Manuscript Oai265-A@2 · Oai265-A.tex · section · c0a34a45

Geometry and interaction convention

Stage 0: the frozen finite-domain package › Geometry and interaction convention · lines 221–312

Oai265-A@2#eq:H, Oai265-A@2#eq:gap, Oai265-A@2#rec:AL, Oai265-A@2#eq:AL

Let $Λ \subset Z^{2}$ be finite. Its graph is the induced graph with edge set

E_{Λ} = {{x, y} : x, y \in Λ, ∣ x - y ∣_{1} = 1} .

The graph distance $d_{Λ}$ is $+ \infty$ between distinct connected components. For $A \subseteq Λ$ , write

A^{c} = Λ ∖ A,

\partial_{Λ} A = {{x, y} \in E_{Λ} : x \in A, y \in A^{c}},

\partial_{in} A = {x \in A : \exists y \in A^{c}, {x, y} \in E_{Λ}},

and

Z_{A} = {x \in Λ : x is an endpoint of an edge in \partial_{Λ} A} .

Thus $Z_{A}$ includes endpoints on both sides of the cut. Its definition is necessary for the numerical factor below.

Fix

q \in N, q \geq 2, R \geq 0, J > 0, Δ > 0.

The Hilbert space is

H_{Λ} = x \in Λ ⨂ C^{q} .

We use the interaction convention

H = \emptyset \neq = X \subseteq Λ diam_{d_{Λ}} X \leq R \sum h_{X}, h_{X} = h_{X}^{†}, ∥ h_{X} ∥_{\infty} \leq J .

There is one aggregated term per support $X$ . An arbitrary number of separately bounded copies on the same support is not permitted to bypass the strength bound. Scalar identity terms can be removed without changing the state or gap.

The ground vector has norm one, is unique up to phase, and satisfies

H Ω = E_{0} Ω, H - E_{0} I \geq Δ (I - ∣Ω ⟩ ⟨ Ω∣) .

All entropy logarithms are natural:

S (ρ) = - Tr (ρ lo g ρ), 0 lo g 0 = 0.

The entropy is dimensionless; changing to thermodynamic units would introduce an independently specified factor $k_{B}$ .

The external statement under audit asserts that a finite constant $C = C (q, R, J, Δ)$ exists such that every instance (eq:H)–(eq:gap), on every finite induced domain, satisfies

S_{Ω} (A) \leq C ∣ \partial_{Λ} A ∣ (A \subseteq Λ) .

The same $C$ is independent of domain size, shape, and region. This is a documentary record, not a theorem proved or assumed here OpenAI265A.

There is no subsystem-gap, frustration-free, translation-invariant, product-path, continuum, or efficient-algorithm hypothesis in this frozen package.

Displayed formulas in this passage

Oai265-A@2#eq:H · line 263
$H = \emptyset \neq = X \subseteq Λ diam_{d_{Λ}} X \leq R \sum h_{X}, h_{X} = h_{X}^{†}, ∥ h_{X} ∥_{\infty} \leq J .$
unlabeled · line 225
$E_{Λ} = {{x, y} : x, y \in Λ, ∣ x - y ∣_{1} = 1} .$
unlabeled · line 231
$A^{c} = Λ ∖ A,$
unlabeled · line 234
$\partial_{Λ} A = {{x, y} \in E_{Λ} : x \in A, y \in A^{c}},$
unlabeled · line 239
$\partial_{in} A = {x \in A : \exists y \in A^{c}, {x, y} \in E_{Λ}},$
unlabeled · line 245
$Z_{A} = {x \in Λ : x is an endpoint of an edge in \partial_{Λ} A} .$
unlabeled · line 255
$q \in N, q \geq 2, R \geq 0, J > 0, Δ > 0.$
unlabeled · line 259
$H_{Λ} = x \in Λ ⨂ C^{q} .$
Oai265-A@2#eq:gap · line 277
$H Ω = E_{0} Ω, H - E_{0} I \geq Δ (I - ∣Ω ⟩ ⟨ Ω∣) .$
unlabeled · line 286
$S (ρ) = - Tr (ρ lo g ρ), 0 lo g 0 = 0.$
Oai265-A@2#eq:AL · line 298
$S_{Ω} (A) \leq C ∣ \partial_{Λ} A ∣ (A \subseteq Λ) .$