Theorem: Isotropic matched operational existence

Fix $c>0$. There are countable substrates $({\Lambda }_n,d_n,1)$, injective location maps $z_n:{\Lambda }_n\to {\mathbb{R}}^2$, physical tick durations $h_n\to 0$, and state-independent periodic partition schedules such that:

1. every block has $d_n$-diameter at most one and a physical spacetime footprint tending to zero;

2. every scheduled dependency path satisfies the exact endpoint bound

   $$\Vert z_n(y)-z_n(x){\Vert }_2\le c(T-s)h_n;$$

3. the metric past envelopes and scheduled pasts converge to the Euclidean past cones, in the uniform local distance-function sense;

4. for every Euclidean unit vector $v$, every $u>0$, and every $x\in {\mathbb{R}}^2$, matched classical relay endpoints converge to

   $$(0,x),(u,x+cuv),$$

   and their readout laws have total-variation distance one;

5. transport of every fixed finite-dimensional quantum register is the identity after logical identification, including every untouched reference;

6. every causal finite circuit in the declared class has a realizing family whose logical complete channels equal its circuit channel exactly, on all sufficiently fine scales;

7. local replacements have matched realizing families with common schedule, resource preparation, routing, unchanged block channels, and common readouts;

8. evaluation, admitted finite composition, and the matched-locality relation extend to the fixed-skeleton operational completion.

The limit operational maximal signalling speed is $c$.

The completion statement concerns the compatibility class explicitly defined here. It is not a completeness theorem for arbitrary growing discrete networks known only through one convergent terminal law.

Proof. Items (i)–(iii) are proved in Theorems thm:metric, thm:routing, and thm:cones. Items (iv)–(v) are proved in Theorems thm:identity and thm:relay. Items (vi)–(vii) are proved in Theorems thm:realization and thm:matched. Item (viii) is proved in Theorems thm:completion and thm:completionmatch. The speed conclusion is Corollary cor:maxspeed.