The source FQFT chain states

$$G_q\to {\rho }_{KM}\to M_{D_H}\to S[\mathrm{\Phi };D_H],$$

and describes the graph as being completed into a Hausdorff measure space. What is not currently supplied is the refinement operator that makes that completion mathematical.

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MIC-D01 — Minimum Refinement Datum

R25 freezes the minimum microscopic object as

$$\mathfrak{R}=((G_n{)}_{n\ge 1},({\ell }_n{)}_{n\ge 1},(d_n{)}_{n\ge 1},(m_n{)}_{n\ge 1},({\mathcal{E}}_n{)}_{n\ge 1},({\iota }_{n,n+1}{)}_{n\ge 1}),$$

where:

1. $G_n=(V_n,E_n)$ is a finite graph at refinement level $n$.
2. ${\ell }_n>0$ is the mesh scale, with ${\ell }_n\to 0$.
3. $d_n$ is a metric on $V_n$.
4. $m_n$ is a positive Borel measure on $V_n$.
5. ${\mathcal{E}}_n$ is a discrete Dirichlet form on $L^2(V_n,m_n)$.
6. ${\iota }_{n,n+1}:V_n\hookrightarrow V_{n+1}$ is an injective nesting or identification map.

A claim of "fractal completion" is undefined until all six components are exhibited.

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MIC-P01 — Scale-First Principle

$$\boxed{{\displaystyle \text{Physics is determined by the scale-indexed system }\mathfrak{R},\text{ not by an unrefined single graph }{G}_q.}}$$

No static property of $G_q$ can be promoted to a continuum field-theory property without specifying how it transforms under $\mathfrak{R}$.

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Discrete-to-Continuum Compatibility Laws

To be admissible, $\mathfrak{R}$ must satisfy three compatibility criteria:

Metric Compatibility (Gromov–Hausdorff)

$$(V_n,d_n)\stackrel{GH}{\to }(M,d_{\mathrm{\infty }}),$$

where $(M,d_{\mathrm{\infty }})$ is a compact metric space of finite Hausdorff dimension $D_H$.

Measure Compatibility (Weak Convergence)

$${\iota }_{n\ast }{m}_n\rightharpoonup m_{\mathrm{\infty }},\mathrm{supp}(m_{\mathrm{\infty }})=M.$$

Spectral Compatibility (Mosco / Norm-Resolvent)

The graph Laplacians $L_n$ associated to $({\mathcal{E}}_n,m_n)$ must converge to a continuum operator $\mathrm{\Delta }$ on $L^2(M,m_{\mathrm{\infty }})$ in the sense of Mosco convergence of Dirichlet forms:

$${\mathcal{E}}_n\stackrel{M}{\to }{\mathcal{E}}_{\mathrm{\infty }},(L_n+I{)}^{-1}\stackrel{s}{\to }(\mathrm{\Delta }+I{)}^{-1}.$$