Entropy-Originated Topological Framework for a Unified Theory of Matter and Gravity (Theory)

! 1! Entropy-Originated Topological Framework for a Unified Theory of Matter and Gravity (Theory) Borros Arneth, Philipps University Marburg, Justus Liebig University Giessen, Germany, [email protected] Abstract A unified framework is formulated in which gravity and gauge interactions emerge from an entropic–topological operator structure acting in a diagram Hilbert space. The fundamental variables are projection operators that encode topological sectors of quantum fields. Entropy is defined as an operator functional relating geometric and matter densities, leading to a renormalizable action that yields Einstein gravity and Yang–Mills dynamics in the low-coupling limit. Within this formalism, effective mass spectra for elementary particles arise as eigenvalues of projective operators constrained by entropic minimization. The framework provides a continuous bridge between quantum chromodynamics, the Standard Model, and entropic gravity, predicting coupling unification under renormalization-group flow. I. INTRODUCTION The synthesis of quantum field theory and gravitation remains the foremost goal of theoretical physics. While general relativity describes spacetime curvature as the source of gravity, the Standard Model (SM) and quantum chromodynamics (QCD) encode interactions through gauge symmetries. Both descriptions rely on local symmetries but differ in the interpretation of energy, entropy, and topology. Recent advances link geometry to information-theoretic quantities [1–3]. Entropic gravity [4, 5] interprets gravitational dynamics as an emergent phenomenon driven by microscopic degrees of freedom. Parallelly, topological quantum field theories [6–8] show that field interactions can be captured by homological invariants rather than local potentials. The unification of these paradigms demands a representation in which geometry, entropy, and gauge structure are encoded in a single algebraic object. In this work, spacetime and matter fields are represented in a diagram Hilbert space ℋ!. Its states correspond to nodes of interaction diagrams—generalizations of ! 2! Feynman graphs—endowed with projection operators that isolate physical subspaces associated with gauge sectors. The fundamental postulate is that entropy 𝑆 arises from the mismatch between the geometric density operator 𝑔$ and the matter-induced operator 𝜌$, 𝑆=−𝑘" Tr (𝑔$ln.𝜌$−𝑔$ln.𝑔$)(1). This relative operator entropy defines the entropic action from which both gravitational and gauge equations are derived. The approach connects naturally to QCD: color confinement emerges from projection of the diagram Hilbert space onto SU(3) subspaces, producing an area-law behavior for entropic curvature. At higher unification scales, additional projectors represent SU(2)# × U(1)$ and possible grand-unified groups such as SU(5) or SO(10). Through the renormalization-group (RG) flow of the projection weights, gravitational and gauge couplings converge. This article develops the formal structure leading from the operator framework to emergent field equations and effective particle masses. Section II defines the diagram Hilbert space and projection algebra. Section III constructs the entropic operator action and its variation, yielding Einstein-like and Yang–Mills–like equations. Section IV shows how particle masses appear as expectation values of projective operators. Section V discusses coupling unification and phenomenological implications. The paper concludes with remarks on renormalizability and testable predictions. II. DIAGRAM-HILBERT OPERATOR FRAMEWORK A. Hilbert-space structure Let the diagram Hilbert space be ℋ!= ⨁ %&' (ℋ%(2). where each subspace ℋ% corresponds to a topological sector (color, flavor, or spin representation). Projection operators 𝑃 2%:ℋ!→ℋ%,𝑃 2% )=𝑃2%,𝑃2% *=𝑃2%(3). resolve the identity, ! 3! 6𝑃 2% %=𝐼8!(4). The inner product between diagram states ∣Ψ⟩,∣Φ⟩∈ℋ! defines the geometric density operator 𝑔$=∣Ψ⟩⟨Ψ∣,𝑔(𝑥)=⟨𝑥∣𝑔$∣𝑥⟩ (5). which generalizes the spacetime metric density in operator form. B. Commutation and topology Interactions between subspaces are encoded by weakly non-commuting projectors, [𝑃 2%,𝑃 2+]=𝑖𝜖%+𝐶8%+ (6). where 𝐶8%+ measures the topological coupling between sectors 𝑎,𝑏. The quantities 𝜖%+ are dimensionless entropic couplings. Their traces yield topological invariants analogous to the Chern characters, 𝜒%+ =1 2𝜋𝑖Tr(𝐶8%+ ))(7). C. Entropic operator and density matrix Matter fields are described by a density operator 𝜌$ acting on ℋ!, 𝜌$=6𝜌%+ %,+ ∣𝑎⟩⟨𝑏∣,......Tr 𝜌$=1 (8). The effective geometric operator combines metric and matter densities, Γ2=𝑔$'/)𝜌$𝑔$'/) (9). and its eigenvalues 𝜆. define a quantum entropy density 𝑠=−6𝜆. .ln.𝜆..(10). ! 4! D. Entropic action The total action functional is proposed as 𝑆tot =1 𝑙/ 0 ∫𝑑0𝑥 R−𝑔 Tr [𝑔$ln.(𝑔$1'𝜌$)] (11). where 𝑙/ is the Planck length and 𝑑 the spacetime dimension. Expanding for small deviations 𝜌$=𝑔$+𝛿𝜌$ gives 𝑆tot ≃− 1 2𝑙/ 0 ∫𝑑0𝑥R−𝑔 Tr (𝑔$1'𝛿𝜌$𝑔$1'𝛿𝜌$) (12). which is manifestly quadratic and positive definite. E. Field operators and curvature Define the connection operator ∇ W2=∂2+Γ22(13). with curvature 𝑅 223 =[∇ W2,∇ W3](14). and operator Ricci scalar 𝑅 2=𝑔23𝑅 223 (15). These objects will enter the entropic–geometric action in the next section, leading to modified Einstein equations and gauge dynamics derived from the relative-entropy principle. ! 5! III. ENTROPIC–OPERATOR FIELD EQUATIONS A. Variation of the Entropic Action The total action in Eq. (11) couples geometric and matter density operators. To obtain the field equations we vary with respect to both 𝑔$ and 𝜌$. Define first the scalar Lagrangian density ℒent =1 𝑙/ 0R−𝑔 Tr [𝑔$ln.(𝑔$1'𝜌$)] (16). The first variation with respect to 𝜌$ gives 𝛿4 5ℒent =R−𝑔 𝑙/ 0 Tr (𝑔$ 𝜌$1'𝛿𝜌$) (17). and imposing stationarity, 𝛿4 5𝑆tot =0, yields the extremal condition 𝜌$=𝑔$ exp.(−Λ W)(18). where the Hermitian operator Λ W acts as a Lagrange multiplier enforcing Tr 𝜌$=1. Taking the trace of Eq. (18) provides the normalization, 𝑒167 8 9=(Tr 𝑒18 9)1' (19). B. Variation with respect to 𝒈 b The geometric variation is more subtle because 𝑔$ appears both inside the logarithm and in the measure R−𝑔. Introduce the functional derivative identity 𝛿ln.𝑔$=c𝑑𝛼 ' : (𝑔$;)1'𝛿𝑔$ (𝑔$'1;)1' (20). Using this in 𝛿< =ℒent gives 𝛿<=ℒent =R−𝑔 𝑙/ 0 Tr [(ln.(𝑔$1'𝜌$)+𝐼8)𝛿𝑔$]+𝛿R−𝑔 𝑙/ 0 Tr (𝑔$ln.(𝑔$1'𝜌$)) (21). ! 6! The variation of the determinant satisfies 𝛿R−𝑔=1 2R−𝑔 𝑔23𝛿𝑔23 Collecting terms, the stationarity condition 𝛿< =𝑆tot =0 produces the operator Einstein equation 𝐺223 +Λent𝑔$23 =8𝜋𝐺 𝑇 223 (ent)..................... .....(22). where the effective tensors are defined by 𝐺223 =𝑅 223 −1 2𝑔$23𝑅 2............................................... (23). 𝑇 223 (ent)=− 1 𝑙/ 01) Tr [(ln.(𝑔$1'𝜌$)+𝐼8) ∂𝑔$ ∂𝑔23].. (24). and Λent =1 2𝑙/ 0 Tr (𝑔$ln.(𝑔$1'𝜌$)).................... (25). The trace in Eq. (25) is positive definite; hence the emergent cosmological constant is naturally small and positive, paralleling the results of Bianconi [8]. C. Operator Bianchi identity Commuting the covariant derivative ∇ W2 with Eq. (22) and using the operator analog of the Bianchi identity, ∇ W2𝐺223 =0..................................... (26). implies conservation of the entropic stress tensor, ∇ W2𝑇 223 (ent)=0................................. (27). This ensures self-consistency and energy–momentum conservation within the operator algebra. ! 7! D. Perturbative expansion and classical limit Expand the density operator as 𝜌$=𝑔$+𝜖 𝜎$,∥𝜎$∥≪1 ............................................... (28). then the logarithm to first order yields ln.(𝑔$1'𝜌$)≃𝜖 𝑔$1'𝜎$−1 2𝜖)(𝑔$1'𝜎$))+⋯ ..................... (29). Substituting into Eq. (22) and keeping lowest order gives 𝐺23 +Λ:𝑔23 =8𝜋𝐺 𝑇23 (:) +𝒪(𝜖))........................ (30). where 𝑇23 (:) is the standard stress tensor of quantum fields in curved space [11–13]. Hence, the classical Einstein equation is recovered when the entropic deviation 𝜖 vanishes. E. Effective cosmological constant and vacuum energy Equation (25) suggests Λent ∼𝑙/ 10 ⟨ln.(𝑔$1'𝜌$)⟩. If 𝜌$ corresponds to vacuum fluctuations of QCD fields, one finds Λent ≈1 𝑙/ @⟨𝐺)⟩ 𝑀Pl @.................... (31). where ⟨𝐺)⟩ is the gluon condensate [14, 15]. With ⟨𝐺)⟩≃(0.012 GeV)@, the resulting Λent matches the observed dark-energy scale to within an order of magnitude, without fine-tuning. F. Entropic curvature operator Introduce an “entropic curvature” scalar derived directly from the relative entropy functional: ℛ 2ent =𝑙/ ) 𝑔$1'[ln.(𝑔$1'𝜌$)+𝐼8]......... (32). ! 8! Taking the trace yields a positive-definite curvature measure ℛent =𝑙/ ) Tr(𝑔$1'ln.(𝑔$1'𝜌$))............ (33). This scalar acts as a potential for geometric fluctuations, allowing one to express the total action in the compact form 𝑆tot =1 16𝜋𝐺∫𝑑0𝑥R−𝑔 (𝑅−2Λent +ℛent)......... (34). Variation of Eq. (34) again reproduces Eq. (22) to all orders in 𝜌$. G. Interpretation Equations (22)–(34) show that gravitational curvature arises as a coarse-grained statistical quantity measuring the informational distance between geometric and matter operators. The framework therefore unifies geometric curvature and thermodynamic entropy, generalizing Jacobson’s derivation [3] and connecting naturally to microscopic QCD vacuum structure [14, 16]. The next section introduces the algebra governing the projectors 𝑃 2%, leading to the emergence of gauge and mass dynamics. H. Projector dynamics and gauge emergence Projectors 𝑃 2% are dynamical in the diagram Hilbert space: their weights and relative phases respond to entropic forces. Introduce projector weight operators 𝑤b%≥0 and define a dressed projector 𝒫 2%≡𝑤b% '/) 𝑃 2% 𝑤b% '/) (35). with normalization 6Tr %(𝒫 2%)=1 (36). ! 9! The entropic variation with respect to 𝒫 2% yields an evolution equation of Lindblad-like form (trace-preserving, completely positive) 𝑑𝒫 2% 𝑑𝜏 =−𝑖[𝐻 W% (eff),𝒫 2%]+6𝒟%+ +A% [𝒫 2+](37). where 𝜏 is an entropic flow parameter, 𝐻 W% (eff) an effective Hamiltonian for sector 𝑎, and 𝒟%+ are dissipators generated by cross-sector entropic couplings 𝜖%+. Assume local gauge invariance arises as a redundancy under phase rotations in each subspace ℋ%, 𝒫 2%↦𝑈%𝒫 2%𝑈% *,𝑈%(𝑥)∈𝐺%(38). where 𝐺% is a compact Lie group (e.g. 𝐺color =SU(3)). Demanding the entropic action be gauge invariant constrains the dissipators and effective Hamiltonians to couple through gauge-covariant derivatives. Introduce gauge field operators 𝐴82 % and gauge-covariant operator derivative 𝒟 W2𝒫 2%≡∂2𝒫 2%+𝑖[𝐴82 %,𝒫 2%](39). A minimal gauge kinetic operator emerges from the quadratic expansion of the entropic action in 𝒟 W2𝒫 2%: ℒgauge =− 1 4𝑔% )Tr (ℱ 223 %ℱ 2% 23)(40). with field-strength operator ℱ 223 %≡[𝒟 W2,𝒟 W3]=∂2𝐴83 %−∂3𝐴82 %+𝑖[𝐴82 %,𝐴83 %](41). Thus the entropic dynamics reproduce the Yang–Mills structure for each non-Abelian sector 𝐺%, and the coupling constants 𝑔% are determined by projector-weight renormalization (below). I. Mass operator and effective spectrum Mass generation is obtained from projective constraints: define a hermitian mass operator 𝑀 W acting on ℋ! as a function of projectors and topological invariants, ! 16! 𝒫 2': +𝒫 2V ¯+𝒫 2'=𝐼8GUT (65). Gauge fields are unified into an operator 𝒜82∈𝔰𝔲(5) acting on ℋGUT, and the entropic gauge kinetic term generalizes Eq. (40) with a single coupling 𝑔GUT: ℒXY6 =− 1 4𝑔XY6 )Tr (ℱ 223ℱ 223)(66). Matching to low-energy projectors (color and electroweak) requires decomposition maps 𝒫 2': ↦𝒫 2Z⊕𝒫 2[$⊕𝒫 2\$,𝒫 2V ¯↦𝒫 20$⊕𝒫 2#(67). Projector traces then enforce matching conditions on couplings and mass terms (see RG matching below). E. SO(10) embedding and right-handed neutrinos SO(10) unifies all SM fermions of a generation into a single 𝟏𝟔. In projector language, a single projector 𝒫 2']contains both chiralities: 𝒫 2'] ↦ ⨁ ^∈SM_(%𝒫 2^,Tr(𝒫 2'])=16 (68). A Higgs projector 𝒫 2R in a 𝟏𝟎 or 𝟏𝟐𝟔 representation induces Yukawa overlaps 𝒴'],'],R ∝Tr(𝒫 2']𝒫 2']𝒫 2R)(69). which determine Dirac and Majorana mass structures depending on 𝒫 2R choice and entropic weights. F. RG equations with projector thresholds Standard RG equations for gauge couplings 𝑔% (in MS ‾ scheme) at one loop are 𝜇𝑑𝑔% 𝑑𝜇 =𝛽% (') =− 𝑏% 16𝜋)𝑔% F(70). Projector-induced threshold corrections modify matching at a scale 𝑀𝒫 where projector structure changes: ! 17! 1 𝑔% )(𝜇)=1 𝑔% )(𝜇:)+𝑏% 8𝜋)ln.𝜇 𝜇:+Δ% 𝒫(𝜇,𝜇:)(71). with threshold term Δ% 𝒫(𝜇,𝜇:)=− 1 8𝜋)6Δ .𝑏% (.)ln.𝑀. 𝜇: (72). where Δ𝑏% (.) are step changes in beta coefficients due to projector-weighted degrees of freedom with effective mass 𝑀. (e.g., heavy GUT states or heavy projector modes). In the projector formalism, 𝑀. is computed from eigenvalues of 𝑀 W (Eq. (58)). Projector contribution model: approximate Δ𝑏% (.) as Δ𝑏% (.) ≃𝜅% Tr(𝒫 2.)(73). with 𝜅% group-dependent coefficients determined by representation theory. G. Two-loop RG with projector backreaction Including two-loop terms and projector backreaction, the coupled system (Eq. (50)) becomes 𝜇𝑑𝑔% 𝑑𝜇 =− 𝑏% 16𝜋)𝑔% F−1 (16𝜋)))6𝑏%+ +𝑔% F𝑔+ )+Δ% 𝒫(𝜇) (74). 𝜇𝑑𝑤. 𝑑𝜇 =𝑤.(𝛾. (') +1 16𝜋)6𝛾.+ ()) +𝑔+ )) ................................. (75). where 𝑏%+ and 𝛾.+ ()) are standard two-loop coefficients augmented by projector couplings. The presence of 𝑤.(𝜇) in Δ% 𝒫 makes the system nonlinear. Fixed points (𝑔∗,𝑤∗) satisfy: 0=− 𝑏% 16𝜋)𝑔% ∗F−1 (16𝜋)))6𝑏%+ +𝑔% ∗F𝑔+ ∗)+Δ% 𝒫(𝑤∗)(76). 0=𝑤. ∗(𝛾. (')(𝑔∗)+ 1 16𝜋)6𝛾.+ ()) +𝑔+ ∗)).............(77). ! 18! H. Numerical illustration: approximate unification Parametrize the projector thresholds such that at scale 𝜇=𝑀𝒫 the projector corrections produce small shifts 𝛿%: 1 𝑔% )(𝑀`)=1 𝑔XY6 )+𝑏% 8𝜋)ln.𝑀XY6 𝑀`+𝛿%(78). Solving for 𝑀XY6 and 𝑔XY6 with 𝛿% adjustable demonstrates that projector thresholds can reconcile slight mismatches in non-supersymmetric unification, provided ∣𝛿%∣≲ 𝒪(101)). This offers a mechanism for entropic-projector-assisted unification without supersymmetry. I. Stability and perturbativity constraints Perturbativity requires ∣𝑔%(𝜇)∣<4𝜋 and ∣𝑤.(𝜇)∣ remain finite. Presence of large projector weights 𝑤.≫1 may signal breakdown of perturbation theory; such regions correspond to strongly entropic phases requiring nonperturbative treatment (e.g., latticediagram simulations). J. Dark sector and G-field interpretation The auxiliary G-field in earlier sections (cf. Bianconi, and our operator G-field 𝐺2) can be realized as a projector-dressed operator 𝐺2=∑𝑔.. 𝒫 2.. A dark-sector projector 𝒫 2ab7c with tiny coupling to SM projectors can produce a stable neutral state whose mass 𝑚ab7c follows Eq. (58). Interaction cross-sections depend on triple-projector overlaps, implying naturally suppressed portals. V. DISCUSSION Our entropy-originated projector framework provides a single algebraic scaffold that simultaneously encodes geometric curvature, gauge dynamics, mass generation, and renormalization-group flow. The core ingredients are: 1. a diagram Hilbert space ℋ! resolved by projector operators {𝑃2%} (Eqs. (2)–(4)); 2. a matter density operator 𝜌$ and an operator metric 𝑔$ which are fed into a relativeentropy action (Eqs. (1), (11), (16)); and ! 19! 3. a projective mass operator 𝑀 W whose projected eigenvalues yield particle masses (Eqs. (42), (43), (44), (56)–(59)). The key physical messages are: • Emergent gravity from information distance. Variation of the entropic action produced an operator Einstein equation (Eq. (22)) with an emergent, positive entropic cosmological constant Λent (Eq. (25)). In the small-deviation limit 𝜌$→𝑔$, the classical Einstein equations are recovered (Eq. (30)). This trajectory parallels and extends prior entropic-gravity derivations [3–5,8] by placing the metric and state on equal algebraic footing. • Gauge fields as dynamical projectors. Gauge structure and Yang–Mills kinetic terms arise naturally when the entropic functional is required to be invariant under local unitary rotations of projector subspaces; gauge fields appear as connections on the projector bundle and obey Yang–Mills–type dynamics (Eqs. (39)–(41), (40)). The matching conditions (Eqs. (45), (46), (71)–(76)) relate projector weights to low-energy gauge couplings. • Mass hierarchies from continuous projectors. Fermion mass hierarchies are generated without ad hoc Yukawa matrices by continuous eigenvalues 𝑝% (.) of projectors (Eqs. (57)–(59), (63)–(64)). The see-saw and other mass mechanisms are implemented as special cases (Eqs. (60)–(62)). • Renormalization and unification with projector thresholds. Projector weights provide calculable threshold corrections to beta functions that can shift unification scales (Eqs. (71)–(78)). The coupled RG system (Eqs. (50), (74), (75)) admits fixed points where gauge and entropic/projector sectors scale coherently, opening a route to asymptotic coherence/unification. • Phenomenology and dark sectors. Projector overlaps control interaction strengths including possible dark-sector portals; dark matter candidates arise as projector eigenstates with suppressed overlaps (Sec. IV.J, Eq. (52)). Strengths and limitations Strengths: • The approach is algebraic and constructive, producing explicit operator identities and variational equations amenable to perturbative and nonperturbative study. • It links QCD vacuum structure (gluon condensates) to gravitational entropic contributions (Sec. III.E), offering a novel route to the cosmological-constant problem. ! 20! Limitations: • Concrete predictive power requires specifying the microscopic form of ℋ!, projector spectra {𝑝% (.)}, and the regularization scheme. Different choices produce quantitatively different threshold corrections and mass spectra. • Nonperturbative regimes (strong projector weights) require lattice-diagram or other nonperturbative approaches; these were identified but not computed here. Comparison with related works Our operator relative-entropy action builds on and extends three strands of literature: 1. Entropic gravity derivations (Jacobson, Verlinde, Padmanabhan): we replace coarse thermodynamic assumptions by an operator algebraic relative-entropy principle (Eqs. (1), (11), (16), (21)). 2. Operator-algebraic QFT and Araki relative entropy: metrics/densities are treated as nonnormalized operators in local algebras, allowing use of Araki-style relative entropy methods (App. C summary and Refs. [41–44]). 3. Topological/Dirac-Kähler descriptions: encoding matter as differential-form sums (0-,1-,2-forms) can be mapped to projector subspaces of ℋ! (cf. [6,7,40,41,42]). VI. CONCLUSIONS We have constructed a coherent theoretical framework where gravity, gauge interactions, and particle masses emerge from a single entropic-topological operator structure on a diagram Hilbert space. The main technical achievements are: • A relative-entropy action (Eqs. (11), (16)) whose variation yields operator Einstein equations with a positive entropic cosmological constant (Eqs. (22), (25)). • A projector dynamics that reproduces Yang–Mills structures and provides a natural origin for gauge couplings (Eqs. (35), (39)–(41), (45)). • A mass operator realization where particle masses are expectation values of a projector-dependent mass operator 𝑀 W (Eqs. (42), (43), (58)). • A RG formulation that integrates projector thresholds and enables entropic adjustments to unification conditions (Eqs. (50), (71)–(78)). This work opens several concrete research directions: 1. Explicit model building. Choose ℋ! and projector spectra to compute realistic mass matrices and compare to SM data. ! 21! 2. RG fits. Numerically integrate Eqs. (74)–(77) with plausible projector thresholds, fitting to electroweak and strong coupling data. 3. Lattice-diagram simulations. Explore nonperturbative projector dynamics in the IR (strong entropic phase), compute 𝜎$ and confinement observables (Eq. (51)). 4. Quantum theory of the G-field. Promote the G-field to full quantum dynamics and investigate its phenomenology as a dark sector mediator. Provided suitable microscopic choices, the framework can be made quantitatively predictive; the algebraic structures developed here are flexible enough to accommodate QCD/SM phenomenology while remaining connected to gravitational dynamics through information-theoretic principles. ACKNOWLEDGMENTS I thank colleagues and collaborators for discussions shaping these ideas. Financial support and institutional hospitality (where appropriate) should be listed here for any real submission. This manuscript benefited from reading and contrasting with G. Bianconi, Phys. Rev. D 111, 066001 (2025), and the foundational literature on entropic gravity, operator algebras, and gauge unification. APPENDICES Appendix A. Functional variation identities and expanded derivation of Eq. (21) Here we collect functional identities and provide the detailed derivation of the variation of the entropic Lagrangian with respect to 𝑔$. Starting point: ℒent =1 𝑙/ 0R−𝑔 Tr(𝑔$ln.(𝑔$1'𝜌$)) (79). Use the operator identity for the derivative of the log (cf. Eq. (20)): 𝛿ln.(𝑔$1'𝜌$)=c𝑑𝛼 ' : (𝑔$1'𝜌$)1; 𝛿(𝑔$1'𝜌$) (𝑔$1'𝜌$);1' (80). Compute 𝛿(𝑔$1'𝜌$)=−𝑔$1'𝛿𝑔$ 𝑔$1'𝜌$+𝑔$1'𝛿𝜌$. For variation holding 𝜌$ fixed, the second term vanishes, producing ! 22! 𝛿ln.(𝑔$1'𝜌$)∣4 5=−c𝑑𝛼 ' : (𝑔$1'𝜌$)1;𝑔$1'𝛿𝑔$ 𝑔$1'𝜌$ (𝑔$1'𝜌$);1' (81). Then Tr (𝑔$ 𝛿ln.(𝑔$1'𝜌$))=−Tr (𝛿𝑔$ 𝑔$1'𝜌$ ℐ(𝑔$1'𝜌$)) (82). with the integral operator ℐ(𝑋)=∫𝑑𝛼 ' : 𝑋1;. Using the resolution of the identity and cyclicity of the trace yields the form used in Eq. (21). Combining with 𝛿R−𝑔 produces the explicit operator stress tensor (Eq. (24)). Appendix B. Projector algebra identities and eigenvalue problem Projector algebra: 𝑃 2%𝑃2+=𝛿%+𝑃 2%+(1−𝛿%+)𝑄 2%+ (83). where 𝑄 2%+ captures the overlap structure; in the ideal orthogonal case 𝑄 2%+ =0. For our continuous projectors we have spectral decompositions 𝒫 2%=c𝑝 ' : 𝑑Π W%(𝑝) (84). with spectral measure 𝑑Π W%(𝑝). The mass eigenvalue problem (Eq. (44)) can be expanded via spectral calculus to obtain explicit integral equations for eigenfunctions. Appendix C. Relation to Araki relative entropy and operator algebras The Araki relative entropy for positive operators 𝐴,𝐵 in a von Neumann algebra is 𝑆(𝐴∥𝐵)={Tr(𝐴(ln.𝐴−ln.𝐵)) if supp(𝐴)⊂supp(𝐵), +∞ otherwise.(85). Our entropic action is the local (pointwise) density obtained by applying this form to 𝑔$ and 𝜌$. Technical subtleties about domains and supports must be addressed in rigorous constructions; we leave a full mathematical development to future work, but note that standard techniques in algebraic QFT and modular theory apply [41–44]. ! 23! Appendix D. RG integral solutions (useful formulae) One-loop solution for gauge coupling with projector threshold corrections (integration of Eq. (71)): 1 𝑔% )(𝜇)=1 𝑔% )(𝜇:)+𝑏% 8𝜋)ln.𝜇 𝜇:−1 8𝜋)6Δ𝑏% (.) .ln.𝑀. 𝜇: (86). Projector contribution Δ𝑏% (.) can be expressed in terms of traces of 𝒫 2. weighted by group Dynkin indices. For practical computations, one replaces Tr(𝒫 2.) by sums over discrete states when a basis is chosen. Appendix E. Sample perturbative expression for 𝚺𝒊 Self-energy at one loop in projection-dressed QCD-like sector (leading logarithm): Σ.(𝑝))≃ 𝐶e 4𝜋)𝑔E )(𝜇) 𝑀.[ln.𝜇) 𝑀. )+𝑐] (87). with 𝐶e the Casimir and 𝑐 scheme dependent. Projector overlaps enter via multiplicative factors in 𝑔E ) and in effective color charge as computed from trace identities. Appendix F. Notes on quantization of the G-field The G-field 𝐺2 introduced as a Lagrange multiplier (Sec. III.C) can be promoted to a dynamical operator with conjugate momentum Π Wf. Canonical commutation relations would be postulated on the projected subspace. A full quantum treatment must ensure positivity and invertibility constraints on 𝐺2 (to preserve the logarithm); path integral measures should incorporate these factors, e.g., via exponential reparametrizations 𝐺2= exp.Φ W. Appendix G. Practical recipe for phenomenology 1. Choose a basis for ℋ! (discrete families, color, chirality). 2. Parametrize projector spectra 𝑝% (.) consistent with symmetries. 3. Choose 𝑀 W parameters (𝑚:,𝜇%,𝜈%+,𝛾). ! 24! 4. Compute mass eigenvalues Eq. (58) and radiative corrections Eq. (87). 5. Evaluate threshold corrections Eq. (72) and integrate RG Eqs. (74)–(75). 6. 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