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

Thm	Name	Statement (informal)
A	Energy isolation	Internal edge-energy of a fruit is at least $(1 - θ)$ of its total volume.
B	Finiteness of doors	$∥ Σ_{τ} (F) ∥ \leq θ \cdot vol (F) / τ$ .
C	Self-interpretation	The door set and residual energy are determined by the fruit's intrinsic data alone (Axiom A5).
D	Metastability	Expected escape time from a fruit under the lazy walk is at least $1/ (2 θ)$ .
E	Curvature localisation	Residual curvature decays exponentially from doors; for general $G$ , deep-interior energy is a contraction of the near-door energy.
F	Spectral stability	Small weight perturbations produce bounded conductance changes; strong fruits persist.
G	Door stability	Doors with margin above $∥ V ∥ \cdot ∥ δ W ∥_{\infty}$ are stable; only a thin boundary layer of candidates can change.
H	Flow stability	The Yang–Mills gradient flow converges to a stable equilibrium; convergence is exponential at non-degenerate minima.

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 $vol_{t} (F) = \sum_{i, j \in F} W_{t} (i, j) + cut_{t} (F, \overset{ˉ}{F})$ . Under the fruit condition (F2) $ϕ_{t} (F) \leq θ$ , so $cut_{t} (F, \overset{ˉ}{F}) \leq θ \cdot vol_{t} (F)$ . Rearranging gives the bound. Equality holds at the threshold and at volume balance.

Theorem B — Finiteness of Doors

Proof idea. $τ \cdot ∣ Σ_{τ} ∣$ is bounded by the sum of door boundary values, which is in turn bounded by $cut_{t} (F, \overset{ˉ}{F}) \leq θ \cdot vol_{t} (F)$ . 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 $Φ_{F} \leq ϕ_{t} (F) \leq θ$ . The Sinclair–Jerrum bound gives a spectral-gap estimate $\tilde{γ}_{F} \leq θ$ ; Aldous–Fill's hitting-time inequality then yields $E_{π_{F}} [T_{esc}] \geq 1/ (2 \tilde{γ}_{F})$ . 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 $L φ^{*} = - B diag (W) α$ ; 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 $λ_{2}$ decays as $e^{- λ_{2} d}$ (Chung–Yau), giving the stated exponential localisation with $β = λ_{2} (F^{\circ})$ . For general $G$ , a discrete harmonic-map and maximum-principle argument on $F_{deep}$ gives the contraction.

Theorem F — Spectral Stability

Proof idea. Perturbation of numerator (cut) and denominator (volume) of $ϕ = cut / vol$ are each bounded by $∣ V ∣^{2} ∥ δ W ∥_{\infty}$ . A direct quotient-rule estimate yields the stated linear bound in $∥ δ W ∥_{\infty}$ 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 $∣ V ∣ \cdot ∥ δ W ∥_{\infty}$ bound directly. Doors with values away from the threshold therefore remain doors.

Theorem H — Flow Stability

Proof idea. Four ingredients: (i) $E$ 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 $λ_{m i n}$ governs the rate, yielding the exponential bound with $γ = λ_{m i n} /2$ after a Grönwall argument.

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, $S U (2)$ sketch) remain in the research archive and will be surfaced as chapters are cleaned for public reading.

The eight theorems at a glance

Thm

Name

Statement (informal)

Energy isolation

Internal edge-energy of a fruit is at least

(1 - θ)

of its total volume.

Finiteness of doors

∥ Σ_{τ} (F) ∥ \leq θ \cdot vol (F) / τ

Self-interpretation

The door set and residual energy are determined by the fruit's intrinsic data alone (Axiom A5).

Metastability

Expected escape time from a fruit under the lazy walk is at least

1/ (2 θ)

Curvature localisation

Residual curvature decays exponentially from doors; for general

G

, deep-interior energy is a contraction of the near-door energy.

Spectral stability

Small weight perturbations produce bounded conductance changes; strong fruits persist.

Door stability

Doors with margin above

∥ V ∥ \cdot ∥ δ W ∥_{\infty}

are stable; only a thin boundary layer of candidates can change.

Flow stability

The Yang–Mills gradient flow converges to a stable equilibrium; convergence is exponential at non-degenerate minima.

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)

Theorem D — Metastability

Proof idea. Define the restricted sub-stochastic chain on

F

and show that its conductance

Φ_{F} \leq ϕ_{t} (F) \leq θ

. The Sinclair–Jerrum bound gives a spectral-gap estimate

\tilde{γ}_{F} \leq θ

; Aldous–Fill's hitting-time inequality then yields

E_{π_{F}} [T_{esc}] \geq 1/ (2 \tilde{γ}_{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 φ^{*} = - B diag (W) α

; 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

λ_{2}

decays as

e^{- λ_{2} d}

(Chung–Yau), giving the stated exponential localisation with

β = λ_{2} (F^{\circ})

. For general

G

, a discrete harmonic-map and maximum-principle argument on

F_{deep}

gives the contraction.

Theorem H — Flow Stability

Proof idea. Four ingredients: (i)

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

λ_{m i n}

governs the rate, yielding the exponential bound with

γ = λ_{m i n} /2

after a Grönwall argument.

The eight theorems at a glance#

Logical dependencies#

Theorem A — Energy Isolation#

Theorem B — Finiteness of Doors#

Theorem C — Self-Interpretation#

Theorem D — Metastability#

Theorem E — Curvature Localisation#

Theorem F — Spectral Stability#

Theorem G — Door Stability#

Theorem H — Flow Stability#

Related reading#

The eight theorems at a glance#

Logical dependencies#

Theorem A — Energy Isolation#

Theorem B — Finiteness of Doors#

Theorem C — Self-Interpretation#

Theorem D — Metastability#

Theorem E — Curvature Localisation#

Theorem F — Spectral Stability#

Theorem G — Door Stability#

Theorem H — Flow Stability#

Related reading#

The eight theorems at a glance

Logical dependencies

Theorem A — Energy Isolation

Theorem B — Finiteness of Doors

Theorem C — Self-Interpretation

Theorem D — Metastability

Theorem E — Curvature Localisation

Theorem F — Spectral Stability

Theorem G — Door Stability

Theorem H — Flow Stability

Related reading

The eight theorems at a glance

Logical dependencies

Theorem A — Energy Isolation

Theorem B — Finiteness of Doors

Theorem C — Self-Interpretation

Theorem D — Metastability

Theorem E — Curvature Localisation

Theorem F — Spectral Stability

Theorem G — Door Stability

Theorem H — Flow Stability

Related reading