FQFT Scale Cascade

Status Boundary

This work is part of the authorial PHYSICA / Science of Fabric Reality corpus. It is presented as a research framework, formalization target, computational model, or theoretical synthesis unless explicitly marked otherwise. It is not presented as accepted physics, external consensus, or experimentally confirmed science.

1. Scientific Purpose

Defines the visual, geometric, and structural requirements for representing the Fractal Quantum Field Theory (FQFT) Multiscale Cascade, coarse-graining operators, and scale-covariance mappings within the formal framework.

2. Concepts Visualized

• Multiscale Operator Cascade · Hierarchical field representations across scale levels $\ell =0,1,\dots ,L$ spanning Planck scale to macroscopic domains.
• Restriction and Prolongation Mappings · Discrete coarse-graining operators ${\mathcal{R}}_{\ell }:{\mathcal{H}}_{\ell }\to {\mathcal{H}}_{\ell +1}$ and reconstruction operators ${\mathcal{P}}_{\ell }:{\mathcal{H}}_{\ell +1}\to {\mathcal{H}}_{\ell }$.
• Renormalization Group Invariants · Scale-covariant flow $\beta (g)=\frac{\partial g}{\partial \ln \mu }$ preserving topological knot invariants across resolutions.

3. Mathematical Symbols & Grounding

• ${\varphi }_{\ell }$: Effective field state at resolution scale $\ell$.
• ${\mathcal{R}}_{\ell },{\mathcal{P}}_{\ell }$: Restriction and prolongation operators satisfying ${\mathcal{R}}_{\ell }\circ {\mathcal{P}}_{\ell }=\mathbf{I}$.
• $D_{\ell }^{\text{rec}}$: Recursive scale-dilation generator.
• ${\kappa }_{\ell }$: Scaling exponent governing localized knot-packet decay.

4. Required SVG Layers & Coordinate Geometry

• Coordinate System: viewBox="0 0 800 500", responsive SVG container with fluid scaling.
• Layer 1: base-topology:
  • Vertically stacked lattice planes representing decreasing spatial resolutions ($\ell =0,1,2,3$).
  • Grid cell widths scaling by factor $2^{\ell }$, bound to var(--vp-c-border).
• Layer 2: interaction-vectors:
  • Downward conical projection arrows indicating coarse-graining (${\mathcal{R}}_{\ell }$).
  • Upward expansion rays indicating prolongation (${\mathcal{P}}_{\ell }$), colored with var(--vp-c-brand-1) and var(--vp-c-brand-2).
• Layer 3: labels-and-nodes:
  • Scale index labels ($\ell =0,\dots ,L$), operator markers, and cutoff thresholds bound to var(--vp-c-text-1).
  • Accessible screen reader title and descriptions.

5. Caption & Interpretation Boundary

Figure: Schematic representation of the multiscale scale cascade in Fractal Quantum Field Theory, illustrating hierarchical lattice coarse-graining and scale-covariant prolongation.

NOTE

This figure serves as a formalization target and visual aid for the authorial model. It must not be interpreted as empirical proof or accepted physics.

6. Canonical Continuations

Direction	Target Resource	Purpose	Formal Code Link
Primary Theory	Fractal Quantum Field Theory	Multiscale field ontology and invariants	Fabrica.FQFT
Formal Lean Module	Formalization Roadmap	Verified scale covariance in Fabrica.FQFT	Fabrica.FQFT
Simulation Model	FQFT Scale Transition Model	Multiscale computational lattice solver	Fabrica.Fabricon
Figures Atlas	Visual Cartography Atlas	Figure program index and taxonomy	Visual Cartography
Formalization Gateway	Formalization Gateway	Lean 4 Interactive Proof Suite	Fabrica.FQFT

EXTERNAL REFERENCE