Parabolic–Transport Compatibility for Branched Spin Structures: Smooth Extension, Root Stacks, Reduced Energy, and Relative Spin Holonomy

Let f:X→ Y be a nonconstant holomorphic map of compact Riemann surfaces with theta-characteristics SX, SY, ramification divisor Rf, degree d, and discrepancy bundle Lf=f^ SY⊗ SX⁻¹, so that Lf⊗ 2 OX(-Rf). The manuscript contributes (i) a hermitian framework, PTCC, comparing two independent normalizations of Lf; (ii) a Dirichlet-type problem of optimal metric correction; (iii) exact fibrewise jet criteria for smooth realizability of the optimum. Classical tools (Bott–Chern, Green, Bismut–Gillet–Soulé, Borel) are applied, not reproved. 1. Smooth compensation without PTCC At a point of local degree e, a metric on Lf is divisorially compatible if hLf=|z|ᵉ⁻¹v with v> 0 smooth. Write f^ωY=λfωX, λf=|z|²⁽ᵉ⁻¹⁾u. Then hcorr,f=hXhLfλf-1/2=hXvu-1/2 is smooth and positive, and the connection ∇X+∇Lf-12∂logλf extends smoothly, the residues -1/2 dz/z cancelling. No PTCC hypothesis is needed. The mechanism is elementary; the claim is the separation of local compensation from transport normalization. 2. The PTCC framework Transport normalization. With φf: SX⊗ Lf~f^ SY and hₜᵣ=φf^(f^hY), the compatible metric hLfPTCC=hₜᵣhX⁻¹λf1/2 gives hcorr,f=hₜᵣ. Parabolic reference. From μf: Lf⊗2 OX(-Rf) set (hLfᵖᵃʳ)²=μf^h-Rfᵈⁱᵛ. It is independent of hX,hY,ωX,ωY, has c₁=-12[Rf], and is flat off Rf. PTCC condition: φf^(f^∇Y1,0)=∇X1,0+∇Lf,par1,0-12∂logλf, equivalently hLfPTCC=C hLfᵖᵃʳ. The potential Φf=log(hLfPTCC/hLfᵖᵃʳ) is smooth on X, the defect is ∂Φf, and EPTCC(f)=∫X i ∂Φf∂Φf , where ∂∂Φf is the smooth Bott–Chern difference of the transported and corrected curvatures. PTCC holds iff Φf is constant iff EPTCC(f)=0. The energy depends on the hermitian and Kähler data, not on the cover alone. With βX=log(hX/hXωX) and βY likewise, Φf=f^βY-βX+const. Hence PTCC is automatic for Kähler-induced spin metrics (hLfPTCC=|f'|). 3. Optimal correction on the target On H¹(M)=W1,2(M, R)/ R with Dirichlet pairing ·,·M, define Tf by d Tfu,vY= u,f^vX. Then Tff^=Id, EX(f^v)=d EY(v), and Pf=f^Tf is the orthogonal projection onto f^ H¹(Y). A target deformation hY↦ eᵇhY replaces Φf by Φf+f^b, and EX(Φf+f^b)= EX((Id-Pf)Φf)+d EY(TfΦf+b). The minimizer is bₘᵢₙ=-TfΦf=TfβX-βY, and the reduced energy is If=infb EX(Φf+f^b)= EX((Id-Pf)βX)=dist²H¹(X)([βX],f^ H¹(Y)). It is independent of the target data and vanishes iff βX is constant on every fibre. A continuous representative of Tfu is Vᵤ(y)=1d∑_p∈ f⁻¹(y)eₚ u(p); the optimum e^VβXhYωY is continuous and smooth iff VβX is. Hence If=0 does not give a smooth optimum. 4. Fibrewise jet criteria In the model w=zᵉ, the cyclic mean τₑu(w)=1e∑ζᵉ=1u(ζ w1/e) satisfies τₑ(zᵉᵐ⁺ʳ zᵉⁿ⁺ʳ)=wᵐ wⁿ|w|2r/e, 0≤ r< e, and kills zᵃ zᵇ unless e a-b. Let f⁻¹(y₀)= pⱼ have local degrees eⱼ and source Taylor coefficients cab⁽ʲ⁾, and let Θy₀=ⱼ r/eⱼ:1≤ r< eⱼ. Set Bmnθ(u;y₀)=1d∑eⱼθ∈ Zeⱼ ceⱼm+eⱼθ, eⱼn+eⱼθ⁽ʲ⁾. Theorem. Vᵤ∈ Cᵏ near y₀ iff Bmnθ(u;y₀)=0 for all θ∈Θy₀ with m+n+2θ≤ k; Vᵤ is smooth iff all Bmnθ vanish. No finite jet decides smoothness. The obstruction is a weighted fibre sum, so points compensate: for e₁=2, u₁=|z₁|² and e₂=4, u₂=-12|z₂|⁴ the contributions ±2d|w| cancel. Since Bmnθ(Φf+f^a)=Bmnθ(Φf) for smooth a, there is a smooth b with Φf+f^b constant iff If=0 and Bmnθ(Φf;y₀)=0 for all fractional indices. The two conditions are independent: the first places the obstruction in f^ H¹, the second kills it on jets. 5. Generic non-attainment and the gcd criterion For ramified f, the space Cf of u∈ C^∞(X) with smooth Tf[u] is closed, of infinite codimension, with empty interior, yet its image is dense in H¹(X). Source metrics without smooth optimum are open dense. With gy= eₚ:p∈ f⁻¹(y), ∃ hX smooth with If=0 and no smooth minimizer gy > 1 for some y∈ Y. A positive reduced energy may still admit a smooth optimum. 6. Orbit variance and towers For Galois f:X→ X/G, If=12|G|∑g∈ G EX(βX-g^βX); for towers, Ig∘ f= If+( f) EY((Id-Pg)TfβX)≥ If. 7. Green formulation, composition, anomaly Let 2i ∂∂Φf=σfωX. Then EPTCC(f)=12 σf ²_ H⁻¹(X) and, with νf=f_(σfωX), If=12( σf ²_ H⁻¹(X)-1d νf ²_ H⁻¹(Y)). For composable maps, Qf,g=h_Lg∘ f/(hLf f^hLg) is smooth positive and Dg∘ f= Df+f^ Dg-∂log Qf,g. Normalized metrics are multiplicative: Φg∘ f=Φf+f^Φg, so PTCC is stable under composition. In general EPTCC(g∘ f)= EPTCC(f)+( f) EPTCC(g)+ APTCC(f,g), with APTCC(f,g)=2Re∫X i ∂Φf∂(f^Φg) obeying a cocycle identity, possibly nonzero when Qf,g=1. On the smooth line Ef= SX⊗ Lf with h₀=hXhLfᵖᵃʳλf-1/2 and h₁=eΦfh₀, a specialization of Bismut–Gillet–Soulé gives log ²Q,h₁ ²Q,h₀=∫XΦf ιh₀+14π EPTCC(f). Thus EPTCC is 4π times the quadratic part of the anomaly; PTCC means equal local index densities, whereas the integrated indices always agree. The raw anomaly is unbounded below under constant rescalings since χ(Ef)< 0; the renormalized one has infimum If/4π. Signs depend on the determinant-line convention; no conical determinant arises. 8. Closed computations For fd=zᵈ, Fubini–Study forms, and hY'=etψhY with ψ=1-|w|²/1+|w|²: $$ EPTCC(f_d;t)=4π/3 d t^2, {_ζΔd,t}{_ζΔd,0}=e-dt^2/3-(d-1)t∏ₖ₌₁ᵈ⁻¹{}_1F_1 ( kd;1;2t).$$ A non-Galois cubic ($S_3$ closure) gives $ I_f≈0.714504$, uncertified. 9. Spin information and conditional spectral results The normalized corrected data are isometric to $(f^{} S_Y,f^{}h_Y,f^{}∇_Y)$, hence independent of $ S_X$; the spin is restored by the relative line $ T= S_X⊗ S_0⁻¹^0(X)[2]$ with its flat unitary connection. For the coupled operators $ DE_f, T=√2(∂E_f⊗ T+∂E_f⊗ T^{})$, a conditional theorem assumes negative source curvature, non-coincident geodesic lengths, $ReHolE_f(γ)≠0$ on primitive geodesics, and an orbital heat-kernel expansion. It asserts injectivity of $ T( DE_f, T)$ through $HT(t)-HT'(t)~ C t-1/2e^{-_{}^2/4t}$, $C≠0$. The delicate point is the orbital amplitude (Laplace formula for parallel transport along geodesics); genericity rests on Baire. Universal injectivity fails: on $y^2=x^5-x$, the automorphism $(x,y)↦(-x,iy)$ exchanges two inequivalent twists, giving exact isospectrality. 10. Scope and corrections Not established: an absolute PTCC invariant; universal spectral separation of spins; a conical determinant formula; PTCC as root-stack descent; certified spectral bounds in genus two. Two corrections: the sign in the Fourier annex must be $+$ for the positive Laplacian, and the genus two metric ι^gFS needs a chosen hermitian product on H⁰(X,A), with the spectral hypotheses unverified. The core needing comparison with prior work on ramified traces is the coefficients Bmnθ, the Cᵏ criterion, intra-fibre co