Open Problems & Unresolved Boundaries

Scientific integrity requires explicit demarcation of open mathematical problems, unproven conjectures, and experimental challenges. This page documents the central unresolved frontiers of the Science of Fabric Reality research program.

┌─────────────────────────────────────────────────────────────────────────────┐
│                       OPEN RESEARCH FRONTIER MATRIX                         │
├───────────────────────────────────┬─────────────────────────┬───────────────┤
│ Open Problem                      │ Domain                  │ Formal Status │
├───────────────────────────────────┼─────────────────────────┼───────────────┤
│ OP-1: Non-Perturbative Continuum  │ Functional Analysis     │ OPEN TARGET   │
│ OP-2: Osterwalder-Schrader Pos.   │ Constructive QFT        │ OPEN TARGET   │
│ OP-3: Fiber Coupling Gate 𝓜(X, F)  │ Differential Geometry   │ UNRESOLVED    │
│ OP-4: Dynamic Effective Dimension │ Spectral Geometry       │ UNRESOLVED    │
│ OP-5: Non-Perturbative Lean Closure │ Interactive Verification│ 28 VERIFIED (B1)│
└───────────────────────────────────┴─────────────────────────┴───────────────┘

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1. Problem OP-1: The Non-Perturbative Continuum Limit

• The Problem: A discrete metric-measure Dirichlet substrate $(X_a,d_a,{\mu }_a)$ is parametrized by microscopic lattice spacing $a$. While the norm-resolvent limit converges for linear scalar operators, the non-perturbative convergence of the non-linear interacting action functional $S_a[\mathrm{\Phi }]$ to a smooth pseudo-Riemannian path integral remains unproven:$$\underset{a\to 0}{lim}\parallel ({\mathcal{L}}_a+m^2{)}^{-1}-(-{\mathrm{\nabla }}^2+m^2{)}^{-1}\parallel =0\stackrel{?}{⟹}\underset{a\to 0}{lim}\int \mathcal{D}{\mathrm{\Phi }}_a{e}^{-S_a[{\mathrm{\Phi }}_a]}=\int \mathcal{D}\mathrm{\Phi }{e}^{-S[\mathrm{\Phi }]}$$
• Required Resolution: Rigorous proof that the weak limit of functional measures ${\mu }_a\rightharpoonup {\mu }_{\text{smooth}}$ preserves local Lorentz covariance without coordinate singularity.

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2. Problem OP-2: Osterwalder–Schrader Reflection Positivity

• The Problem: For Euclidean Schwinger functions $S_n(x_1,\dots ,x_n)$ to reconstruct physical Lorentzian Wightman distributions on a relativistic Hilbert space ${\mathcal{H}}_{\text{phys}}$, they must satisfy reflection positivity across spacelike hyperplanes:$$\sum _{j,k}⟨\mathrm{\Theta }{f}_j,f_k⟩\ge 0$$
• Current Status: On discrete 5-regular graphs or non-integer Hausdorff metric-measure spaces, spatial reflection $\mathrm{\Theta }$ cannot be defined as a simple coordinate sign inversion $(t\mapsto -t)$. A generalized reflection automorphism on discrete graph states is defined heuristically, but reflection positivity is not analytically proved.

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3. Problem OP-3: Microscopic Fiber Coupling Gate $\mathcal{M}(X,\mathcal{F})$

• The Problem: In Fractal Quantum Field Theory (FQFT), the state space is modeled as a base spacetime manifold $X$ coupled to an internal spectral fiber $\mathcal{F}$. While the uncoupled product $X\times \mathcal{F}$ yields standard kinetic Laplacians, the dynamical coupling gate $\mathcal{M}(X,\mathcal{F})$ governing non-linear gauge interactions across scale boundaries remains an active formalization target.

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4. Problem OP-4: Dynamic Derivation of Effective Dimension $D_H^{\ast }$

• The Problem: The calibrated lepton mass formula uses an effective dimensional parameter $D_H^{\ast }\approx 27.6$. While this value yields zero-residual retrodictions for electron, muon, and tau masses, it is currently introduced as a phenomenological fit parameter.
• Required Resolution: Deriving $D_H^{\ast }$ analytically as the global ground-state minimum of an invariant lattice action on 5-regular Ramanujan graphs without empirical tuning.

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5. Problem OP-5: Non-Perturbative Formal Proof Embedding

• Current Progress: The formal Lean 4 proof suite in docs/04-mathematics/formalization-roadmap has achieved 32 machine-verified theorems across 11 formal modules without sorry or admit stubs, with 1 quarantined axiom FQFT_closure (Formalization Batch 1).
• The Active Challenge: Embedding infinite-dimensional continuous operator domain closures (e.g. non-local KP-field self-adjoint maximal domains and non-perturbative path measure limits) into machine-checked Lean 4 Mathlib structures remains an active mathematical frontier.
• Roadmap: Active formalization targets are tracked in the Mathematics Formalization Roadmap.

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6. Canonical Continuations

Direction	Target Resource	Purpose
Results Ledger	Scientific Results Ledger →	32 machine-verified Lean 4 theorems and negative null benchmark ($p_{\text{null}}=0.62$)
Formal Proofs	Formalization Batch 1 Specifications →	Direct Lean 4 code specifications across all 11 verified modules
Proof Roadmap	Lean 4 Formalization Roadmap →	32 machine-verified theorem records and lemma dependency DAGs
Reproducibility	Computational Reproducibility →	RFC 8785 JCS verification receipts and reproducible proof environment
Review Portal	SFR Review Portal & Verification Tracks →	Independent critique tracks and formal claim challenge submission