World

Prerequisites: Chapter 7 (Existence).

8.1 Definition of the World#

Definition 8.1 (Instantaneous world). $\boxed{\mathfrak{W}_t := \bigl(\mathcal{W}_t,\;\mathfrak{F}_t,\;\Sigma_t\bigr)}$

• $\mathcal{W}_t$: the relational field (Definition 3.6) --- the substrate.
• $\mathfrak{F}_t$: the fruit set (Definition 4.6) --- the cohesion layer.
• $\Sigma_t:\mathfrak{F}_t\to\mathcal{P}(V)$: the door function $F\mapsto\Sigma_\tau(F,t)$ --- the connection layer.

Definition 8.2 (Full world). $\boxed{\mathfrak{W} := \{\mathfrak{W}_t\}_{t\in\mathbb{T}}.}$

Definition 8.3 (Existence functor). $\mathrm{Ex}_t:\mathfrak{F}_t\to\bigsqcup_{F\in\mathfrak{F}_t}\mathcal{M}(F,\Sigma_t(F)),\qquad \mathrm{Ex}_t(F):=\mathrm{Existence}(F,t).$

8.2 Three-Layer Structure#

Theorem 8.4 (Decomposition of the world).

Layer 1 (Substrate): $\mathcal{W}_t$ --- the relational field.

Layer 2 (Cohesion): $\mathfrak{F}_t$ --- the fruit set.

Layer 3 (Connection): $\Sigma_t$ --- the door function.

Determination is sequential: $\mathcal{W}_t\Rightarrow\mathfrak{F}_t\Rightarrow\Sigma_t$.

Proof.

• $\mathcal{W}_t$: the gauge class of the raw data.
• $\mathfrak{F}_t$: determined by the scalar invariants $W_t,d_t$ of $\mathcal{W}_t$ (Definition 4.5 uses only $\phi_t$ and $\mathrm{vol}_t$).
• $\Sigma_t(F)$: determined by each $F\in\mathfrak{F}_t$ together with $\mathcal{D}(F,t)$ (Theorem C).

The dependency is $\mathcal{W}_t\to\mathfrak{F}_t\to\Sigma_t$, with no backward dependencies. $\square$

8.3 Gauge Invariance of the World#

Theorem 8.5 (Gauge invariance).

(i) $\mathcal{W}_t$ is a gauge class by definition.

(ii) $\mathfrak{F}_t$ is gauge-invariant (Proposition 4.7(ii)).

(iii) $\Sigma_t(F)$ is gauge-invariant (Theorem G).

Therefore $\mathfrak{W}_t$ does not depend on the choice of gauge representative.

8.4 Spectral Stability (Theorem F)#

Theorem F (Spectral stability). Let $\|\delta W\|_\infty:=\max_{i,j}|W'_t(i,j)-W_t(i,j)|$.

(i) $|\phi'_t(F)-\phi_t(F)|\le C_1\cdot\|\delta W\|_\infty\cdot|V|^2/\mathrm{vol}_t(F)$.

(ii) If $\phi_t(F)\le\theta-\epsilon$, then $F\in\mathfrak{F}'_t$ provided $\|\delta W\|_\infty<\delta_0(\epsilon,F)$.

Proof.

(i) Track the perturbation of numerator and denominator separately.

The fractional perturbation of $\phi_t(F)=\mathrm{cut}/\mathrm{vol}$:

The numerator is bounded by $\|\delta W\|_\infty\cdot|V|^2\cdot C\cdot\mathrm{vol}_t(F)$. The denominator $\ge\mathrm{vol}_t(F)^2/2$ for small perturbations.

Hence $|\phi'-\phi|\le C_1\cdot\|\delta W\|_\infty\cdot|V|^2/\mathrm{vol}_t(F)$, where:

(ii) Set $\delta_0:=\epsilon\cdot\mathrm{vol}_t(F)/(C_1\cdot|V|^2)$. Then $|\phi'-\phi|<\epsilon$, so $\phi'_t(F)<\theta$. The volume condition $\mathrm{vol}_t(F)\le\frac{1}{2}\mathrm{vol}_t(V)$ is preserved similarly. $\square$

Explicit constant. $C_1\le 4$ suffices in all cases. For fruits with $\phi_t(F)\ll\theta$, the stability margin $\epsilon=\theta-\phi_t(F)$ is large.

8.5 Reading the World#

The complete procedure for interpreting a world:

1. Observe the raw data $(V,w_t,g_t)$ (gauge-dependent representation).
2. Identify fruits: compute $W_t,d_t$, then $\phi_t(S)$ for candidate subsets; collect $\mathfrak{F}_t$.
3. Detect doors: for each $F\in\mathfrak{F}_t$, compute $b_{F,t}(i)$ and determine $\Sigma_\tau(F,t)$.
4. Extract existence: on $F^\circ=F\setminus\Sigma$, minimise $\mathcal{E}_{F^\circ}(h)$ and record $[A_\infty]$.
5. Record: $\mathrm{Existence}(F,t)=([A_\infty],\Sigma,\mathbf{e})$.

Key point: the world is unchanged throughout this process. The interpreter observes and records; interpretation does not alter the world (Principle V).

8.6 Discrete--Continuous Correspondence#

A detailed dictionary is given in Appendix C.