The official semantic authority and concept registry for the Science of Fabric Reality (SFR) corpus and Fractal Quantum Field Theory (FQFT) research program. Every entry declares an explicit epistemic term-class to prevent conflation between standard mathematical literature and authorial theoretical constructs.

SCIENTIFIC REFERENCE SPINE

GlossaryCURRENT

Canonical terminology & formal definitions

Sources

Bibliographic authority & source traces

Theorems

Axioms, lemmas & proof obligations

Evidence

Falsifiability & pre-registered ledgers

Data

Numerical testbeds & simulation records

Review

Academic discovery & formal critiques

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1. Epistemic Term-Class Firewall

To maintain absolute scientific transparency, every concept in this registry is partitioned into one of five mutually exclusive classes:

Term Class	Definition & Scientific Scope	Epistemic Boundary	Registry Count
ESTABLISHED_MATHEMATICS	Proven theorems, definitions, and constructs from standard peer-reviewed mathematical literature.	Full mathematical validity; does not constitute validation of FQFT.	11
R26_REVISION_CONSTRUCT	Candidate architectural completions introduced in R26 (FQFT-v2) to separate global Dirichlet spacetime from internal spectral fibers.	Candidate mathematical architecture; not empirical discovery.	5
FORMALIZATION_TARGET	Conjectured operators, coupling terms, or dynamical equations targeted for proof-assistant formalization.	Open theoretical gate; proof/derivation currently unclosed.	1
AUTHORIAL_DEFINITION	Foundational definitions, axioms, or ontology formulated by Ivan Pasev in the SFR corpus.	Theoretical research framework; requires independent critique.	2
Total Tracked Concepts	Unified machine-readable registry (/data/sfr_canonical_glossary.json)	Rigorous Epistemic Partition	19

Mandatory Epistemic Boundary

External sources referenced in this glossary provide mathematical background, comparative vocabulary, and analytical methods. Comparator sources do not constitute external validation or empirical confirmation of authorial physics.

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2. Established Mathematical Foundations (ESTABLISHED_MATHEMATICS) — 11 Terms

Ramanujan Graph

• Term Class: ESTABLISHED_MATHEMATICS
• Notation: $G=(V,E)$ with degree $d$, $\lambda (G)=\underset{|{\lambda }_i|<d}{max}|{\lambda }_i|\le 2\sqrt{d-1}$
• Domain: Spectral Graph Theory / Number Theory
• Definition: A $d$-regular graph whose non-trivial adjacency matrix eigenvalues are bounded in absolute value by the Alon–Boppana limit $2\sqrt{d-1}$. Ramanujan graphs are optimal spectral expanders exhibiting maximal spectral gaps and rapid random-walk mixing.
• Primary Sources: Lubotzky, Phillips, & Sarnak (1988), Combinatorica 8(3), 261–277 (doi:10.1007/BF02126799); Margulis (1988).
• Role in FQFT: Internal 5-regular Ramanujan fibers (e.g. $K_6$) provide discrete spectral comparators for gauge and charge generation without macroscopic spacetime collapse.

Expander Graph

• Term Class: ESTABLISHED_MATHEMATICS
• Notation: $(G_n{)}_{n\ge 1}$, Cheeger constant $h(G)\ge c>0$, spectral gap $\mathrm{\Delta }\lambda \ge ϵ>0$
• Domain: Combinatorics / Functional Analysis
• Definition: A sequence of sparse graphs with uniformly bounded vertex degrees that maintain high vertex and edge expansion, ensuring rapid diffusion and absence of topological bottlenecks.
• Primary Source: Hoory, Linial, & Wigderson (2006), Bull. Amer. Math. Soc. 43(4), 439–561 (doi:10.1090/S0273-0979-06-01126-8).
• Role in FQFT: Proved in theorem MIC-T01 that naive global fixed-degree expander sequences under uniform metric shrinkage ${\ell }_n{d}_{G_n}$ cannot converge to finite-dimensional compact fractals.

Kesten–McKay Law

• Term Class: ESTABLISHED_MATHEMATICS
• Notation: ${\mu }_d(x)=\frac{d\sqrt{4(d-1)-x^2}}{2\pi (d^2-x^2)}{\mathbf{1}}_{[-2\sqrt{d-1},2\sqrt{d-1}]}(x)$
• Domain: Random Matrix Theory / Free Probability / Graph Limits
• Definition: The spectral density of the adjacency operator on the infinite $d$-regular tree $T_d$, and the asymptotic eigenvalue distribution of large $d$-regular graphs with few short cycles. For degree $d=5$, the support is $[-4,4]$, and the corresponding graph Laplacian $L=5I-A$ has continuous spectral support on $[1,9]$.
• Primary Sources: Kesten (1959), Trans. Amer. Math. Soc. 92, 336–354 (doi:10.1090/S0002-9947-1959-0109367-6); McKay (1981), Linear Algebra Appl. 40, 203–216 (doi:10.1016/0024-3795(81)90150-6).
• Role in FQFT: Models local internal spectral statistics on $T_5$ fibers (MICRO-27), strictly partitioned from macroscopic spacetime geometry.

Dirichlet Form

• Term Class: ESTABLISHED_MATHEMATICS
• Notation: $(\mathcal{E},\mathcal{D}(\mathcal{E}))$ on $L^2(X,m)$
• Domain: Constructive Analysis / Potential Theory
• Definition: A closed, symmetric, bilinear form on a Hilbert space $L^2(X,m)$ that satisfies the Markov property: if $u\in \mathcal{D}(\mathcal{E})$, then $v=(0\vee u)\wedge 1\in \mathcal{D}(\mathcal{E})$ and $\mathcal{E}(v,v)\le \mathcal{E}(u,u)$. Generates a non-positive self-adjoint Laplacian $\mathrm{\Delta }$ and symmetric sub-Markovian semigroup $P_t=e^{t\mathrm{\Delta }}$.
• Primary Sources: Fukushima, Oshima, & Takeda (2011), Dirichlet Forms and Symmetric Markov Processes; Kigami (2001), Analysis on Fractals, Cambridge Univ. Press.
• Role in FQFT: Canonical foundation for the macroscopic spacetime carrier ${\mathfrak{R}}_X$, ensuring scale-recursive spectral convergence.

Hausdorff Dimension ($D_H$)

• Term Class: ESTABLISHED_MATHEMATICS
• Notation: ${\dim }_H(X)=inf\{d\ge 0:{\mathcal{H}}^d(X)=0\}$
• Domain: Geometric Measure Theory / Metric Geometry
• Definition: The critical exponent at which the $d$-dimensional Hausdorff measure of a metric space $X$ transitions from $\mathrm{\infty }$ to $0$, measuring metric scaling volume.
• Primary Source: Falconer (2014), Fractal Geometry: Mathematical Foundations and Applications, Wiley.

Spectral Dimension ($d_s$)

• Term Class: ESTABLISHED_MATHEMATICS
• Notation: $d_s=2\underset{t\to 0}{lim}\frac{\ln p_t(x,x)}{\ln t}=2\underset{\lambda \to \mathrm{\infty }}{lim}\frac{\ln N(\lambda )}{\ln \lambda }$
• Domain: Mathematical Physics / Diffusion on Fractals
• Definition: The fundamental analytical dimension governing on-diagonal heat kernel decay $p_t(x,x)\sim t^{-d_s/2}$ and eigenvalue Weyl asymptotics $N(\lambda )\sim {\lambda }^{d_s/2}$ on non-smooth metric spaces.
• Primary Source: Rammal & Toulouse (1983), J. Physique Lettres 44(1), 13–22; Hambly & Kumagai (1999).

Walk Dimension ($d_w$)

• Term Class: ESTABLISHED_MATHEMATICS
• Notation: $d_w=\frac{2D_H}{d_s}$, mean displacement $\mathbb{E}[|X_t-X_0|]\sim t^{1/d_w}$
• Domain: Stochastic Processes / Anomalous Diffusion
• Definition: The scaling exponent relating diffusion time to spatial displacement on self-similar geometries, satisfying the Einstein relation on fractals with anomalous diffusion.
• Primary Source: Barlow (1998), Diffusions on Fractals, Springer LNM 1690; Hambly & Kumagai (1999).

Norm-Resolvent Convergence

• Term Class: ESTABLISHED_MATHEMATICS
• Notation: $\parallel (A_n-zI{)}^{-1}-(A-zI{)}^{-1}\parallel \to 0$ as $n\to \mathrm{\infty }$ for $z\in \mathbb{C}\setminus \mathbb{R}$
• Domain: Operator Theory / Spectral Analysis
• Definition: The strongest standard operational topology for sequences of unbounded self-adjoint operators. Ensures uniform semigroup convergence $\parallel e^{-t{A}_n}-e^{-tA}\parallel \to 0$ and continuity of the spectrum without spectral pollution.
• Primary Source: Reed & Simon (1980), Methods of Modern Mathematical Physics: Functional Analysis, Academic Press; Post (2006), J. Funct. Anal. 238(2), 522–566.

Quasi-Unitary Equivalence

• Term Class: ESTABLISHED_MATHEMATICS
• Notation: $J_n:{\mathcal{H}}_n\to {\mathcal{H}}_{\mathrm{\infty }},\parallel J_n^{\ast }{J}_n-I\parallel \to 0$
• Domain: Operator Theory / Spectral Geometry
• Definition: Asymptotic unitary equivalence between operator sequences on varying Hilbert spaces with vanishing metric and projection defect.
• Primary Source: Post (2006), J. Funct. Anal. 238(2), 522–566.

Tree Boundary & Visual Metric

• Term Class: ESTABLISHED_MATHEMATICS
• Notation: $\partial T_d$, visual ultrametric $d_{\alpha }(\xi ,\eta )=e^{-\alpha |\xi \wedge \eta |}$ ($\alpha >0$)
• Domain: Non-Archimedean Analysis / Hyperbolic Geometry
• Definition: The boundary at infinity $\partial T_d$ of a rooted $d$-regular tree, consisting of infinite equivalence classes of geodesic rays. The visual metric $d_{\alpha }$ induces an ultrametric Cantor topology with exact Hausdorff dimension ${\dim }_H(\partial T_d)=\ln (d-1)/\alpha$.
• Primary Source: Casson & Bleiler (1987); Kigami (2001).

Benjamini–Schramm Local Weak Convergence

• Term Class: ESTABLISHED_MATHEMATICS
• Notation: $(G_n,o_n)\stackrel{BS}{\to }(G_{\mathrm{\infty }},o)$
• Domain: Graph Limits / Probability Theory
• Definition: Convergence of finite graph sequences in local distribution: for every fixed radius $r\ge 1$ and rooted finite graph $H$, the probability that the $r$-ball around a uniformly chosen root vertex $o_n$ is isomorphic to $H$ converges to the measure on the limit $(G_{\mathrm{\infty }},o)$.
• Primary Source: Benjamini & Schramm (2001), Electron. J. Probab. 6, no. 23, 1–13 (doi:10.1214/EJP.v6-96).

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3. Candidate Revision Constructs (R26_REVISION_CONSTRUCT) — 5 Terms

Local/Global Split

• Term Class: R26_REVISION_CONSTRUCT
• Notation: ${\mathcal{H}}_n=L^2(V_n,m_n;{\mathcal{H}}_F)$, $L_n^{LG}=L_n^X\otimes I+{\mu }_F^{2}I\otimes L_F$
• Domain: FQFT Mathematical Physics (FQFT-v2)
• Definition: A candidate architectural completion resolving expander metric collapse by factoring the field bundle into a global Dirichlet carrier ${\mathfrak{R}}_X$ (governing spacetime diffusion and continuum limits) and a fixed internal spectral fiber $\mathfrak{F}$ (governing microscopic gauge/charge structure).
• First Canonical Use: FQFT Microscopic Architecture R26 (MICRO-25).
• Status: Candidate Revision Architecture. Proved mathematically compatible (FQFT-MIC-T06); not empirically selected by nature.

Internal Spectral Fiber ($\mathfrak{F}$)

• Term Class: R26_REVISION_CONSTRUCT
• Notation: $\mathfrak{F}=({\mathcal{H}}_F,L_F)$, finite dimension $N_F=\dim {\mathcal{H}}_F$
• Domain: Microscopic Field Bundle Theory
• Definition: A localized, non-propagating finite-dimensional Hilbert space and self-adjoint generator $L_F$ acting pointwise on field values at each spatial vertex, decoupled from spatial coordinate scaling.
• First Canonical Use: MICRO-26.

Finite Fiber ($K_6$)

• Term Class: R26_REVISION_CONSTRUCT
• Notation: $K_6$, $\mathrm{Spec}(L_{K_6})=\{0,6^{\times 5}\}$
• Domain: Microscopic Field Bundle Theory
• Definition: Realization of the internal spectral fiber via a finite 5-regular Ramanujan graph on 6 vertices with discrete non-zero eigenvalues.
• First Canonical Use: MICRO-26.

Internal Spectral Scale (${\mu }_F$)

• Term Class: R26_REVISION_CONSTRUCT
• Notation: ${\mu }_F\in {\mathbb{R}}^{+}$ (mass / energy dimension $[M{]}^1$)
• Domain: Dimensional Field Analysis
• Definition: The fundamental physical coupling constant converting dimensionless internal graph eigenvalues ${\nu }_a$ into physical energy/mass scales: $\mathrm{Spec}(L_n^{LG})=\{{\lambda }_{j,n}+{\mu }_F^2{\nu }_a\}$.
• Status: Open Dynamical Gate (FQFT-MIC-G04). Source notes do not provide a derived value for ${\mu }_F$.

Local/Global Split Compiler (${\mathcal{C}}_{LG}$)

• Term Class: R26_REVISION_CONSTRUCT
• Notation: ${\mathcal{C}}_{LG}={\mathfrak{R}}_X⊞\mathfrak{F}$
• Domain: Mathematical Compiler Architecture
• Definition: A typed categorical functor that constructs field Hilbert spaces and tensor-sum kinetic operators from a Dirichlet refinement sequence $(X_n,{\mathcal{E}}_n)$ and a fixed fiber $({\mathcal{H}}_F,L_F)$, proving finite-fiber norm-resolvent stability.
• First Canonical Use: MICRO-25.

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4. Formalization Targets & Open Gates (FORMALIZATION_TARGET) — 1 Term

Spacetime-Fiber Interaction Operator ($\mathcal{M}$)

• Term Class: FORMALIZATION_TARGET
• Notation: $\mathcal{M}(X,F):\mathcal{D}(\mathcal{M})\subset \mathcal{H}\to \mathcal{H}$
• Domain: Dynamical Field Coupling
• Definition: The hypothetical non-zero mixing operator perturbing the baseline uncoupled tensor sum $L^{LG}={\mathrm{\Delta }}_X\otimes I+{\mu }_F^{2}I\otimes L_F+\mathcal{M}(X,F)$.
• Status: Open Coupling Gate (FQFT-MIC-G05, FQFT-MIX-26). Any predictive FQFT particle model must formally specify $\mathcal{M}$ and compute its renormalization group flow.

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5. Foundational Authorial Definitions (AUTHORIAL_DEFINITION) — 2 Terms

Fabricon

• Term Class: AUTHORIAL_DEFINITION
• Notation: ${\mathfrak{f}}_0$
• Domain: Science of Fabric Reality (SFR) Foundations
• Definition: The elementary irreducible relational unit in the Science of Fabric Reality corpus, defining the minimal localized observable cell of structural invariance.
• Primary Source: Ivan Pasev (2025), Principia Fabrica, GILC Press.

Pasev Gauge Principle

• Term Class: AUTHORIAL_DEFINITION
• Notation: ${\mathcal{G}}_{\text{Pasev}}$
• Domain: Invariant Field Cybernetics
• Definition: The structural principle stating that physical laws are invariant under recursive multi-scale relabelings of relational networks that preserve the underlying Dirichlet energy form.
• Primary Source: Ivan Pasev (2025), Teoria Fabrica Realica (TFR).