World

Prerequisites: Chapter 7 (Existence).

8.1 Definition of the World#

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

• ${\mathcal{W}}_t$: 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). ${\mathrm{Ex}}_t:{\mathfrak{F}}_t\to {⨆}_{F\in {\mathfrak{F}}_t}\mathcal{M}(F,{\Sigma }_t(F)),{\mathrm{Ex}}_t(F):=\mathrm{Existence}(F,t).$

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 ${\varphi }_t$ 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 $\Vert \delta W{\Vert }_{\infty }:={\max }_{i,j}|W_t^{\prime }(i,j)-W_t(i,j)|$.

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

(ii) If ${\varphi }_t(F)\le \theta -ϵ$, then $F\in {\mathfrak{F}}_t^{\prime }$ provided $\Vert \delta W{\Vert }_{\infty }<{\delta }_0(ϵ,F)$.

Proof.

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

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

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

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

(ii) Set ${\delta }_0:=ϵ\cdot {\mathrm{vol}}_t(F)/(C_1\cdot |V{|}^2)$. Then $|{\varphi }^{\prime }-\varphi |<ϵ$, so ${\varphi }_t^{\prime }(F)<\theta$. 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 ${\varphi }_t(F)\ll \theta$, the stability margin $ϵ=\theta -{\varphi }_t(F)$ 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 ${\varphi }_t(S)$ 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.