Part II · Main Theorems A–H (summary)

Part II of the book collects the complete proofs of the eight theorems that pin down the behaviour of fruits, doors, and the optimal gauge under perturbation and flow. The proofs themselves involve spectral-graph techniques (Cheeger, Sinclair–Jerrum), Łojasiewicz convergence, and a discrete harmonic-map analysis; the full text lives in the research archive. This page collects the statements and one-paragraph proof ideas.

The eight theorems at a glance#

Logical dependencies#

              ┌─────────────┐
              │  Theorem A  │  Energy isolation
              └──────┬──────┘
         ┌──────────┼─────────────┬──────────────┐
         ▼          ▼             ▼              ▼
    Thm B,C      Thm D         Thm F          (Ch 7)
   (Doors)   (Metastability) (Stability)         │
         │                                       ▼
         ▼                                   Thm H
    Thm E + G                            (Flow stability)

A is the foundation. B, C, D, F are first-tier consequences. E and G require the sequential protocol and perturbation analysis introduced in Ch 6. H stands somewhat apart, drawing on compactness and Łojasiewicz rather than the other theorems.

Theorem A — Energy Isolation#

Proof idea. The volume decomposes as $\mathrm{vol}_t(F) = \sum_{i,j \in F} W_t(i,j) + \mathrm{cut}_t(F, \bar F)$. Under the fruit condition (F2) $\phi_t(F) \le \theta$, so $\mathrm{cut}_t(F, \bar F) \le \theta \cdot \mathrm{vol}_t(F)$. Rearranging gives the bound. Equality holds at the threshold and at volume balance.

Theorem B — Finiteness of Doors#

Proof idea. $\tau \cdot |\Sigma_\tau|$ is bounded by the sum of door boundary values, which is in turn bounded by $\mathrm{cut}_t(F, \bar F) \le \theta \cdot \mathrm{vol}_t(F)$. The second inequality follows by the same argument applied to $\sum e_p$.

Theorem C — Self-Interpretation#

Proof idea. The door measure $b_{F, t}(i) = d_t(i) - \sum_{j \in F} W_t(i, j)$ uses only quantities available from inside $F$: the degree sum $d_t(i)$ and the internal weights. The door set is a level set of $b$; its energies are $e_p = b_{F, t}(p)$. No exterior information is required.

Theorem D — Metastability#

Proof idea. Define the restricted sub-stochastic chain on $F$ and show that its conductance $\Phi_F \le \phi_t(F) \le \theta$. The Sinclair–Jerrum bound gives a spectral-gap estimate $\tilde\gamma_F \le \theta$; Aldous–Fill's hitting-time inequality then yields $\mathbb{E}_{\pi_F}[T_{\mathrm{esc}}] \ge 1 / (2 \tilde\gamma_F)$. The bound is tight up to constants (barbell graph).

Theorem E — Curvature Localisation#

Proof idea. For $U(1)$, the optimal gauge solves a linear system $L \varphi^* = -B \, \mathrm{diag}(W) \, \alpha$; the residuals lie in the projection onto the cycle space, whose source terms sit near the doors. The discrete Green's function on a graph with spectral gap $\lambda_2$ decays as $e^{-\sqrt{\lambda_2} \, d}$ (Chung–Yau), giving the stated exponential localisation with $\beta = \sqrt{\lambda_2(F^\circ)}$. For general $G$, a discrete harmonic-map and maximum-principle argument on $F_{\mathrm{deep}}$ gives the contraction.

Theorem F — Spectral Stability#

Proof idea. Perturbation of numerator (cut) and denominator (volume) of $\phi = \mathrm{cut} / \mathrm{vol}$ are each bounded by $|V|^2 \|\delta W\|_\infty$. A direct quotient-rule estimate yields the stated linear bound in $\|\delta W\|_\infty$ with explicit constant $C_1$.

Theorem G — Door Stability#

Proof idea. Both degree and internal-weight terms of $b_{F, t}$ are linear in $W$; the perturbation inherits a $|V| \cdot \|\delta W\|_\infty$ bound directly. Doors with values away from the threshold therefore remain doors.

Theorem H — Flow Stability#

Proof idea. Four ingredients: (i) $\mathcal{E}$ is real-analytic on the compact manifold $G^{F^\circ}$; (ii) the flow exists globally by compactness; (iii) the Łojasiewicz inequality applies, giving finite arc-length and hence convergence (Simon, 1983); (iv) at a non-degenerate minimum the Hessian's smallest eigenvalue $\lambda_{\min}$ governs the rate, yielding the exponential bound with $\gamma = \lambda_{\min} / 2$ after a Grönwall argument.

Related reading#

• Part I · Chapter 9 dependency context — Theorem A is introduced here.
• Part I · Chapter 6 — setup for Theorems B, C, G.
• Part I · Chapter 7 — optimal gauge for Theorems E, H.
• Full proofs and the worked Part II examples (5-node, 10-node, $SU(2)$ sketch) remain in the research archive and will be surfaced as chapters are cleaned for public reading.