Relative Topology, Principal Bundle Reduction, and Index Theory on Perforated Information Manifolds: Unified Framework Toward S(U(3)\times U(2)), Three-Generation Index, and Yukawa--Winding

Relative Topology, Principal Bundle Reduction, and Index Theory on Perforated Information Manifolds: Unied Framework Toward S(U(3) ×U(2)) , Three-Generation Index, and YukawaWinding Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Abstract Full-rank density matrix manifold Dfull N={ρ > 0,trρ= 1} is open convex contractible, Uhlmann principal bundle admits global square-root section w=√ρ on full domain, thus absolute integer-valued topological invariants are absent on full domain. This paper turns to perforated relative topology : In N= 5 case, removing tubular neighborhood of threetwo level gap closing set Σ3|2={λ3=λ4} from Dfull 5 yields perforated domain Dexc . On Dexc construct rank 3/2 subbundles (E3,E2) via Riesz spectral projection , realizing principal bundle structure group reduction U(5) →U(3) ×U(2) ; further utilizing determinant balancing yields S(U(3) ×U(2)) reduction. We prove general group isomorphism SU(m)×U(n)∼ =SU(m)×SU(n)×U(1)Zlcm(m,n), with (m, n) = (3,2) deriving (SU(3) ×SU(2) ×U(1))/Z6 . Through relative K -theory boundary map unifying projectionChern class with massclutching ( det b Φ winding), on two-dimensional transverse S1 obtain Ind(DA+ Φ) = wind det b Φ = ⟨c1(LΦ),[S1]⟩. In CP2 spin c /Dolbeault calibration compute index = 3 as three-generation prototype. This paper provides complete proofs of all core propositions and theorems, with two protocollevel reproducible experimental/numerical schemes (purication interference loop and photonic Diracmass vortex). Appendices include: unied contour and global smoothness, group isomorphism gcd/lcm normalization and root selection, rigorous proof of relative K -theory and Chern character commutative diagram, Fredholm construction for Callias/AnghelBunke index theorem, and one-page arithmetic derivation of minimal charge 1/6 when Γ = Z6 . Keywords : Uhlmann principal bundle; perforated relative topology; Riesz projection; principal bundle reduction; S(U(3)×U(2)) ; Z6 quotient; relative cohomology/ K -theory; Dolbeault/spin c index; Callias/AnghelBunke index; determinant line bundle; line operator spectrum; reproducible experimental protocol 1 Notation, Assumptions, and Scope  Mixed state manifold : Dfull N={ρ∈Herm+ N:ρ > 0,trρ= 1} , this paper xes N= 5 .  Eigenvalue order : λ1≥ ··· ≥ λ5 ; spectral gap function g(ρ) := λ3−λ4 . 1  Perforated domain : Take δ > 0 , dene Dexc := {ρ∈ Dfull 5:g(ρ)≥2δ} . Boundary Y:= ∂Tubε(Σ3|2) equivalent to g= 2δ tubular boundary.  Riesz projection : Fix unied contour family γ3 (see Lemma 1.2 and Appendix A), let P3(ρ) = 1 2πi Iγ3 (z−ρ)−1dz, P2=I−P3.  Uhlmann principal bundle : P={√ρ U :ρ∈ Dfull 5, U ∈U(5)} , right action w·V=wV ; π(w) = ww† yields U(5) -principal bundle P→ Dfull 5 .  Regular/ordinary process (verication checklist) : Along path full rank, generator local CPTP and C1 , optional continuous purication gauge, and avoiding Σ3|2 ( g≥2δ ). This checklist only motivational: full domain lacks integer global classes; this paper focuses relative quantization on perforated domain.  Normalization : de Rham pairing uniformly takes 1 2πi factor; on CP2 hyperplane class H normalized as RCP1H= 1 , RCP2H2= 1 . 2 Main Results (Statements) Theorem 1 (A: Group Isomorphism, gcd/lcm Normalization) . Let g= gcd(m, n) , ℓ= lcm(m, n) = mn/g . Homomorphism φ:SU(m)×SU(n)×U(1) →S(U(m)×U(n)), φ(A, B, z) = diagzn/gA, z−m/gB is surjective with ker φ≃Zℓ . Thus SU(m)×U(n)∼ =SU(m)×SU(n)×U(1)Zℓ. Special case (m, n) = (3,2) ⇒ℓ= 6 . Proposition 2 (B: Partition Uniqueness) . Under constraint simple factors exactly SU(3) , SU(2) retaining only one U(1) , unique feasible partition of U(5) is 5 = 3 + 2 . Theorem 3 (C: Relative Bridging) . Assume mass end term Φ invertible on Y , take unitization b Φ : Y→U(N) . Then relative K -theory boundary image ∂[det b Φ] ∈K0(X, Y ) equals projection line bundle [det E3]−[det E2] ; on two-dimensional link ⟨c1(LΦ),[S2]⟩=⟨c1(det E3),[S2]⟩ ∈ Z. Theorem 4 (D: Callias/AnghelBunke) . If outer region invertibility Φ2≥cI , [∇,Φ] ∈L∞ , Φ∈ W1,2 loc etc. hold, then Ind(DA+ Φ) = deg b Φ|Sd−1 ∞∈πd−1(U). By Bott periodicity πk(U) = Z ( k odd), 0 ( k even), obtain: index possibly nonzero only when transverse dimension d is even ; when d= 2 Ind = 1 2πi ITr(b Φ−1db Φ) = wind det b Φ = ⟨c1(LΦ),[S1]⟩. 2 Theorem 5 (E: CP2 Index) . Td(TCP2) = 1 + 3 2H+H2 , ch(O(1)) = 1 + H+1 2H2 , thus index  DO(1) = 3 . Corollary 6 (F: SM Global Group) . By Theorem A, Proposition B, obtain S(U(3) ×U(2)) ∼ =SU(3) ×SU(2) ×U(1) Z6 . Appendix E further provides electric/magnetic charge lattice and arithmetic derivation of minimal charge step 1/6 for line operator spectrum when Γ = Z6 . 3 Degeneration Set Geometry and Unied Contour Proposition 7 (2.1: Codimension 3 and S2 -Link) . In three-dimensional transverse slice maintaining (λ2, λ5) gap with no additional symmetry, Σ3|2={λ3=λ4} is codimension 3 regular subset, its small sphere boundary link homotopic to S2 . Proof essentials : Restrict Hamiltonian to near-degenerate 2-dimensional eigensubspace, obtain h=xσx+yσy+zσz ; degeneracy condition (x, y, z) = (0,0,0) yields three independent real constraints. See Appendix A.3. Lemma 8 (2.2: Unied Contour; Global C∞ ) . For any compact K⊂ Dexc , there exist δ > 0 and nite cover {Uj} with closed curve family {γj} such that: ∀ρ∈Uj , γj has distance ≥δ from complement spectrum; thus P3,2 is C∞ on Uj and can be smoothly patched. Details in Appendix A.1A.2. 4 Principal Bundle Reduction to S(U(3) ×U(2)) Theorem 9 (3.1: Reduction = Section) . Let P→X be U(5) -principal bundle, G=P×U(5)Gr3(C5) . Section σ from P3 exists if and only if P admits U(3) ×U(2) -reduction PH⊂P . Proof : Standard principal bundle theory, Appendix B.4. Proposition 10 (3.2: Gauge Nature of Determinant Balancing) . Background trivial bundle C5 with xed volume form yields gauge isomorphism det E3⊗det E2≃C , reducing to S(U(3) ×U(2)) . Theorem 11 (3.3: Group Isomorphism; Theorem A for m= 3 , n= 2 ) . S(U(3) ×U(2)) ∼ =(SU(3) ×SU(2) ×U(1))/Z6. Proof : See Appendix B.1; particularly note root selection step in surjectivity: given (g3, g2) , take z∈U(1) satisfying z6= det g3 , let A=z−2g3∈SU(3) , B=z3g2∈SU(2) . Kernel isomorphic to Z6 . Proposition 12 (3.4: Partition Uniqueness; Proposition B) . Partition 5 = 3+2 is unique satisfying simple factors SU(3) , SU(2) with only one U(1) ; (4 + 1) lacks SU(2) , (3 + 1 + 1) and (2 + 2 + 1) both retain two U(1) 's. Details in Appendix B.2. 3 5 Two Characterizations of Relative Topology and Their Equivalence (Theorem C) 5.1 Relative K -Theory and Boundary Map For pair (X, Y )=(Dexc, ∂Tubε) , long exact sequence ··· → K1(Y)∂ −−→ K0(X, Y )→K0(X)→ ··· . If Φ invertible on Y , then unitization b Φ : Y→U(N) denes [b Φ] ∈K1(Y) , its boundary ∂[b Φ] ∈K0(X, Y ) . 5.2 Commutative Diagram and de Rham Representative Odd Chern character ch1:K1(Y)→H1(Y;Q) with 1-dimensional representative ch1([b Φ]) = 1 2πiTr(b Φ−1db Φ). Commutative diagram exists (Appendix C.1): K1(Y)∂ −→ K0(X, Y ) ↓ch1↓ch H1(Y)∂ −→ H2(X, Y ) Thus ch∂[b Φ]=∂h1 2πiTr(b Φ−1db Φ)i∈H2(X, Y ). 5.3 Equivalence Proposition (Theorem C) Compared with E3 , E2 from Riesz projection, utilizing naturality and clutchinggluing argument (Appendix C.2), obtain ∂[det b Φ] = [det E3]−[det E2]∈K0(X, Y ), thus on two-dimensional link ⟨c1(LΦ),[S2]⟩=⟨c1(det E3),[S2]⟩ . 6 Spin c /Dolbeault Index on CP2 (Theorem E) Take H=c1(O(1)) , RCP2H2= 1 . Td(TCP2) = 1 + 1 2c1+1 12(c2 1+c2) = 1 + 3 2H+H2,ch(O(1)) = eH= 1 + H+1 2H2. Top-dimensional coecient 1 + 3 2+1 2= 3 , thus index DO(1) = 3 . Kodaira vanishing ensures χ=h0= 3 . 4 7 Callias/AnghelBunke Index = Degree; Two-Dimensional Winding Formula (Theorem D) 7.1 Fredholm Conditions Let M complete, Dirac-type operator DA with self-adjoint end term Φ . If there exist R , c > 0 such that on M\BR , Φ2≥cI , with [∇,Φ] ∈L∞ , Φ∈W1,2 loc , then DA+ Φ is Fredholm (Appendix D.1). 7.2 Index = Degree and Parity Boundary homomorphism and Bott isomorphism yield Ind(DA+ Φ) = deg b Φ|Sd−1 ∞∈πd−1(U), πk(U) = (Z, k odd 0, k even . For two-dimensional transverse Ind = 1 2πi IS1 Tr(b Φ−1db Φ) = wind det b Φ = ⟨c1(LΦ),[S1]⟩, consistent with zero-mode counting. Sign convention: S1 takes counterclockwise orientation. 8 Alignment with GSM Line Operator Spectrum and Minimal Charge 1/6 By Theorem 3.3: GSM ∼ =(SU(3) ×SU(2) ×U(1))/Z6 . Kernel generator can take (ω−1 3I3,−I2, ei2π/6), ω3=ei2π/3. Action on (t, s, q) (respectively SU(3) triality, SU(2) parity, U(1) integer charge) is ω−t 3·(−1)s·ei2πq/6. Necessary and sucient condition for descending to quotient group: q≡2t+ 3s(mod 6) . Thus normalized hypercharge Y=q/6 has minimal fractional step 1/6 . Appendix E provides one-page derivation and example table for electric/magnetic charge lattice, Dirac pairing integer matrix, and θ period. 9 Protocol-Level Experimental and Numerical Schemes (Overview) E1 Purication Interference (Image around Σ3|2 ) : Discretize unied contour, readout 2πϕrel , where ϕrel =RS2 link Fdet E3/(2π)∈Z . Sampling Nshots ≳30 , phase noise δϕ ≲0.25 rad can stably determine integer. Failure cases: path grazes Σ3|2 , non-smooth purication; countermeasures: enlarge contour radius, increase purity gap and repeat sampling. E2 Photonic DiracMass Vortex : Encode mass phase eikθ , outer region |m| → m∞>0 . Zero-mode count |k| , near-eld intensity centralization, band-gap midpoint energy form ngerprint. Robust region: phase error ≤10◦ , coupling mismatch ≤5% . Appendix F provides parameter table and passing standard. 5 10 Discussion and Outlook  Relative vs absolute : Full domain contractible → absolute integer class vanishes; perforated → relative class quantization.  Dimensional eect : det only fully detects in two-dimensional transverse; higher dimensions require stable U group generators.  Group theory bridging : Spectral splitting induced S(U(3) ×U(2)) works synergistically with line spectrum dictionary, yielding minimal charge step 1/6 .  Follow-up : Multi-defect superposition relative class addition, robust window under noise non-equilibrium, systematic generalization with higher-order ( r -block) splitting. A Spectral Geometry and Unied Contour (Corresponding to 2) A.1 Spectral Gap Lower Bound and Contour Selection Let gap(ρ) = min{λ3−λ4, λ2−λ3, λ4−λ5} . On X=Dexc , gap >0 continuous; for any compact K⊂X , let δ= minKgap >0 . For each ρ∈K take circle γρ centered at λ3+λ4 2 with radius δ/2 , it encloses upper spectrum cluster with distance ≥δ/2 from complement spectrum. A.2 Riesz Projection C∞ Dependence By resolvent estimate |(z−ρ)−1| ≤ 2/δ and smoothness of z7→ (z−ρ)−1 , P3(ρ) = 1 2πi Hγρ(z−ρ)−1dz is C∞ in ρ . Using nite cover {Uj} with partition of unity patching, obtain global C∞ projection eld P3,2 . A.3 Codimension 3 and S2 -Link At λ3 & λ4 near-degeneracy, take E=E34 ⊕E⊥ , eective Hamiltonian h=ασz+ℜβ σx+ℑβ σy ; degeneracy ⇔(α, ℜβ, ℑβ) = (0,0,0) , three independent real equations thus codimension 3. Take normal small ball B3 , its boundary S2 is link. B Group Isomorphism and Minimal Partition (Corresponding to 3) B.1 Complete Proof of Theorem A Homomorphism φ(A, B, z) = diag(zn/gA, z−m/gB), g = gcd(m, n), ℓ =mn g. Kernel : φ(A, B, z) = I⇒A=z−n/gIm , B=zm/gIn . By A∈SU(m)⇒z−nm/g = 1 ⇒zℓ= 1 . Map κ:µℓ→ker φ, κ(z)=(z−n/gIm, zm/gIn, z) is group isomorphism, thus ker φ≃Zℓ . 6 Surjectivity (root selection) : Given (g3, g2)∈S(U(m)×U(n)) (i.e., det g3det g2= 1 ), take z∈U(1) satisfying zℓ= det g3. Let A=z−n/gg3∈SU(m), B =zm/gg2∈SU(n). Then det A=z−nm/g det g3=z−ℓdet g3= 1,det B=znm/g det g2=zℓdet g2= 1, with φ(A, B, z)=(g3, g2) . Thus obtain stated isomorphism. B.2 Partition Uniqueness Table Partition Simple part S -constrained U(1) count Conclusion 4+1 SU(4) 1 No SU(2) 3+1+1 SU(3) 2 Violates one U(1)  2+2+1 SU(2) ×SU(2) 2 Same 3+2 SU(3) ×SU(2) 1 Unique satisfying B.3 Generalization S(U(k)×U(ℓ)) ∼ =(SU(k)×SU(ℓ)×U(1))/Zlcm(k,ℓ) ; explicit form of kernel generator depends on embedding normalization, but quotient group isomorphism class invariant. C Relative K -Theory and Chern Character (Corresponding to 4) C.1 Commutative Diagram For pair (X, Y ) , odd Chern character ch1:K1(Y)→H1(Y;Q) yields ch1([u]) = 1 2πiTr(u−1du). Even Chern character ch : K0(X, Y )→Heven(X, Y ;Q) with de Rham boundary operator ∂ form commutative diagram K1(Y)∂ −→ K0(X, Y ) ↓ch1↓ch H1(Y)∂ −→ H2(X, Y ) whose commutativity follows from naturality and MayerVietoris patching. 7 C.2 Bridging Equality Let b Φ : Y→U(N) be unitized mass, ∂[det b Φ] ∈K0(X, Y ) . On other hand, Riesz projection yields E3 , E2 , thus [det E3]−[det E2]∈K0(X, Y ) . Using homotopy extension making b Φ stably compatible with spectral splitting morphism, through commutative diagram pairing in H2(X, Y ) to link S2 's integer equality, thus two relative classes equal. C.3 Explicit Pairing in Two-Dimensional Transverse If Y 's piecewise link is S1 , then IS1 1 2πiTr(b Φ−1db Φ) = ZS2 Fdet E3 2π∈Z. D Callias/AnghelBunke (Corresponding to 6) D.1 Fredholm Construction Take outer region cuto χ and parametrix Q=χΦ−1 . Have (DA+ Φ)Q=I−K1, Q(DA+ Φ) = I−K2, where K1,2 relatively compact (by [∇,Φ] ∈L∞ , Rellich compact embedding and outer region invertibility). Thus DA+ Φ is Fredholm. D.2 Boundary Map and Degree Homotope outer region to direction-only dependent Φ∞(θ) , index equals boundary map ∂[b Φ∞]∈ e K0(Sd)∼ =Z . Bott isomorphism yields Ind = deg(b Φ∞)∈πd−1(U). D.3 Two-Dimensional Single Vortex Example Φ(r, θ) = U(θ)H(r) , U(θ) = diag(eikθ,1,1) , H(r→ ∞)→m0I . Then b Φ = U , Ind = k . Taking counterclockwise orientation as positive, k→ −k index changes sign. E Line Operator Spectrum and Minimal Charge 1/6 (Corresponding to 7) E.1 Kernel Generator and Congruence By Theorem 3.3, Γ≃Z6 generator can take g∗= (ω−1 3I3,−I2, ei2π/6), ω3=ei2π/3. Action on (t, s, q) is ω−t 3(−1)sei2πq/6 . Quotient descent condition: ω−t 3(−1)sei2πq/6= 1 ⇐⇒ q≡2t+ 3s(mod 6). Let Y=q/6⇒Y≡t/3 + s/2 (mod Z) , thus minimal fractional unit 1/6 . 8 E.2 Electric/Magnetic Charge Lattice and Dirac Pairing (Schematic) Denote (e;m) as electric/magnetic charge vector, central gluing yields congruence constraint matrix C satisfying (e;m)7→ (e;m)+Cn ( n∈Zr ) equivalence. Dirac pairing integer matrix Ω well-dened integrality on quotient; θ period undergoes equivalence contraction after quotient group identication. Example table: fundamental representations 3 and 2 's (t, s) values bring Y 's fractional parts {1/3,1/2} , synthesizing with U(1) phase yields minimal step 1/6 span. F Experimental and Numerical Checklist (Corresponding to 8) F.1 E1 Purication Interference  Input : Loop C , δ , sampling Nshots , (T1, T2) .  Steps : Puricationevolutioninterference readoutphase unwrapcontour integral.  Output : ϕrel ∈Z .  Passing standard : |err(ϕrel)|<0.25 can determine integer; if fails, enlarge loop radius and Nshots . F.2 E2 Photonic Vortex  Input : Array size, coupling J , mass amplitude m∞ , vortex number k .  Steps : Phase map encodingexcitationnear-eld imagingspectral localizationzero-mode counting.  Output : Zero-mode count |k| .  Passing standard : Band gap > noise bandwidth, central peak signicant with energy near midpoint. F.3 Numerical Script Essentials  Grid (Nθ, Nr) take Nθ≥64 ;  Riesz projection performs contour quadrature along xed radius δ circle;  Wilson-loop's c1 consistent with wind det b Φ , error ∼ O(h2) . End of Main Text and Appendices 9