_{\min}(\mathbf H_\pi) \) of the π‑locking Hessian.

Acceptance band (tunable): candidates typically have \(R_\pi \lesssim 10^{-2}\), low variance, and small positive \( \lambda_{\min} \).

Diagram: Genesis of the Breach

  1. Flat 2D sheet: coherent layer under tension.
  2. Strain builds: long‑wave modes soften as \(L \to L_{\text{crit}}\).
  3. Ignition: many modes fire (staccato); rims nucleate.
  4. Circularization: rims tend toward constant curvature (π‑locking).
  5. 3D foam: volumetric structures appear; mixed 1D/2D residues persist.
2D sheet breaching into 3D foam via supercritical instability
Fig. 1 — Supercritical breach of a 2‑D sheet into 3‑D. Many long‑wave modes ignite; rims circularize under tension.

Artifacts & Repos

References (context/analogy)

  1. Mattei & Milton (2017), “Field patterns without blow up,” New J. Phys. — cited for analogy only; not PFT.
  2. Caloz et al. (2020), “Spacetime Metamaterials — Part I,” IEEE TAP — general context on time‑varying media.

Trademarks: Pattern Field Theory™ (PFT™) and related marks are claimed trademarks. © James Johan Sebastian Allen.