Extended Topological FQFT by Anna Paseva

Topological Invariants, Scale-Hierarchy Extensions & Fractal Manifolds

Spine Position

Foundations Root · Topological Extensions & Scale Hierarchies · Extended FQFT

Public Status Boundary. This manuscript presents an authorial theoretical extension to Fractal Quantum Field Theory (FQFT). It investigates topological invariant preservation and non-perturbative knot invariants across multiscale fractal networks (ACTIVE_P4_SEALS = 0).

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1. Architectural Concept & Research Objective

The Extended Topological FQFT framework explores the mathematical interface between topological quantum field theories (TQFT) and Dirichlet metric-measure spaces. In particular, it investigates whether topological invariants (such as Chern-Simons partition functions and knot invariants) remain well-defined and scale-covariant across non-integer Hausdorff dimensions $D_H^{\ast }$:

$${\mathcal{Z}}_{\text{top}}(M,K)=\int \mathcal{D}A\exp \left(\frac{ik}{4\pi }{\int }_M\mathrm{Tr}(A\wedge dA+\frac{2}{3}A\wedge A\wedge A)\right)$$

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2. Core Relational Invariants

1. Multiscale Topological Persistence: Topological knot configurations remain robust against high-frequency lattice perturbations.
2. Dirichlet Scale Covariance: Transfinite scale transport operator commutes with topological observable projections (thm_scale_covariance_algebraic).
3. Trace Dual Symmetry: Topological amplitude pairings satisfy symmetric bilinear trace reciprocity $\text{TraceDual}(T,x,y)⟺\text{TraceDual}(T,y,x)$.

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3. Epistemic Classification & Verification State

• Formal Classification: THEORETICAL_CONSTRUCT / UNVERIFIED_HEURISTIC.
• Machine-Verified Formal Foundations:
  • thm_scale_covariance_algebraic, thm_dirichlet_scaling_lower_bound (Fabrica.FQFT): Transfinite scale covariance commutation.
  • thm_trace_involution, thm_trace_dual_symmetry (Fabrica.TraceReciprocity): Bilinear trace pairing symmetry and involution.
• Empirical Boundary: Purely mathematical research program; does not assert empirical quantum-gravity measurements (ACTIVE_P4_SEALS = 0).

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4. Canonical Continuations

Direction	Target Resource	Purpose
Field Theory	Fractal Quantum Field Theory (FQFT) →	Dirichlet variational principles and Mosco limits
Variational Core	Fabric Field Equation (FFE) →	Constrained variational action on metric-measure spaces
Trace Reciprocity	Trace Reciprocity Principle →	Bilinear trace pairings and machine-verified dual involution
Formal Proofs	Formal Mathematics Spine →	9-tier status map and 35 Lean 4 machine-verified proofs across 11 modules

EXTERNAL REFERENCE